Polymerization kinetics of single microtubules and
microtubule bundles under force and confinement
Dissertation
zur Erlangung des Grades eines
Doktors der Naturwissenschaften
der Fakultät Physik
der Technischen Universität Dortmund
vorgelegt von
Björn Zelinski
aus Bochum
Dezember, 2013
Erster Gutachter Prof. Dr. Jan Kierfeld
Zweiter Gutachter Dr. Kai P. Schmidt
Kontakt zum Autor bjoern.zelinski@tu-dortmund.de
3"Nur Steine . . . " Tim Fischer
4
Abstract
Microtubules are, next to actin filaments, one of the main components of the cytoskele-
ton of eukaryotic cells. The static and dynamical properties are essential for numerous
intracellular processes. Their characteristic polymerization behaviour, called dynamic
instability, is the key factor of the temporal and spatial organization of the microtubule
cytoskeleton. In dynamic instability, microtubules stochastically alternate between phases
of polymerization and phases of fast depolymerization. Polymerizing microtubules can
generate forces in the piconewton range, which are use in positioning of cellular organelles
or chromosome separation. The interplay of dynamic instability and force generation leads
to complex polymerization kinetics. In this thesis we investigate the force generation of
polymerizing microtubules in the presence of dynamic instability. We establish a coarse-
grained model for the polymerization dynamics of single microtubules and include realistic
force-velocity relations and velocity-dependent catastrophe models. In the first part of
this thesis, the polymerization dynamics of single microtubules are investigated, based on
the coarse-grained model, in three different scenarios, which mimic typical cellular envi-
ronments. The scenarios are as follows: (i) A single microtubule is confined by a rigid and
fixed wall, (ii) a single microtubule polymerizes under constant force, and (iii) a single
microtubule polymerizes against a elastically coupled obstacle. For all three scenarios
we calculate stochastic length distributions analytically and characterize the polymeriza-
tion dynamics as a function of the growth parameters, like the tubulin concentration, the
rescue rate or the obstacle stiffness.
In scenario (i) we introduce a realistic model for wall-induced catastrophes and find ex-
ponentially increasing or decreasing stationary length distributions. In scenario (ii) we
determine the critical force fc which provides the transition from unbounded to bounded
growth. The critical force is smaller than the stall force and increases logarithmically
with the tubulin concentration and rescue rate. In scenario (iii) we calculate the poly-
merization force for growth against the elastic obstacle in the absence and in the presence
of rescue events. For a vanishing rescue rate the maximal polymerization force grows
logarithmically with the tubulin concentration and is always smaller than the stall force.
In the presence of rescue events, we analytically calculate steady-state length distribu-
tions as a function of the growth parameters, and the average polymerization force. In
the steady state the average polymerization force equals the critical force fc for growth
6under constant force. In addition, we establish a dynamical mean field theory and com-
pare mean-field results for the average polymerization force to full stochastic calculations.
Based on the dynamical mean field theory, we investigate the relaxation dynamics of a
single microtubule if growth parameters are perturbed.
The relaxation dynamics into the new steady state is governed by an exponential decay
with a characteristic time scale, which is a function of the perturbed growth parameters.
Finally we show that results are robust with respect to changes in the force-velocity
relation and the catastrophe model.
In the second part of this thesis, we extend the single microtubule model to bundles of
parallelly growing microtubules and investigate the polymerization dynamics of micro-
tubule bundles growing against an elastically coupled obstacle. This scenario is close to
the in vitro experiments by Laan et al.. We establish a dynamical mean field theory, char-
acterize the cooperative polymerization dynamics and the generated force, as a function
of the growth parameters and the obstacle stiffness. In the absence of rescue events and
a stiff obstacle, the maximal polymerization force growth linear in the number of micro-
tubule within the bundle. This is in agreement with results from in vitro experiments
by Laan et al.. For a soft obstacle, we find a crossover to a logarithmic dependency on
the microtubule number. For a non-vanishing rescue rate and a soft obstacle, we find
an oscillatory polymerization dynamics, governed by collective catastrophes and collective
rescues. The steady-state polymerization force oscillates around its average value and
is a linear function of the microtubule number and the tubulin concentration. Finally
we investigate the validity of the dynamical mean field theory, based on semi-stochastic
calculations.
Zusammenfassung
Mikrotubuli sind, neben Actinfilamenten, der Hauptbestandteil des Zytoskeletts eukary-
otischer Zellen. Die statischen und dynamischen Eigenschaften von Mikrotubuli sind
existenziell für viele zellinterne Prozesse. Ihr charakteristisches Polymerisationsverhalten,
die dynamische Instabilität, ist von zentraler Bedeutung für die zeitliche und räumliche
Organisation des Mikrotubuli-Zytoskeletts. Unter der dynamischen Instabilität versteht
man das stochastische Alternieren von Phasen mit moderater Polymerisation und Phasen
schneller Depolymerisation. Polymerisierende Mikrotubuli können Kräfte im Piconewton-
Bereich erzeugen, welche zur Positionierung von Zellorganellen oder zur Trennung von
Chromosomen genutzt werden. Das Zusammenspiel von dynamischer Instabilität und
Krafterzeugung bewirkt eine komplexe Polymerisationsdynamik.
In dieser Arbeit untersuchen wir die Krafterzeugung durch polymerisierende Mikro-
tubuli unter dem Einfluss der dynamischen Instabilität. Dazu etablieren wir ein
idealisiertes Polymerisationsmodell für einzelne Mikrotubuli, welches realistische Kraft-
Geschwindigkeits- und geschwindigkeitsabhängige Katastrophenmodelle beinhaltet.
Im ersten Teil dieser Arbeit wird die Polymerisationsdynamik einzelner Mikrotubuli,
anhand des idealisierten Modells, in drei unterschiedlichen Szenarien untersucht, welche
ein typisches zelluläres Umfeld widerspiegeln. Die drei Szenarien sind: (i) Ein einzelner
Mikrotubulus polymerisiert in einer durch starre Wände begrenzten Umgebung, (ii) ein
einzelner Mikrotubulus polymerisiert unter einer konstanten Kraft und (iii) ein einzelner
Mikrotubulus polymerisiert gegen ein elastisches Hindernis, welches eine längenabhängige
Kraft erzeugt.
Für alle drei Szenarien bestimmen wir analytische Ausdrücke für die stochastischen Län-
genverteilungen und charakterisieren die Polymerisationsdynamik in Abhängigkeit der
Wachstumsparameter, wie der Tubulinkonzentration, der Rettungsrate und der Härte
des elastischen Hindernisses. In Szenario (i) präsentieren wir ein realistisches Modell für
Katastrophen, welche durch die starren räumlichen Begrenzungen erzeugt werden. Wir
finden exponentiell steigende oder fallende stationäre Längenverteilungen. In Szenario (ii)
bestimmen wir die kritische Kraft Fc, welche den Übergang von ungebundenem zu gebun-
denem Wachstum bestimmt. Die kritische Kraft ist kleiner als die stall force and wächst
logarithmisch mit der Tubulinkonzentration. In Szenario (iii) berechnen wir die mittlere
Kraft, die ein einzelner Mikrotubulus erzeugt. In Abwesenheit von Rettungen wächst
8die maximale Polymerisationskraft logarithmisch mit der Tubulinkonzentration und ist
kleiner als die sogenannte "stall force" Fstall, bei der die Wachstumsgeschwindigkeit ver-
schwindet. Für eine von Null verschiedene Rettungsrate bestimmen wir analytische Aus-
drücke für die stationären Längenverteilungen und die mittlere Kraft als Funktionen der
Wachstumsparameter. Im stationären Zustand ist die mittlere Kraft gleich der kritischen
Kraft Fc. Zusätzlich zu stochastischen Berechnungen etablieren wir eine dynamische mean
field Theorie und vergleichen Resultate aus beiden Theorien miteinander. Basierend auf
der dynamischen mean field Theorie, untersuchen wir das Relaxationsverhalten einzel-
ner Mikrotubuli, nachdem Wachstumsparameter gestört wurden. Wir finden eine ex-
ponentielle Relaxationsdynamik mit einer charakteristischen Zeitskala, welche durch die
gestörten Wachstumsparameter dominiert wird. Zum Abschluss zeigen wir, das die erhal-
tenen Resultate robust gegenüber Änderungen in der Kraft- Geschwindigkeits-Relation
und des Katastrophenmodelles sind.
Im zweiten Teil dieser Arbeit untersuchen wir die Polymerisationsdynamik von Bündeln
aus parallelen Mikrotubuli, die gegen ein elastisches Hindernis wachsen. Dieses Szenario
ähnelt den in vitro Experimenten, die von Laan et al. durchgeführt wurden. Wir etablieren
eine dynamische mean field Theorie, charakterisieren die kooperative Polymerisationsdy-
namik und bestimmen die erzeugte Kraft als Funktion der Wachstumsparameter und der
Elastizität des Hindernisses. In Abwesenheit von Rettungen und für ein steifes Hinder-
nis, steigt die maximale Polymerisationskraft linear mit der Anzahl an Mikrotubuli in
Übereinstimmung mit Resultaten von Laan et al. Für ein weiches Hindernis findet ein
Übergang zu einer logarithmischen Abhängigkeit statt. Für eine von Null verschiedene
Rettungsrate und ein weiches Hindernis finden wir eine oszillatorische Polymerisations-
dynamik, welche durch kollektive Katastrophen und kollektive Rettungen dominiert wird.
Im stationären Zustand oszilliert die Polymerisationskraft stabil um einen Mittelwert
und wächst linear mit der Anzahl an Mikrotubuli und der Tubulinkonzentration. Zum
Abschluss untersuchen wir die Gültigkeit der dynamischen mean field Theorie anhand
semi-stochastischer Rechnungen.
Contents
Title page 1
Abstract 5
Zusammenfassung 7
Contents 9
Acronyms 13
Thesis overview 15
1 Introduction and motivation 17
1.1 Microtubules and actin filaments . . . . . . . . . . . . . . . . . . . . . . . 17
1.2 Nuclear positioning in fission yeast by dynamic microtubule bundles . . . . 26
1.3 Membrane deformation by polymerizing microtubules . . . . . . . . . . . . 27
1.4 Optical trap experiments on microtubule bundles . . . . . . . . . . . . . . 28
2 Coarse-grained microtubule model 33
2.1 Single microtubule dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Force-dependent growth velocity . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Force-dependent catastrophe rate . . . . . . . . . . . . . . . . . . . . . . . 39
3 Single microtubule simulations 47
3.1 Single microtubule simulation model . . . . . . . . . . . . . . . . . . . . . 47
3.1.1 Confinement by rigid walls . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.2 Constant force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.3 Elastic force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.3.1 Vanishing rescue rate . . . . . . . . . . . . . . . . . . . . . 49
3.1.3.2 Non-zero rescue rate . . . . . . . . . . . . . . . . . . . . . 50
4 Single microtubule dynamics under force and confinement 51
4.1 Single microtubule dynamics under confinement by rigid walls . . . . . . . 52
10 CONTENTS
4.2 Single microtubule dynamics under a constant force . . . . . . . . . . . . . 55
4.3 Single microtubule dynamics under an elastic force . . . . . . . . . . . . . 58
4.3.1 Vanishing rescue rate . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.2 Non-zero rescue rate . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.2.1 Stochastic approach . . . . . . . . . . . . . . . . . . . . . 62
4.3.2.2 Mean field approach . . . . . . . . . . . . . . . . . . . . . 68
4.3.2.3 Dilution dynamics . . . . . . . . . . . . . . . . . . . . . . 71
4.4 Experimental and linear catastrophe model . . . . . . . . . . . . . . . . . . 73
4.4.1 Experimental catastrophe model by Janson et al. . . . . . . . . . . 74
4.4.1.1 Vanishing rescue rate . . . . . . . . . . . . . . . . . . . . . 74
4.4.1.2 Non-zero rescue rate . . . . . . . . . . . . . . . . . . . . . 75
4.4.2 Linear catastrophe model . . . . . . . . . . . . . . . . . . . . . . . 76
4.4.2.1 Vanishing rescue rate . . . . . . . . . . . . . . . . . . . . . 77
4.4.2.2 Non-zero rescue rate . . . . . . . . . . . . . . . . . . . . . 77
4.5 Generalized force-velocity relation . . . . . . . . . . . . . . . . . . . . . . . 78
4.5.1 Vanishing rescue rate . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5.2 Non-zero rescue rate . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5 First conclusion 81
6 Microtubule ensembles 83
6.1 Model for microtubule ensemble dynamics . . . . . . . . . . . . . . . . . . 83
7 Simulation of microtubule ensembles 87
7.1 Microtubule ensemble simulation model . . . . . . . . . . . . . . . . . . . . 87
7.1.1 Vanishing rescue rate . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.1.2 Non-zero rescue rate . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8 Microtubule ensemble dynamics 91
8.1 Microtubule bundle dynamics at zero rescue rate . . . . . . . . . . . . . . . 92
8.2 Microtubule bundle dynamics at non-zero rescue rate . . . . . . . . . . . . 94
8.2.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.2.2 Dynamical mean field theory . . . . . . . . . . . . . . . . . . . . . . 104
8.2.3 Limit cycle oscillations and absence of bifurcations . . . . . . . . . 105
8.2.4 Robustness with respect to catastrophe models . . . . . . . . . . . 109
8.2.5 Improved mean-field theory including stochastic fluctuations . . . . 113
8.2.6 Semi-stochastic approach . . . . . . . . . . . . . . . . . . . . . . . . 115
9 Final conclusion 127
10 Outlook 129
CONTENTS 11
A Literature values for MT growth parameters and simulation parameters131
Bibliography 133
Teilpublikationen 145
Danksagung 147
12 CONTENTS
Acronyms
OPDF overall probability density function
MT microtubule
MTs microtubules
MAPS microtubule-associated proteins
TUB tubulin
GTP guanosine triphosphate
GDP guanosine diphosphate
MFPT mean first-passage time
RNA ribonucleic acid
MTOC microtubule organizing center
ATP adenosine triphosphate
ADP adenosine diphosphate
14 CONTENTS
Thesis overview
In chapter 1 we provide the necessary biological background to this thesis. We focus on
the polymerization kinetics of microtubules and actin filaments and discuss three repre-
sentative examples of force generation by polymerizing microtubules, which provide the
basic motivation of this thesis.
In chapter 2 we present a coarse-grained model for microtubule polymerization under
a resisting force and in the presence of dynamic instability. We discuss force-velocity
relations and force-dependent catastrophe rates.
Chapter 3 deals with stochastic simulations of single microtubules, based on the coarse-
grained model presented in chapter 2. Simulation parameters and parameter ranges are
specified.
In chapter 4 we present the results for single microtubule polymerization dynamics in
three different confinement scenarios: (i) polymerization in confinement by fixed rigid
walls, (ii) polymerization under a constant force, and (iii) polymerization against an
elastically coupled obstacle. For all three scenarios, we calculate stochastic microtubule
length distributions, as a function of growth parameters. We address the relaxation
dynamics after growth parameters are perturbed and show that results are robust with
respect to changes in the force-velocity relation and the catastrophe model.
In chapter 6 the single microtubule model is extended to bundles of N parallelly growing
microtubules. We introduce force-sharing and focus on growth against an elastic obstacle.
This scenario is close to recent in vitro experiments.
Chapter 7 deals with stochastic simulations of microtubule bundles, based on the model
for microtubule bundles discusses in chapter 6.
In chapter 8 we present results for microtubule bundles growing against an elastic obstacle.
We establish a dynamical mean field theory and characterize the cooperative polymeriza-
tion dynamics, as a function of the growth parameters. Finally we test the validity of the
dynamical mean field theory, on the basis of semi-stochastic calculations.
16 CONTENTS
Chapter 1
Introduction and motivation
The cytoskeleton of a living cell is a complex and highly dynamical network of biological
filaments. Its static and dynamical properties are the basis for almost all mechanical
processes inside a living cell. Cell motility, positioning of cellular organelles, cell division
and shape generation are intracellular processes, governed by the cytoskeleton. The cy-
toskeleton is composed of cooperating networks of actin, intermediate, and microtubule
filaments. But also regulating proteins and molecular motors are parts of the cytoskele-
ton. The polymerization dynamics of microtubules and actin filaments, in cooperation
with molecular motors, are the key factors of the spatial and temporal organization and
reorganization of the cytoskeleton and are, therefore, of particular importance. Continu-
ous research and new experimental techniques, like TIRF-microscopy or optical tweezers,
have revealed numerous properties of single microtubule and actin filaments. But in living
cells the cooperation of single filaments or the cooperation of different filament types is
of central importance. Examples of the cooperation of single filaments are the mitotic
spindle in the case of microtubule filaments, and the contractile ring in the case of actin
filaments. Neuronal growth cones are a typical example for the cooperation of differ-
ent filament types. The investigation and characterization of cooperative polymerization
dynamics are subject of ongoing research and are also the main objects of this thesis.
In the following section we provide the biological background for this thesis. We focus
on the polymerization dynamics of microtubules and actin filaments and their regulation.
For a detailed biological introduction we refer to [1]. An introduction to biophysics can
be found in [2]. A recently published review article on microtubules can be found in Ref.
[3].
1.1 Microtubules and actin filaments
Microtubules (MTs) assemble from the α/β-tubulin heterodimer [1]. The heterodimer, in
the following called monomer, is ∼ 8 nm large [4, 5] and consists of a α- and β-tubulin
monomer, bound by a non-exchangeable GTP-nucleotide [6] (see Fig. 1.1). At the β-
18 Introduction and motivation
tubulin monomer a second, exchangeable GTP-nucleotide is bound, that can either be
in the GTP- (non-hydrolyzed) or in the GDP-form (hydrolyzed). Tubulin monomers
attach to each other in a "head-to-tail" fashion and form protofilaments. The α-end of
the new tubulin monomer is bound by the β-end of the previous monomer via the GTP-
nucleotide. MTs usually consist of 13 protofilaments, which form a hollow cylindrical tube
with an outer diameter of ∼ 25nm (see Fig. 1.1). Protofilaments are laterally bound by
non-covalent bonds, which are less stable than bonds between single tubulin monomers
within the protofilaments. The exact bond type and their strength are subject of current
research [7, 8]. MTs are typically 10− 50 µm long [9] and exhibit a persistence length
of ∼ 6 mm, which is larger than their typical length [2]. Under typical conditions MTs
can be seen as stiff and rigid polymers. The tubulin monomers within each protofilament
point in the the same direction and create a structural polarity. Since all protofilaments
within the MT are aligned parallelly, the MT itself shows a structural polarity. One can
distinguish two different ends: the "−"-end consists of α-tubulin while the "+"-end consists
of β-tubulin. Tubulin monomers in solution show different affinities to both MT ends (see
Fig. 1.1)[9]. The addition of new monomers is much faster at the "+"-end compared to
the "−"-end, which can, in general, be regarded as static. MT nucleation can be initiated
by an increase of the concentration of free tubulin above a critical concentration. In vivo,
nucleation is supported by different types of proteins [1]. In centrosomes, a special type of
microtubule-organizing center (MTOC), γ-tubulin promotes the nucleation of MTs [2, 10].
The α-end of tubulin monomers binds to the γ-tubulin complex, which is tightly bound to
the MTOC. The "−"-end of the new nucleated MT is fixed at the MTOC and is protected
against depolymerization, while the "+"-end freely polymerizes in the cytosol. From the
MTOC, often located close to the cell nucleus, MTs grow through the cell periphery
towards the cell membrane. In in vitro experiments, nucleation of new MTs can also be
initiated by an increase of the temperature [11, 12].
Filamentous actin (F-actin) assembles from actin monomers. Actin monomers are ∼ 5nm
long and possess a ATP-/ADP-nucleotide binding side [1, 2]. As tubulin monomers, actin
monomers attach to each other in a "head-to-tail" fashion and form protofilaments with
a distinct structural polarity. As in the case of MTs we find a fast-growing "+"-end and
a slow-growing "−"-end with the nucleotide binding side pointing towards the "−"-end
(see Fig. 1.2). F-actin consists of two protofilaments, that form a right-handed helix of
5−9nm diameter [1], and exhibits a typical length of ∼ 5 µm [13]. Its persistence length is
∼ 15 µm, which is smaller than the typical filament length. F-actin filaments can therefore
be seen as a thin and flexible, compared to the rather stiff and cylindrical MT structure
(compare Figs. 1.1 and 1.2). The nucleation of F-actin can also be initiated by an increase
of the free actin concentration above a critical concentration [1]. As in the case of MTs,
there are several proteins, which provide actin nucleation in vivo[1]. The Arp2/3-complex
resembles the "−"-end of an actin filament and serves as a seed for new actin filaments.
1.1 Microtubules and actin filaments 19
Figure 1.1: The structure of a microtubule and its subunits. (A) The subunit
of each protofilament is the tubulin heterodimer, formed by a very tightly linked
pair of α- and β-tubulin monomers. The GTP molecule in the α-tubulin monomer
is so tightly bound that it can be considered an integral part of the protein. The
GTP molecule in the β-tubulin monomer, however, is less tightly bound and has an
important role in filament dynamics. Both nucleotides are shown in red. (B) One
tubulin subunit (α−β heterodimer) and one protofilament are shown schematically.
Each protofilament consists of many adjacent subunits with the same orientation.
(C) The microtubule is a stiff hollow tube formed from 13 protofilaments aligned in
parallel. (D) A short segment of a microtubule viewed in an electron microscope. (E)
Electron micrograph of a cross section of a microtubule showing a ring of 13 distinct
protofilaments. Picture and caption taken from Ref. [1].
20 Introduction and motivation
Figure 1.2: The structure of an actin monomer and actin filament. (A) The actin
monomer has a nucleotide (either ATP or ADP) bound in a deep cleft in the center
of the molecule. (B) Arrangement of monomers in a filament. Although the filament
is often described as single helix of monomers, it can be thought of as consisting of
two protofilaments, held together by lateral contacts, which wind around each other
as two parallel strands of a helix, with a twist repeating every 37 nm. All the subunits
within the filament have the same orientation. (C) Electron micrograph of negatively
stained actin filaments. Picture and caption taken from Ref. [1].
The Arp2/3 complex also generates a characteristic branching of actin filaments. The
branching of actin filaments in the presence of the Arp2/3 complex is explained in more
detail later in the text. The Formin-complex actively binds free actin monomers from
solution and serves as a nucleation seed [1]. In addition, the Formin-complex remains
associated with the "+"-end of the actin filament and supports the incorporation of new
actin monomers into the filament. There are no analogues to the Arp2/3 complex and
Formin proteins for MTs. Most actin filaments are located close to the cell membrane
and form, in cooperation with actin regulation proteins, the cell cortex [1].
The combination of structural polarity, different monomer affinities and hydrolysis of the
NTP-nucleotide leads to complex polymerization kinetics of MTs and actin filaments.
For GTP-tubulin in solution hydrolysis proceeds very slow. In filamentous form tubu-
lin acts as a GTPase [3, 14], which catalyzes the hydrolysis of the exchangeable GTP-
nucleotide. Within the filament, GTP-tubulin hydrolyzes significantly faster to GDP-
tubulin, compared to GTP-tubulin in solution[1]. In the process of hydrolysis, a phos-
phate group is released from the GTP-nucleotide and the GDP-nucleotide remains bound
within the filament. Thus, hydrolysis leads to two different types of filament structures:
one consisting of GTP-tubulin and one consisting of GDP-tubulin (see Fig. 1.3).
Since GTP-tubulin shows a higher affinity for the "+"-end of a MT, new GTP-tubulin
monomers are primarily incorporated into the "+"-end, compared to the "−"-end. In
addition, the "−"-end is often bound to a γ-tubulin complex, which prevents the assembly
1.1 Microtubules and actin filaments 21
DDDDD D T T T
DT T T T
T T T T T
−
−
−
+
+
+
catalyzed hydrolysis
fast assemble
no or slow
assemble
GTP-cap
Figure 1.3: Sketch of MT polymerization and hydrolysis of GTP-tubulin. (+): "+"-
end of the MT. (-): "−"-end of the MT. T denotes GTP-tubulin monomers (green).
D denotes GDP-tubulin monomers (red). From top to bottom: New GTP-tubulin
monomers are attached to the "+"-end of the MT with high affinity. No or only
few GTP-tubulin monomers are attached to the "−"-end (top). Within the filament,
GTP-tubulin is hydrolyzed to GDP-tubulin. Hydrolysis is catalyzed by the GTPase
activity of the tubulin monomer (center). As hydrolysis proceeds, only GDP-tubulin
remains at the "−"-end of the MT, whereas a cap of GTP-tubulin is formed at the
"+"-end (bottom).
of GTP-tubulin monomers into the "−"-end. As hydrolysis proceeds within the filament,
the loss of GTP-tubulin at the "+"-end is balanced by new GTP-tubulin from solution,
while the amount of GDP-tubulin at the "−"-end increases. This results in a large patch
of GDP-tubulin at the "−"-minus end and a patch of GTP-tubulin at the "+"-end, which
is often referred as the GTP-cap (see Fig. 1.3). If hydrolysis overtakes the assembly
of new GTP-tubulin monomers, the GTP-cap is lost and the whole filament consists of
GDP-tubulin only (see Fig. 1.3). The interplay of incorporation of GTP-tubulin, catalyzed
hydrolysis and the loss of the GTP-cap causes the characteristic polymerization behaviour
of MTs, called dynamic instability, which we discuss in greater detail in the following
paragraph.
MTs undergo a stochastic transition between periods of moderate growth and fast de-
polymerization, called dynamic instability [9]. In phases of growth, new GTP-tubulin
monomers are incorporated in the "+"-end and the MT growth on average. Depending
on the concentration of free tubulin we find growth velocities of up to ∼ 5 µm/min. Typ-
ical values of growth velocities are listed in table A.1. Periods of growth are ended by
a complex process called catastrophe. Under typical conditions we find a wide range,
(10−4 . . .10−2) s−1, of catastrophe frequencies [15, 16]. A detailed description of catas-
trophes follows in the next paragraph. In a period of shrinkage, large parts of single
protofilaments curl outwards and depolymerize. The MT rapidly loses length with an
22 Introduction and motivation
average velocity of ∼ 10 µm/min and switches back to a period of growth, in a process
called rescue. Again there is a wide range, (10−2 . . .10−1) s−1, of rescue frequencies. Typ-
ical values of the velocity of shrinkage and the rescue frequency are collected in table A.1.
Electron micrographs of a growing and a shrinking MT are shown in Fig. 1.4. In Fig.
1.5 a typical MT length trajectory, obtained from experiments, is shown. It is generally
accepted, that the hydrolysis of GTP- to GDP-tubulin within the MT causes dynamic
instability. However, the exact mechanism is still subject to ongoing research. In the
following, we discuss two possible mechanisms that lead to catastrophes. In a period of
growth, new GTP-tubulin monomers are incorporated into the "+"-end of a MT and are
hydrolyzed to GDP-tubulin on a characteristic timescale. The characteristic timescale is
still not know in detail, but current research suggest values of ∼ 1 s [7, 17, 18]. Thus a
MT consists of a section of GTP-tubulin, located at the "+"-end and called GTP-cap, and
a body of GDP-tubulin. In a purely chemical mechanism, bonds between protofilaments
are weakened by the hydrolysis of GTP-tubulin, while the remaining GTP-cap stabilizes
the MT structure. If the GTP-cap is lost or reduced to a critical size, the MT structure
is destabilized, protofilaments begin to depolymerize and a catastrophe is initiated [19].
A sketch of the stabilizing GTP-cap is shown in Fig. 1.4.
Another mechanism leading to catastrophes is based on the structural plasticity of the
MT lattice [20]. In the context of the structural plasticity mechanism, it is assumed that
GDP-tubulin prefers a curved configuration compared to a rather straight configuration of
GTP-tubulin. This curvature generates additional stress within the MT. As in the pure
chemical mechanism, the GTP-cap stabilizes the MT structure and its loss triggers a
catastrophe. However, the kinetics governing the catastrophe mechanism can be complex
and are not fully understood. A possible kinetics is the formation of cracks, propagating
through the MT lattice. A critical amount of cracks can destabilize the MT structure and
initiate a catastrophe [21, 22].
Mechanisms that induce rescue events are less investigated, compared to catastrophe
mechanisms. Current research results indicate, that remaining patches of non-hydrolyzed
GTP-tubulin within the MT structure can induce rescue events [23, 24]. However, a basic
theoretical description of rescue events is still missing.
Dynamic instability is the key player in the spatial and temporal organization of the
MT cytoskeleton. In Secs. 1.2 and 1.3 we discuss two representative examples of cellular
processes, which are governed by the reorganization of the MT cytoskeleton.
We compare the polymerization dynamics of actin filaments to MT polymerization and,
therefore, focus on the "+"-end of an actin filament and do not discuss polymerization
kinetics at the "−"-end. The "−"-end of an actin filament can be bound to stabilizing
proteins, such like the Arp2/3 complex, and can in such a case be regarded as static.
A detailed description of actin polymerization, including polymerization kinetics at both
filament ends, can be found in [1] and [2].
1.1 Microtubules and actin filaments 23
Figure 1.4: In an intact microtubule, protofilaments made from GDP-containing
subunits are forced into a linear conformation by many lateral bonds within the mi-
crotubule wall, given a stable cap of GTP- containing subunits. Loss of the GTP-cap,
however, allows the GDP-containing protofilaments to relax into their more curved
conformation.This leads to a progressive disruption of the microtubule. Next to the
drawing of a growing and shrinking microtubule, electron micrographs show actual
microtubules in each of these two states, as observed in preparation in vitreous ice.
Note the particular curling, disintegration GDP-containing protofilaments at the end
of the shrinking microtubule. Picture and caption taken from Ref. [1].
Figure 1.5: Life history of a microtubule undergoing dynamic instability inside a
living cell. The rates of growth and rapid shortening are constant for extended periods,
and the transitions are abrupt and stochastic. From Cassimeris et al. [25]. Picture
and caption taken from Ref. [14].
24 Introduction and motivation
DD T T T T
DDDD D D D
−
−
+
+
catalyzed hydrolysis
ATP-cap
fast
polymerization
fast
depolymerization
Figure 1.6: Sketch of actin polymerization and hydrolysis of ATP-Actin. (+): "+"-
end of the actin filament. (-): "−"-end of the actin filament. T denotes ATP-actin
monomers (green). D denotes ADP-actin monomers (red). Top: New ATP-tubulin
monomers are attached to the "+"-end of the actin filament and form an ATP-cap,
while hydrolysis proceeds within the filament. The actin filament growth on average.
Bottom: The ATP-cap is lost and ADP-actin monomers depolymerize from the "+"-
end with a high rate. The actin filament shrinks on average.
In filamentous form actin acts as an ATPase and catalyzes the hydrolysis of ATP-actin
to ADP-actin. As in the case of MTs, the combination of fast attachment of ATP-actin
monomers at the "+"-end and hydrolysis leads to an increase of ADP-actin at the "−"-
end and to an ATP-cap at the "+"-end (see Fig. 1.6). If monomer attachment is fast
compared to hydrolysis, the actin filament growth on average. We find average growth
velocities of ∼ 0.2 . . .0.5 µm/min [2, 26, 27]. If hydrolysis overtakes monomer attachment,
the ATP-cap is lost. ADP-actin monomers depolymerize from the "+"-end with a rate of
∼ 7 s−1 and the actin filament begins to shorten [2]. With a monomer size of ∼ 5nm and a
depolymerization rate of ∼ 7s−1, we find an average velocity of shrinkage of ∼ 2 µm/min
[2]. This dynamic transition from average growth to average shrinkage, resembles the
dynamic instability of MTs. The loss of the ATP-cap and the subsequent loss of length
are similar to MT catastrophes and phases of fast shrinkage. But it should be noted that
in the case of actin, single ADP-monomers depolymerize from the "+"-end, whereas in the
case of MTs, large parts of protofilaments curl outwards and break into smaller pieces.
The underlying mechanism of shrinkage might be different in actin filaments and MTs.
Actin filaments do not show rescue events and depolymerize completely. This is another
difference to MT dynamical instability, where phases of shrinkage are ended by rescue
events.
The MT cytoskeleton and the actin cytoskeleton are both strongly regulated by proteins.
There is a multitude of well known regulating proteins. In the following, we focus on pro-
teins which modulate the polymerization kinetics of MTs and actin filaments, and support
their spatial arrangement within the cell. The polymerization dynamics of MTs is affected
by several regulating proteins. Stathmin proteins bind free GTP-tubulin monomers from
1.1 Microtubules and actin filaments 25
solution and prevent their assembly. As a consequence, the velocity of growth decreases
and the catastrophe frequency increases, since no new stabilizing GTP-tubulin monomers
are incorporated into the MT-tip [28]. Kinesin-8/-13 proteins (unconventional kinesins)
destabilize the "+"-end and induce catastrophes [1, 22, 29]. In contrast, XMAP125 pro-
teins stabilize the "+"-end and accelerate the assembly of new tubulin monomers [1]. Tau
and MAP2 proteins bind to the MT and increases the frequency of rescue events (see
table A.1). In addition, both proteins support the formation of MT bundles. Along with
MTs, MAP2 proteins form bundles of widely spread MTs, whereas Tau proteins promote
the formation of closely packed MT bundles. As mentioned above, the γ-tubulin complex
serves as a nucleation seed and protects the "−"-end against depolymerization. MT regu-
lating proteins modulate basic properties of the dynamic instability, but do not completely
suppress the dynamic instability. In addition to MT regulating proteins, chemical drugs
are know that influence the polymerization kinetics of MTs. Two examples are nocodazol,
which binds tubulin monomers and prevent their incorporation, and taxol, which bind to
MTs and stabilizes them. A large list of MT regulating proteins can be found in [1].
Not only MTs are regulated by intracellular proteins, but also the polymerization kinetics
of actin filaments. Profilin proteins bind free actin monomers from solution and support
their incorporation into the "+"-end of the actin filament. The velocity of growth is
increased. In contrast, thymosin proteins bind actin monomers, but prevent assembly.
Also Formin proteins, mentioned above, promote the assembly of new actin monomers.
Cofilin proteins bind to existing actin filaments and destabilize the filament. The Arp2/3
complex, bound to preexisting actin filaments, nucleates new actin filaments at a 70°
angle relative to the original filament. This branching results in a dense, dynamic network
of actin filaments. Filamin proteins connect two single actin filaments at a 90° angle,
whereas Fimbrin and α-Actinin proteins support the formation of bundles, which consist
of parallel actin filaments. A large list of actin regulating proteins can be found in [1].
We now discuss three representative examples for the spatial and temporal organization
of polymerizing MTs in a confining environment, the complex interplay of force generation
and dynamic instability, and in the case of in vitro experiments, the connection to cellular
processes.
26 Introduction and motivation
1.2 Nuclear positioning in fission yeast by dynamic
microtubule bundles
The fission yeast schizosaccharomyces pombe bacterium exhibits an elliptic form, with a
length of ∼ 10 µm and a width of ∼ 3 µm. The stiff cell membrane, which can not be
deformed by the internal cytoskeleton, encloses a cell nucleus of ∼ 1µm diameter [see Fig.
1.7(a)]. In cell division it is necessary that the cell nucleus maintains its position close to
the cell center, since the nucleus determines the future division plane. A deflection from
its central position during cell division leads to incomplete chromosome separation and
thus to cell death. To ensure a precise positioning of the cell nucleus, schizosaccharomyces
pombe possesses a repositioning mechanism, based on dynamic MT bundles [30, 31, 32].
Starting from the poles of the nucleus, 3− 4 MT bundles grow along the length axis of
the bacterium, directed towards the cell membrane. The bundles consist of 2− 3 MTs
with their minus ends connected to a MTOC and their plus ends undergoing dynamic
instability (see Fig. 1.7). After a catastrophe, MTs shrink back to the nucleus and are
renucleated. Within the bundles, MTs are parallelly arranged and stabilized by MAPS
[33], whereas bundles are aligned anti-parallel to each other. If the cell nucleus is displaced
from its central position by cell movement or by centrifugation [34], MTs on the deflection
side contact the cell membrane before opposing MTs do. Contact MTs polymerize against
the cell membrane and generate a force, which drives the nucleus back to its central po-
sition [see Fig. 1.7(b)]. Under typical condition MTs do not buckle considerably, so that
polymerization can be seen as the only source of force generation. In in vivo experiments,
MTs push the cell nucleus with a velocity of ∼ 0.5 µm/min through the cytosol [34], cor-
responding to a constant force of ∼ 1 pN, which can be estimated from Stokes law [35].
In addition, experimental results show a decreased velocity of growth (∼ 1.3 µm/min) of
pushing MTs compared to force-free growing MTs (∼ 2 µm/min) [34, 36]. MTs in contact
with the cell membrane also exhibit an increased catastrophe frequency [36], indicating a
connection of decreased growth velocity and increased catastrophe frequency. The repo-
sitioning is stopped, if the opposing MTs are also in contact with the cell membrane and
counterbalance the pushing force. The nucleus is again placed in the center of the cell
and undergoes oscillations of small amplitude, caused by MT length fluctuations. The
repositioning of the cell nucleus by dynamic MT bundles could also be reproduced in simu-
lations [37]. However, several questions remain to be answered: Do rescue events of single
MTs influence the repositioning, although they are not observed in experiments? Do MT
bundles generate higher forces? To answer these questions, it is necessary to investigate
polymerization dynamics of single MTs and MT bundles in a confining environment and
under a resisting force.
1.3 Membrane deformation by polymerizing microtubules 27
(a) (b)
pushing MTs
MTOC
growing MT
shrinking MT
Force
Figure 1.7: (a): Fluorescence microscopy images of schizosaccharomyces pombe.
Schizosaccharomyces pombe strain expressing GFP-tubulin and the nuclear pore
marker Nup85-GFP. Picture and caption taken from Ref. [37]. (b): Sketch of nu-
clear positioning mechanism. The dashed black line marks the center of the cell.
Polymerizing MTs (green) drive the nucleus (blue) back to its central position. For
reasons of clarity, we only show one MT per bundle.
1.3 Membrane deformation by polymerizing micro-
tubules
In vitro polymerizing MTs are able to generate polymerization forces high enough to
deform elastic lipid membranes. Fygenson et al. investigated the deformation of phos-
pholipid vesicles by enclosed MTs [38]. Tubulin solution is introduced into a vesicle and
MT nucleation is initiated by an increase of temperature. Inside the vesicle, 1− 3 MTs
polymerize towards the lipid membrane and start to deform the nearly spherical vesicle
(diameter ∼ 10 µm) into an ellipsoid (see Fig. 1.8). As MT polymerization continues,
two thin membrane tubes form around the MTs. The vesicle assumes a shape resembling
the Greek later φ (see Fig. 1.8). Membrane tubes are typically 2− 10 times larger than
the initial diameter of the vesicle. For large deformations (> 20 µm), Fygenson et al.
observed MTs buckling under their own polymerization force. After catastrophes, MTs
depolymerize completely and the elastic vesicle assumes its initial shape. A shape analysis
of the deformed vesicle yields polymerization forces up to ∼ 4 pN, which are in the same
range as forces generated in fission yeast (see Sec. 1.2). In vitro, polymerizing MTs can
deform lipid membranes and generate deformations in the µm range. In vivo, membrane
deformation by polymerizing MTs may play a major role during neuronal outgrowth or
in guidance of neuronal growth cones [39, 40]. However, cell membranes are compounds
of lipid membranes, membrane proteins, and the underlying cell cortex, and are, in gen-
eral, less elastic than pure lipid membranes. It remains to be answered, if single MTs
can deform cell membranes in vivo or if bundles of simultaneously polymerizing MTs are
required to generate sufficient forces. In addition, it is unclear if membrane deformation
can be regulated by MT growth parameters, like the tubulin concentration. To answer
these questions, it is necessary to investigate the force generation of single MTs and MT
28 Introduction and motivation
Figure 1.8: A phospholipid vesicle deformed by 1 to 3 microtubules (observed float-
ing freely prior to vesicle deformation). The number at lower right is the length of
the long axis in microns. This vesicle fortuitously stuck to the glass slide (spot below
center), restricting its rotational diffusion. The final frame has a different vesicle since
the other never lay in the focal plane as a φ shape. Picture and caption taken from
Ref. [38].
bundles, growing against an elastic lipid membrane or, more generally spoken, growing
against an elastic obstacle.
1.4 Optical trap experiments on microtubule bundles
We now focus on a special type of in vitro experiments, namely optical trap experi-
ments, whose results are the basic motivation for this thesis. In the following we present
the fundamental design of optical traps, discuss current research results and illustrate
commonalities of experiments with optical traps and theoretical modelling. There is an
extensive amount of literature on optical traps. We restrict our discussion on fundamental
works and present review articles for further reading [41, 42, 43, 44].
Ashkin et al. performed the fundamental works on optical traps in the late 1980s [45] and
early 1990s [46]. Optical traps consist of a controllable device of optical lenses, multiple
lasers [47] and one or more dielectric particles [48]. The laser light is highly focused
and a particle is placed in the focus. The laser light and the particle interact in two
different ways [42]. Incident photons transfer momentum to the particle and, according to
Newtons law, this momentum transfer results in a force acting on the particle. This force
is termed scattering force. The scattering interaction and the dipole interaction generate
a harmonic potential, in which the focus corresponds to the potential minimum. The
dielectric particle is soundly trapped within the harmonic potential, with the minimum
of the harmonic potential, i.e. the focus, as its equilibrium position. If the particle is
1.4 Optical trap experiments on microtubule bundles 29
deflected from its equilibrium position, or if the position of the focus is changed by the
controllable lens system, the harmonic potential exerts a resiting force, directed towards
the focus. The resisting force increases linear with the displacement distance from the
focus. Its amplitude is determined by the steepness of the harmonic potential. The
fundamental properties of the harmonic potential, like its steepness or its spatial extent,
can be adjusted by varying the intensity of the laser light or the spatial form of the focus.
This allows a precise force control. Common realizations are approximately constant forces
and spring-like elastic forces, linearly increasing with the displacement distance from the
focus [17]. In experiments the steepness of the harmonic potential is determined from
fluctuations of the dielectric particle around its equilibrium position. The displacement
distance of the particle from its equilibrium position and its return dynamics can be
measured with high accuracy. Spatial and temporal resolutions of current optical traps
are in the range of nanometers and microseconds [43].
Optical traps are widely used to study the effect of in- or external forces in biological pro-
cesses. They are successfully applied to the investigation of force generation by molecular
motors [49], folding kinetics of RNA [50] or polymerization kinetics of biopolymers [51].
In the following we focus on experiments on MTs [17, 52], and in particular on the force
generation by MT bundles [53].
In optical trap experiments on MTs, a MT seed is attached to a silica bead. The bead is
placed in a microchamber and trapped in the focus of the optical trap [see Fig. 1.9(a)].
When GTP-tubulin solution is introduced, the MT starts to polymerize until its tip
reaches a wall of the microchamber. The polymerizing MT now deflects the bead from its
equilibrium position. The displacement is measured and offers valuable information on
the current MT length, the incorporation dynamics of new monomers [17, 52], its velocity
of growth and on the magnitude of the polymerization force, generated by the MT [17].
The MT polymerizes until its growth is stalled by the resisting force or a catastrophe
occurs. In general, no rescue events are observed in this type of experiments and the
experiment must be restarted with a new MT seed. Both the stall force and the time
spent in the growing state, that is the time spent polymerizing against the resisting force,
are fundamental quantities. They contain detailed information on force-velocity relations
and on mechanisms that lead to catastrophes [17].
Only recently it became possible to study bundles of parallelly growing MT in optical traps
[53]. Laan et al. investigated the force generation by bundles of parallel MTs growing
simultaneously against an elastic force, realized by an optical trap [see Fig. 1.9(b)]. The
experiments by Laan et al. yield two main results:
• Polymerizing MT bundles generate considerably higher forces, compared to a single
MT. Laan et al. showed that the polymerization force of a MT bundle growths
strictly linear in the number of MTs within the bundle, so that N MTs produce
the N-fold polymerization force of a single MT under same experimental conditions.
30 Introduction and motivation
At a tubulin concentration of ∼ 25 µM, Laan et al. found a polymerization force
of ∼ 2.7 pN for a single MT. Under the same experimental conditions, two MTs
generated a polymerization force of ∼ 5.5 pN and three MTs ∼ 8.1 pN, which are
approximately multiples of ∼ 2.7 pN. The force was shared equally between all MTs
within the bundle.
• During the experiments Laan et al. observed cooperative dynamics within the MT
bundle, so called collective catastrophes. At forces close to the maximal polymer-
ization force of the bundle, a catastrophe of a single MT lead to a cascade of catas-
trophes. The force acting on the remaining MTs was increased and within no time,
each MT within the bundle switched into a state of shrinkage and the whole bundle
entered a state of collective shrinkage. Laan et al. termed this collective behaviour
a collective catastrophe.
In stochastic simulations, both the linear dependence of the polymerization force on the
MT number and collective catastrophes could be reproduced by Laan et al. However, in
the performed simulations Laan et al. used an artificially increased catastrophe rate and
investigated a rather small set of parameter values. In addition an underlying theoretical
description of MT-bundle polymerization dynamics is still missing.
We establish a detailed theoretical model for the cooperative polymerization dynamics of
MT bundles growing against an elastic force. The presented model reproduces experi-
mental results, puts them on a solid theoretical basis and allows us to investigate a wide
set of parameters. We also include rescue events, which have not been considered by Laan
et al. Rescue events of single MTs lead to a new type of collective behaviour, so called
collective rescues [54]. Although rescue events have not been observed in optical trap ex-
periments so far, they play a crucial role in vivo and due to continuous improvements of
experimental techniques, it is only a matter of time until rescue events become accessible
in optical trap experiments.
The last paragraph illustrates another advantage of experiments with optical traps. This
type of experiment is performed under well controlled experimental conditions and can
easily be reproduced. In addition, the composition of optical trap experiments can be
reduced down to basic components, like a rigid wall or a spring-like resisting force. These
basic components can be described by theoretical models, without too much computa-
tional effort. On the other hand optical trap experiments can be designed according to
basic theoretical models and offer a good approach to merge theoretical modelling and in
vitro experiments.
1.4 Optical trap experiments on microtubule bundles 31
(a) (b)
Figure 1.9: (a) Differential-interference-contrast micrograph of an experiment show-
ing the bead, microtubule, and barrier. Picture and caption taken from Ref. [17]. (b)
Schematic picture of a MT bundle nucleated by an axoneme growing against a barrier.
A construct made of a bead attached to the axoneme is maintained in position by a
"keyhole" trap: A single strong trap holds the bead while many shallow traps form a
line trap constraining the axoneme’s movement along a single direction. Picture and
caption taken from Ref. [53].
32 Introduction and motivation
Chapter 2
Coarse-grained microtubule model 1
In this chapter we present a coarse-grained single MT model, which includes the dynamic
instability and introduce the basic notation. Section 2.1 deals with single MT dynamics
in the absence of force. In Section 2.2 the polymerization velocity under an opposing
force is discussed, based on the Brownian ratchet model and is compared to a more
generalized force-velocity relation by Kolomeisky et al. In Section 2.3 we first introduce
the cooperative catastrophe model by Flyvbjerg et al. in the absence of force. Later the
catastrophe rate is connected to an opposing force via the force-velocity relation (see Sec.
2.2). In addition, we present a catastrophe model based on experimental data, and a
linear catastrophe model. Their differences and similarities are discussed.
2.1 Single microtubule dynamics
The MT dynamics in the presence of its dynamic instability is described in terms of prob-
ability densities and switching rates [56, 57]. In the growing state, a MT polymerizes
with average velocity v+. Typically we find v+ = (0.7−8)×10−8 m/s = (0.4−5) µm/min
(Table A.1). The MT stochastically switches from a state of growth (+) to a state of
shrinkage (−) with the catastrophe rate ωc. In the shrinking state, it rapidly depolymer-
izes with an average velocity v− ' 3×10−7m/s (Table A.1). With the rescue rate ωr the
MT stochastically switches from a state of shrinkage back to a state of growth. We model
catastrophes and rescues as Poisson processes such that 〈τ+〉= 1/ωc and 〈τ−〉= 1/ωr are
the average times spent in the growing and shrinking states, respectively.
The stochastic time evolution of an ensemble of independent MTs, growing along the
x-axis, can be described by two coupled master equations for the probabilities densities
p+(x,t) and p−(x,t) of finding a MT with length x at time t in a growing or shrinking
1Parts of the text have been published in reference [55] and are ©2012 American Physical Society.
http://publish.aps.org/copyrightFAQ.html
34 Coarse-grained microtubule model
state,
∂tp+(x,t) =−ωcp+(x,t)+ωrp−(x,t)−v+∂xp+(x,t) (2.1)
∂tp−(x,t) = ωcp+(x,t)−ωrp−(x,t)+v−∂xp−(x,t). (2.2)
In the following, we will always use a reflecting boundary at x= 0: A MT shrinking back
to zero length undergoes a forced rescue instantaneously. This corresponds to
v+p+(0, t) = v−p−(0, t). (2.3)
A more refined model including a nucleating state, instead of a reflecting boundary
condition at x = 0, has been considered in ref. [58]. For a constant and fixed catastrophe
rate ωc, eqs. (2.1) and (2.2), together with the boundary condition (2.3), can be solved
analytically on the half-space x > 0, and we can determine the overall probability density
function (OPDF) of finding a MT with length x at time t, P (x,t) ≡ p+(x,t)+ p−(x,t)
[56, 57]. The solution exhibits two different phases: a phase of bounded growth and a
phase of unbounded growth.
In the phase of bounded growth the average length loss during a period of shrinkage,
v−〈τ−〉 = v−/ωr, exceeds the average length gain during a period of growth, v+〈τ+〉 =
v+/ωc. The steady-state solution of P (x,t) assumes a simple exponential form P (x) =
|λ|−1e−x/|λ| with an average length 〈x〉= |λ| and a characteristic length parameter
λ≡ v+v−
v+ωr−v−ωc
, (2.4)
with λ−1 < 0 for bounded growth [56]. The transition to the regime of unbounded growth
takes place at λ−1 = 0, where the average length gain during growth equals exactly the
average length loss during shrinkage,
v+ωr = v−ωc, (2.5)
such that 〈x〉 diverges.
In the regime of unbounded growth with λ > 0, the average length gain during a period
of growth is larger than the average length loss during a period of shrinkage. There is
no steady state solution, and for long times P (x,t) asymptotically approaches a Gaussian
distribution
P (x,t)≈ 1
2
√
piDJ t
exp
[
−(x−Jt)
2
4DJ t
]
(2.6)
centered on an average length which approaches linear growth 〈x〉 ≈ Jt with a mean
2.1 Single microtubule dynamics 35
0
0.01
0.02
0.03
0.04
0 25 50 75 100
P
(x
)[
µm
−
1 ]
x [µm]
Pλ<0(x)∼e−x/|λ|
Pλ>0(x)∼e
− (x−Jt)
2
4DJt
t = 5000 s
J > 0
Figure 2.1: OPDF P (x,t) = p+(x,t)+p−(x,t) as solution of eqs. (2.1) and (2.1). Red
line: Steady-state OPDF P (x) for bounded growth. We set v+ = 2×10−8 m/s, v− =
3× 10−7 m/s, ωc = 0.0025 s−1, and ωr = 0.05 s−1, corresponding to a characteristic
length parameter λ≈−4×10−5 [eq. (2.4)]. Green line: OPDF P (x,t) for unbounded
growth and t = 3600 s from eq. (2.6). We set v+ = 2×10−8 m/s, v− = 3×10−7 m/s,
ωc = 0.0025 s−1, and ωr = 0.1 s−1, corresponding to an average growth velocity J ≈
1.2×10−8 m/s [eq. (2.7)].
velocity J and with diffusively growing width 〈x2〉−〈x〉2 ≈ 2DJ t with a diffusion constant
DJ [56, 57].
The average growth velocity is given by
J = v+ωr−v−ωc
ωr+ωc
(2.7)
because the asymptotic probabilities to be in a growing or shrinking state are pi+ =
ωr/(ωc+ωr) and pi− = ωc/(ωc+ωr), respectively. The diffusion constant DJ is
DJ =
ωcωr(v++v−)2
(ωc+ωr)3
. (2.8)
The transition between the two growth phases can be achieved by changing one of the
four parameters of MT growth, ωc, ωr, v+, or v−. In the following, we will use catastrophe
models, where the catastrophe rate ωc is a function of the growth velocity v+, which in
turn is determined by the GTP-tubulin concentration via the GTP-tubulin on-rate ωon
(assuming a fixed off-rate ωoff). Moreover, experimental data suggest that v− is fixed to
values close to ∼ 10−7 m/s (Table A.1). As a consequence, there are two tunable control
parameters left, the GTP-tubulin concentration or, equivalently, the tubulin on-rate ωon
and the rescue rate ωr.
These are the control parameters we will explore for single MT dynamics in the context
36 Coarse-grained microtubule model
∆x
D
ωon ωoff
d
x
F
Figure 2.2: Sketch of the Brownian ratchet model. A single MT (green) polymer-
izes along the x-axis under an opposing force F , with GTP-tubulin on-rate ωon and
GTP-tubulin off-rate ωoff. The particle diffuses freely along the x-axis with diffusion
coefficient D. A new GTP-tubulin dimer is incorporated into the MT tip, only if a
gap of size ∆x = d is created by the diffusing particle.
of this thesis.
These parameters are also targets for regulation by MAPS, such as OP18/stathmin, which
reduces ωon by binding to GTP-tubulin dimers [28] or MAP4, which increases the rescue
rate ωr [59] (see Sec. 1.1).
Typical values for MT growth parameters can be found in appendix A, Table A.1.
2.2 Force-dependent growth velocity
In a growing state, GTP-tubulin dimers are attached to any of the 13 protofilaments with
the GTP-tubulin on-rate ωon, which is directly related to the GTP-tubulin concentration.
We explore a regime ωon =30, ...,100s−1, see Table A.1. GTP-tubulin dimers are detached
with the rate ωoff = 6s−1 [60] such that we can typically assume ωon  ωoff.
In the absence of force or restricting boundaries, the average velocity of growth is given
by
v+(0) = d(ωon−ωoff) . (2.9)
Here d denotes the effective dimer size d≈ 8nm/13≈ 0.6nm.
To include force generation by polymerizing MTs into the coarsed-grained single MT
model, we introduce a special mechanism, the Brownian ratchet model, which couples
MT polymerization to force generation [61]. A sketch of the Brownian ratchet model is
presented in Fig. 2.2.
A single MT polymerizes in x-direction. A particle, which undergoes diffusive motion
with diffusion coefficient D, is placed in front of the MT. F is an opposing force and
acts along the x-axes. The diffusive motion of the particle generates a gap of size ∆x
between the MT tip and the particle surface. Only if the gap size equals the size of
a dimer, thus ∆x = d, a new dimer can be incorporated into the MT tip. Then the
2.2 Force-dependent growth velocity 37
MT elongation is governed by the GTP-tubulin on-rate ωon and the diffusion coef-
ficient D of the particle. Depending on the ration 2D/(ωond2) we can distinguish to cases:
diffusion-limited-polymerization: 2D/(ωond2) 1
In the diffusion-limited case the average time to diffuse a distance d, 2D/d2, is small
compared to the inverse GTP-tubulin on-rate ω−1on . The inverse GTP-tubulin on-rate ω−1on
is larger than the average time it takes the particle take to diffuse a distance ∆x = d
under an opposing force F . As soon as a gap of size ∆x = d is created by the diffusing
particle, a GTP-tubulin dimer drops in and is incorporated into the MT tip. The inverse
GTP-tubulin on-rate ω−1on nearly equals the average time the particle take to diffuse a
distance ∆x = d under an opposing force F . This is a mean-first passage time problem,
and can analytically be treated in the context of Kramers rate theory [62, 63]. Detailed
calculations can be found in various textbooks [2, 62, 63], so we restrict our discussion to
the final result, the growth velocity v+(F ) under force. We find
v+(F ) 
D
d
(
Fd
kBT
)2
[exp(Fd/kBT )−1− (Fd/kBT )]−1 . (2.10)
The growth velocity is a linear function of the diffusion coefficient and is independent
of the GTP-tubulin on-rate. For small forces, Fd/(kBT )  1, v+ approaches
v+ ≈ 2D/d = const. The diffusion coefficient can be calculated from the Einstein
relation and Stokes law, and is given by D = kBT/(6piηr) [2]. Here r is the radius
of the spherical particle and η the dynamic viscosity of water. In optical trap
experiments we typically have r = 1µm [47, 48], η = 10−3Ns/m2, T = 300K and
we find 2D/(ωond2) ≈ 104 − 105  1 for ωon = (10 − 100)s−1. The condition of
diffusion-limited-polymerization is not fulfilled under typical experimental conditions.
We, therefore, focus on the cases 2D/(ωond2) 1, the reaction-limited-polymerization.
reaction-limited-polymerization: 2D/(ωond2) 1
In the reaction-limited case, the diffusion coefficient is very large. The particle is in ther-
mal equilibrium with its surrounding and the gap size ∆x explores all possible positions.
A new GTP-tubulin dimer is only seldom attached to the MT tip and the probability to
find a gap of size ∆x follows a Boltzmann distribution p(∆x)∼ exp(∆xF/kBT ). Since a
new GTP-tubulin dimer can only be incorporated into the MT tip if ∆x = d, the GTP-
tubulin on-rate is modified by an additional Boltzmann factor and the force-dependent
growth velocity becomes:
v+(F ) = d [ωon exp(−Fd/kBT )−ωoff] . (2.11)
Here Fd is the work that has to be done against the force F to incorporate a single GTP-
tubulin dimer of size d; kB is the Boltzmann constant and T = 300 K the temperature.
38 Coarse-grained microtubule model
Equation (2.11) can explicitly be shown by establishing reaction-diffusion equations for
the gap size ∆x [2, 61]. In the reaction-limited case, their solution is given by eq. (2.11).
In the following we use the dimensionless force
η ≡ F/F0 with F0 = kBT/d, (2.12)
in terms of which the force-dependent growth velocity is given by
v+(η) = d
[
ωone−η−ωoff
]
. (2.13)
The characteristic force F0 has a value F0 = kBT/d ≈ 7pN. In experiments a charac-
teristic force F0 ≈ 0.8pN has been measured [53]. It is still unknown what causes the
difference between theoretical and experimentally measured values. Throughout this the-
sis we will use F0 ≈ 7pN, unless we compare our theoretical results with experimentally
measurements by Laan et al.[53]. In this case we will present results for F0 ≈ 7pN and
the experimental measured value F0 ≈ 0.8pN.
The dimensionless stall force
ηstall = ln(ωon/ωoff) (2.14)
is defined by the condition of vanishing growth velocity v+(ηstall) = 0. We typically have
ηstall ' 1.5, ...,3 or Fstall ' 10, ...,20pN for ωon = 30, ...,100s−1. The stall force is the
maximal force that the MT can generate in the absence of catastrophes. In the context
of this thesis, we will investigate how the forces that can be generated in the presence of
the dynamic instability compare to this stall force.
There are further models, which connect the polymerization velocity v+ to a resisting
force F [64, 65, 66]. However, most of these different approaches yield no explicit force-
velocity relations, and can not directly be used for further calculations. Furthermore, the
Brownian ratchet model resembles the basic setting of current optical trap experiments
[17, 52, 53]. Therefore, we will focus on the Brownian ratchet model, and the reaction-
limited case. In addition, we will show later, that our results do not qualitatively change,
if a more generalized force-velocity relation, outlined in the next paragraph, is employed.
In their investigation of experimental data Kolomeisky et al. used the generalized growth
velocity
v+ (η,θ) = d{ωon exp(−θη)−ωoff exp[(1− θ)η]} , (2.15)
which depends on the dimensionless “load distribution factor” θ [67]. The load distribution
factor θ ∈ [0,1] determines whether the on- or off-rates are affected by external force, while
keeping the ratio of overall on- and off-rate unaffected. Under force both the tubulin on-
rate ωon and the tubulin off-rate ωoff now acquire an additional Boltzmann-like factor.
For θ = 1, we obtain again v+(η) as given by eq. (2.13). The dimensionless stall force is
2.3 Force-dependent catastrophe rate 39
vh
vh vh
Ti
m
e
GTP-cap
GTP-cap
ωon
ωoff
vectorial hydrolysis
with velocity vh
random hydrolysis
with rate r
Figure 2.3: Sketch of the cooperative hydrolysis model by Flyvbjerg et al. New
GTP-tubulin dimers attach/detach to the MT tip with rate ωon and ωoff respectively.
In random hydrolysis GTP-tubulin (green) stochastically hydrolyzes to GDP-tubulin
(red) with rate r. In vectorial hydrolysis GTP-tubulin with adjacent GDP-tubulin
hydrolyzes and hydrolysis fronts move through the MT with average velocity vh. A
catastrophe is initiated when the GTP-cap reaches zero length.
unaffected by θ and is still given by ηstall = ln(ωon/ωoff).
2.3 Force-dependent catastrophe rate
In this thesis, we focus on purely chemical catastrophe events (see Sec. 1.1), where catas-
trophes are initiated by the loss of the stabilizing GTP-cap, and in particular on the
cooperative hydrolysis model proposed by Flyvbjerg et al. [18, 68]. We will not discuss
more elaborate multistep catastrophe models [29, 69, 70], nor include aspects of "struc-
tural plasticity" into the description [20]. Within this model, GTP-tubulin is hydrolyzed
by a combination of random and vectorial hydrolysis. A similar cooperative model has
recently been proposed for the hydrolysis dynamics of F-actin [27, 71].
In the following we present a detailed description of the catastrophe model by Flyvbjerg
et al. in the absence of force. Later we connect the resulting catastrophe rate to an
opposing force.
We consider a section of a single MT, consisting entirely of GTP-tubulin and located at
the plus end of the growing MT. In the following this section is called GTP-cap. New
GTP-tubulin dimers are attached to the GTP-cap with GTP-tubulin on-rate ωon, which
is connected to the growth velocity v+ via eq. (2.9). In random hydrolysis, GTP-tubulin
is stochastically hydrolyzed to GDP-tubulin with a rate r∼ 3.7×106m−1s−1 at a random
site within the GTP-cap. The cap is split up into a new cap and a remaining fraction of
40 Coarse-grained microtubule model
GTP-tubulin, now located at the interior of the MT (see Fig. 2.3). In a second hydrolysis
mechanism, called vectorial hydrolysis, only GTP-tubulin with adjacent GDP-tubulin is
hydrolyzed. This results in hydrolysis fronts, propagating, through the GTP-cap and the
remaining fraction of GTP-tubulin, with average velocity vh ∼ 4.2×10−9m/s.
Since the remaining fraction of GTP-tubulin is hydrolyzed from both sides and no GTP-
tubulin can be added to this part of the MT, it disappears fast compared to the new
GTP-cap. In contrast the GTP-cap growth with average velocity v = v+− vh. A catas-
trophe is initiated when the stabilizing GTP-cap vanishes, that is its length is decreased
down to zero. The cap length is stochastically reduced by fluctuation, caused by random
hydrolysis, and negative average velocity v = v+− vh < 0. Within the cooperative hy-
drolysis model, the inverse catastrophe rate ω−1c is given by the mean first-passage time
(MFPT) to the state of zero cap length.
We provide an analytical expression for the MFPT in the state of vanishing cap length,
as a function of the hydrolysis parameters r and vh, and the growth velocity v+, strictly
following references [68] and [18].
The stochastic time evolution of an ensemble of independent GTP-caps is described by a
linear master equation for the probability density p(x,t) to find a GTP-cap of length x
at time t
∂tp(x,t) = ∂xj(x,t) (2.16)
with
j(x,t) = vp(x,t)−D∂xp(x,t)− rxP (x,t). (2.17)
Here rx denotes the rate with which GTP-caps are split up into two new fragments,
vp(x,t) the average elongation rate and D∂xp(x,t) the rate with which the cap length is
changed by fluctuations, parametrized by the diffusion coefficient D = 0.5d(v++vh). The
number of GTP-caps of length longer than x at time t is
P (x,t) =
∫ ∞
x
p(x,t)dx (2.18)
and P (0, t) the overall number of GTP-caps at time t. From eq. (2.16) we find
∂tP (0, t) = j(0, t) = vp(0, t)−D∂xp(0, t). (2.19)
Since p(x,t) = 0 for x < 0 the boundary condition at x = 0 reads p(0+, t) = 0 and the
catastrophe rate ωc, that is the temporal alteration of the total number of GTP-caps
based on the overall number of GTP-caps within the ensemble, is than given by
ωc ≡−
∂tP (0, t)
P (0, t)
= D∂xp(0, t)
P (0, t)
. (2.20)
To obtain an explicit expression for the catastrophe rate ωc, P (0, t) and ∂tP (0, t) have
2.3 Force-dependent catastrophe rate 41
to be determined. Therefore eq. (2.16) is integrated on the interval x ∈ [0,∞[. Since
j(∞, t) = 0 ∀t this results in
∂tP (x,t) =
(
−v∂x+D∂2x− rx
)
P (x,t). (2.21)
Introducing
x0 ≡ (D/r)1/3 (2.22)
t0 ≡
(
Dr2
)−1/3
(2.23)
γ ≡ v
2D2/3r1/3
= vt0
2x0
(2.24)
and the dimensionless variables
ξ ≡ x/x0 (2.25)
τ ≡ t/t0 (2.26)
˜P (ξ,τ)≡ exp[−vx/(2D)]P (x,t) = exp(−ξγ)P (x,t) (2.27)
we find
∂τ ˜P (ξ,τ) =
(
∂2ξ − ξ−γ2
)
˜P (ξ,τ). (2.28)
Equation (2.28) is a linear partial differential equation and is solved by separation of
variables. The same type of partial differential equation describes the dynamics of a
quantum mechanical particle in an electric field [72]. Its general solution is
˜P (ξ,τ) =
∫
dα c(α)exp(−ατ)Ai
(
ξ+γ2−α
)
, (2.29)
where Ai(x) is the first Airy function [73, 74], α the separation constant and c(α) a
coefficient function. Both, α and c(α), have to be determined from boundary conditions.
With p(0, t) = ∂xP (0, t) = 0 and eqs. (2.25) and (2.27) we obtain
∫
dα c(α)exp(−ατ)
[
γAi
(
γ2−α
)
+Ai′
(
γ2−α
)]
= 0 (2.30)
from eq. (2.29).
Here Ai′(x) denotes the first derivative of Ai(x) with respect to x. From eq. (2.30) follows
that c(α) is only nonzero for a discrete set of values αk, k = 0,1,2, . . . , which are subject
to the condition
42 Coarse-grained microtubule model
0
2
4
6
8
10
-3 -2 -1 0 1 2 3
α k
(γ
)
γ
α0
α1
α2
α3
α4
Figure 2.4: First five graphs of αk(γ) for k = 0,1,2,3,4 as a function of γ.
γ =−
Ai′
(
γ2−αk
)
Ai(γ2−αk)
. (2.31)
Equation (2.31) is solved numerically for different values of γ. Here we want to point
out, that γ = γ(v+) is a function of the growth velocity v+ [see eq. (2.24)]. All further
calculations can also be performed in terms of v+. However, we maintain the notation by
Flyvbjerg et al. to simplify further calculations. The first five graphs of αk = αk(γ) are
depicted in Fig. 2.4, as a function of γ.
We find αk(γ) < αk+1(γ) for a fixed γ and α0(γ)  αk(γ) ∀k for γ ≥ 1. The set of
functions Ai(x+αk)/Ai′ (αk) forms an orthonormal basis on x ∈ [0,∞[ and eq. (2.29)
can be expanded into this basis [74]. With initial condition P (ξ,0)∝ δ(ξ) we obtain
P (ξ,τ)∝
∞
∑
k=0
exp[−αk(γ)τ + ξγ]
Ai
(
ξ+γ2−αk
)
Ai′ (γ2−αk)
. (2.32)
From eq. (2.32) and eq. (2.26) follows
∂tP (0, t)∝
∞
∑
k=0
αk(γ)
t0
exp[−αk(γ)τ ]
Ai
(
γ2−αk
)
Ai′ (γ2−αk)
(2.33)
and finally
2.3 Force-dependent catastrophe rate 43
ωc =−
∂tP (0, t)
P (0, t)
= t−10



∞
∑
k=0
exp[−αk(γ)τ ]
Ai
(
γ2−αk
)
Ai′ (γ2−αk)



×



∞
∑
k=0
αk(γ)exp[−αk(γ)τ ]
Ai
(
γ2−αk
)
Ai′ (γ2−αk)



−1
. (2.34)
As exp[−α0(γ)τ ]< exp[−αk(γ)τ ] , for k , 0, τ > 0 and a fixed value of γ, each summand
in eq. (2.34) with k , 0 is quickly decreased as τ increases. On long times scales τ  1,
that is t t0, the catastrophe rate ωc approaches
ωc =
α0(γ)
t0
. (2.35)
Since MTs exhibit stable growth on long time scales, up to several minutes [15], and
t0 = [0.5d(v++vh)]−1/3 ≈ (10−30)s (Table A.1), we typically have τ = t/t0  1 and the
catastrophe rate ωc = ωc(v+) as a function of γ, and, therefore, as a function of the growth
velocity, is given by eq. (2.35). An asymptotic expansion of α(γ) in terms of γ−1 yields
α0 = (2γ)−1+O(γ−3) for γ > 1 and we obtain from eq. (2.35)
ωc ≈Dr/v ∝ v−2/3+ . (2.36)
The catastrophe rate decreases as a power law over a wide range of growth velocities.
To calculate the catastrophe rate ωc(v+) as a function of the growth velocity we solved
eq. (2.31) numerically and obtained a high order polynomial for α0(v+). This polynomial
is used in combination with eq. (2.35) to compute ωc(v+). In Fig. 2.5 the catastrophe rate
is depicted as a function of v+. It is clearly visible that ωc(v+) increases as the growth
velocity v+ decreases.
The velocity-dependence of the catastrophe rate ωc(v+) as calculated from eq. (2.35) gives
rise to a force-dependence ωc(η) =ωc[v+(η)]. We assume that this is the only effect of force
on the catastrophe rate [75]. As a result, the catastrophe rate increases exponentially,
when v+(η) is decreased by applying a force η = F/F0, but a finite value is maintained at
v+(η) = 0, which is ωc(v+ = 0)≈ 2.9s−1. The catastrophe rate ωc(η) as a function of the
dimensionless load force η is shown in Fig. 2.5.
For qualitative approximations, the force-dependence of the catastrophe rate can be de-
scribed by an exponential increase above the characteristic force F0,
ωc(η)∼ ωc(η=0)eη. (2.37)
Different catastrophe models have been proposed and have been shown to describe exper-
44 Coarse-grained microtubule model
0
0.01
0.02
0 0.5 1 1.5 2
ω c
(v
+
)
[s−
1 ]
v+ [µm/min]
ωc,Flyv
ωc,Jans
ωc,lin
(a)
0
0.05
0.1
0 0.5 1 1.5 2 2.5
ω c
(η
)
[s−
1 ]
η
ωc,Flyv
ωc,Jans
ωc,lin
(b)
Figure 2.5: (a): The catastrophe rate ωc(v+) as a function of the growth velocity
v+ for the Flyvbjerg model (solid line), the Janson model (dashed line) and the linear
catastrophe model (dotted line). (b): The catastrophe rate ωc(η) as a function of the
dimensionless load force η = F/F0 with F0 ∼ 7pN and ωon = 50 s−1 for the Flyvbjerg
model (solid line), the Janson model (dashed line) and the linear catastrophe model
(dotted line). In contrast to the linear catastrophe rate both, the Flyvbjerg and the
Janson catastrophe rate, increase exponentially for forces larger than the characteristic
force F0.
imentally available data on single-MT catastrophe rates . One alternative phenomenolog-
ical catastrophe model has been proposed by Janson et al. based on experimental data
for the inverse catastrophe rate, i.e., the average catastrophe time 〈τc〉 = 1/ωc [75]. The
experimental data show that 〈τc〉 increases linearly with the growth velocity v+ such that
the catastrophe rate, which we will refer as ωc,Jans, is given by
ωc,Jans =
1
a+ bv+
. (2.38)
with a' 20 s and b' 1.4×1010 s2m−1 [75].
Also within the catastrophe model by Janson et al., the catastrophe rate ωc,Jans becomes
force dependent via the force-dependence of the growth velocity [see eq. (2.13)], and the
resulting catastrophe rate is a nonlinear and increasing function of the dimensionless force
η (see Fig. 2.5), which increases exponentially above the characteristic force F0.
Both the Flyvbjerg and Janson catastrophe models describe available experimental data
on single MTs, as has been shown in Refs. [18, 68] and [75], respectively.
In both models the catastrophe rate ωc decreases as a power law over a wide range of
growth velocities,
ωc ∝ v−2/3+ (Flyvbjerg) and ωc,Jans ∝ v−1+ (Janson). (2.39)
Because of v+ ∼ dωone−F/F0 for large velocities, ωc increases exponentially with force F
above the characteristic force F0 in both catastrophe models. This can also be seen in
the comparison in Fig. 2.5.
2.3 Force-dependent catastrophe rate 45
It is possible to consider other types of catastrophe models where the catastrophe rate ωc =
ωc(v+) is a decreasing function of the growth velocity but does not increase exponentially
with force F above the characteristic force F0. One particularly simple example is a
catastrophe rate which decreases linearly with velocity,
ωc,lin(v+) = ˜a−˜bv+. (2.40)
The coefficients ˜a and ˜b are determined from experimental data [75], under condition
˜a > ˜bv+ to ensure ωc,lin(v+) > 0 ∀v+. Janson et al. obtained ωc,min(v+)' 1,8×10−3 s−1
for v+ ' 3,8× 10−8 m/s and ωc,max(v+) ' 0.05 s−1 for v+ ' 0 m/s, which are the minor
and major catastrophe rates measured. A linear fit with eq. (2.40) results in ˜a = 0.05 s
and ˜b= 1×106 m−1. This choice of ˜a and ˜b, however, leads to λ < 0 and to bounded MT
growth, for all parameter values given in Table A.1. We, therefore, set ωc,max =0.1×ωc,max
and obtain ˜a = 0.005 s and ˜b = 8× 104 m−1 from the linear fit. The linear catastrophe
rate ωc,lin is shown in Fig. 2.5(a) as a function of the growth velocity v+ and in Fig.
2.5(b) as a function of the dimensionless load force η. In contrast to the catastrophe
rates by Flyvbjerg et al. and Janson et al., the linear catastrophe rate does not increase
exponentially with forces larger than the characteristic force F0.
46 Coarse-grained microtubule model
Chapter 3
Single MT simulations 1
In this chapter we present the single MT simulation model. The stochastic equations of
motion for a single MT and their numerical implementation are described2. We spec-
ify simulation parameters, parameter ranges and define ensemble- and time-averages of
observables.
3.1 Single microtubule simulation model
In the simulations we solve the stochastic Langevin-like equations of motion for the length
x(t) of a single MT using numerical integration with fixed time steps ∆t and including
stochastic switching between growth and shrinkage. In a growing state x(t) is increased
by v+∆t, while in a state of shrinkage, it is decreased by v−∆t. The equations of motion,
at time t+∆t, are
x(t+∆t) = x(t)+v+(x)∆t (growing state) (3.1)
x(t+∆t) = x(t)−v−∆t (shrinking state) (3.2)
At t=0 we set x(0) = 0, that is each MT has zero length, and if x(t)≤ 0 the MT undergoes
a forced rescue event, corresponding to a reflecting boundary at x= 0 (see Sec. 2.1) In the
growing state, v+ is calculated from eq. (2.9) for zero force and from eq. (2.11) under force.
If the more generalized force-velocity relation by Kolomeisky et al. is embedded in the
simulation, v+ is calculated from eq. (2.15). As mentioned in Sec. 2.2, v− is independent
of force and takes the constant value v− = 3×10−7m/s (see Table A.1).
In each time step a uniformly distributed random number χ∈ [0,1] is compared to ωr,c∆t.
If χ < ωr,c∆t the MT changes its state of growth. We use two nested xor-shift random
1Parts of the text have been published in reference [55] and are ©2012 American Physical Society.
http://publish.aps.org/copyrightFAQ.html
2All numerical implementations are performed in C++
48 Single microtubule simulations
number generators with time-dependent seed, to generate uniformly distributed random
numbers [76]. The catastrophe rate ωc is calculated from eq. (2.35) in the Flyvbjerg
model, from eq. (2.38) in the Janson model and from eq. (2.40) in the linear catastrophe
model.
To assure ωr,c∆t≤ 1, we use a time step ∆t= 0.1s. During the simulations all parameters
of growth are fixed, see Table A.2, except for ωon, which is varied in the range ωon =
(30−100)s−1, and ωr, which is varied in a range ωr = (0.03−0.2)s−1, see Table A.1.
We simulate an ensemble of Nens = 1000−10000 (see Table A.2) independent MTs. Time-
dependent ensemble averages 〈O〉(t) of an observable O(t) are taken over the stochastic
trajectories and are calculated from
〈O〉(t) = N−1ens
Nens
∑
j=1
Oj(t). (3.3)
In addition we perform time averages via
〈O〉= ∆T−1
∆T
∑
j=1
〈O〉(t). (3.4)
Here ∆T denotes the measurement time and we typically have ∆T = 50000 time steps.
In order to determine the characteristic observables for the dynamics of single MTs we
have to anticipate numerical results from Chap. 4. We present different types of stochastic
trajectories and clarify how observables are determined from numerical data. All shown
results are discussed in detail in the following Chapter.
3.1.1 Confinement by rigid walls
For a fixed combination of ωon and ωr we run Nens independent simulations. In the
steady state, determined by visual inspection of sample trajectories, the average length
is calculated via eq. (3.4). In addition, we count each MT in the growing state and with
x(t) = L, and obtain the number of MTs in the growing state and stuck to the boundary
wall.
3.1.2 Constant force
For a constant force we determine the critical force Fc, which provides the transition be-
tween unbounded and bounded growth, from the condition of a vanishing average velocity
〈V (Fc)〉 ≈ 0. For a fixed combination of ωon and ωr, we run Nens independent simulations.
In the simulations we start with a small force F , trace the ensemble-averaged length 〈x〉(t)
as a function of time (see Fig. 3.1) and perform a linear fit with
3.1 Single microtubule simulation model 49
0
20
40
60
80
100
120
2000 3000 4000 5000
〈x
〉(
t)
[µ
m
]
t [s]
〈V (F )〉> 0
〈V (F )〉> 0
〈V (Fc)〉 ≈ 0
F =1.0 pN
F =3.5 pN
F =6.3 pN≈Fc
Figure 3.1: Sample trajectories for growth against a constant force F . We set
ωon = 70 s−1 and x0 = 1 µm. Symbols represent the ensemble-averaged length 〈x〉(t)
as a function of time for F = 1 pN(), 3.5 pN(), and 6.3 pN(). Solid lines are
linear fits of eq. (3.5) to the data points.
〈x〉(t) = x0+ 〈V (F )〉t. (3.5)
Here x0 denotes the length of the ensemble at t= 0 and 〈V (F )〉 the ensemble-averaged ve-
locity. We set x0 = 1 µm. This choice is arbitrary and does not influence the fit procedure.
It turns out, that the choice of x0 = 1 µm reduces the impact of the reflecting boundary at
x= 0 for forces close to the critical force Fc. The simulation time is significantly reduced.
In independent simulation runs, we successively increase the constant force F , obtain the
average velocity 〈V (F )〉 from the linear fit and determine the the critical force Fc from
the condition 〈V (Fc)〉 ≈ 0 (see Fig. 3.1), for a given combination of ωon and ωr.
3.1.3 Elastic force
3.1.3.1 Vanishing rescue rate
For vanishing rescue rate ωr = 0 we set x0 = 0. Figure 3.2(a) shows the stochastic time
evolution of a single MT. The MT polymerizes against the elastic obstacle until the
maximal polymerization force Fmax is reached. At the maximal polymerization force
Fmax a catastrophe occurs and the dynamics stops, due to missing rescue events. In
the simulations, a catastrophe is initiated if the probability ωc(F )∆t to switch into a
state of shrinkage, exceeds the uniformly distributed random number χ (see Sec. 3.1).
The maximal polymerization force Fmax is therefore determined from the condition χ <
ωc(Fmax)∆t. Fmax is a stochastic quantity, so we run Nens independent simulations ,
50 Single microtubule simulations
0
2
4
6
8
10
12
14
0 20 40 60 80 100 120 140
F
(t
)[
pN
]
t [s]
χ<ωc(Fmax)∆t
Fmax
(a)
0
2
4
6
8
0 5000 10000 15000 20000
F
(t
)[
pN
]
t [s]
(b)
〈F 〉
Figure 3.2: Single MT sample trajectories. (a): Vanishing rescue rate ωr = 0.
Polymerization force F (t) as a function of time t. We set x0 = 0, ωon = 70 s−1 and
k = 10−5 N/m. The vertical, dashed line marks the maximal polymerization Fmax.
The stochastic quantity Fmax is determined from the condition χ < ωc(Fmax)∆t. (b):
Non-zero rescue rate ωr = 0.05 s−1. Polymerization force F (t) as a function of time
t. We set x0 = 10 µm, ωon = 70 s−1 and k = 10−7 N/m. The dashed line represents a
single stochastic trajectory and the solid line is the time-dependent ensemble average,
obtained via eq. (3.3). The average steady-state force 〈F 〉 (t≈ 15000 s) is calculated
from eq. (3.4) and marked by a vertical dashed line.
perform an ensemble average via eq. (3.3) and obtain the ensemble averaged maximal
polymerization force 〈Fmax〉.
3.1.3.2 Non-zero rescue rate
For a non-vanishing rescue rate ωr > 0 we set x0 = 10−5m. Figure 3.2(b) shows the
stochastic time evolution of a single MT growing against the elastic obstacle. In addition
the time-dependent ensemble averaged polymerization force 〈F (t)〉 is depicted. The time-
dependent ensemble average is performed over Nens independent simulations via eq. (3.3).
In the steady state, determined by visual inspection of sample trajectories, 〈F (t)〉 is time
averaged over ∆T time steps via eq. (3.4). We obtain the average steady state force 〈F 〉.
Chapter 4
Single microtubule dynamics under
force and confinement 1
In this chapter we present the results for single MT dynamics in three different confinement
scenarios, which mimic cellular environments.
Section 4.1 deals with a single MT confined in a box with rigid walls. We introduce a
model for wall-induced catastrophes and calculate the OPDF as a function of the growth
parameters and the confinement size.
In Sec. 4.2 the MT growth under constant force. We find, as in the absence of force, a
regime of bounded and a regime of unbounded growth. Both regimes are characterized by
the force-dependent characteristic length parameter λ(f) and the critical force fc, which
provides the transition between the two regimes, is determined.
In Sec. 4.3 we investigate MT growth against an elastic obstacle. We start with vanishing
rescue rate ωr, which corresponds to typical experimental conditions, and estimate the
average maximal polymerization force by a dynamical mean field theory. Afterwards we
focus on the case of a non-vanishing rescue rate. An analytical expression for the OPDF
in the steady state is presented and discussed. We calculate the average polymerization
force in the steady state as a function of the growth parameters and discuss the influence
of the obstacle stiffness on length distributions and polymerization forces. In addition
we present a dynamical mean field theory, compare mean field results to full stochastic
calculations and discuss the validity of the mean field approximation. Within the mean
field theory we derive a time evolution of the time-dependent average polymerization
force. Analogue to dilution experiments, we investigate the dynamics of the MT after a
sudden change in the GTP-tubulin concentration. Finally we show that our results are
robust with respect to changes in the force-velocity relation and in the catastrophe model.
Here we restrict our discussion to mean field results.
1Parts of the text have been published in reference [55] and are ©2012 American Physical Society.
http://publish.aps.org/copyrightFAQ.html
52 Single microtubule dynamics under force and confinement
0 L
ωc,L
ωc(v+)ωr
v+
v− ∆
Figure 4.1: Schematic representation of the confinement and possible MT configu-
rations. From top to bottom: A MT growing with v+; MT shrinks with v−. MT in a
state of growth and stuck to the boundary wall with v+ = 0 and ωc,L the dashed line
marks the interval ∆, in which the flow of probability from the boundary back into
the confinement can be measured. .
4.1 Single microtubule dynamics under confinement
by rigid walls
A single MT is confined to a one-dimensional box of fixed length L with rigid boundary
walls at x = 0 and x = L as shown schematically in Fig. 4.1 [77, 78]. There is no force
acting on the MT but within the box catastrophes are induced upon hitting the rigid
walls. We propose the following mechanism for these wall-induced catastrophes: When
the MT hits the boundary at x = L, its growth velocity v+ has to reduce to zero, which
leads to an increase of the catastrophe rate to ωc,L ≡ ωc(v+ = 0). Since ωc,L is finite,
wall-induced catastrophes are not instantaneous but the MT sticks for an average time
1/ωc,L to the boundary before the catastrophe, which is in contrast to previous studies
[79]. Throughout this section we use the catastrophe model by Flyvbjerg et al. For the
average time spent at the boundary before a catastrophe, we find ω−1c,L ≈ 0.29s. The
catastrophe rate at the wall, ωc,L, is much higher than the bulk catastrophe rate ωc(v+).
For ωon = 50s−1 we find ωc,L/ωc ' 2300.
To include the mechanism of wall-induced catastrophes into the description by master
equations, we introduce the probabilities Q+ and Q− of finding the MT stuck to the
boundary in a growing state and in a shrinking state, respectively. The stochastic time
evolution of Q+(t) and Q−(t) is given by:
∂tQ+(t) =−ωc,LQ+(t)+ωrQ−(t)+v+p+(L) (4.1)
∂tQ−(t) = +ωc,LQ+(t)−ωrQ−(t)−
v−
∆
Q−(t). (4.2)
4.1 Single microtubule dynamics under confinement by rigid walls 53
The quantity v+p+(L) is the flow of probability from the interior of the confining box onto
its boundary and is given by the solution of eq. (2.1) and (2.2) for x=L, while (v−/∆)Q−
is the probability current from the boundary back into the interior, where ∆ denotes a
small interval in which the flow v−Q− can be measured. This implies that there is a
boundary condition v−p−(L,t) = (v−/∆)Q− for the backward current density at x = L,
in addition to the reflecting boundary condition [eq. (2.3)] at x = 0. An identical model
for wall-induced catastrophes has recently been introduced in Ref. [58].
In the steady state and in the limit ∆≈ 0 we find
Q+ ≈
v+
ωc,L
p+(L) (4.3)
Q− ≈ 0, (4.4)
and v−p−(L,t) = (v−/∆)Q− = v+p+(L). Equation (4.3) shows that there is a non-zero
probability Q+ of finding a MT in a state of growth and stuck to the boundary, which is
given by the flow of probability from the interior of the confining box onto its boundary
divided by the average time being stuck to the boundary. In contrast, eq. (4.4) states
that there is no MT in a shrinking state and stuck to the wall. This is intuitively clear
since a MT undergoing a catastrophe begins to shrink instantaneously. In the steady
state, we solve eqs. (2.1), (2.2) and (4.3) simultaneously with the additional normalization
∫ L
0 [p+(x)+p−(x)]dx+Q+ = 1. We find v+p+(x) = v−p−(x) and
P (x) = Nex/λ
(
1+ v+
v−
)
(4.5)
Q+ = N
v+
ωc,L
eL/λ (4.6)
with λ from eq. (2.4) and a normalization
N−1 = λ
(
1+ v+
v−
)
(
eL/λ−1
)
+ v+
ωc,L
eL/λ. (4.7)
Equation (4.5) shows that we find an exponential OPDF P (x) in confinement with the
same characteristic length |λ|. If the growth is unbounded in the absence of confinement,
which corresponds to λ−1 > 0, the OPDF is exponentially increasing; if the growth is
bounded in the absence of confinement, which corresponds to λ−1 < 0, the OPDF remains
exponentially decreasing in confinement. The same result has been obtained in Ref. [79]
within a discrete growth model. In independent in vivo experiments, both exponentially
increasing [80] and exponentially decreasing OPDFs [56] have been found.
In the following we focus on the case λ−1 > 0 of exponentially increasing OPDFs. In the
54 Single microtubule dynamics under force and confinement
steady state, the average length of a MT within the confining box is given by
〈x〉=
∫ L
0
xP (x)dx+Q+L
= N
{(
1+ v+
v−
)
λ2
[
1+ eL/λ
(L
λ
−1
)]
+L v+
ωc,L
eL/λ
}
. (4.8)
In the limit of instantaneous wall-induced catastrophes, Q+ ≈ 0, we obtain
〈x〉
L
≈ 1
1− e−L/λ
− λ
L
, (4.9)
i.e., the average MT length 〈x〉/L depends on the two control parameters ωr and ωon
only via the ratio L/λ. This scaling property is lost if wall-induced catastrophes are not
instantaneous because eq. (4.8) then exhibits additional v+- and thus ωon-dependencies.
Within our model the increased catastrophe rate at the boundary gives rise to an increased
overall average catastrophe rate
ωc,eff = ωc(v+)+Q+[ωc,L−ωc(v+)], (4.10)
for which we find ωc,eff ' 0.03s−1 for L = 1µm and ωc,eff ' 0.006s−1 for L = 10µm as
compared to ωc ' 0.0015s−1 for these conditions.
We set the length of the confining box to L = 1µm and L = 10µm, which are typical
length scales in experiments [17, 81] and cellular environments [34], and we calculate 〈x〉
and Q+ as functions of ωon and ωr. The parameter regimes displayed in Figs. 4.2 and
4.3 correspond to regimes L/λ  1 for L = 10µm and L/λ  1 for L = 1µm. Results
obtained from stochastic simulations agree with analytical findings (Figs. 4.2 and 4.3).
It is clearly visible that the size L of the confinement has a significant influence on 〈x〉,
mainly via the ratio L/λ.
The probability Q+ to find the MT at the wall increases with increasing rates in the
range of Q+ ≈ 0, ...,0.03 and exhibits only a weak dependency on L, see Figs. 4.3. Even
for maximum rates, the probability of finding a MT in a growing state and stuck to
the wall is limited to several percent, due to the large catastrophe rate ωc,L at x = L.
Therefore, in most cases wall-induced catastrophes can be viewed as instantaneous, and
the approximation (4.9) works well. For increasing on-rate ωon or rescue rate ωr, the ratio
L/λ approaches L/λ ≈ Lωr/v− from below. According to the approximation (4.9), the
mean length 〈x〉 then increases and approaches 〈x〉/L≈ 1/(1− e−Lωr/v−)−v−/Lωr from
below. For L=10µm, we have L/λ 1 and the length distribution is exponential, P (x)∼
ex/λ. The ratio 〈x〉/L saturates at a high value 〈x〉/L≈ 0.7, ...,0.9 [Figs. 4.2 (a),(c)]. For
L/λ  1 the MT length distribution becomes very narrow around the maximal length
L. In contrast, for L = 1µm, we have L/λ  1, and L is too small to establish the
characteristic exponential decay of the length distribution. The length distribution P (x)
4.2 Single microtubule dynamics under a constant force 55
0.25
0.5
0.75
1
25 50 75 100
〈x
〉/
L
ωon [s−1]
L = 10µm
(a)
0.48
0.52
0.56
0.6
25 50 75 100
〈x
〉/
L
ωon [s−1]
L = 1µm
(b)
0.25
0.5
0.75
1
0.05 0.1 0.15 0.2 0.25 0.3
〈x
〉/
L
ωon [s−1]
L = 10µm
(c)
0.48
0.5
0.52
0.54
0.56
0.58
0.05 0.1 0.15 0.2 0.25 0.3
〈x
〉/
L
ωon [s−1]
L = 1µm
(d)
Figure 4.2: The average length 〈x〉/L as a function of ωon and ωr for confinement
by fixed rigid walls. Data points are results from stochastic simulations, lines are
analytical results (eq. 4.8). Top row: The average length 〈x〉/L as a function of ωon for
different values of ωr = 0.03s−1(),0.05s−1(),0.1s−1(),0.2s−1(•) and 0.3s−1(4).
(a) L=10µm. (b) L=1µm. Lower row: The average length 〈x〉/L as a function of ωr
for different values of ωon = 25s−1(),50s−1(),75s−1(N),100s−1(). (c) L= 10µm.
(d) L = 1µm.
is almost uniform, and the ratio 〈x〉/L≈ 0.5, ...,0.6 deviates only slightly from the result
〈x〉/L = 1/2, characteristic for a broad uniform distribution [Figs. 4.2(b),(d)].
4.2 Single microtubule dynamics under a constant
force
In the second scenario a constant force η is applied to the MT and the right boundary
is removed, so that the MT is allowed to grow on x ∈ [0,∞[. According to eq. (2.13) the
growth velocity under force is smaller, but it remains constant for fixed η. With eq. (2.35)
this results in a higher, but also constant, catastrophe rate ωc[v+(η)] > ωc[v+(0)]. Since
v− and ωr are independent of force, the stochastic dynamics of the MT is described by
eqs. (2.1) and (2.2) with the same solutions P (x,t) as in the absence of force, but with
a decreased velocity of growth v+(η) and an increased catastrophe rate ωc(η) [56, 57].
Throughout this section we use the catastrophe model by Flyvbjerg et al. In particular,
we still find two regimes, a regime of bounded growth and a regime of unbounded growth.
56 Single microtubule dynamics under force and confinement
0
0.004
0.008
0.012
25 50 75 100
Q
+
ωon [s−1]
L = 10µm
(a)
0
0.005
0.01
0.015
0.02
25 50 75 100
Q
+
ωon [s−1]
L = 10µm
(b)
0
0.005
0.01
0.015
0.05 0.1 0.15 0.2 0.25 0.3
Q
+
ωr [s−1]
L = 10µm
(c)
0
0.005
0.01
0.015
0.02
0.025
0.05 0.1 0.15 0.2 0.25 0.3
Q
+
ωr [s−1]
L = 10µm
(d)
Figure 4.3: The probability Q+ to find the MT at the wall as a function of ωon
and ωr for confinement by fixed rigid walls. Data points are results from stochastic
simulations, lines are analytical results (eq. 4.3). Top row: Q+ as a function of ωon for
different values of ωr = 0.03s−1(),0.05s−1(),0.1s−1(),0.2s−1(•) and 0.3s−1(4).
(a) L = 10µm. (b) L = 1µm. Lower row: Q+ as a function of ωr for different values
of ωon = 25s−1(),50s−1(),75s−1(N),100s−1(). (c) L = 10µm. (d) L = 1µm.
4.2 Single microtubule dynamics under a constant force 57
In the regime of bounded growth P (x,t) is again exponentially decreasing, and the force-
dependent average length is 〈x(η)〉= |λ(η)| with the corresponding force-dependent length
parameter
λ(η)≡ v+(η)v−
v+(η)ωr−v−ωc(η)
(4.11)
as compared to eq. (2.4) in the absence of force. In the regime of unbounded growth
〈x(η)〉 increases linearly in time with the force-dependent mean velocity J(η) = [v+(η)ωr−
v−ωc(η)]/[ωr +ωc(η)], cf. eq. (2.7). The MT length distribution P (x,t) assumes again a
Gaussian form [eq. (2.6)] where also the diffusion constant DJ(η) follows the same eq.
(2.8) with force-dependent growth velocity v+(η) and catastrophe rate ωc(η).
In the presence of a constant force η, the transition between bounded and unbounded
growth is governed by the force-dependent length parameter λ(η). The regimes of bounded
and unbounded growth are now separated by the condition λ−1(η) = 0, which is shifted
compared to the case η = 0, see Fig. 4.5. The inverse length parameter λ−1(η) is a
monotonously decreasing function of force η and changes sign from positive to negative
values for increasing force η. Therefore λ−1(ηc) = 0 or
v+(ηc)ωr = v−ωc(ηc), (4.12)
defines a critical force for the transition from unbounded to bounded growth. A single MT
exhibiting unbounded growth (λ−1(0) > 0) in the absence of force undergoes a transition
to bounded growth with λ−1(η) < 0 by applying a supercritical force η > ηc. On the
other hand, starting with a combination of on-rate ωon and rescue rate ωr and a force η,
which results in bounded growth with λ−1(η) < 0, the MT can still enter the regime of
unbounded growth by increasing ωon or ωr so that the force η becomes subcritical, η < ηc
or λ−1(η) > 0.
Rewriting condition (4.12) as v+(ηc) = v−ωc(ηc)/ωr > 0 and using that v+(η) decreases
with η, it follows that the critical force is always smaller than the stall force, ηc < ηstall,
which satisfies v+(ηstall) = 0, and it approaches the stall force only for vanishing catastro-
phe rate. Qualitatively, we can obtain an explicit result for the critical force ηc by using
the approximations of an exponentially decreasing growth velocity, v+(η) ≈ v+(0)e−η,
which is valid for ωon  ωoff [see eq. (2.13)], and an exponentially increasing catastrophe
rate above the characteristic force F0, eq. (2.37), in the condition (4.12) for the critical
force. This leads to
ηc ∼
1
2
ln
(
v+(0)ωr
v−ωc(0)
)
∼ 1
2
ln
(
ωondωr
v−ωc(0)
)
(4.13)
which shows that the critical force grows approximately logarithmically with on-rate ωon
(note that the catastrophe rate in the absence of force decreases with ωon as ωc(0)∝ 1/ωon
[18]) and rescue rate ωr. A negative ηc for small on-rates and rescue rates signals that
the MT is for all forces η > 0 in the bounded phase. In Fig. 4.4 we show exact results for
58 Single microtubule dynamics under force and confinement
0
0.5
1
1.5
2
25 50 75 100
η c
ωon [s−1]
B
UB
ωr=0.03s−1
ωr=0.05s−1
ωr= 0.1s−1
ωr= 0.2s−1
Figure 4.4: Critical force ηc as a function of ωon for ωr =
0.03s−1(),0.05s−1(),0.1s−1(),0.2s−1(•). Data points represent results from
simulations, lines represent solutions of eq. (4.12) for a fixed combination of ωon and
ωr. B: bounded growth. UB: unbounded growth.
the critical force ηc as a function of the on-rate ωon and for different rescue rates ωr from
solving condition (4.12) numerically and from stochastic simulations. Agreement between
both methods is good.
The condition λ−1(η) = 0 specifies the boundary between bounded and unbounded growth
at a given force η. In Fig. 4.5, the resulting phase boundary is shown as a function of
ωon and ωr. There is good agreement between numerical solutions of λ−1(η) = 0 and
stochastic simulations. With increasing force, the boundary between the two regimes of
growth shifts to higher values of ωon and ωr, and forces up to F ∼ 1.4 ·F0 can be overcome
by a single MT in the parameter regimes of ωon and ωr considered.
4.3 Single microtubule dynamics under an elastic
force
In the third scenario, an elastically coupled barrier is placed in front of the MT as shown
in Fig. 4.6, which models the optical traps used in Refs. [17, 53] or the elastic cell cortex
in vivo. If the barrier is displaced from its equilibrium position x0 by the growing MT
with length x > x0, it causes a force F (x) = k(x−x0) resisting further growth. For x < x0
there is no force. We use x0 = 0µm in the case of vanishing rescue rate, x0 = 10µm in
the case of finite rescue rate and a spring constant k in the range 10−7N/m (soft) to
10−5N/m (stiff as in the optical trap experiments in [53]).
An elastic force F (x) = k(x−x0) represents the simplest and most generic x-dependent
4.3 Single microtubule dynamics under an elastic force 59
0
0.1
0.2
0.3
25 50 75 100
ω r
[s−
1 ]
ωon [s−1]
B
UB
η=0
η=0.3
η=0.6
η=0.9
η=1.2
η=1.4
Figure 4.5: Phase boundary between bounded (B) and unbounded growth (UB)
as a function of ωon and ωr for MT growth under constant force. Data points for
η =0(),0.3(),0.6(),0.9(•),1.2(4),1.4(N) represent results from simulations, lines
represent solutions of λ−1(η) = 0 [see eq. (4.11)] for a constant force η.
force. Whereas for a confinement of fixed length or a constant force, the MT length x
was the only stochastic variable, the force F (x) itself is now coupled to x and becomes
stochastic as well. Therefore, not only are the MT length distributions of interest but also
the maximal and average polymerization forces which are generated during MT growth.
4.3.1 Vanishing rescue rate
We first discuss growth in the absence of rescue events, ωr = 0. This situation corresponds
to optical trap experiments [17, 53], which are performed on short time scales and no rescue
events are observed. In a state of growth the MT grows against the elastic obstacle with
velocity v+[η(x)] and η(x) increases. For simplicity we suppress the x-dependency in the
notation in the following. At a maximal polymerization force ηmax, the MT undergoes
a catastrophe and starts to shrink back to zero and the dynamics stops due to missing
rescue events. No steady state is reached. Since switching to the state of shrinkage is a
stochastic process, the maximal polymerization force ηmax is a stochastic quantity which
fluctuates around its average value. We calculate the average maximal polymerization
force 〈ηmax〉 within a mean field approach.
Because no steady state is reached in the absence of rescue events, we have to use a
dynamical mean field approach, which is based on the fact that the MT growth velocity
dx/dt = v+(η) is related to the time evolution of the force by dη/dt = (k/F0)dx/dt.
In order to obtain the average maximal polymerization force, we perform an ensemble
60 Single microtubule dynamics under force and confinement
F (x)=k (x−x0)
v+ [η(x)]
v−
x0 x
ωr ωc [η(x)]
Figure 4.6: Schematic representation of a single MT growing against an elastic
obstacle with spring constant k. From top to bottom: MT shrinks (red) with v−.
MT growth (green) under force F (x) = k (x−x0) with η(x)≡ F (x)/F0, v+[η(x)], and
force-dependent catastrophe rate ωc[η(x)].
average over many realizations of the stochastic growth trajectory and obtain
d
dt
〈η〉= k
F0
〈v+(η)〉. (4.14)
Since the probability density function p(η), which determines the ensemble average, is
not known, we use a mean-field like approximation to perform further calculations. In
the following, we assume a strongly localized probability density distribution function,
centered at the average polymerization force 〈η〉, and with most of the probability weight
localized at or close to the average maximal polymerization force 〈η〉. In addition we
neglect all higher correlations. This type of approximation allows us to replace global
averages by local averages. To be more precise, we set 〈v+(η)〉 ≈ v+(〈η〉). The average
growth velocity under force is replaced by the growth velocity under the average force 〈η〉
and we obtain, in mean field approximation, the following equation of motion for 〈η〉,
d
dt
〈η〉= k
F0
v+(〈η〉) (4.15)
With the initial condition 〈η〉(0) = 0 we find a time evolution
〈η〉(t) = ln
[
(1−ωon/ωoff)e−t/τ +ωon/ωoff
]
(4.16)
≈ ηstall+ln[1− exp(−t/τ)] (4.17)
with a characteristic time scale τ = F0/dkωoff ≈ (102...104)s for k ≈ 10−5...10−7N/m. For
4.3 Single microtubule dynamics under an elastic force 61
long times t τ , eq. (4.16) approaches the dimensionless stall force 〈η〉 = ηstall, see eq.
(2.14), which is the maximal polymerization force in the absence of catastrophes. The
approximation (4.17) holds for ωon/ωoff  1.
MT growth is ended, however, by a catastrophe, and the average time spent in the growing
state is t = 1/ωc(〈ηmax〉). Together with eq. (4.16), this gives a self-consistent mean field
equation for the average maximal polymerization force 〈ηmax〉,
〈ηmax〉= ln
[
(1−ωon/ωoff)e−1/ωc(〈ηmax〉)τ +ωon/ωoff
]
. (4.18)
The average maximal polymerization force 〈ηmax〉 is always smaller than the stall force
ηstall as can be seen from eqs. (4.16) and (4.17). Since ωon/ωoff  ωcτ  1 for realistic
forces and parameter values, eq. (4.18) can be approximated by
〈ηmax〉 ≈ ln
(
ωon
ωoffτωc(〈ηmax〉)
)
= ηstall− ln [τωc(〈ηmax〉)] . (4.19)
For a catastrophe rate increasing exponentially above the characteristic force F0, eq.
(2.37), we find
〈ηmax〉 ∼
1
2
ln
(
ωondk
F0ωc(0)
)
, (4.20)
i.e., the average maximal polymerization force grows logarithmically in ωon (note that the
catastrophe rate in the absence of force decreases as ωc(0)∝ 1/ωon [18]), see Fig. 4.7 for
k = 10−5N/m. Within a slightly different catastrophe model obtained from experimental
data and discussed in section 4.4.1, this logarithmic dependence can be shown exactly.
Fig. 4.7 shows 〈ηmax〉 as a function of ωon. Analytical results from eq. (4.18) agree with
numerical findings from stochastic simulations. The average maximal polymerization
force 〈ηmax〉 increases with increasing k, see eq. (4.20), but it remains smaller than the
stall force ηstall. Stochastic simulations show considerable fluctuations of 〈ηmax〉, which
are caused by broad and exponentially decaying probability distributions for ηmax and
which we quantify by measuring the standard deviation 〈η2max〉−〈ηmax〉2. For increasing
k, probability distributions become more narrow and mean field results approach the
simulation results for 〈ηmax〉.
62 Single microtubule dynamics under force and confinement
0
1
2
3
25 50 75 100
〈η
m
ax
〉
ωon [s−1]
ηstallk = 10
−5N/m
k = 10−6N/m
k = 10−7N/m
Figure 4.7: Average maximal polymerization force 〈ηmax〉 for an elastic obstacle and
in the absence of rescues as a function of ωon for different values of k = 10−5N/m(N),
10−6N/m(), and 10−7N/m(•). Data points represent results from simulations, solid
lines are solutions of eq. (4.18). Error bars represent the standard deviation of the
stochastic quantity 〈ηmax〉. Dashed line: dimensionless stall force ηstall = ln(ωon/ωoff).
4.3.2 Non-zero rescue rate2
4.3.2.1 Stochastic approach
For a non-zero rescue rate ωr, phases of growth, in which η(x) increases and which last
1/ωc(η) on average, are ended by catastrophes which are followed by phases of shrinkage.
Shrinking phases last 1/ωr on average, and during shrinkage the elastic obstacle is relaxed
and η(x) decreases. After rescue, the MT switches back to a state of growth. In contrast to
the case without rescue events, the system can attain a steady state. In this steady state,
the average length loss during shrinkage, v−/ωr, equals the average length gain during
growth, v+(η)/ωc(η), and the MT oscillates around a time-averaged stall length 〈x〉, which
is directly related to the time-averaged polymerization force by 〈η〉 = (k/F0)(〈x〉−x0).
In the following, the steady state dynamics and the average polymerization force are
characterized. We start with an analysis of the full master equations focusing on the
stationary state followed by a dynamical mean field theory, which can also be applied to
dilution experiments.
In the presence of a x-dependent force η(x), the master equations for the time evolution
2Parts of the results of this section have been achieved in cooperation with Nina Müller during her
bachelor thesis [82].
4.3 Single microtubule dynamics under an elastic force 63
of p+(x,t) and p−(x,t) become
∂tp+(x,t) =−ωc(x)p+(x,t)+ωrp−(x,t)−∂x[v+(x)p+(x,t)] (4.21)
∂tp−(x,t) = ωc(x)p+(x,t)−ωrp−(x,t)+v−∂xp−(x,t), (4.22)
which differ from eqs. (2.1) and (2.2) by the x-dependence of the growth velocity and the
catastrophe rate. Both growth velocity v+(x) = v+[η(x)] and catastrophe rate ωc(x) =
ωc{v+[η(x)]} become x-dependent via their force-dependence. Therefore, also the force-
dependent length parameter λ(η) from eq. (4.11) becomes x-dependent via its force-
dependence, λ(x) = λ[η(x)]. Eqs. (2.1) and (2.2) are supplemented by a reflecting bound-
ary condition v+(0)p+(0, t) = v−p−(0, t) at x = 0, similar to eq. (2.3).
For the steady state, eqs. (4.21) and (4.22) are solved on the half-space x> 0 with reflecting
boundary conditions at x = 0, and we can calculate the overall MT length distribution
P (x) = p+(x)+p−(x) explicitly,
P (x) = N
(
1+ v−
v+(x)
)
ex0/λ(0) exp
[
∫ x
x0
dx′/λ(x′)
]
(4.23)
with a normalization
N−1 =
∫ ∞
0
dx
(
1+ v−
v+(x)
)
ex0/λ(0)e
∫ x
x0
dx′/λ(x′), (4.24)
where λ(x) = λ(η=0) in the force-free region x < x0 and λ(x) = λ[η(x)] for x > x0 and,
likewise, v+(x) = v+(η=0) for x < x0 and v+(x) = v+[η(x)] for x > x0. This implies
ex0/λ(0)e
∫ x
x0
dx′/λ(x′) = ex/λ(0) and, thus, a simple exponential dependence of P (x) for x <
x0. A similar OPDF has been found for dynamic MTs in the presence of MT end-tracking
molecular motors [83].
With increasing length x, also the force η(x) increases and, thus, v+[η(x)] decreases and
ωc[η(x)] grows exponentially. If x becomes sufficiently large that the condition λ−1[η(x)]<
0 holds, the distribution P (x) starts to decrease exponentially. In this length regime
the MT undergoes a catastrophe with high probability. Because the distribution always
decreases exponentially for sufficiently large x, a single MT growing against an elastic
obstacle is always in the regime of bounded growth regardless of how large the values of
ωon and ωr are chosen. This behavior is a result of the linearly increasing force, which
gives rise to arbitrarily large forces for increasing x in contrast to growth under constant
or zero force, where a MT can either be in a phase of bounded or unbounded growth as
mentioned above.
The behavior is also in contrast to length distributions in confinement between fixed rigid
walls, where we found a transition between exponentially decreasing and increasing length
distributions: The elastic obstacle typically leads to a non-monotonic length distribution
64 Single microtubule dynamics under force and confinement
with a maximum in the region x>x0 (as long as the on-rate ωon and rescue rate ωr are suf-
ficiently large and the obstacle stiffness k sufficiently small). While rescue events and an
exponential decrease in the growth velocity v+[η(x)] cause P (x) to increase exponentially
for small MT length, catastrophes are responsible for an exponential decrease for large x.
The interplay between rescues and catastrophes gives rise to strongly localized probability
distributions with a maximum. Figs. 4.8 (a-d) show the steady state distribution P (x)
obtained from eq. (4.23) for different values of ωon and ωr. We chose k = 10−7N/m and
x0 = 10µm. In the steady state, a stable length distribution with a well defined average
length 〈x〉 =
∫∞
0 P (x)xdx is maintained although the MT is still subject to dynamic in-
stability. The length distributions drop to zero for large x, where λ−1(x)∼−ωc(x)/v+(x)
and ωc(x)/v+(x) increases exponentially with increasing force.
The most probable MT length xmp maximizes the stationary length distribution [eq.
(4.23)]. Because v−  v+(x) and using the approximation of an exponentially decreasing
growth velocity, v+[η(x)] ≈ v+(0)e−η(x), which is valid for ωon  ωoff [see eq. (2.13)], we
obtain a condition λ−1(xmp) =−∂xη(xmp) =−k/F0 or
v+(ηmp)ωr−v−ωc(ηmp) =−(k/F0)v−v+(ηmp) (4.25)
for the corresponding most probable force ηmp = (k/F0)(xmp−x0).
For an exponentially increasing catastrophe rate above the characteristic force F0, eq.
(2.37), we find
ηmp ∼
1
2
ln
[
v+(0)ωr
v−ωc(0)
(
1+ kv−
F0ωr
)]
(4.26)
We can distinguish two limits:
(i) For a soft obstacle with kv−/F0ωr  1 the most probable force ηmp is identical to the
critical force ηc for MT dynamics under constant force, see eq. (4.13), because the right
hand side in the condition (4.25) for ηmp can be neglected and we exactly recover condition
(4.12) for ηc. The most probable MT length thus “self-organizes” into a “critical” state
with ηmp ≈ ηc, and a MT pushing against a soft elastic obstacle generates the same force
as if growing against a constant force. This force grows logarithmically in the on-rate ωon
and the rescue rate ωr.
(ii) For a stiff obstacle with kv−/F0ωr  1, on the other hand, the most probable force
is larger than the critical force, ηmp  ηc, and the MT growing against a stiff obstacle
generates a higher force. This limit can also be realized for vanishing rescue rate ωr,
and for kv−/F0ωr  1 we indeed recover the maximal pushing force in the absence of
rescue events, i.e. ηmp ≈ 〈ηmax〉 from eq. (4.20) with v+(0) ≈ ωond. This force grows
logarithmically in the on-rate ωon. Furthermore, if ηmp becomes negative for small on-
rates and rescue rates [leading to λ−1(0) < −k/F0, see eq. (4.26)] the stationary length
distribution has no maximum, see for example Figs. 4.8(a,b) at the lowest on-rates.
4.3 Single microtubule dynamics under an elastic force 65
With respect to the MT’s ability to generate force the two limits can be interpreted
also in the following way: F0 is the characteristic force above which the catastrophe
rate increases exponentially. For kv−/F0ωr  1, the average length loss during a period
of shrinkage, v−/ωr, is much smaller than the length F0/k, which is the displacement
x− x0 of the elastic obstacle under the characteristic force F0. Therefore, the MT tip
always remains in the region x > x0 under the influence of the force for a soft obstacle
with kv−/F0ωr  1, whereas, for a stiff obstacle kv−/F0ωr  1, it typically shrinks back
into the force-free region x < x0 before the next rescue event . The force generation by
the MT can only be enhanced by rescue events if rescue takes place under force in the
regime x > x0. Therefore, we find an increased polymerization force ηmp ≈ ηc  〈ηmax〉
as compared to the force 〈ηmax〉 without rescue events discussed in the previous section
only in the limit kv−/F0ωr  1, i.e., for a soft obstacle or sufficiently large rescue rate.
In the limit kv−/F0ωr  1 of a stiff obstacle, the MT only generates the same force as in
the absence of rescues, ηmp ≈ 〈ηmax〉.
By comparing the condition (4.12) or v+(ηc) = v−ωc(ηc)/ωr for the critical force ηc, the
condition (4.25) or v+(ηmp) = v−ωc(ηmp)/ωr(1+ kv−/F0) < v−ωc(ηmp)/ωr for the most
probable force ηmp, and the condition v+(ηstall) = 0 for the stall force, see eq. (2.14), it
follows that
ηc ≤ ηmp  ηstall (4.27)
i.e., force generated against an elastic obstacle is between critical and stall force but
typically well below the stall force, which is the maximal polymerization force in the
absence of catastrophes. Therefore, the stall length xstall = (F0/k) ln(ωon/ωoff) + x0 is
always much larger than the most probable MT length xmp at the maximum of the
stationary length distribution, see Fig. 4.8(a). This shows that the dynamic instability
reduces the typical MT length significantly compared to simple polymerization kinetics.
In order to quantify the width of the stationary distribution P (x) we expand the expo-
nential in eq. (4.23) up to second order about the maximum at xmp. To do so we first
expand λ−1(x) up to first order:
λ−1(x)≈− k
F0
[
v+(xmp)ωr+v−ωc(xmp)
v+(xmp)v−
]
(x−xmp) (4.28)
where we used v+[η(x)] ≈ v+(0)e−η(x), which is valid for ωon  ωoff [see eq. (2.13)], and
where we approximated the catastrophe rate by an exponential ωc[η(x)]≈ ωc(0)eη(x) ac-
cording to eq. (2.37) resulting in ω′c[η(x)]≈ kωc[η(x)]/F0. The prime denotes a derivative
with respect to the length x. Using the expansion [eq. (4.28)] in eq. (4.23), we obtain an
approximately Gaussian length distribution
P (x)≈ N
(
1+ v−
v+(x)
)
ex0/λ(0) exp
[
(xmp−x0)2
2σ2
]
exp
[
−(x−xmp)
2
2σ2
]
(4.29)
66 Single microtubule dynamics under force and confinement
with a width
σ2 = F0
k
[
v+(xmp)v−
v+(xmp)ωr+v−ωc(xmp)
]
≈
(F0
k
)2(
1+ 2F0ωr
kv−
)−1
(4.30)
where we used the saddle point condition (4.25) in the last approximation and the expo-
nential approximations v+[η(x)]≈ v+(0)e−η(x) and ωc[η(x)]≈ ωc(0)eη(x). Again we have
to distinguish the two limits of soft and stiff obstacles:
(i) For a soft obstacle with kv−/F0ωr  1 we find σ2 ≈ F0v−/2kωr. This shows that the
width of the length distribution decreases with increasing ωr but is roughly independent
of the on-rate ωon, as can also be seen in the series of numerical results shown in Figs.
4.8. A closer inspection shows that the width of the stationary length distribution P (x)
is slightly decreasing with the on-rate ωon.
(ii) For a stiff obstacle with kv−/F0ωr  1, on the other hand, we find σ2 ≈ (F0/k)2,
which only depends on obstacle stiffness. All in all, σ2 is monotonously decreasing for
increasing stiffness k.
For a soft obstacle kv−/F0ωr  1, high rescue rates thus lead to a sharply peaked length
distribution P (x) and suppress fluctuations of the MT length around x = xmp and we
expect 〈x〉 ≈ xmp to a very good approximation. This property of a sharp maximum in
P (x) will make the mean field approximation that is discussed in the next section very
accurate.
If the obstacle stiffness k is increased the most probable MT length xmp = x0+ηmpF0/k
approaches x0, and a considerable probability weight is shifted to MT lengths x below x0
(see Fig. 4.9). The average length approaches and finally drops below x0. This signals
that the force generated by the MT is no longer sufficient to push the obstacle out of
its equilibrium position x0. The obstacle now serves as a fixed rigid boundary and P (x)
approaches the results from eqs. (4.6) and (4.7). The dynamics of a single MT within
confinement can therefore be seen as a special case of the dynamics in the presence of an
elastic obstacle, i.e., for small ωon and ωr or for large spring constants k.
So far we have quantified the generated force by the most probable force ηmp. The gener-
ated force can also be quantified by the average steady-state force 〈η〉=
∫∞
0 η(x)P (x)dx.
Using the stationary distribution [eq. (4.23)] with normalization [eq. (4.24)] we can calcu-
late 〈η〉; results are shown in Fig. 4.10 in comparison with the most probable force ηmp,
which is determined numerically from the maximum of P (x), and the stall force ηstall in
the absence of dynamic instability from eq. (2.14). For 〈η〉, there is excellent agreement
with stochastic simulations over the complete range of parameter values. The results
clearly show that the dynamic instability reduces the ability to generate polymerization
forces since, even for large values of ωon and ωr, the average force 〈η〉 is always smaller
4.3 Single microtubule dynamics under an elastic force 67
0
0.04
0.08
0.12
0 50 100 150
P
(x
)[µ
m
−
1 ]
x [µm]
ωr=0.03s−1
25s−1
50s−1 75s−1
100s−1
(a)
xstall(ωon=25s−1)
0
0.02
0.04
0.06
0.08
0 50 100 150
P
(x
)[µ
m
−
1 ]
x [µm]
ωr=0.05s−1
25s−1
50s−1 75s−1 100s−1
(b)
0
0.02
0.04
0.06
0.08
0 50 100 150
P
(x
)[
µm
−
1 ]
x [µm]
ωr=0.1s−1
25s−1 50s−1 75s−1 100s−1
(c)
0
0.04
0.08
0.12
0 50 100 150
P
(x
)[µ
m
−
1 ]
x [µm]
ωr=0.2s−1
25s−1 50s−1 75s−1100s−1
(d)
Figure 4.8: Stationary MT length distribution P (x) in the steady state for growth
against an elastic obstacle with ωon = 25s−1,50s−1,75s−1,100s−1 and different values
of ωr. We set k = 10−7N/m and x0 = 10−5m. (a) ωr = 0.03s−1. (b) ωr = 0.05s−1. (c)
ωr =0.1s−1. (d) ωr =0.2s−1. Dashed line represents x0. In picture (a) the stall length
xstall for ωon = 25s−1, obtained from simple polymerization kinetics, is indicated by
an arrow.
0
0.1
0.2
0.3
0.4
0.5
0 20 40 60 80
P
(x
)[µ
m
−
1 ]
x [µm]
ωon = 50s−1
ωr = 0.05s−1
k = 10−7N/m
k = 10−6N/m
k = 10−5N/m
Figure 4.9: Steady state distribution P (x) from eq. (4.23) for ωon = 50s−1, ωr =
0.05s−1 and different values of the spring constant k. Dashed line represents x0.
68 Single microtubule dynamics under force and confinement
0
0.5
1
1.5
2
2.5
3
25 50 75 100
〈η
〉
ωon [s−1]
k = 10−7N/m
ηstall
Figure 4.10: Average steady state force 〈η〉 as a function of ωon for growth against
an elastic obstacle with ωr = 0.03s−1(),0.05s−1(),0.1s−1(),0.2s−1(•) and k =
10−7N/m. Solid lines: 〈η〉 =
∫∞
0 η(x)P (x)dx with P (x) given by eqs. (4.23) and
(4.24). Dashed lines: 〈η〉 calculated from mean field equation (4.32). Dotted lines:
most probable force ηmp, measured in simulations, for ωr = 0.03s−1 and ωr = 0.05s−1.
Also shown is the dimensionless stall force ηstall obtained from simple polymerization
kinetics [eq. (2.14)].
than the stall force. Nevertheless forces up to F ∼ 1.5F0 can be obtained in the steady
state for realistic parameter values. Comparing 〈η〉 and ηmp we find 〈η〉 ≤ ηmp, and both
forces become identical, 〈η〉 ≈ ηmp, in the limit of large rescue rates or a soft obstacle
kv−/F0ωr  1. Comparing different combinations of ωon and ωr and the corresponding
forces, one finds that the influence of the on-rate ωon on force generation is more signifi-
cant than the influence of the rescue rate ωr. For ωon = 100s−1, a four fold increase of the
rescue rate ωr gives rise to an increase of 〈η〉 by a factor of ∼ 1.5, while for ωr = 0.1s−1, a
four fold increase of the on-rate ωon results in an amplification of the force 〈η〉 by a factor
of ∼ 9. These results can be explained within a mean field theory presented in the next
section.
4.3.2.2 Mean field approach
In the following, we show that we can reproduce many of the results for the average
polymerization force 〈η〉 for non-zero rescue rate using a simplified mean field approach.
Since the switching between the two states of growth is a stochastic process, the length
x and the force η(x) are stochastic variables. Therefore, the velocity of growth v+[η(x)]
and the catastrophe rate ωc[η(x)] also become stochastic variables which, in the steady
state, fluctuate around their average values. Within the mean field approach we neglect
these fluctuations and use 〈v+[η(x)]〉= v+(〈η〉) and 〈ωc[η(x)]〉= ωc(〈η〉). In the mean field
4.3 Single microtubule dynamics under an elastic force 69
0
0.5
1
1.5
25 50 75 100
〈η
〉
ωon [s−1]
k = 10−5N/m
Figure 4.11: Average steady state force 〈η〉 as a function of ωon for growth against
an elastic obstacle with ωr = 0.03s−1(),0.05s−1(),0.1s−1(),0.2s−1(•) and k =
10−5N/m. Solid lines: 〈η〉 =
∫∞
0 η(x)P (x)dx with P (x) given by eqs. (4.23) and
(4.24). Dashed lines from bottom to top: 〈η〉 calculated from mean field equation
(4.32) for ωr = 0.03s−1,0.05s−1,0.1s−1, and 0.2s−1.
approximation, the average time in the growing state is given by 1/ωc(〈η〉) and the average
growth velocity is v+(〈η〉). The average time in a shrinking state is 1/ωr. Therefore, the
mean field probabilities to find the MT growing or shrinking are p+ = ωr/[ωr +ωc(〈η〉)]
and p− = ωc(〈η〉)/[ωr +ωc(〈η〉)], respectively. This results in the following mean field
average velocity v of a single MT under force:
v(〈η〉) = v+(〈η〉)ωr−v−ωc(〈η〉)
ωr+ωc(〈η〉)
. (4.31)
In the steady state the barrier is pushed so far that 〈η〉 stalls the MT. We require
v(〈η〉) = 0 and obtain the condition
v+(〈η〉)ωr = v−ωc(〈η〉) (4.32)
for the stationary state. This condition corresponds to a force, where the average length
gain during growth, v+(〈η〉)/ωc(〈η〉), equals the average length loss during shrinking,
v−/ωr. From the mean field equation (4.32), the average steady state force, 〈η〉 can be
calculated as a function of ωr and ωon. The average length 〈x〉 can be obtained from
the relation 〈η〉= (k/F0)(〈x〉−x0). Results obtained from the mean field equation (4.32)
match numerical results from stochastic simulations very well as shown in Fig. 4.10.
The mean field condition (4.32) is identical to the condition (4.12) for the critical force
70 Single microtubule dynamics under force and confinement
ηc for MT dynamics under constant force such that
〈η〉= ηc, (4.33)
which can be interpreted as “self-organization” of the average MT length or the average
force to the “critical” state. Therefore, the curves presented in Fig. 4.10 for 〈η〉 are
identical to the curves shown in Fig. 4.4 for ηc.
This also allows us to take over the results we derived for the critical constant force
ηc. Using the approximation of an exponentially decreasing growth velocity, v+[η(x)] ≈
v+(0)e−η(x), which is valid for ωon  ωoff [see eq. (2.13)], and an exponentially increasing
catastrophe rate above the characteristic force F0, see eq. (2.37), we find
〈η〉 ∼ 1
2
ln
(
v+(0)ωr
v−ωc(0)
)
. (4.34)
which is identical to the result (4.12) for ηc.
Comparing with the stall force and the most probable force, we use relation (4.27) and
find
〈η〉= ηc ≤ ηmp  ηstall. (4.35)
In the limit of a soft obstacle, kv−/F0ωr  1, the average force 〈η〉 approaches the most
probable force 〈η〉 ≈ ηmp, whereas the mean field average force 〈η〉 is always smaller than
the stall force ηstall in the absence of dynamic instability from eq. (2.14).
Finally, we discuss the limits of validity of the mean field approximation. The mean field
approximation is based on the existence of a pronounced maximum in the stationary MT
length distribution P (x), which contains most of the weight of the probability density
P (x). It breaks down if this maximum broadens or vanishes, such that a considerable
amount of probability density is shifted below x0 into the regime of force-free growth.
Then the MT typically shrinks into the force-free region x< x0 during phases of shrinkage
such that the growing phase explores the whole range of forces starting from η = 0 up to
η > 〈η〉, and the approximation of a constant average force η ≈ 〈η〉 during growth is no
longer fulfilled. For small spring constants k or large values of ωr, the length distribution
P (x) assumes a Gaussian shape with width σ, see eqs. (4.29) and (4.30). When k is
increased for a fixed combination of ωon and ωr, the average length 〈x〉 approaches x0 as
〈x〉−x0 ∝ 1/k, whereas the width σ of the length distribution only decreases as σ∝ 1/
√
k
in the regime of a soft obstacle kv−/F0ωr  1, as can be seen from eq. (4.30). Therefore,
an increasing amount of probability density is shifted below x0, where no force is acting
on the MT ensemble (see Figs. 4.8(a) and 4.9). The mean field approximation is only
valid for spring constants k which fulfill 〈x〉−x0  σ/2 for given parameters ωon and ωr.
4.3 Single microtubule dynamics under an elastic force 71
With 〈η〉= (k/F0)(〈x〉−x0) this is equivalent to a condition
〈η〉  kσ
2F0
≈ 1
2
(
1+ 2F0ωr
kv−
)−1/2
(4.36)
according to eq. (4.30). This condition can only be fulfilled in the limit of a soft obstacle
with kv−/F0ωr  1. For the validity of the mean field approximation we therefore recover
the condition that the average length loss during a period of shrinkage, v−/ωr, is much
smaller than the typical displacement F0/k of the elastic obstacle under the characteristic
force F0. Then the MT tip always remains in the region x > x0 under the influence of the
force.
4.3.2.3 Dilution dynamics
Within the mean field approach we can also derive an analytical time evolution of the
average time-dependent force 〈η〉(t). The time evolution is based on eq. (4.31), which
gives a mean field approximation for the average MT velocity v(〈η〉) as a function of the
average force. On the other hand, the average MT growth velocity is related to the time
derivative of the average force by
d
dt
〈η〉= k
F0
d
dt
〈x〉= k
F0
v(〈η〉). (4.37)
Using eq. (4.31) for v(〈η〉), this gives a mean field equation of motion for 〈η〉(t) similar to
eq. (4.15) in the absence of rescue events. Integrating this equation numerically we obtain
mean field trajectories for the average force 〈η〉(t) as a function of time t. Figs. 4.12 shows
such trajectories for k = 10−7N/m and a initial condition 〈η〉(0) = 0 at t= 0. Also shown
in Figs. 4.12 are results from stochastic simulations, which show excellent agreement with
the mean field trajectories.
We now address the question of how fast a single MT responds to external changes of
one of its growth parameters. Here we focus on fast dilution of the tubulin concentration,
which is directly related to the tubulin on-rate ωon. In vivo, the tubulin concentration can
be changed by tubulin binding proteins like stathmin [28], while in in vitro experiments,
the tubulin concentration can be diluted within seconds [84]. In the following we give a
mean field estimate of the typical time scale, which governs the return dynamics of the MT
back to a new steady state after the tubulin on-rate is suddenly decreased. In the initial
steady state the average velocity v(〈η〉i) vanishes and the average polymerization force 〈η〉i
(and, thus, the average length 〈x〉i) can be calculated from the condition v+(〈η〉i)ωr =
v−ωc(〈η〉i), cf. eq. (4.32), for a given combination of ωon and ωr. If ωon is suddenly
decreased this leads to a sudden decrease in the growth velocity to ˜v+(η)< v+(η) and an
increase of the catastrophe rate to ˜ωc(η) > ωc(η), resulting in a negative average velocity
72 Single microtubule dynamics under force and confinement
0
0.5
1
1.5
0 2500 5000 75001000012500
〈η
〉(
t)
t [s]
ωon=100s−1
ωon=70s−1
ωon=50s−1
ωon=30s−1
ωr=0.05s−1
(a)
0
0.5
1
1.5
0 2500 5000 75001000012500
〈η
〉(
t)
t [s]
ωr=0.2s−1
ωr=0.05s−1
ωr=0.03s−1
ωon=50s−1
(b)
Figure 4.12: (a): Average force 〈η〉(t) as a function of time t for k = 10−7N/m,
ωr = 0.05s−1, and ωon = 30,50,75,100s−1. Symbols: time-dependent average force
〈η〉(t) measured in simulations. Solid lines: time-dependent average force trajectories
calculated from eq. (4.37). (b): Average force 〈η〉(t) as a function of time t for
k = 10−7N/m, ωon = 50s−1, and ωr = 0.03,0.05,0.2s−1. Symbols: time dependent
average force 〈η〉(t) measured in simulations. Solid lines: time-dependent average
force trajectories calculated from eq. (4.37).
v(〈η〉) = [˜v+(〈η〉)ωr−v−˜ωc(〈η〉)]/[ωr+ ˜ωc(〈η〉)]< 0 according to eq. (4.31). Consequently,
the MT starts to shrink with an average velocity v(〈η〉) < 0. This relaxes the force
from the elastic obstacle, i.e., 〈η〉(t) starts to decrease from the initial value ηi ≡ 〈η〉i.
With decreasing average force 〈η〉(t), the average growth velocity v(〈η〉(t)) increases again
(because ˜v+ increases and ˜ωc decreases) until the steady state condition ˜v+(〈η〉f)ωr =
v−˜ωc(〈η〉f) holds again and a new steady state force 〈η〉f < 〈η〉i is reached (see Fig. 4.13).
The relaxation dynamics to the new steady state after tubulin dilution is therefore gov-
erned by the average velocity v(〈η〉) given by eq. (4.31). To extract a characteristic
relaxation time scale, we expand the average velocity v(〈η〉) to first order around the final
steady-state polymerization force ηf ≡ 〈η〉f, which is the solution of eq. (4.32) with ωr and
the decreased tubulin on-rate ωon, which takes its dilution value. Using v(ηf) = 0 one
finds in first order
v(〈η〉)≈−
[
v+(ηf)ωr+v−ω′c(ηf)
ωr+ωc(ηf )
]
(〈η〉−ηf) (4.38)
where the prime denotes the derivative with respect to the force η. In the last approxi-
mation we used the mean field condition eq. (4.32) and v+[η(x)]≈ v+(0)e−η(x), which is
valid for ωon  ωoff [see eq. (2.13)]. This expansion is only valid for average forces close to
the new average polymerization force ηf. Using this expansion, the time evolution (4.37)
of the average force after dilution exhibits an exponential decay
〈η〉(t) = ηf+(ηi−ηf)e−t/τd (4.39)
4.4 Experimental and linear catastrophe model 73
0.5
0.75
1
0 10000 20000 30000
〈η
〉(
t)
t [s−1]
ηf
ηi
〈η〉(t)∼e−t/τ
Figure 4.13: Average force 〈η〉(t) as a function of time t. Symbols are results
obtained from simulations. We set k = 10−7N/m, ωr = 0.05s−1 and ωon = 75s−1. At
t = 20000s, ωon is diluted down to ωon = 50s−1. Dashed line represents a fit with an
exponential decay (4.39) to the simulated data with fit parameter τd ≈ 1762s. Solid
lines indicate the average force in the initial state ηi before dilution and in the new
final state ηf after dilution.
with a characteristic dilution time scale
τd =
F0
k
ωr+ωc(ηf)
v+(ηf)ωr+v−ω′c(ηf)
≈ F0
k
ωr+ωc(ηf)
2v−ωc(ηf)
(4.40)
where we approximated the catastrophe rate by an exponential ωc[η(x)]≈ ωc(0)eη(x) ac-
cording to eq. (2.37), and we used the mean field condition eq. (4.32). In the limit
ωc(ηf) ωr, i.e., at forces ηf  1, we obtain the simple result τd ≈ F0/2v−k. In general,
the relaxation time τd is proportional to the square σ2 of the width of the stationary
distribution, cf. eq. (4.30). A narrow length distribution gives rise to fast relaxation to
the new average force.
A fit of eq. (4.39) to simulated data results in τd ≈ 1762s (see Fig. 4.13), while eq. (4.40)
yields τd ≈ 2766s. Results for τd match well in their order of magnitude, but slightly differ
in their absolute values, due to the exponential approximation of v+(η) and ωc(η).
4.4 Experimental and linear catastrophe model
So far we have employed the catastrophe rate derived by Flyvbjerg et al., to which we
will refer as ωc,Flyv in the following. This expression for the catastrophe rate was based
on theoretical calculations of the inverse passage time to a state with a vanishing GTP-
cap, see Sec. 2.3 and eq. (2.35). In order to investigate the robustness of our results with
74 Single microtubule dynamics under force and confinement
respect to changes of the catastrophe model, we now investigate two alternative expression
for the catastrophe rate, the experimental catastrophe rate ωc,Jans by Janson et al. (see
Sec. 2.3) and the linear catastrophe rate ωc,lin (see Sec. 2.3). Throughout this section,
we focus on the third confinement scenario of an elastic obstacle, and we compare results
from the different catastrophe models for zero rescue rate ωr = 0 and non-zero rescue rate
ωr > 0.
We start with the catastrophe model by Janson et al. (Sec. 4.4.1) and subsequently discuss
the linear catastrophe model (Sec. 4.4.2).
4.4.1 Experimental catastrophe model by Janson et al.
4.4.1.1 Vanishing rescue rate
We begin with the case ωr =0 without rescue events, and we calculate the average maximal
polymerization force within the experimental catastrophe model using the self-consistent
mean field eq. (4.18), which holds independently of the choice of catastrophe model (see
Sec. 4.3.1). As for the catastrophe by Flyvbjerg et al., we have ωc,Jansτ  1 for realistic
parameter values and v+(〈η〉) < −b/a, and eq. (4.18) can be solved explicitly for 〈ηmax〉
in this limit. We find an average maximal polymerization force
〈ηmax〉 ≈ ln
([
(
A2+B
)1/2
−A
])
(4.41)
with
A≡ (ωon/ωoff−1)adωoff− (ωon/ωoff−1)b− τ
2τ
B ≡ (ωon/ωoff−1)adωon
τ
.
Since ωon/ωoff  1, eq. (4.41) can be approximated by
〈ηmax〉 ≈ ln(ωon/ωmax) (4.42)
with
ωmax ≡
2τωoff
[
(adωoff− b)2+4adωoffτ
]1/2
− [adωoff− b]
(4.43)
and τ =F0/(dkωoff) as in Sec. 4.3.1. For realistic parameter values, we have τ  adωoff≥ b,
and recover the expression (4.20) derived using the Flyvbjerg catastrophe model:
〈ηmax〉 ≈
1
2
ln
(
ω2onad
ωoffτ
)
≈ 1
2
ln
(
ωondk
F0ωc,Jans(0)
)
. (4.44)
4.4 Experimental and linear catastrophe model 75
0
1
2
25 50 75 100
〈η
m
ax
〉
ωon [s−1]
ηstall
k=10−7N/m
k=10−6N/m
k=10−5N/m
(a)
ωc,Jans
ωc,Flyv
0
1
2
25 50 75 100
〈η
〉
ωon [s−1]
ηstall
ωr=0.03s−1
ωr=0.2s−1
(b)
ωc,Flyv
ωc,Jans
Figure 4.14: (a): Average maximal polymerization force 〈ηmax〉 as a function of
ωon and ωr = 0 for k = 10−7N/m, 10−6N/m and 10−7N/m (top to bottom). Dotted
line: dimensionless stall force ηstall. Solid lines: 〈ηmax〉 obtained with ωc,Flyv [eq.
(4.18)]. Dashed lines: 〈ηmax〉 obtained with ωc,Jans [eq. (4.41)]. (b): Average steady
state force 〈η〉 as a function of ωon. We set k = 10−7N/m, ωr = 0.2s−1 (top) and
ωr = 0.03s−1(bottom). Solid lines: 〈η〉 obtained with ωc,Flyv [eq. (4.32)]. Dashed
lines: 〈η〉 obtained with ωc,Jans [eq. (4.45)]. Dotted line: dimensionless stall force
ηstall.
In Fig. 4.14 (a), 〈ηmax〉 as obtained from eq. (4.18) with the Flyvbjerg catastrophe model
and eq. (4.41) with the experimental catastrophe model are shown as a function of ωon.
Results match qualitatively and quantitatively well, although they are obtained from two
different catastrophe models. The maximal polymerization force 〈ηmax〉 always remains
smaller than the stall force ηstall.
4.4.1.2 Non-zero rescue rate
We calculate the OPDF P (x) in the steady state from eq. (4.23) with the experimental
catastrophe rate ωc,Jans(η) given by eq. (2.38). In Fig. 4.15 results for ωr = 0.2s−1, k =
10−7N/m and ωon = 25,50,75,100s−1 are shown.
For a soft spring constant k = 10−7N/m the OPDFs are sharply peaked. In Sec. 4.3.2 we
discussed the characteristics of this sharply peaked OPDFs in detail within the context
of the catastrophe model by Flyvbjerg et al. In particular, for a soft spring constant
k, results from full stochastic calculations can be reproduced by a simplified mean field
theory (see Sec. 4.3.2.2). This mean field approach is based on a pronounced maximum
in the stationary MT length distribution P (x). As we find sharply peaked OPDFs for the
experimental catastrophe rate ωc,Jans by Janson et al., we restrict our further analysis to
mean field results.
For the experimental catastrophe rate [eq. (2.38)], the mean field equation (4.32) can be
solved explicitly, and the average steady-state force 〈η〉 is given by
〈η〉= ln(ωon/ωav) , (4.45)
76 Single microtubule dynamics under force and confinement
0
0.02
0.04
0.06
0.08
0.1
0 50 100 150
P
(x
)[µ
m
−
1 ]
x [µm]
ωr=0.2 s−1k = 10−7 N/m
25s−1 50s−1 75s−1 100s−1
Figure 4.15: Stationary MT length distribution P (x) for growth against an elastic
obstacle from eq. (4.23) and with the experimental catastrophe rate ωc,Jans [eq. (2.38)].
We set k = 10−7N/m, x0 = 10−5m (dashed line) and ωr = 0.2s−1. Results for ωon =
25s−1,50s−1,75s−1,100s−1 are shown.
with
ωav ≡


(
b
2ad
)2
+ v−
ωrad2


1/2
− b
2ad
+ωoff (4.46)
Again 〈η〉< ηstall since ωav >ωoff. Fig. 4.14 (b) show 〈η〉 as a function of ωon. For realistic
parameter values, we have v−/ωr  b2/a and (v−/ωrad2)1/2  ωoff, and we recover the
expression (4.34) derived using the Flyvbjerg catastrophe model:
〈f〉 ≈ 1
2
ln
(
ω2onωrad2
v−
)
≈ 1
2
ln
(
v+(0)ωr
v−ωc,Jans(0)
)
. (4.47)
In Fig. 4.14(b), results for 〈η〉 from both catastrophe models are shown as a function
of on-rate ωon. The average steady state force obtained from ωc,Flyv is always slightly
larger than 〈η〉 obtained from ωc,Jans, since ωc,Jans(η) > ωc,Flyv(η) for forces smaller than
or comparable to F0. Otherwise, both results agree qualitatively and quantitatively well.
4.4.2 Linear catastrophe model
Now results obtained with the linear catastrophe rate ωc,lin are discussed. All calculations
are carried with the catastrophe rate parameters ˜a= 0.005 s and ˜b= 8×104 m−1. Since ˜a
and ˜b are determined from experimental data, which do not support a linear catastrophe
rate, the choice of ˜a and ˜b is rather arbitrary. We want to stress out that, another choice
of ˜a and ˜b may affect the validity of approximations and lead to essential differences in
the final results.
4.4 Experimental and linear catastrophe model 77
0
0.02
0.04
0.06
0.08
0 50 100 150
P
(x
)[µ
m
−
1 ]
x [µm]
ωr=0.2s−1k=10−7N/m
25s−1 50s−175s−1
100s−1
(a)
0
1
2
25 50 75 100
〈η
〉
ωon [s−1]
(b)
ηstall
ωr=0.03s−1
ωr=0.2s−1
ωc,Flyv
ωc,lin
Figure 4.16: (a): Stationary MT length distribution P (x) for growth against an
elastic obstacle from eq. (4.23) and with the linear catastrophe rate ωc,lin (see eq.
(2.40)). We set k = 10−7N/m, x0 = 10−5m (dashed line) and ωr = 0.2s−1. Results for
ωon = 25s−1,50s−1,75s−1,100s−1 are shown. (b): Average steady state force 〈η〉 as a
function of ωon. We set k = 10−7N/m, ωr = 0.2s−1 (top) and ωr = 0.03s−1(bottom).
Solid lines: 〈η〉 obtained with ωc,Flyv [eq. (4.32)]. Dashed lines: 〈η〉 obtained with
ωc,lin [eq. (4.48)]. Dotted line: dimensionless stall force ηstall.
4.4.2.1 Vanishing rescue rate
We start with ωr = 0, and we calculate the average maximal polymerization force within
the linear catastrophe model using the self-consistent mean field eq. (4.18), which holds
independently of the choice of catastrophe model (see Sec. 4.3.1). Equation (4.18) is
solved numerically. For all investigated sets of parameter values, we find 〈ηmax〉 > ηstall.
The average maximal polymerization force exceeds the stall force. This is an artifact of
the linear catastrophe model, which does not increase exponentially for forces larger than
the characteristic force F0, but remains constant. Due to this fact, eq. (4.18) only yields
a non-physical fixed point. We do not further discuss the linear catastrophe model in
the absence of rescue events, but we want to stress out, that a different choice of the
catastrophe parameters ˜a and ˜b may lead to more reasonable results.
4.4.2.2 Non-zero rescue rate
We calculate the OPDF P (x) in the steady state from eq. (4.23) with the linear catastro-
phe rate ωc,lin(η) given by eq. (2.40). In Fig. 4.16 results for ωr = 0.2s−1, k = 10−7N/m
and different values of ωon are shown.
Again we find sharply peaked OPDFs and restrict further discussions to mean field results
(see Sec. 4.3.2 and Sec. 4.4.1.2).
For the linear catastrophe rate [eq. (2.40)], the mean field equation (4.32) can be solved
explicitly, and the average steady-state force 〈η〉 is given by
〈η〉= ln(ωon/ωlin) , (4.48)
78 Single microtubule dynamics under force and confinement
with
ωlin =
˜av−
d
(
ωr+˜bv−
) +ωoff. (4.49)
Again 〈η〉<ηstall since ωlin >ωoff. Fig. 4.16(b) shows 〈η〉 as a function of ωon. For realistic
parameter values, we have ωon/ωoff > 1 and ˜bv−/ωr < 1 and, as with the Flyvbjerg and
the Janson model, we recover expression (4.34)
〈η〉 ≈ ln
(
v+(0)ωr
v−ωc,lin(0)
)
, (4.50)
except from a factor 1/2.
In Fig. 4.16(b), results for 〈η〉 obtained with ωc,lin and ωc,Flyv are shown as a function of
on-rate ωon. The average steady state force obtained from ωc,lin is always slightly smaller
than 〈η〉 obtained from ωc,Flyv, but approaches 〈η〉 obtained from ωc,Flyv for large rescue
rates. Again both results agree qualitatively and quantitatively well.
4.5 Generalized force-velocity relation
We now discuss the influence of the force-velocity relation on the MT dynamics. A change
in the force-velocity relation directly modifies the velocity of growth v+(η), but it also
affects the catastrophe rate ωc[v+(η)], which are both crucial parts of the MT dynamics. In
the following, we employ the more generalized force-velocity relation by Kolomeisky et al.
(see Sec. 2.2). Throughout this section, we focus on the third confinement scenario of an
elastic obstacle, and show that our results are robust with respect to this generalization.
We discuss both cases of vanishing and non-vanishing rescue rate.
4.5.1 Vanishing rescue rate
We use the generalized force-velocity relation v+(η,θ) given by eq. (2.15) and the catas-
trophe rate ωc,Flyv(η) by Flyvbjerg et al. in order to calculate the average maximal poly-
merization force 〈ηmax〉 from the self-consistent mean field eq. (4.18). In Fig. 4.17 (a),
〈ηmax〉 is shown as a function of the load distribution factor θ for k = 10−5N/m and
different values of ωon. At θ = 1, the maximal force 〈ηmax〉 equals the maximal polymer-
ization force obtained with v+(η) from eq. (2.13). With decreasing θ, 〈ηmax〉 increases
but remains below the dimensionless stall force. The growth velocity v+(η,θ) increases
with decreasing θ for a fixed force η and, therefore, the maximal polymerization force
〈ηmax〉 increases. For high tubulin on-rates ωon = (75− 100)s−1 and small θ ≈ 0, . . . ,0.2,
the maximal polymerization force 〈ηmax〉 approaches the dimensionless stall force.
4.5 Generalized force-velocity relation 79
1
2
3
0 0.2 0.4 0.6 0.8 1
〈η
m
ax
〉
θ
(a)
100s−175s
−1
50s−1
30s−1
ηstall(ωon=100s−1)
0
1
2
3
0 0.2 0.4 0.6 0.8 1
〈η
〉
θ
(b)
100s−1
75s−1
50s−1
30s−1
ωr=0.05s−1
ηstall(ωon=100s−1)
Figure 4.17: (a): Solid lines: Average maximal polymerization force 〈ηmax〉 as a
function of θ for k = 10−5N/m and different values of ωon. Dotted line: Dimensionless
stall force ηstall for ωon = 100s−1. Dashed line: 〈ηmax〉 for ωon = 50s−1 and v+(θ = 1)
corresponding to eq. (2.11). (b): Solid lines: Average steady state force 〈η〉 as a
function of θ for k = 10−7N/m, ωr = 0.05s−1 and different values of ωon. Dotted line:
Dimensionless stall force ηstall for ωon = 100s−1. Dashed line: 〈η〉 for ωon = 50s−1 and
v+(θ = 1) corresponding to eq. (2.11).
4.5.2 Non-zero rescue rate
We calculate the OPDF P (x) in the steady state from eq. (4.23) with the force-velocity
relation v+(η,θ) [eq. (2.15] and the catastrophe rate ωc,Flyv(η) by Flyvbjerg et al. [eq.
(2.35)]. In Fig. 4.18 results for ωon = 50s−1, ωr = 0.05s−1 and k = 10−7N/m are shown.
Again we find sharply peaked OPDFs and restrict further discussions to mean field results
(see Sec. 4.3.2 and Sec. 4.4.1.2).
For non-zero rescue rate, the average steady state force 〈η〉 is calculated from the mean
field eq. (4.32), where we use the force-velocity relation v+(η,θ) [eq. (2.15)] and the
catastrophe rate ωc,Flyv(η) by Flyvbjerg et al. In Fig. 4.17(b), results for 〈η〉 are shown
as a function of θ for k = 10−7N/m, ωr = 0.05s−1 and different values of ωon. At θ = 1,
〈η〉 equals the average steady state force obtained with a velocity v+(η) taken from eq.
(2.13). The average steady state force 〈η〉 increases with decreasing θ, as explained above.
For high tubulin on-rates ωon = (75− 100)s−1 and small θ ≈ 0, . . . ,0.2 , also the average
steady state force 〈η〉 again approaches the dimensionless stall force but remains smaller.
80 Single microtubule dynamics under force and confinement
0
0.01
0.02
0.03
0.04
0.05
0 50 100 150
P
(x
)[µ
m
−
1 ]
x [µm]
ωon=50s−1
ωr=0.05s−1
k=10−7N/m
θ = 0
θ = 0.5
θ = 1
Figure 4.18: Stationary MT length distribution P (x) for growth against an elastic
obstacle from eq. (4.23), with the catastrophe rate ωc,Flyv [eq. (2.35)] and the gener-
alized force-velocity relation by Kolomeisky et al. [eq. (2.15)]. We set k = 10−7N/m
and x0 = 10−5m (dashed line). Results for ωon = 50s−1, ωr = 0.05s−1 and θ = 0,0.5,1
are shown.
Chapter 5
First conclusion 1
We studied single MT dynamics in three different confining scenarios: (i) confinement by
fixed rigid walls (Sec. 4.1), (ii) an open system under constant force (Sec. 4.2), and (iii)
MT growth against an elastic obstacle with a force that depends linearly on MT length
(Sec. 4.3). These three scenarios represent generic confinement scenarios in living cells or
geometries, which can be realized experimentally in vitro. For all three scenarios, we are
able to quantify the MT length distributions as a function of growth parameters and the
load force.
The parameter λ, see eq. (2.4) governs the MT length distributions in confinement by
fixed rigid walls. For confinement by rigid walls we introduced a realistic model for wall-
induced catastrophes, which in most cases can been seen as instantaneous. There is a
transition from exponentially increasing (λ> 0) to exponentially decreasing (λ< 0) length
distributions if λ changes sign. The average MT length is increasing for increasing on-rate
ωon and increasing rescue rate ωr, as shown in Figs. 4.2. Wall-induced catastrophes lead
to an overall increase in the average catastrophe frequency, which we quantify within the
model.
The parameter λ(η), see eq. (4.11), governs the MT length distributions for growth under
constant force. For MT growth under a constant force, there exists a transition between
bounded and unbounded growth as in the absence of force, with exponentially decaying
and Gaussian-shaped OPDFrespectively. This transition takes place where the parameter
λ(η) changes sign and determines the critical force ηc. We determined the critical force
ηc as function of the on-rate ωon and the rescue rate ωr (see Fig. 4.4). Under force, the
transition to unbounded growth is shifted to higher values of ωon or higher rescue rates
(see Figs. 4.5).
In scenario of an elastic obstacle, stochastic MT growth also gives rise to a stochastic force.
For this model, we also quantify the average polymerization force generated by the MT in
the presence of the dynamic instability. MT growth under a MT length-dependent linear
1Parts of the text have been published in reference [55] and is ©2012 American Physical Society.
http://publish.aps.org/copyrightFAQ.html
82 First conclusion
elastic force allows for regulation of the generated polymerization force by experimentally
accessible parameters such as the on-rate, the rescue rate or the stiffness of the elastic
barrier. The force is no longer fixed but a stochastically fluctuating quantity because the
MT length is a stochastic quantity.
For zero rescue rate, i.e., in the absence of rescue events, we find that the average maximal
polymerization force 〈ηmax〉 before a catastrophe depends logarithmically on the tubulin
concentration and is always smaller than the stall force in the absence of dynamic insta-
bility as shown in Fig. 4.7. The average maximal polymerization force and its logarithmic
dependency on the tubulin concentration should be accessible in in vitro experiments.
For a non-zero rescue rate, we find a steady state length distribution, which becomes
increasingly sharply peaked for increasing rescue rate and is tightly controlled by micro-
tubule growth parameters, see Figs. 4.8. Interestingly, the average microtubule length
self-organizes such that the average steady state polymerization force 〈η〉 equals the crit-
ical force for the boundary of bounded and unbounded growth, 〈η〉 = ηc. Because of
the sharply peaked MT length distribution, the average polymerization force 〈η〉 can be
calculated rather accurately within a mean field approach as can be seen in Figs. 4.10
and 4.11. The average polymerization force is always smaller than the stall force in the
absence of dynamic instability. Results from mean field theory remain valid, as long as
the OPDF is sharply peaked. With increasing stiffness k of the elastic barrier, the average
MT length approach results from confinement between rigid walls.
Within this mean field approach, we can also describe the dynamics of the average force,
see Figs. 4.12. This might be useful in modeling dilution experiments, where the response
to sudden changes in the on-rate is probed. For this type of experiment, we estimate typ-
ical polymerization force relaxation times. After dilution we find an exponential decaying
return dynamics to the new steady state with a characteristic timescale τd, governed by
growth parameters and the stiffness of the elastic barrier.
We show that our findings are robust against changes of the catastrophe model (Figs. 4.14)
as long as the catastrophe rate increases exponentially and the growth velocity decreases
exponentially above a characteristic force. For vanishing and non-vanishing rescue rate
the catastrophe models by Flyvbjerg et al. and Janson et al. yield similar results. With
the linear catastrophe model, results could only be reproduced for non-vanishing rescue
rate (Figs. 4.16). For zero rescue rate, the dynamical mean field theory does not provide
reasonable results, due to the missing exponentially increase of the catastrophe rate above
the characteristic force.
Finally we show that results are also robust against variations of the relation between
force and polymerization velocity in the growing phase (Figs. 4.17), which are obtained
by introducing a load distribution factor.
Chapter 6
Microtubule ensembles 1
In this section we present the model for MT ensemble dynamics. We focus on the growth
against an elastic obstacle (see Sec. 4.3), introduce force sharing between simultaneously
pushing MTs and extend the basic notation.
6.1 Model for microtubule ensemble dynamics
We consider an ensemble of N parallel MTs, directed along the x-direction. The ensemble
is growing in a positive x-direction and pushing against an elastic barrier, as shown in
Fig. 6.1. Each single MT is modelled as described in Chap. 2. The cooperative dynamics
is governed by the number n+ ≤N of leading MTs.
The elastic barrier is modelled as a spring with equilibrium position x0 and a spring
constant k. Barrier displacement by the n+ leading MTs with their tips positioned at
x > x0 causes a force F = F (x) = k(x−x0) resisting further growth; for x < x0 there is
a force-free region (see Sec. 4.3). We assume that the force F is equally shared between
all n+ leading MTs such that each leading MT is subject to a force F/n+. Force-sharing
is the only coupling between the MTs. In the presence of rescue events, i.e., for nonzero
rescue rate, we force MTs shrinking to x = 0 to undergo rescue. This corresponds to a
reflecting boundary condition at x = 0.
Under a shared force F/n+, the growth velocity of a MT reduces to
v+(F/n+) = d
(
ωone−F/n+F0 −ωoff
)
(6.1)
with the characteristic force F0 = kBT/d [see eq. (2.11)]. In the following we change the
notation compared to Sec. 2.2. From now on we use F/F0 instead of η to describe the force
acting on a single MT. Although this is just a change of name, this notation simplifies
the calculation of scaling relationships (see Secs. 8.2.3 and 8.2.5).
1Parts of the text have been published in reference [54] and are ©2012 American Physical Society.
http://publish.aps.org/copyrightFAQ.html
84 Microtubule ensembles
F (x)=k (x−x0)
v+ [F (x)/n+]
v−
v+(0)
x0 x
ωr
ωr
ωc [F (x)/n+]
ωc(0)
Figure 6.1: Schematic representation of an ensemble with N = 4 MTs growing
against an elastic obstacle with spring constant k. From top to bottom: n+ = 2
simultaneously pushing MTs (green) with velocity v+ [F (x)/n+] under a shared force
F (x)/n+. MT in a state of shrinkage (red) with velocity v−. MT growing (green)
under zero force with velocity v+(0).
6.1 Model for microtubule ensemble dynamics 85
The force-dependent growth velocity also gives rise to a catastrophe rate
ωc = ωc [v+(F/n+)] increasing with force. All non-leading MTs grow with the
higher zero force velocity v+(0) > v+(F/n+) in their growing state and switch to the
state of shrinkage with the zero force catastrophe rate ωc(0)  ωc(F/n+). Therefore,
non-leading MTs which grow force free and fast, “catch up” leading MTs, which grow
under force with reduced velocity. This mechanism supports a state of collective growth,
where a relatively large number n+ of MTs are pushing cooperatively. We assume that
the shrinking velocity v− is independent of force.
This model for the dynamics of the MT ensemble is very similar to the model underlying
the simulations in Ref. [53]. In particular, we use the same rules for the coupling between
MTs by the load force. The most important difference is that we include rescue events in
the single MT dynamics, which have not been considered in Ref. [53], and, in addition,
we do not employ an artificially increased catastrophe rate (see Sec. 1.4).
86 Microtubule ensembles
Chapter 7
Simulation of microtubule ensembles
1
In this chapter we present the simulation model for microtubule ensemble dynamics. The
single MT simulation model (see Chap. 3) is extended to an ensemble of N MTs, parallelly
growing against an elastic obstacle (see Chap. 6). As in Chap. 3, we specify simulation
parameters, parameter ranges and characteristic observables.
7.1 Microtubule ensemble simulation model
In the simulation we solve the Langevin-like equations of motion for the length xi(t)(i =
1, . . . ,N) of each single MT within the ensemble. There are N equations of motions,
coupled via force-sharing of the n+ leading MTs. At t = 0 all MTs have zero length
xi(0) = 0 and are in a state of growth, such that n+ = N . Each single MT is simulated
as described in Chap. 3.
In each time step, we have to determine the number n+ of the leading force-sharing MTs.
This is done by regarding all growing MTs within a distance v+(F/n+)∆t of the leading
MTs as leading for the next time step.
In a growing state and under the shared force F/n+ the velocity of growth v+(F/n+),
for the n+ leading MTs, is calculated from eq. (6.1). All non-leading MTs grow with the
higher zero force velocity v+(0) given by eq. (2.9). In a state of shrinkage each MT shrinks
with average velocity v− = 3×10−7m/s (see Table A.1).
The catastrophe rate ωc is calculated from eq. (2.35) in the Flyvbjerg model and from eq.
(2.38) in the Janson model.
The number N of MTs within the ensemble is varied in the range of N = 2, . . . ,20 and
we simulate Nens = 100− 1000 independent MT ensembles. Compared to simulations
of single MTs (see Chap. 3), the number of independent MT ensembles is decreased to
1Parts of the text have been published in reference [54] and are ©2012 American Physical Society.
http://publish.aps.org/copyrightFAQ.html
88 Simulation of microtubule ensembles
0
10
20
30
40
0 50 100 150 200
F
(t
)[
pN
]
t [s]
Fmax
n+(Fmax)=0
(a)
0
1
2
3
4
5
0 10 20 30
n +
(F
)
F [pN]
Fmax
n+(Fmax)=0
(b)
Figure 7.1: Sample trajectories for N = 5 MTs for vanishing rescue rate ωr. We set
x0 = 0, ωon = 70s−1 and k = 10−5N/m. (a): Polymerization force F (t) as a function
of time t. Dots represent the position x(t) of the elastic obstacle, which is equivalent
to the position of the n+ leading MTs. Solid lines are trajectories of single MTs. At
the maximal polymerization force Fmax (t≈ 175s) all MTs are in a state of shrinkage
and the dynamics stops due to missing rescue events. (b): n+(F ) as a function of
the polymerization force F . The maximal polymerization force is obtained from the
condition n+(Fmax) = 0 (F ≈ 34pN).
reduce simulation time. This does not lead to significant measurement inaccuracies in the
numerical results.
As outlined in Chap. 3 we vary ωon in the range ωon = (30−100)s−1 and ωr in the range
ωr = (0.03−0.2)s−1.
The most important observables are the position of the elastic obstacle x(t), which is
equivalent to the position of the n+ leading MTs, and corresponds to a force F (t) =
(x(t)− x0), and the number n+ of simultaneously pushing MTs between the resisting
force is shared equally.
In order to determine the characteristic observables for the dynamics of MT ensembles,
we have to anticipate numerical results from Secs. 8.1 and 8.2. We present two types of
stochastic trajectories and clarify how observables are determined from numerical data.
Both types of stochastic trajectories are described in detail later (see Sec. 8.1 for ωr = 0
and Sec. 8.2 for ωr > 0).
7.1.1 Vanishing rescue rate
For vanishing rescue rate ωr = 0 we set x0 = 0. Figure 7.1(a) shows the stochastic time
evolution of N = 5 MTs. The MT bundle polymerizes against the elastic obstacle until
the maximal polymerization force Fmax is reached. At Fmax all MTs within ensemble have
switched into a state of shrinkage, such that n+ = 0 [see Fig. 7.1(b)] and the dynamics
stops due to missing rescue events. The maximal polymerization force Fmax is therefore
determined by the condition n+(Fmax) = 0. Since Fmax is a stochastic quantity we perform
ensemble-averages, via eq. (3.3), over Nens independent simulation runs.
7.1 Microtubule ensemble simulation model 89
22
24
26
28
30
32
34
45000 50000 55000
F
(t
)[
pN
]
t [s]
(a)
tstart,i
tend,i
tstart,i+1 0
2
4
6
8
10
45000 50000 55000
n +
(t
)
t [s]
(b)
tstart,i tend,i tstart,i+1
n+=1
Figure 7.2: (a) Polymerization force F (t) as a function of time t for N = 10 MTs and
non-zero rescue rate ωr =0.05s−1. We set x0 =10−5m, ωon =70s−1 and k=10−7N/m.
Vertical dashed lines mark the begin (t≈ 47000s) and the end (t≈ 52000s) of a phase
of collective growth. (b) The number of simultaneously pushing MTs n+(t) as a
function of time, corresponding to the polymerization force F (t) shown in Fig. 7.2(a).
Vertical dashed lines mark the begin (t≈ 47000s) and the end (t≈ 52000s) of a phase
of collective growth. Horizontal dashed line represents n+ = 1.
7.1.2 Non-zero rescue rate
For a non-vanishing rescue rate ωr > 0 we set x0 = 10−5m. Figure 7.2(a) shows the
stochastic time evolution of N = 10 MTs. In contrast to Sec. 7.1.1 rescue events lead
to a steady state of the cooperative dynamics. In the steady state the MT ensemble
alternates between phases of collective growth and phases of collective shrinkage [see
Fig. 7.2(a)]. This gives rise to oscillations in the polymerization force F (t) = (x(t)−x0)
around its mean value Fs,N. First we perform a time-dependent ensemble average over
Nens independent simulations runs via eq. (3.3). In the steady state, which is determined
by visual inspection of sample trajectories, we run an additional time-average via eq. (3.3)
and obtain the ensemble averaged polymerization force Fs,N in the steady state.
From numerical data we can also determine the average length of phases of collective
growth and shrinkage. Phases of collective growth are characterised by a stable number
of simultaneously pushing MTs with n+ > 1. In a collective catastrophe all MT switch into
a state of shrinkage and n+ drops down to zero. The system enters a state of collective
shrinkage with n+ = 0−1.
We therefore define the begin tstart of a phase of collective growth by n+(tstart) > 1 and
the end tend by n+(t) = 0 [see Fig. 7.2(b)]. The length of a phase of collective growth is
than given by T+ = tend− tstart. Since the ensemble alternates between collective growth
and shrinkage, the length of a phase of collective shrinkage is calculated from the time
difference of two successive periods of collective growth and is given by T− = tstart,i+1−
tend,i. Here i denotes the sequence of phases of collective growth (see Fig. 7.2).
In phases of collective shrinkage and for high forces F (t), n+ is subject to considerable
fluctuations. Especially for large values of ωon and ωr, the begin of a phase of collective
90 Simulation of microtubule ensembles
growth can not be captured by the condition n+(tstart)> 1 since even fluctuations leading
to n+ = 2− 3 do not initiate a phase of collective growth. The condition n+(tstart) > 1
is adjusted to n+(tstart) > 2− 3 depending on the parameter set. This is done by visual
inspection of sample trajectories. However, for most parameter values n+ > 1 is sufficient
to indicate the begin of a phase of collective growth.
With tstart and tend we can now determine the mean number n+ of simultaneously pushing
MTs averaged over a phase of collective growth by
n+ = T−1+
tend
∑
tstart
n+(t). (7.1)
Since n+ is still a stochastic quantity we also perform an ensemble average of typically
1000-10000 successive phases of growth and obtain the time- and ensemble averaged num-
ber 〈n+〉 of pushing MTs in the steady state.
Chapter 8
Microtubule ensemble dynamics 1
In this chapter we present the results for MT ensemble dynamics for zero and non-zero
rescue rate.
Sec. 8.1 deals with MT bundles polymerizing against an elastic force in the absence of
rescue events. This scenario is close to the optical trap experiments by Laan et al.[53]
(see Sec. 1.4). We develop a dynamical mean field theory and quantify the cooperative
dynamics of MT in terms of the mean force 〈F 〉 and the mean number of simultaneously
pushing MTs 〈n+〉. The maximal polymerization force Fmax is calculated as a function
of the number of MTs N , the stiffness of the barrier k, and the tubulin on-rate ωon. In
addition we show that our results are robust with respect to changes in the catastrophe
model and are in good agreement with available experimental data for small N or a stiff
elastic barrier [53].
Sec. 8.2 deals with MT bundles polymerizing against an elastic force in the presence of
rescue events. We present results from full stochastic simulations (Sec. 8.2.1) and discuss
the cooperative dynamics, in particular collective catastrophes and collective rescues.
In Sec. 8.2.2 the dynamic mean-field theory is extended by rescue events of single MTs.
The oscillatory mean-field dynamics is discussed, based on the nullclines of the mean-
field equations (Sec. 8.2.3). We calculate the critical mean-field force Fc, the average
polymerization force Fs,N and the average number of pushing MTs as functions of the
system parameters. Additionally we show that our results are robust with respect to
changes in the catastrophe model (Sec. 8.2.4). In Sec. 8.2.5 fluctuations in the number of
pushing MTs are included into the dynamical mean-field theory. In Sec. 8.2.6 we develop
a semi-stochastic approach to the stochastic dynamics of the leading MTs and investigate
the validity and limits of the dynamical mean-field theory.
1Parts of the text have been published in reference [54] and are ©2012 American Physical Society.
http://publish.aps.org/copyrightFAQ.html
92 Microtubule ensemble dynamics
8.1 Microtubule bundle dynamics at zero rescue rate
We start the analysis with the case of zero rescue rate because this case is conceptually sim-
pler to understand as rescue events are absent, and there are only collective catastrophes
to be discussed. Furthermore, this case is particularly important because experimental
data are available: In recent experiments, Laan et al. [53] showed that MT ensembles
exhibit phases of collective growth followed by collective catastrophes, where all leading
MT nearly simultaneously undergo a catastrophe. The experiments were performed on
short time scales such that no rescue events occur. It was also observed that the maximal
polymerization force before catastrophes grows linearly in N . We quantify these features
based on a dynamical mean-field theory.
In an ensemble of N MTs the dynamic instability of individual MTs leads to stochastic
fluctuations in the number n+ of leading MTs. The force F changes by filament growth
according to F˙ = kx˙ with x˙ = v+(F/n+) if the ensemble grows (n+ ≥ 1) and x˙ =−v− if
all MTs shrink (n+ = 0). In a state of collective growth, a stable mean number of MTs
are pushing cooperatively, while the force F is increasing by growth against the elastic
barrier. If the number n+ of pushing MTs is reduced by an individual catastrophe, the
force on the remaining n+− 1 leading MTs increases and, thus, their catastrophe rate
ωc(F/n+) increases. A cascade of individual catastrophes, a collective catastrophe, can be
initiated until a state n+ = 0 is reached with all MTs shrinking. This is the final absorbing
state of the system in the absence of rescue events.
The stochastic dynamics of n+ in a growing phase in the absence of rescue events is
described by a one-step master equation with backward rate rn+ = n+ωc(F/n+) for de-
creasing n+ by one, which derives from the catastrophe rate of individual MTs under
force sharing. In a mean-field approach, we replace the stochastic variables F and n+
by their time-dependent mean values 〈F 〉 and 〈n+〉 , averaged over many realizations of
the stochastic n+- and F -dynamics, and neglect all higher-order correlations, e.g. set
〈F/n+〉= 〈F 〉/〈n+〉. In the growing phase, we then obtain two coupled mean-field equa-
tions,
d〈n+〉/dt =−〈n+〉ωc (〈F 〉/〈n+〉) , (8.1)
d〈F 〉/dt = kv+ (〈F 〉/〈n+〉) . (8.2)
In the mean-field approximation we can calculate the maximal polymerization force Fmax,
again averaged over many realizations, that is reached during the mean first-passage time
from n+ = N to n+ = 0 by solving
d〈F 〉
d〈n+〉
= d〈F 〉/dt
d〈n+〉/dt
=− kv+
(〈F 〉/〈n+〉)
〈n+〉ωc (〈F 〉/〈n+〉)
. (8.3)
8.1 Microtubule bundle dynamics at zero rescue rate 93
with initial conditions 〈F 〉= 0 for 〈n+〉= N in order to find 〈F 〉= Fmax at 〈n+〉 ≈ 0.
Above the characteristic force F0, the ratio v+(F )/ωc(F ) decays exponentially because
v+(F ) decreases exponentially and ωc[v+(F )] increases exponentially. Therefore, we can
solve in two steps: (i) As long as the shared force is small compared to F0, 〈F 〉/〈n+〉F0,
we neglect the force and find 〈F 〉 ≈ kv+(0)/ωc(0) ln(N/〈n+〉). (ii) For 〈F 〉/〈n+〉  F0,
on the other hand, the catastrophe frequency increases exponentially, and we can assume
that d〈F 〉/d〈n+〉 ≈ 0 and 〈F 〉 remains constant.
The boundary between regimes (i) and (ii) is determined by the condition 〈F 〉/〈n+〉= F0:
Regime (i) applies for 〈n+〉 > n0 with n0 = αW (N/α), where W (x) is the Lambert W -
function, which is the solution of x = WeW (for W ≥−1). The parameter
α≡ kv+(0)/ωc(0)F0 (8.4)
is a dimensionless measure for the stiffness of the elastic barrier. Because 〈F 〉 remains
constant for 〈n+〉< n0, the resulting maximal polymerization force is given by
Fmax = n0F0 = F0αW (N/α), (8.5)
with a logarithmic asymptotic F max ≈ F0α ln(N/α) for large N  α [85] or a soft barrier
and a quasilinear behavior Fmax ≈ F0N(1−N/α), which is independent of α to leading
order, for small N  α or a stiff barrier. The mean-field result from eq. (8.5) agrees with
numerical solutions of the mean field dynamics as given by eqs. (8.1) and (8.2) and full
stochastic simulations both for soft and stiff barriers, as can be seen in Fig. 8.1.
The parameter α can also be interpreted as a measure for the relative speed of the initial
〈n+〉 and 〈F 〉 dynamics according to the mean-field equations (8.1) and (8.2), which allows
us to give simple arguments for the maximal polymerization force Fmax: For α  N
(the case of a soft barrier), the 〈n+〉 dynamics is fast compared to the 〈F 〉 dynamics.
Therefore, 〈n+〉 decays approximately force free in a time tc ∼ 1/ωc(0) lnN from 〈n+〉=N
to 〈n+〉= 1. During this time, the force reaches a value Fmax ∼ kv+(0)tc ∼ F0α lnN . For
α  N (the case of a stiff barrier), the 〈n+〉-dynamics is initially slow compared to the
〈F 〉-dynamics and 〈n+〉 ≈N until the characteristic force F0 per MT is reached and the
catastrophe rate increases exponentially. Up to this point, essentially N MTs share the
force such that F increases up to Fmax ∼ F0N until catastrophes set in. This takes a
time tc ∼ NF0/kv+(0) and ∆〈n+〉 ∼ ˙〈n+〉tc ∼ N/α  1 is indeed small such that the
assumption 〈n+〉 ≈N is consistent.
In the experiments in Ref. [53], the spring stiffness was k ' 10−5 N/m, which gives α' 27
such that these experiments were performed in the quasilinear regime of a stiff barrier,
where we predict Fmax ≈ F0N for all experimentally accessible N (see the upper lines in
Fig. 8.1). This linear increase is in agreement with the experimental results but the ratio
Fmax/N is only of the order of 3 pN experimentally, while F0 ' 7 pN. This hints at a
94 Microtubule ensemble dynamics
0
2
4
6
8
10
12
14
16
2 4 6 8 10
F m
ax
/F
0
N
Fmax(N)
F0αW (N/α)
k=10−7 N/m
k=10−5 N/m
◦,4 ωc,Jans
•,N ωc,Flyv
Figure 8.1: Fmax/F0 as a function of N for zero rescue rate, ωon = 70 s−1, and
k = 10−7 N/m (α ' 0.27, soft barrier) and k = 10−5 N/m (α ' 27, stiff barrier). We
compare results for the catastrophe models of Flyvbjerg (blue, solid symbols) and
Janson (red, open symbols). Data points are simulation results; error bars represent
the standard deviation of the stochastic quantity Fmax/F0. Solid lines are numerical
solution of the mean field dynamics [see eq. (8.1) and eq. (8.2)]. Dashed lines are
analytical estimates according to eq. (8.5).
lower value for F0 in the force-polymerization velocity relation for MTs; experimentally,
a value F0 ' 2 pN has been measured in Ref. [86], which is indeed compatible with the
experimental results of Ref. [53].
In Fig. 8.1, we compare mean-field theory and simulation results for the Flyvbjerg and
the Janson catastrophe model for both soft and stiff barriers. For both models, we find
agreement between mean-field theory and simulations and, moreover, both models give
comparable values for generated forces. This demonstrates that results for the cooperative
force generation at zero rescue rate are robust with respect to details of the single-MT
catastrophe model. The essential feature entering the mean-field theory is the exponential
increase of the catastrophe frequency with force above the characteristic force F0.
8.2 Microtubule bundle dynamics at non-zero rescue
rate
We now consider force generation in the presence of rescue events. Rescue events were
not included in the simulation model in Ref. [53]. Also experiments in Ref. [53] were
performed on short time scales such that no rescue events occurred. However, rescue
events are an essential part of MT dynamics, and their influence on force generation and
MT dynamics needs to be addressed.
8.2 Microtubule bundle dynamics at non-zero rescue rate 95
0
5
10
15
20
15000 16000 17000 18000 19000 20000
x(
t)
[µ
m
]
t [s]
ωc,Flyv k=10−5 N/m
x0
xobs(t)
xi(t)
Figure 8.2: Typical simulation trajectories for N = 10 MTs, ωon = 70 s−1, ωr =
0.05 s−1 and a stiff obstacle with k = 10−5 N/m. We use the Flyvbjerg catastrophe
model with F0 ' 7pN. Dashed line represents x0 = 10 µm. Solid lines: Positions of
all MTs as a function of time t. Blue, filled dots: The obstacle position xobs(t) is the
position of the leading MTs (only every 1000th data point shown). In the force free
region x < x0, MTs decouple.
In the presence of rescue events, the dynamics will not change considerably for N α, i.e.,
for a stiff barrier because this limit corresponds to a slow 〈n+〉-dynamics, which cannot
benefit from additional rescue events. Moreover, because the 〈F 〉-dynamics is fast, rescues
will not happen before the force-free region x < x0 is reached, where MTs decouple (see
Fig. 8.2).
Therefore, we focus on the influence of rescue events for N  α or a soft barrier corre-
sponding to a fast 〈n+〉 dynamics.
We begin our discussion with simulation results and discus characteristic steady-state
trajectories.
8.2.1 Simulation results
In the regime of a soft barrier, the collective dynamics becomes strongly modified. Apart
from collective catastrophes also collective rescue events occur: After a collective catastro-
phe the system is in a state n+ = 0 with all MTs shrinking. Individual rescue events lead
to n+ = 1, but a single MT bearing the whole force undergoes an immediate catastrophe
again with high probability. Therefore, a cascade of rescue events a collective rescue is
necessary to increase n+ back to a number sufficient to maintain stable collective growth.
Snapshots from stochastic simulations, corresponding to characteristic phases of collective
growth are shown in Fig. 8.5.
Alternating collective catastrophes and collective rescue events give rise to oscillations in
96 Microtubule ensemble dynamics
100
150
200
250
300
35000 40000 45000 50000
x(
t)
[µ
m
]
t [s]
(a)
ωc,Flyv
xobs(t)
xi(t)
0
2
4
6
8
10
35000 40000 45000 50000
n +
(t
)
t [s]
(b)
n+=1
ωc,Flyv
Figure 8.3: Typical simulation trajectories for N = 10 MTs, ωon = 70 s−1, ωr =
0.05 s−1 and for the Flyvbjerg catastrophe model with F0 ' 7pN. (a): Positions of
all MTs as a function of time t (solid lines); the obstacle position xobs(t) (blue, filled
dots) is the position of the leading MT (only every 1000th data point shown). (b):
The number n+ of leading MTs as a function of time t. Collective catastrophes and
collective rescue events can clearly be recognized: In a collective catastrophe n+ drops
to n+ = 1 and x(t) of the leading MTs starts to shrink; after a collective rescue n+
starts to increase again to values n+ > 1, and x(t) of the leading MTs start to grow.
8.2 Microtubule bundle dynamics at non-zero rescue rate 97
0
10
20
30
40
50
35000 37500 40000
x(
t)
[µ
m
]
t [s]
(a)
ωc,Jansxobs(t)
xi(t)
0
2
4
6
8
10
35000 37500 40000
n +
(t
)
t [s]
(b)
n+=1
ωc,Jans
Figure 8.4: Typical simulation trajectories for N = 10 MTs, ωon = 70 s−1, ωr =
0.05 s−1 and for the Janson catastrophe model with F0 ' 0.8pN. (a): Positions of
all MTs as a function of time t (solid lines); the obstacle position xobs(t) (blue, filled
dots) is the position of the leading MT (only every 100th data point shown). (b):
The number n+ of leading MTs as a function of time t. Collective catastrophes and
collective rescue events can clearly be recognized: In a collective catastrophe n+ drops
to n+ = 1 and x(t) of the leading MTs starts to shrink; after a collective rescue n+
starts to increase again to values n+ > 1, and x(t) of the leading MTs start to grow.
98 Microtubule ensemble dynamics
(a) (b) (c) (d)
Figure 8.5: Snapshots from stochastic simulations for N = 10 MTs, ωon = 70 s−1,
ωr = 0.05 s−1 and k = 10−7 N/m. MT configurations corresponding to characteristic
phases of the collective dynamics for growth against an elastic obstacle (blue): (a)
MT ensemble shortly after an collective rescue. All MTs are in a state of growth
(green), with n+ = 7 simultaneously pushing MTs and three MTs growing under no
force. (b) MT ensemble in a state of collective growth with n+ = 6 simultaneously
pushing MTs, three MTs shrinking (red) and one MT growing under no force. (c) MT
ensemble shortly after an collective catastrophe, with nine MTs shrinking and, caused
by a stochastic rescue event, n+ = 1 pushing MTs. (d) MT ensemble in a state of
collective shrinkage, with eight MTs shrinking and two MTs growing under no force.
the steady-state polymerization force or, equivalently, the position xobs of the obstacle.
Such oscillations with alternating collective catastrophes and collective rescue events can
clearly be seen in the stochastic simulation trajectories for the positions x of the MTs [see
Figs. 8.3(a) and 8.4(a)] and the number n+ of leading MTs [see Figs. 8.3(b) and 8.4(b)]
as a function of time t. The simulation trajectories also show that this phenomenon is
robust with respect to the catastrophe model and can be observed both for the Flyvbjerg
and the Janson catastrophe models, which are shown in Figs. 8.3 and 8.4 respectively,
and exhibit qualitatively very similar behavior. Similar oscillations have been observed
in the simulations in Ref. [53] in the presence of MT renucleation instead of MT rescue
and for a constant force.
Here we want to stress that there are major differences between oscillations presented here
and observed by Laan et al. Laan et al. used a small constant force in their simulations,
whereas we use an elastic force, that allows the system to find its steady-state force
without external restrictions. We find oscillatory behavior at much larger forces then
used by Laan et al. Another major difference is renucleation of MTs instead of rescue
events of single MTs. From our point of view renucleation and rescue events do not lead
to the same type of oscillatory behavior. In the absence of rescue events MTs shrink back
to zero length after a catastrophe and restart to grow after a characteristic time, that
is the inverse nucleation rate. Since renucleation of single MTs is independent of other
nucleation events, single MTs decouple and renucleation does not lead to synchronous
MT growth. However, spontaneous renucleation of two or more MTs is possible and
8.2 Microtubule bundle dynamics at non-zero rescue rate 99
causes a temporary synchronization of MT growth. This synchronization is not based on
a cooperative mechanism and is lost on long time scales. In contrast, including rescue
events we find stable synchronous MT oscillations on long time scales (see Figs. 8.3 and
8.4). Rescue events and an elastic resisting force enable the system to organize itself into a
stable cooperative dynamics, in particular a stable oscillation of phases with simultaneous
growth, ended by collective catastrophes, and phases of simultaneous shrinkage, ended by
collective rescues.
In the simulations we measure the polymerization force Fs,N = 〈F 〉, which is averaged
over time and many realizations as a function of MT number N and on-rate ωon. This
time-averaged polymerization force is also the stall force of the MT ensemble. The results
are shown in Fig. 8.6(a) for the Flyvbjerg catastrophe model and in Fig. 8.6(b) for the
Janson catastrophe model. The main finding of the simulations is an approximately linear
increase of the polymerization force Fs,N with the number N of MTs [see Fig. 8.6(a)
and 8.6(b)]. This shows that for large MT ensembles, rescue events give rise to much
higher polymerization forces as compared to the logarithmic N dependence derived in the
previous section in the absence of rescue events for a soft barrier (N  α). Simulations
also show an approximately linear increase of the polymerization force with the on-rate
ωon [see Figs. 8.7(a) and 8.7(b)].
We also show numerical results for the time-averaged pushing fraction ν+ = 〈n+〉/N of
MTs as a function of the on-rate ωon in Figs. 8.8(a) and 8.8(b). The pushing fraction
increases with on-rate, which demonstrates an increasing tendency of MTs to push syn-
chronously at higher on-rates, where larger forces are generated.
Simulation results for the Flyvbjerg model and the Janson model show a very similar
linear increase for the polymerization force Fs,N with N [see Figs. 8.6(a) and 8.6(b)] and
very similar results for the time-averaged pushing fraction ν+ of MTs [see Figs. 8.8(a)
and 8.8(b)], which is in accordance with the qualitatively similar simulation trajectories
shown in Figs. 8.3 and 8.4 for both catastrophe models. This further supports that our
results are robust with respect to the catastrophe model.
The absolute values of typical forces in Figs. 8.6(a), 8.6(b), 8.7(a), and 8.7(b) and, sim-
ilarly, between typical MT lengths in Figs. 8.3(a) and 8.4(a) differ, however, between
the two catastrophe models. The reason is that the basic force scale of the problem is
the characteristic force F0, above which the catastrophe rate increases exponentially, as
will be shown below. We have chosen the theoretical value F0 = kBT/d ' 7 pN for the
Flyvbjerg model and the much smaller value F0 = 0.8 pN according to Ref. [53] with the
Janson model. In units of the characteristic force F0, typical forces are very similar [see
Figs. 8.6(a), 8.6(b), 8.7(a), and 8.7(b) right scale].
Here we want to refer to a closely related biophysical model system: a cluster of molecular
bonds. In the following, we illustrate similarities of the dynamics of molecular bonds under
a constant force and the cooperative dynamics of a MT ensemble, and point out major
100 Microtubule ensemble dynamics
0
10
20
30
40
0 5 10 15 20
0
1
2
3
4
5
F s
,N
[p
N
]
F s
,N
/F
0
N
(a)
ωc,Flyv
Fc/2
Fc/2
Fc/2NFstall
(
ωon=70s−1
)
0
1
2
3
4
0 5 10 15 20
0
1
2
3
4
5
F s
,N
[p
N
]
F s
,N
/F
0
N
Fc/2
Fc/2
Fc/2
NFstall
(
ωon=70s−1
)
(b)
ωc,Jans
Figure 8.6: Average polymerization force Fs,N as a function of microtubule number
N for ωr = 0.05 s−1 and a soft barrier with k = 10−7 N/m. Data points: Simulation
results for different on-rates ωon = 30 s−1 (), 50 s−1 (•), and 70 s−1 (N) Solid lines:
Mean field estimate Fs,N = Fc/2, see eq. (8.22) (neglecting Fmin). Dashed lines:
Numerical mean field solution including stochastic effects. Black solid line: N -fold
single-MT stall force Fs,N = NFstall for ωon = 70 s−1. (a): Simulation and analytical
results using the Flyvbjerg catastrophe model with F0 ' 7 pN. (b): Simulation and
analytical results using the Janson catastrophe model with F0 ' 0.8 pN.
8.2 Microtubule bundle dynamics at non-zero rescue rate 101
0
20
40
60
80
25 50 75 100
0
2
4
6
8
10
F s
,N
[p
N
]
F
/F
0
ωon [s−1]
ωr = 0.05 s−1
(a)
ωc,Flyv
N = 5
N = 10
N = 15
N = 20
0
2
4
6
8
25 50 75 100
0
2
4
6
8
F s
,N
[p
N
]
F
/F
0
ωon [s−1]
ωr = 0.05 s−1
(b)
ωc,Jans
N = 5
N = 10
N = 15
N = 20
Figure 8.7: Average polymerization force Fs,N as a function of the on-rate ωon
for ωr = 0.05 s−1 and a soft barrier with k = 10−7 N/m. Data points: Simulation
results for different MT number N = 5 (),10 (•),15 (N),20 (). Solid lines: Mean
field estimate Fs,N = Fc/2, see eq. (8.22) (neglecting Fmin). Dashed lines: Numerical
mean field solution including stochastic effects. (a): Simulation and analytical results
using the Flyvbjerg catastrophe model with F0 ' 7 pN. (b): Simulation and analytical
results using the Janson catastrophe model with F0 ' 0.8 pN.
102 Microtubule ensemble dynamics
0.25
0.5
0.75
1
25 50 75 100
ν +
ωon [s−1]
ν+(fs,N)
ν+(fc/2)
(a)
ωc,Flyv
ωr = 0.05 s−1
ωr = 0.1 s−1
ωr = 0.2 s−1
0.25
0.5
0.75
1
25 50 75 100
ν +
ωon [s−1]
ν+(fs,N)
ν+(fc/2)
ωc,Jans
(b)
ωr = 0.05 s−1
ωr = 0.1 s−1
ωr = 0.2 s−1
Figure 8.8: Time-averaged pushing fraction ν+ = 〈n+〉/N of MTs as a function of
ωon for soft barrier with k = 10−7 N/m. Data points: Simulation results for different
rescue rates ωr = 0.05 s−1 (•), 0.1 s−1 (N), and 0.2 s−1 (). Error bars represent the
standard deviation of the stochastic quantity ν+. For reasons of clarity we only show
error bars for ωr = 0.1 s−1. All other standard deviations are of the same magnitude.
Solid lines: Solution of eq. (8.16) for f = fc/2. Dashed lines: Solution of eq. (8.16)
for f = Fs,N/N with Fs,N from the numerical mean-field solution including stochastic
effects. (a): Simulation and analytical results using the Flyvbjerg catastrophe model
with F0 ' 7 pN. (b): Simulation and analytical results using the Janson catastrophe
model with F0 ' 0.8 pN.
8.2 Microtubule bundle dynamics at non-zero rescue rate 103
differences between both systems. Erdmann et al. investigated the stochastic dynamics of
clusters of molecular bonds under a constant force [87, 88]. They considered a cluster of
N parallel bonds under a constant force F . Each bond can either be open or closed. The
force F is shared equally between all closed bonds, so that each closed bond is subject to
the force F/i, where i is the number of closed bonds. In the context of MT ensembles, a
closed bond corresponds to one of the leading n+ MTs and an open bond to a non-leading
MT. The dissociation rate of a single bond under force is kdiss = k0 exp(F/F0i), which
is equivalent to the exponential approximation of the catastrophe rate ωc(F/F0n+) ∼
ωc(F = 0)exp(F/F0n+) for F/F0 > 1 and n+ leading MTs [see eq. (2.37)]. The rebinding
of a single bond corresponds to a rescue event of a single MT. After rebinding, single
bonds instantaneously attach to the adhesion cluster. In contrast, single MTs have to
catch up to the leading MTs, which continue to grow against the elastic obstacle. The
inverse "catch-up" time is larger than the bare rescue rate and is a function of the force
(see Sec. 8.2.2). Without rebinding the number i of closed bonds decays constantly
until the absorbing boundary at i = 0 is reached. No steady state is reached. In the
absence of rescue events, a decay of the number n+ of simultaneously pushing MTs toward
the absorbing boundary at n+ = 0 is also observed in the dynamics of a MT ensemble.
As in the case of molecular bonds no steady state is reached (see Sec. 8.1). Under a
constant force both systems are described by the same linear one-step master equation
[see eq. (8.36)]. In both systems the stochastic dynamics is governed by the number of
bonds/MTs, the rebinding/rescue rate, and the dissociation/catastrophe rate. However,
for a MT ensemble growing against an elastic obstacle, the dynamics of the resisting
force F crucially affects the ensemble dynamics. This leads us to the major difference
between both systems. The force acting on the bond cluster is constant, whereas in the
case of the MT ensemble the force increases linearly with the length x(t) of the leading
MTs. The stochastic dynamics of the MT ensemble is coupled to the dynamics of the
resisting force via F˙ = kx˙(t). For a soft obstacle and in the presence of rescue events,
this leads to a remarkable difference in the cooperative dynamics between both systems.
Under a constant force and even with rebinding the bond cluster exhibits a finite lifetime,
before its complete dissociation. If all bonds are open, that is i = 0, the dynamics stops.
This resembles a MT ensemble in a collective phase of growth, followed by a collective
catastrophe. Here the complete dissociation of the bond cluster corresponds to a collective
catastrophe with n+ = 0 pushing MTs. But in contrast to the bond cluster, the MT
ensemble dynamics does not stop, since n+ corresponds to a collective phase of shrinkage.
In a phase of collective shrinkage, the resisting force is reduced via F˙ ∼ −kv−, until a
collective rescue induces a new phase of collective growth. The coupling of the stochastic
MT dynamics to the force dynamics allows the system to organize itself into a steady
state (see Sec. 8.2.1). To conclude, both systems are closely related to each other, but
the additional force dynamics in the case of MT ensembles leads to major differences in
104 Microtubule ensemble dynamics
τ Time t
n+
n+
n+−1
Po
sit
io
n
x(
t)
Figure 8.9: Illustration of time scale τ from eq. (8.7). Blue solid line: The obstacle
position x(t) is the position of the n+ leading MTs. Red solid line: Single MT
shrinking after catastrophe. Green solid line: Single MT growing force-free after
rescue without additional catastrophe and rescue events [see eq. (8.7)]. Dashed black
line: Single MT growing force-free after rescue with additional catastrophe and rescue
events [see eq. (8.6)]. These more complex “catch-up” processes are not included in
the description.
the cooperative dynamics, and allows the MT ensemble to reach a steady state.
8.2.2 Dynamical mean field theory
We will show that all simulation results and the robustness with respect to the catastrophe
model can be explained based on a dynamical mean-field theory.
In the presence of rescue events, the mean-field equation (8.1) for 〈n+〉 becomes modified
in the growing phase. The one-step master equation for n+ in a growing phase also
contains a forward rate gn+ = (N −n+)τ−1 for increasing n+ by one. This forward rate
is determined by a rescue and “catch-up”process for the (N −n+) MTs, which are not
pushing: The time τ denotes the mean time that it takes for a MT to rejoin the group
of n+ pushing MTs after undergoing an individual catastrophe followed by rescue and
force-free growth at a velocity v+(0) that is larger than the velocity v+(F/n+) of the
leading MTs under force (see Fig. 8.9).
After a rescue time 1/ωr the trailing MT has to “catch-up” a distance [v+(F/n+)+v−]/ωr
to the leading MTs, which kept growing with velocity v+(F/n+).
The force-free growing MT is still subject to dynamic instability, thus stochastic catastro-
phe and rescue events, and its dynamics is characterised by the force-free characteristic
length parameter λ(0) [see eq. (2.4)]. For λ < 0 the trailing MT shrinks, on average, back
to zero length and does not return to the leading MT . We therefore focus on λ(0) > 0,
which corresponds to an average force-free velocity of growth 〈v(0)〉 = J(0) > 0 [see eq.
(2.7)].
8.2 Microtubule bundle dynamics at non-zero rescue rate 105
Given a velocity difference 〈v(0)〉−v+(F/n+) to the leading MTs under force, the “catch-
up” requires a time
τ ≈ ω−1r [1+(v+(F/n+)+v−)/(〈v(0)〉−v+(F/n+))] (8.6)
which is larger than the bare rescue time 1/ωr (see Fig. 8.9). We require 〈v(0)〉>v+(F/n+)
and focus on ωr  ωc(0) and v+(0)ωr  v−ωc(0). 〈v(0)〉 can than be approximated by
〈v(0)〉 ≈ v+(0) [see eq. (2.7)] and from eq. (8.6) follows:
τ ≈ ω−1r [1+(v+(F/n+)+v−)/(v+(0)−v+(F/n+))] . (8.7)
Geometrically this can be interpreted as a single rescue event followed by force-free growth
with velocity v+(0) and no further catastrophe and rescue events. In Fig. 8.9, a sketch of
a “catch-up” process is depicted.
Including “catch-up” processes into the description, results in a modified mean-field equa-
tion for 〈n+〉,
d〈n+〉/dt =−ωc (〈F 〉/〈n+〉)〈n+〉+ 〈τ〉−1 (N −〈n+〉) (8.8)
where we have to apply the mean field averaging also to 〈τ〉 in eq. (8.7):
〈τ〉 ≈ ω−1r
[
1+ v+(〈F 〉/〈n+〉)+v−
v+(0)−v+(〈F 〉/〈n+〉)
]
. (8.9)
Typically 〈τ〉 is by a factor of 10 larger than the bare rescue time 1/ωr.
8.2.3 Limit cycle oscillations and absence of bifurcations
For the further analysis of the mean-field dynamics it is advantageous to introduce new
variables, the average force per MT f and the average fraction ν+ of pushing MTs,
f ≡ 〈F 〉/N, ν+ ≡ 〈n+〉/N (8.10)
with 〈F 〉/〈n+〉= f/ν+. Using these variables, the mean-field equations become
dν+/dt =−ν+ωc (f/ν+)+(1−ν+)/〈τ〉 (8.11)
df/dt = kv+ (f/ν+)/N (8.12)
From eqs. (8.11) and (8.12) we can estimate typical timescales of the corresponding dy-
namics and test the assumption of a fast ν+-dynamics compared to the f -dynamics. With
ωc(f/ν)∼ ωc(0)exp(f/ν) [eq. (2.37)] and τ−1(f/ν)∼ ωr/10 , the general solutions of eqs.
(8.11) and (8.12) are given by:
106 Microtubule ensemble dynamics
0
0.25
0.5
0.75
0 1 2 3 4 5 6
0 0.2 0.4 0.6 0.8
ν +
f [pN]
f/F0
A B
C
f/Fstall
v−
v+(f)
⇑
⇑
⇑
⇑
⇓
⇓
⇓
⇓
⇓⇓
fcfmin fmax
ωc,Flyv
Figure 8.10: Nullclines of the mean-field equations for the Flyvbjerg model for
ωon = 70 s−1, ωr = 0.05 s−1, and k = 10−7 N/m (soft barrier). The nullclines of ν+ ≡
〈n+〉/N are solutions of eq. (8.16) and assume a loop shape as a function of f ≡〈F 〉/N .
Open arrows indicate the mean-field flow of 〈n+〉. The nullcline for f is a straight
line ν+ = f/Fstall (dotted). The critical force is fc = Fc/N ' 5 pN for the Flyvbjerg
model. Black arrows indicate the stable mean-field limit cycle; dashed line indicates
effect of stochastic fluctuations.
t
τf
= F−10
∫
[
exp(f/ν+)−
ωoff
ωon
]−1
df + const. (8.13)
t
τν+
≈
∫
[
ωr
10ωc(0)
(1−ν+)− exp(f/ν+)ν+
]−1
dν++ const. (8.14)
with τf = F0N/(kdωon) and τν+ = ωc(0). For k = 10−7 N/m, F0 ' 7 pN and typical values
of N and ωon, we find τf ≈ 104 s and τν+ ≈ 103 s. In general, the ν+-dynamics is by a factor
of 10 faster than the f -dynamics, which agrees to the assumption of a fast ν+-dynamics.
We first discuss the nullclines of f and ν+, i.e., the contours in the f -ν+ plane along which
df/dt = 0 and dν+/dt = 0 is satisfied, respectively.
The nullclines of f require v+(f/ν+) = 0, which leads to a straight line,
f = ν+Fstall (8.15)
in the f -ν+ plane, where the slope is given by the single-MT stall force
Fstall = F0 ln(ωon/ωoff) (see Fig. 8.10).
The nullclines of ν+ are given by
0 = g(f,ν+)≡−ν+ωc(f/ν+)+
1
〈τ〉(f/ν+)
(1−ν+) (8.16)
8.2 Microtubule bundle dynamics at non-zero rescue rate 107
or equivalently
ν+ =
[
ωc
(
f/ν+
)
τ
(
f/ν+
)
+1
]−1
(8.17)
and are independent of N . The shape of the nullclines depends on the functional de-
pendence of the catastrophe rate on the force and, thus, on the catastrophe model. We
will first focus on the Flyvbjerg catastrophe model, for which the nullclines of ν+ have a
characteristic loop shape, as shown in Fig. 8.10, which exhibits two solution branches: a
stable upper branch ν+,u corresponding to a collectively growing state with 〈n+〉=Nν+,u
pushing MTs and an unstable lower branch ν+,l. For a soft barrier, the 〈n+〉-dynamics
is fast, and the force increases slowly during collective growth, while ν+ = ν+,u is tracing
the stable upper branch of the nullcline. The force per MT can increase up to a critical
value fc (with a corresponding value νc for ν+), where stable and unstable branch join
and where the nullcline has vertical slope df/dν+ = 0 in the f -ν+ plane. The critical force
Fc = fcN represents the maximal load force for which the MT ensemble can maintain a
stable state of collective growth. For 〈F 〉 > Fc the number 〈n+〉 of pushing MTs has to
flow spontaneously to a state 〈n+〉= 0.
The critical force fc per MT can be obtained from two conditions: (i) the nullcline equation
g(f,ν+) = 0, i.e., eq. (8.16), and (ii) taking a total derivative with respect to ν+ and using
the condition of a vertical slope df/dν+ =0 we arrive at the second condition ∂∂ν+ g(f,ν+)=
0. Because ωc(F ) increases exponentially above F0, see eq. (2.39), dωc/dF ∼ ωc/F0 is a
good approximation. The effective rescue time 〈τ〉= 〈τ〉(f/ν+) has a much weaker force
dependence, which we neglect. These approximations give
0≈−ωc(f/ν+)
(
1− f
ν+F0
)
− 1
〈τ〉(f/ν+)
f
ν+F0
≈ 1+ 1
ωc(f/ν+)〈τ〉(f/ν+)
(8.18)
It turns out that (for both the Flyvbjerg and the Janson catastrophe models) ωc〈τ〉 ≥ 1
holds over the entire range of forces. In order to estimate fc, we assume ωc〈τ〉  1. This
leads to an estimate fν+ ≈ F0 in eq. (8.18), which can be used in the arguments of ωc and
〈τ〉. Solving the eqs. (8.16) and (8.18) for fc and νc we find analytical estimates,
fc ≈ F0
1
ωc(F0)〈τ〉(F0)
(8.19)
νc ≈
1
1+ωc(F0)〈τ〉(F0)
. (8.20)
According to eqs. (8.15) and (8.16), the nullclines for f and ν+ and, thus, the critical
values fc and νc are strictly independent of N . Therefore, the critical total force Fc =
Nfc has to be strictly linear in the number of MTs. The critical force is the maximal
polymerization force that can be generated during polymerization in the presence of rescue
108 Microtubule ensemble dynamics
events. For a soft barrier (N  α), rescues thus lead to a significant increase in the
maximal polymerization force with a linear N dependence compared to the logarithmic
dependence derived above in the absence of rescues. Moreover, the estimate (8.19) for fc
predicts an increase of the generated force with the on-rate ωon because this increases v+
and, thus, reduces ωc and an increase with the rescue rate ωr because this decreases 〈τ〉
[see eq. (8.6)].
In order to analyze the system for fixed points, we compare the lower branch ν+,l of the
nullcline of ν+ with the nullcline of f [see eq. (8.15)]. The lower branch is governed by the
exponential increase ωc(F )∼ ωc(0)exp(cF/F0) with force (with c = 2/3 in the Flyvbjerg
and c = 1 in the Janson catastrophe model) resulting in ωc(0)exp(cf/ν+F0) ∼ 1/〈τ〉ν+
or f/ν+ ≈ F0 ln(1/ωc(0)〈τ〉ν+)/c. This is always at lower forces than the nullcline [eq.
8.15)] of f because Fstall = F0 ln(ωon/ωoff) F0 ln(1/ωc(0)〈τ〉ν+)/c. This inequality can
be violated only at very high rescue rates ωr giving rise to a small 〈τ〉. We obtained that
ωr  1/s is necessary to obtain a fixed point. Only if this fixed point exists and is stable,
it can undergo a Hopf bifurcation on lowering the rescue rate. We conclude that, for
realistic parameter values ωr ∼ 0.05 s−1, we are always far from a Hopf bifurcation.
The system rather oscillates in a stable limit cycle: After rescue (A in Fig. 8.10), the
pushing force f increases with the MT growth velocity because of f˙ = kv+/N , while
ν+ = ν+,u is tracing the stable branch of the nullcline. At the critical force level fc, a
collective catastrophe occurs (B in Fig. 8.10), where the ensemble is quickly driven to
collective shrinking with 〈n+〉= 0 or 1 and f˙ =−kv−/N .
During shrinking the force level is reduced until an individual rescue event can initiate a
collective rescue at a force Fmin (C in Fig. 8.10). During rescue 〈n+〉 increases quickly
back to its stable fixed point value (A in Fig. 8.10) closing the limit cycle.
The collective rescue force Fmin can be calculated from the condition that the lower
unstable branch of the nullcline given by eq. (8.16) intersects the line 〈n+〉 = 1, leading
to the condition
N = ωc(Fmin)〈τ〉(Fmin)+1. (8.21)
Collective rescue typically happens at rather small force Fmin  F0 such that we find an
essentially linear N dependence Fmin ∼N +O(1).
The collective mean-field dynamics thus oscillates between forces Fmin and Fc. The re-
sulting time-averaged polymerization force
Fs,N = 〈F 〉 ≈ (Fmin+Fc)/2 (8.22)
is also linear in N . This is in agreement with the simulation results [see Fig. 8.6(a)].
Because Fc  Fmin the result Fc ≈ NF0/[ωc(F0)〈τ〉(F0)] from eq. (8.19) determines the
dependence of the polymerization force Fs,N on the on-rate ωon and the rescue rate ωr. The
estimate for Fc predicts an increase of the generated force with the on-rate ωon because
8.2 Microtubule bundle dynamics at non-zero rescue rate 109
this increases v+ and, thus, reduces ωc. For the velocity dependence (2.39) and assuming
v+ ∝ ωon (for ωon  ωon), eq. (8.19) gives Fc ∝ ω2/3on for the Flyvbjerg catastrophe model
(and Fc ∝ ωon for the Janson catastrophe model), which is in qualitative agreement with
the simulation result of an approximately linear increase of the polymerization force with
the on-rate ωon in Fig. 8.7(a) and 8.7(b). From the result (8.19), we also predict an in-
crease of the polymerization force with the rescue rate ωr because this decreases 〈τ〉. The
pronounced increase of Fs,N with the on-rate ωon demonstrates that for an MT ensem-
ble, the polymerization force can be sensitively regulated by changing the concentration
of available monomers. We also find the collective stall force Fs,N always remains much
smaller than the N -fold single-MT stall force, Fs,N NFstall [see Figs. 8.6(a) and 8.6(b)
] in contrast to force-sharing filaments without dynamic instability, where Fs,N = NFstall
holds exactly [66]. A further confirmation of the mean-field theory is provided by sim-
ulation results for the time-averaged pushing fraction ν+ = 〈n+〉/N in Figs. 8.8(a) and
8.8(b). Mean field results for ν+ evaluated using the nullcline eq. (8.16) for f = Fs,N/N
show good agreement with the simulations results.
The oscillatory limit cycle dynamics, which gives rise to collective catastrophe and rescue
oscillations, is robust against perturbations because the system is far from a bifurcation
for realistic rescue rates. Only for very high rescue rates ωr  1/s, does a stable fixed
point exist, which becomes unstable in a Hopf bifurcation on lowering the rescue rate.
Similar collective catastrophes and rescues are also observed in in vitro bulk polymeriza-
tion experiments [89, 90]. In these experiments many MTs synchronously polymerize in a
solution with GTP-tubulin concentration cGTP. All MTs share the available concentration
cGTP and grow with a velocity v+(cGTP), which decreases if GTP-tubulin is consumed.
Here, collective catastrophes and rescues are caused by sharing the concentration cGTP of
available GTP-tubulin, resulting in similar collective oscillations as force-sharing induces
in the present system.
Finally, we want to note that the collective dynamics for N  1 that we described here
differs markedly from the dynamics of a single MT (N = 1) (see Sec. 4.3)[55]. For a single
MT rescue does not happen at a particular force level Fmin but after an average time 1/ωr
set by the individual rescue rate. The resulting N = 1 mean-field equation for the average
force 〈F 〉 is v−/ωr = v+(〈F 〉)/ωc(〈F 〉) [see eq. (4.32)] and equals shrinking and growing
distance between individual rescue and catastrophe events.
8.2.4 Robustness with respect to catastrophe models
An essential requirement for the existence of an oscillatory limit cycle is the loop shape
of the nullclines of 〈n+〉 according to the stationary mean-field equation (8.16) (see Fig.
8.10). Results presented so far have been derived from the Flyvbjerg model. We obtain
a very similar loop-shaped nullcline also with the catastrophe model by Janson et al. [see
110 Microtubule ensemble dynamics
0
0.25
0.5
0 0.1 0.2 0.3 0.4 0.5
0 0.2 0.4 0.6
ν +
f [pN]
f/F0
f/Fstall
ωc,Jans
C
A
B
v−
v+(f)
fcfmin fmax
⇑
⇑
⇑
⇓
⇓ ⇓
⇓
⇓
Figure 8.11: Nullclines of the mean-field equations for the Janson model for ωon =
70 s−1, ωr = 0.05 s−1, and k = 10−7 N/m (soft barrier). The nullclines of ν+ ≡ 〈n+〉/N
are solutions of eq. (8.16) and assume a loop shape as a function of f ≡ 〈F 〉/N .
Open arrows indicate the mean-field flow of 〈n+〉. The nullcline for f is a straight
line ν+ = f/Fstall (dotted). The critical force is fc ' 0.4 pN for the Janson model.
Black arrows indicate the stable mean-field limit cycle; dashed line indicates effect of
stochastic fluctuations.
Fig. 8.11].
The condition for a loop-shape nullcline is the existence of a critical force Fc, where the
two solution branches of eq. (8.16) merge in a point with a vertical tangent. From eq.
(8.16), we can derive the necessary condition
0 < τ−1 < ωc(F )−F
d
dF
ωc(F ) (8.23)
for the existence of a critical force Fc. Therefore, we expect the same type of oscillatory
limit cycle for collective catastrophe and rescue oscillations for a large class of catastrophe
models which fulfill the two following conditions:
(i) The catastrophe rate ωc = ωc(v+) is a function of the growth velocity only.
(ii) The resulting force dependence fulfills condition (8.23), which gives rise to a catas-
trophe rate increasing exponentially with force above the characteristic force F0.
Whereas the Flyvbjerg and Janson catastrophe models and, more generally, all models
with ωc ∝ v−ε+ (ε > 0) fulfill condition (8.23), it is violated for the linear catastrophe model
eq. (2.40).
This explains that the mean-field result of an oscillatory limit cycle is robust with respect
to variations of the catastrophe models: We expect qualitatively similar behavior for all
8.2 Microtubule bundle dynamics at non-zero rescue rate 111
catastrophe rates ωc(F ), which are exponentially increasing with force above a character-
istic force F0, for example, in the standard catastrophe models by Flyvbjerg et al. [18, 68]
or by Janson et al. [75]. This explains the robustness observed in the simulation results
as shown in Figs. 8.4 and 8.3 , and Figs. 8.6, 8.7 and 8.8.
The condition (8.23) is violated for the linear catastrophe model [see eq. (2.40]. Accord-
ingly, we do not expect to find an oscillatory limit cycle with collective catastrophe and
rescue events. For this type of catastrophe model the nullclines are indeed no longer loop
shaped, and the mean-field theory rather predicts a stable fixed point [see Fig. 8.12(b)].
Simulations confirm that collective catastrophe and rescue oscillations are absent for the
linear catastrophe model, and we find a rather stationary position x of the obstacle and,
thus, a stationary polymerization force [see Fig. 8.12(a)].
We investigate the stability of the intersection point by a linear stability analysis [91].
The coordinates of the intersection point are ν∗+ and f∗, and are determined numerically
from eqs. (8.15) and (8.16) as a function of the growth parameters. From eqs. (8.11) and
(8.12) we obtain the following Jacobian matrix
J =


A B
C D


(ν∗+,f∗)
(8.24)
with
A = ∂
∂ν+
[
−ν+ωc(f/ν+)+(1−ν+)τ−1(f/ν+)
]
=−ωc(f/ν+)− τ−1(f/ν+)+
[
(1−ν+)ωr
(v+(0)+v−)ν+
+˜b
]
fωond
F0ν+
exp
(
−f
F0ν+
)
, (8.25)
B = ∂
∂f
[
−ν+ωc(f/ν+)+(1−ν+)τ−1(f/ν+)
]
=
[
(1−ν+)ωr
(v+(0)+v−)ν+
−˜b
]
ωond
F0
exp
(
−f
F0ν+
)
, (8.26)
C = ∂
∂ν+
[
k
N
v+(f/ν+)
]
= kfdωon
NF0ν2+
exp
(
−f
F0ν+
)
, (8.27)
D = ∂
∂f
[
k
N
v+(f/ν+)
]
= kfdωon
NF0ν+
exp
(
−f
F0ν+
)
. (8.28)
With α = tr(J) and β = det(J), the eigenvalues of the Jacobian matrix (8.24) are given
by
Γ1,2 =
α±
√
α2−4β
2
∣
∣
∣
∣
∣
∣
(ν∗+,f∗)
. (8.29)
For a fixed set of growth parameters, both eigenvalues are evaluated at the intersection
point of eq. (8.15) and eq. (8.16) with ν+ = ν∗+ and f = f∗. Results are depicted in Fig.
8.13.
112 Microtubule ensemble dynamics
750
775
800
510000 515000 520000
x(
t)
[µ
m
]
t [s]
ωc,lin
(a)
xobs(t)
xi(t)
0
0.25
0.5
0.75
0 0.2 0.4 0.6 0.8 1 1.2 1.4
ν +
f/F0
(b)
f/Fstall
ωc,Flyv
ωc,lin
stable fixed point
Figure 8.12: (a): Typical simulation trajectories for N = 10 MTs, ωon = 70 s−1,
ωr = 0.05 s−1 and for the linear catastrophe model. (a): Positions of all MTs as a
function of time t (solid lines); the obstacle position xobs(t) (blue, filled dots) is the
position of the leading MTs (only every 100th data point shown). (b): Nullclines of
the mean-field equations for the linear model (dashed line) (2.40) with ˜a = 0.005 s−1
and ˜b = 8× 104 m−1 and for the Flyvbjerg model (solid line). We use ωon = 70 s−1,
ωr = 0.05 s−1, and k = 10−7 N/m (soft barrier). For the linear catastrophe model,
the nullcline for ν+ ≡ 〈n+〉/N is not loop-shaped. A stable fixed point exists at the
intersection with the nullcline for f , which is the straight line ν+ = f/Fstall.
8.2 Microtubule bundle dynamics at non-zero rescue rate 113
-2.5
-2
-1.5
-1
-0.5
25 50 75 100
Γ 1
/1
0−
5
[s−
1 ]
ωon [s−1]
(a)
-0.02
-0.016
-0.012
-0.008
-0.004
25 50 75 100
Γ 2
[s−
1 ]
ωon [s−1]
(b)
Figure 8.13: Eigenvalue Γ1,2 of the Jacobian matrix [eq. (8.24)] as a function of ωon
and ωr for N = 5 (red) and N = 10 (blue). Γ1,2 are calculated from eq. (8.29). (a):
Eigenvalue Γ1 as a function of ωon for N = 5 (red), ωr = 0.05 s−1 (×) and ωr = 0.1 s−1
(). Eigenvalue Γ1 as a function of ωon for N = 10 (blue), ωr = 0.05 s−1 (×) and
ωr =0.1 s−1 (). (b): Eigenvalue Γ2 as a function of ωon for N = 5 (red), ωr =0.05 s−1
(×) and ωr = 0.1 s−1 (). Eigenvalue Γ2 as a function of ωon for N = 10 (blue),
ωr = 0.05 s−1 (×) and ωr = 0.1 s−1 ().
For realistic parameter values (see Table A.1), both eigenvalues Γ1 and Γ2 are less than
zero. The intersection point of the nullclines [see eq. (8.15) and eq. (8.16)] is stable with
respect to small perturbations in ν+ and f . However, both eigenvalues are functions of
the catastrophe parameters ˜a and ˜b. A different choice of ˜a and ˜b may lead to a change
of sign in one of the two eigenvalues and, therefore, to an unstable intersection point.
8.2.5 Improved mean-field theory including stochastic fluctua-
tions
The dynamical mean-field theory explains all simulation results qualitatively. In order to
obtain quantitative agreement with stochastic simulations, we have to take into account
that the maximal force Fmax for a collective catastrophe is typically smaller than the
critical mean-field force Fc (see Figs. 8.6 and 8.7) because of additional stochastic fluctu-
ations of n+, which reduce n+ and decrease the time spend in a collective state of growth,
that is the first passage-time to a shrinking state n+ = 0 (see Figs. 8.10 and 8.11).
This reduction is governed by the mean first passage time T for a discrete one-step process
with forward rates gn+ and reverse rates rn+ ,
gn+ = τ−1(N −n+)
rn+ = ωc(F/n+)n+, (8.30)
starting from the stable n+ = 〈n+,u〉= Nν+,u on the upper branch of the nullcline (with
114 Microtubule ensemble dynamics
a reflecting boundary at n+ = N) to reach the absorbing boundary n+ = 0 [62],
T (fmax,N,〈n+,u〉) =
〈n+,u〉
∑
λ=1
N
∑
µ=λ
∏µ−1
i=λ gi
∏µ
j=λ rj
=
〈n+,u〉
∑
λ=1
N
∑
µ=λ
1
rµ
exp

ln


µ−1
∏
i=λ
gi
ri



 .
(8.31)
We want to note, that n+ = 0 only acts as an absorbing boundary to calculate the MFPT
from the stable upper branch of the nullcline to a state with n+ =0. At n+ =0 a collective
state of growth ends and a collective state of shrinkage begins. The overall MT dynamics
does not stop at n+ = 0.
The mean first passage time T depends on f , N , and 〈n+,u〉 via the forward and reverse
rates.
The force fmax is the load force that is reached by collective growth with a velocity
v+(f/ν+,u) during the MFPT T . Therefore, the maximal ensemble force fmax is deter-
mined self-consistently by the condition
T (fmin,fmax,N,ν+,u) =
N
k
∫ fmax
fmin(N)
dfv−1+ (f/ν+,u). (8.32)
This equation has to be solved together with the fixed point equation for ν+,u [eq. (8.17)]
to obtain fmax as a function of N and the remaining system parameters.
To gain insight into the N -dependence of the maximal ensemble force fmax, we can de-
termine the N -dependence of T by approximating sums by integrals and introducing
functions
g(ν+) ≡ gn+/N = τ−1(f/ν+)(1−ν+)
r(ν+) ≡ rn+/N = ν+ωc(f/ν+). (8.33)
We finally find
T ≈N
∫ ν+,u
0
dx
∫ 1
x
dy 1
r(y)
exp
[
N
∫ y
x
da ln
(
g+(a)
r+(a)
)]
. (8.34)
It is important to note that T is of the form T ∼N exp[Nu(fmax,ν+,u)] with some function
u. T is essentially linear in N for small N but starts to increase faster than linear for
larger N . Because ν+,u is independent of N according to the fixed point equation (8.17)
for ν+,u, and using the condition (8.32) determining fmax, this leads to an approximately
constant fmax for small N < 10, which starts to increase for larger N .
In Fig. 8.6 we show the time averaged MT ensemble force Fs,N as a function of MT number
N and different on-rates ωon. The improved mean-field results are in qualitative and
quantitative agreement with results from full stochastic simulations. This is in contrast
8.2 Microtubule bundle dynamics at non-zero rescue rate 115
to pure mean-field results (see Sec. 8.2.3), which reproduce simulation results qualitatively
well, but differ quantitatively. For large values of N and ωon, the improved mean-field
results overestimate simulations results, due to the exponential increases of the MFPT
for large N and ωon [see eq. (8.34)].
Fig. 8.7 shows Fs,N as a function of the on-rate ωon for different MT numbers N . Again
we find good agreement between improved mean-field and simulation results.
In Fig. 8.8 the pushing fraction ν+ = 〈n+〉/N of MTs is depicted. Also for this quantity,
our improved mean-field theory agrees well with the full stochastic simulation results. In
addition we find no considerable difference between pure and improved mean-field results.
With the improved mean-field theory, we can also get quantitative results for the average
time spend in a growing state, which is roughly half of the average period of collective
steady state oscillations. In a period of growth, the system traces the upper part of the
nullcline with average velocity v+(f/ν+), starting at fmin. In mean field approximation
the average time spend in a collective state of growth, between fmin and the force f , is
than given by
T (fmin,f,N,ν+,u) =
N
k
∫ f
fmin(N)
dfv−1+ (f/ν+,u) (8.35)
with ν+(f) following eq. (8.17). This condition is equivalent to eq. (8.32) with f = fmax.
We calculate T (f) for the three characteristic mean-field forces fc, fmax, and Fs,N and
compare them to growth durations T+ measured in simulations. Results are shown in Figs.
8.14 and 8.15 for N = 5 and N = 10. We find good agreement between improved mean-
field results and stochastic simulations. In both cases T (fmax) and T (Fs,N ) are within the
range of the measured average duration of growth, while T (fc) exceeds measured values
of T+ considerably.
8.2.6 Semi-stochastic approach
In the previous sections, we discussed MT bundle dynamics in the presence of rescue
events. For a soft barrier, we find oscillations of the polymerization force around its mean
value. In the steady state the MT bundle alternates between phases of collective growth
and shrinkage, characterised by the number n+ of simultaneously pushing MT . In a phase
of collective growth we have a stable population n+ > 1 of MTs pushing against the elastic
barrier, while in a state of collective shrinkage we have n+ ≈ 0. Based on a dynamical
mean field theory we discussed the steady-state dynamics in terms of the average force
〈F 〉 and the average number 〈n+〉 of pushing MTs (see Sec. 8.2.3 and 8.2.5). Within the
mean-field theory we assume a fast n+-dynamics compared to a slow F -dynamics, so that
n+ takes its average steady-state value for a fixed (or slow varying) average force 〈F 〉 (see
8.2.5). In addition to the assumption of different timescales of the n+- and F -dynamics,
we employ several further approximations to establish the mean field theory. In a first
step we use 〈gn+(F/n+)〉 ≈ gn+(〈F/n+〉) and 〈rn+(F/n+,n+)〉 ≈ rn+(〈F/n+〉,〈n+〉) [see
116 Microtubule ensemble dynamics
0
0.04
0.08
0.12
0.16
0 1000 2000 3000 4000 5000
P
(T
+
)
T+ [s]
T (Fs,N ) T (Fmax)
T (fc)≈9600 s
〈T+〉
Figure 8.14: Probability P (T+) to find a phase of collective growth with length T+,
measured in simulations (see Sec. 7.1.2) for N =5, ωon =70 s−1 and ωr =0.05 s−1. Ver-
tical colored lines: T (F ) calculated via eq. (8.32) for Fmax (blue line) and Fs,N (green
line). Dashed black line: Average duration of growth 〈T+〉 measured in simulations.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 2000 4000 6000 8000 10000 12000 14000
P
(T
+
)
T+ [s]
T (Fs,N ) T (Fmax)
T (fc)≈24000 s
〈T+〉
Figure 8.15: Probability P (T+) to find a phase of collective growth with length T+,
measured in simulations (see Sec. 7.1.2) for N = 10, ωon = 70 s−1 and ωr = 0.05 s−1.
Vertical lines: T (F ) calculated via eq. (8.32) for Fmax (blue line) and Fs,N (green
line). Dashed black line: Average duration of growth 〈T+〉 measured in simulations.
8.2 Microtubule bundle dynamics at non-zero rescue rate 117
eq. (8.8)], and later 〈F/n+〉 ≈ 〈F 〉〈1/n+〉 and 〈1/n+〉 ≈ 1/〈n+〉 [see. eq. (8.8) and eqs.
(8.11) and (8.12)]. These approximations provide the essential elements 〈n+〉 and 〈F 〉
of the mean-field theory. However, the approximations are a priori not justified, since
the underlying OPDF which determines the averages, is not known. In the following we
present a semi-stochastic approach to the microscopic dynamics of the n+ leading MTs in
phases of collective growth and investigate the validity of the mean-field approximations,
bases on the semi-stochastic n+-dynamics.
The stochastic dynamics of the n+ leading MTs is determined by the probability pn+(t)
to find n+ leading MTs at time t. In order to calculate the probability pn+(t) to find n+
leading MTs for a given set of parameters, we assume a fast n+-dynamics and treat F
as a free tunable parameter. Again, this is a crucial approximation to the coupled n+-
and F -dynamics, but trajectories from simulations indicate, that, shortly after collective
rescues, n+ reaches its steady-state value, while F increases only slowly over much longer
times scales (see Figs. 8.3 and 8.3). In addition, in Sec. 8.2.3 we find, that the macroscopic
timescales of the 〈n+〉- and 〈F 〉-dynamics differ by a factor of 10, corresponding to a ten
fold faster 〈n+〉-dynamics compared to the 〈F 〉-dynamics [see eqs. (8.13) and (8.14)].
The stochastic time evolution of the n+ ≡ n+(F ) with forward rates gn+ ≡ gn+(F,n+) and
backward rates rn+ ≡ rn+(F,n+), given by eqs. (8.30), is described by a linear, one-step
master equation for the probability pn+(t,F ) to find n+ MTs at time t and for a fixed
force F on the discrete state space n+ ∈ [0,N ],
∂tpn+(t) = rn++1 pn++1(t)+gn+−1 pn+−1(t)− (rn+ +gn+) pn+(t), (8.36)
with n+ = 0,1, . . .N and reflecting boundaries at n+ = 0 and n+ = N [62, 63]. At the
boundaries n+ = 0 and n+ = N we have to adjust the forward and backward rate. For
n+ = 0, we set g0 ≡ ωrN and r0 ≡ 0, corresponding to all MTs being in a collective state
of shrinkage. At n+ = N , we set gN ≡ 0 and rN ≡ ωc(F/N)N , corresponding to all MTs
being in a collective state of growth.
In the steady state eq. (8.36) is solved with reflecting boundary at n+ =0 and we calculate
the probability pn+ ≡ pn+(F ) to find n+ leading MT in the steady state explicitly [62],
pn+ =
gn+−1 gn+−2 · · ·g1 g0
rn+ rn+−1 · · ·r2 r1
p0 (8.37)
with p0 fixed by the normalization condition
p0 =

1 +
N
∑
n+=1
g0 g1 · · ·gn+−1
r1 r2 · · ·rn+


−1
. (8.38)
In Figs. 8.16 and 8.17, the steady-state probability pn+ calculated from eqs. (8.37) and
(8.38), is depicted for N = 5 and N = 10 and different values of the dimensionless force
118 Microtubule ensemble dynamics
F/F0. Here we use the catastrophe model by Flyvbjerg et al. with F0 ' 7 pN. For
forces F/F0 < Fs,N/F0 smaller than the time averaged ensemble force Fs,N , pn+ assumes
a broad, Gaussian-shaped distribution, centered on an average 〈n+〉 > 0 and p0 ≈ 0,
corresponding to a stable population of leading MTs and, therefore, to phase of stable
collective growth [see Fig. 8.16(a) and Fig. 8.17(a)]. We find such small forces, which
satisfy F/F0 < Fs,N/F0, shortly after collective rescue events. At F/F0 ≈ Fs,N/F0, pn+
still assumes a broad, Gaussian-shaped distribution, but with a non-vanishing probabil-
ity p0 > 0. With increasing force, probability weight is shifted to smaller values of n+.
However, a stable population of leading MTs is maintained. If the force is increased
to values F/F0 > Fs,N/F0 we find a sudden change in the probability distribution pn+ .
Most of the probability weight is now centered at n+ = 0 and p0  pn+ ∀n+ > 0, so that
p0 ≈ 1 [see Fig. 8.16(d) and Fig. 8.17(d)]. No stable population with n+ > 0 is main-
tained, corresponding to a collective catastrophe. By tuning F , we can drive pn+ from
a broad, Gaussian-shaped distribution (small forces) to a strongly localized distribution
(large forces), with most of the probability weight centered at n+ = 0. This transition in
the probability distribution for the n+ leading MTs can be interpreted as the transition
from a phase of collective growth to a phase of collective shrinkage, as observed in full
stochastic simulations (see Figs. 8.3 and 8.3). Shortly after a collective rescue, we find
small forces and pn+ is Gaussian shaped. The force F is slowly increased by the polymer-
izing MTs and if a force F > Fs,N is reached, pn+ is reduced to p0 ≈ 1 and a collective
catastrophe is initiated. In a state of collective shrinkage the force is decreased until the
collective rescue force Fmin  Fs,N is reached. The ensemble enters a state of collective
growth, pn+ again assumes a Gaussian shape, centered at 〈n+ > 1〉 and the cycle is closed.
The semi-stochastic approach to the n+-dynamics reproduces collective oscillations in the
polymerization force qualitatively.
We want to stress out, that the transition of the steady-state probability distribution
pn+ , from a Gaussian-shaped form to a strong localized form at n+ = 0 with increasing
force, resembles the transition of a single MT growing against a constant force, from the
regime of unbounded into the regime of bounded growth with increasing force (see. Sec.
4.2). For small forces and in the regime of unbounded growth, the probability P (x,t)
to find a MT with length x at time t is Gaussian shaped, while for large forces the MT
enters the regime of bounded growth and P (x) assumes an exponentially decaying form,
with most of the probability weight localized at small MT length. Both, the shape of
probability distributions and their transition with increasing force, are also found in the
steady-state probability pn+ given by eqs. (8.37) and (8.38). In Sec. 4.3 we found that,
for a single MT growing against an elastic obstacle, the average dimensionless force 〈η〉 in
the steady state approximately equals the critical dimensionless force ηc, which provides
the transition from unbounded to bounded growth. For a single MT, growth under a
constant force and growth against an elastic obstacle are therefore connected via ηc ∼ 〈η〉.
8.2 Microtubule bundle dynamics at non-zero rescue rate 119
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
p n
+
n+
(a)
F/F0 = 0.7
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
p n
+
n+
(b)
F/F0 = 1.4≈ Fs,N/F0
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
p n
+
n+
(c)
F/F0 = 2.1
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
p n
+
n+
(d)
F/F0 = 2.5
Figure 8.16: Steady-state probability pn+ to find n+ leading MTs for N = 5, ωon =
70 s−1, ωr = 0.05 s−1 and different values of F/F0. (a): F/F0 = 0.7. (b): F/F0 = 1.4.
(c): F/F0 = 2.1. (d): F/F0 = 2.5. We use the catastrophe model by Flyvbjerg et al.
with F0 ' 7pN.
Based on the results of the semi-stochastic approach, we suspect a similar connection of
growth under constant force and growth against an elastic obstacle for MT ensembles with
N > 1. However, a further investigation of the analogy of the dynamics of single MTs
and MT ensembles needs much more effort and a deeper understanding of the underlying
stochastic processes. This is beyond the scope of this thesis and had to be postponed.
We now want to address the question, if the semi-stochastic approach also reproduces
results from full stochastic simulations quantitatively. In the steady state, we measure
the probability P (n+) to find n+ leading MT in a phase of collective growth. In Figs. 8.18
and 8.19 we compared P (n+) with the steady-state probability pn+ from eqs. (8.37) and
(8.38) with F = Fs,N . For N = 5 results from simulations agree well with results obtained
from eqs. (8.37) and (8.38). Both probability distributions show similar shapes and yield
nearly the same average number 〈n+〉 of leading MTs. However, there are small differences
in the absolute values of the probability distributions. The steady-state probability pn+
exhibits a non-vanishing part for n+ = 0. The probability P (0) can not be measured in
simulations, since n+ = 0 determines the begin of a collective phase of shrinkage and is ,
therefore, not part of a collective phase of growth. For N = 10 probability distributions
are again similar shaped, but pn+ is slightly shifted to higher values of n+ compared to
results from simulations (see Fig. 8.19) and we find P (n+)≈ pn++0.5. The shift to larger
120 Microtubule ensemble dynamics
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10
p n
+
n+
(a)
F/F0 = 1.4
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10
p n
+
n+
(b)
F/F0 = 2.9≈ Fs,N/F0
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10
p n
+
n+
(c)
F/F0 = 3.2
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10
p n
+
n+
(d)
F/F0 = 3.6
Figure 8.17: Steady-state probability pn+ to find n+ leading MTs for N = 10, ωon =
70 s−1, ωr = 0.05 s−1 and different values of F/F0. (a): F/F0 = 1.4. (b): F/F0 = 2.9.
(c): F/F0 = 3.2. (d): F/F0 = 3.6. We use the catastrophe model by Flyvbjerg et al.
with F0 ' 7pN.
8.2 Microtubule bundle dynamics at non-zero rescue rate 121
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
P
(n
+
)
n+
〈n+〉sim 〈n+〉(Fs,N )
pn+(Fs,N )
Simulation
Figure 8.18: Probability P (n+) to find n+ leading MTs in a collective state of
growth, measured in simulations, for N = 5, ωon = 50 s−1 and ωr = 0.05 s−1. Solid
boxes: P (n+) from simulations. Dashed boxes: Steady-state probability pn+(Fs,N )
from eqs. (8.37) and (8.38) with F = Fs,N . For reasons of clarity we reduce the bin
width from one to one-half. Solid vertical line: 〈n+〉 measured in simulations. Dashed
vertical line: 〈n+〉=
∑N
0 n+pn+(Fs,N ).
values of n+ can also be seen in the average number 〈n+〉 of leading MTs. Nevertheless
both distributions exhibit a large overlap and possess similar characteristics.
It is clearly visible, that the semi-stochastic approach to the n+-dynamics reproduces
results from simulations not only qualitatively but also quantitatively. Based on the
steady-state probability distribution pn+ from eqs. (8.37) and (8.38), we now want to
investigate the quality and limits of mean-field approximations, such as 〈ωc(F/n+)〉 ≈
ωc(〈F/n+〉) or 〈(F/n+)〉 ≈ 〈F 〉/〈n+〉 (see Sec. 8.2.2).
In a first step we construct, from the microscopic probability distribution pn+ [eqs. (8.37)
and (8.38)], a macroscopic equation of motion for the number n+ of leading MTs and find
[62]
d
dt
〈n+〉= 〈gn+〉−〈rn+〉= 〈τ−1(F/n+)(N −n+)〉−〈ωc(F/n+)n+〉, (8.39)
with
〈u(n+)〉 ≡
N
∑
n+=0
pn+u(n+). (8.40)
and some function u(n+). The steady-state solution of eq. (8.39) is given by:
〈τ−1(F/n+)(N −n+)〉= 〈ωc(F/n+)n+〉. (8.41)
Equation (8.39) is equivalent to eq. (8.8) before mean-field approximations. The same
applies to eq. (8.41) and eq. (8.16). It is clearly visible, both equations differ by the mean-
122 Microtubule ensemble dynamics
0
0.2
0.4
0.6
0 1 2 3 4 5 6 7 8 9 10
P
(n
+
)
n+
〈n+〉sim 〈n+〉(Fs,N )
pn+(Fs,N )
Simulations
Figure 8.19: Probability P (n+) to find n+ leading MTs in a collective state of
growth, measured in simulations, for N = 10, ωon = 50 s−1 and ωr = 0.05 s−1. Solid
boxes: P (n+) from simulations. Dashed boxes: Steady-state probability pn+(Fs,N )
from eqs. (8.37) and (8.38) with F = Fs,N . For reasons of clarity we reduce the bin
width from one to one-half. Solid vertical line: 〈n+〉 measured in simulations. Dashed
vertical line: 〈n+〉 calculated from pn+ with F = Fs,N .
field approximations, where global averages are replaced by local averages, neglecting all
correlations of higher order.
We start the analysis with 〈rn+〉= 〈ωc(F/n+)n+〉 and perform the following sequence of
successive mean-field approximations:
〈ωc(F/n+)n+〉 → 〈ωc(F/n+)〉〈n+〉 → ωc(F 〈1/n+〉)〈n+〉 → ωc(F/〈n+〉)〉〈n+〉. (8.42)
In each approximation step, the different averages are calculated via eq. (8.40) as a
function of the dimensionless force F/F0 with F0 ' 7pN and N = 10, ωon = 70 s−1 and
ωr = 0.05 s−1. Results are depicted in Fig. 8.20.
For forces up to F/F0 ≈ 2 there is no visible difference between all approximation steps.
The average backward rate rn+ = 〈ωc(F/n+)n+〉 increases for forces F/F0 > 2 and takes
a constant value for F/F0 ≈ 4, corresponding to v+ ≈ 0 and n+ ≈ 0. The approximation
steps ωc(〈F/n+〉)〈n+〉 and ωc(F/〈n+〉)〉〈n+〉 also show a characteristic increase with in-
creasing force, but exhibit a divergence at n+ ≈ 0. This divergence is caused by vanishing
n+ and an exponential increase of ωc(F/n+) for large forces (see Fig. 8.22). Both approxi-
mation steps differ also qualitatively. However, the semi-stochastic average backward rate
〈ωc(F/n+)n+〉 and the final mean-field approximation result ωc(F/〈n+〉)〈n+〉, which is a
basic part of the dynamical mean-field theory used in Sec. 8.2.2 and Sec. 8.2.3, are nearly
identical for small F/F0 and exhibit a characteristic increase for large values of F/F0. In
contrast to the semi-stochastic results, the approximation step rn+ ≈ ωc(F 〈1/n+〉)〈n+〉
8.2 Microtubule bundle dynamics at non-zero rescue rate 123
0
0.02
0.04
0.06
0.08
0 1 2 3 4 5
0 5 10 15 20 25 30 35
r n
+
[s−
1 ]
F/F0
F [pN]
〈ωc(F/n+)n+〉
〈ωc(F/n+)〉〈n+〉
ωc(F 〈1/n+〉)〈n+〉
ωc(F/〈n+〉)〈n+〉
Figure 8.20: Backward rate rn+ = 〈ωc(F/n+)n+〉 as a function of F/F0 (bottom
x-axis) and F (top x-axis) for N =10, ωon =70 s−1 and ωr =0.05 s−1. Solid line: Semi-
stochastic results for 〈rn+〉= 〈ωc(F/n+)n+〉. Dashed and dotted lines: Different types
of mean-field approximations of 〈rn+〉.
does not increase with increasing force, but decreases and approaches zero for F/F0 ≈ 4.
For large forces 〈1/n+〉 remains finite, while 〈n+〉 ≈ 0.
We now investigate the average forward rate gn+ = 〈τ−1(F/n+)(N −n+)〉, in the same
way as in the case of the average backward rate. The following sequence of successive
mean-field approximations is performed:
〈τ−1(F/n+)(N −n+)〉 → 〈τ−1(F/n+)〉(N −〈n+〉)〉 →
τ−1(F 〈1/n+〉)(N −〈n+〉)→ τ−1(F/〈n+〉)(N −〈n+〉). (8.43)
Results for N = 10, ωon = 70 s−1 and ωr = 0.05 s−1 are shown in Fig. 8.21.
We find only small differences between the semi-stochastic result 〈gn+〉 and all mean-field
approximation steps for forces F/F0. 2. With increasing force mean-field approximations
and semi-stochastic results begin to differ. While the semi-stochastic result 〈gn+〉 and
the approximation step 〈gn+〉 ≈ τ−1(F 〈1/n+〉)(N−〈n+〉) decrease and approach zero for
forces F/F0 & 3, the approximation step 〈τ−1(F/n+)〉(N −〈n+〉)〉 and the final mean-
field approximation, used in Sec. 8.2.2 and Sec. 8.2.3, diverge at forces F/F0 & 3. The
average forward rate 〈gn+〉 from semi-stochastic calculations and the final mean-field
approximation 〈gn+〉 ≈ τ−1(F/〈n+〉)(N − 〈n+〉) do not exhibit the same characteristic
behavior for large forces. This is in contrast to the average backward rate 〈rn+〉.
For forces F/F0 . 2, the average forward rate 〈gn+〉 and the average backward rate 〈rn+〉
equal each other qualitatively and quantitatively , since both rates are connected via the
124 Microtubule ensemble dynamics
0
0.01
0.02
0.03
0 1 2 3 4 5
0 5 10 15 20 25 30 35
g n
+
[s−
1 ]
F/F0
F [pN]
〈τ−1(F/n+)(N −n+)〉
〈τ−1(F/n+)〉(N−〈n+〉)
τ−1(F 〈1/n+〉)(N−〈n+〉)
τ−1(F/〈n+〉)(N −〈n+〉)
Figure 8.21: Forward rate 〈gn+〉 = 〈τ−1(F/n+)(N −n+)〉 as a function of F/F0
(bottom x-axis) and F (top x-axis) for N = 10, ωon = 70 s−1 and ωr = 0.05 s−1. Solid
line: Semi-stochastic results for 〈gn+〉. Dashed and dotted lines: Different types of
mean-field approximations of 〈gn+〉.
steady-state solution [eq. (8.41)] of eq. (8.39), but at larger force F/F0 both rates show
different behavior. The average backward rate 〈rn+〉 increases for forces F/F0 > 2, while
〈gn+〉 decreases and tends to zero. The steady-state condition is no longer fulfilled and
the ensemble undergoes a collective catastrophe, as expected for large forces.
At larger forces F/F0 & 2 results for 〈rn+〉 and 〈gn+〉, obtained with the semi-stochastic
approach and results from the final mean-field approximation step, exhibit major dif-
ferences (see Figs. 8.20 and 8.21). However, in Sec. 8.2.3 we found, that characteristic
forces, as the critical force Fc, describe the dynamics of the MT bundle only qualitatively,
and results for the average polymerization force Fs,N obtained from pure mean-field cal-
culations exceed results from stochastic simulations (see Fig. 8.6). In order to achieve
quantitative agreement we introduced an improved mean-field theory (see Sec. 8.2.5).
The essential part of the improved mean-field theory, is the calculation of the maximal
ensemble force Fmax from the MFPT starting at the upper stable branch of the nullcline
with 〈n+,u〉> 0 to an absorbing state with n+ = 0. Results from the improved mean-field
theory agree well with results from simulation and the average polymerization force Fs,N
is much smaller, compared to Fs,N from the pure mean-field theory (see Fig. 8.6). For
ωon = 70 s−1 and ωr = 0.05 s−1, the dimensionless maximal ensemble force is of the order
Fmax/F0 ≈ 4, which is in the range of forces, where 〈rn+〉 and 〈gn+〉, calculated within the
semi-stochastic approach, and the final mean-field approximations do significantly differ
(see Figs. 8.20 and 8.21). To test the validity and limits of the improved mean field theory,
we compute the average fraction ν+(f)stoch of pushing MTs via eq. (8.40) and compare it
to results from the improved mean-field theory. Results are depicted in Fig. 8.22.
8.2 Microtubule bundle dynamics at non-zero rescue rate 125
0
0.25
0.5
0.75
0 1 2 3 4 5 6 7
0 1 2 3 4 5
ν +
F/F0
f [pN]
Fmax/F0 Fc/F0
ν+(f)stoch
ν+(f)MF
Figure 8.22: Average fraction ν+ of pushing MTs as a function of F/F0 (bottom
axis) and f = F/N (top axis) for ωon = 70 s−1 and ωr = 0.05 s−1. Solid line: ν+(f)MF
as solution of mean field equation (8.16). Dashed line: ν+(f)stoch from semi-stochastic
calculations. Solid vertical line: Maximal ensemble force Fmax from improved mean-
field calculations (see Sec. 8.2.5). Dashed vertical line: Critical mean-field force Fc
from pure mean-field calculations (see Sec. 8.2.3)
With increasing force ν+(F/F0)stoch follows the stable upper branch of the nullcline of
eq. (8.16). At F/F0 ≈ 2, ν+(F/F0)stoch starts to decrease and rapidly approaches zero for
forces F/F0 ≈ 4, whereas the stable upper branch yields ν+(F/F0)MF > 0 until the critical
force Fc/F0 is reached. The difference of ν+(F )stoch and ν+(F/F0)MF is in accordance
with results from the previous paragraph. The semi-stochastic approach and the dynamic
mean-field theory, based on the final approximation step, are in good agreement for forces
F/F0 . 2 only and differ significantly for larger forces. Within the improved mean-field
theory we calculate the MFPT as a function of the force F , starting on the stable upper
branch of the nullcline ν+,u(F ). For F/F0 > 2 the starting point of the MFPT is not
confirmed by the semi-stochastic approach. However, the final absorbing state of the
MFPT as a function of force, ν+(F ) = 0, and ν+(F )stoc = 0 are equivalent to each other,
since the characteristic behavior of the MFPT as a function of the force F is determined by
the underlying probability distribution pn+(t,F ), with pn+(t,F ) as solution of eq. (8.36)
(for a detailed description see [62]). The MFPT final absorbing state ν+(F ) = 0 and
ν+(F )stoch = 0 are, therefore, fixed by the same characteristic force Fmax. Since Fmax is
not only determined by the MFPT but also from the condition of stable growth along the
upper branch of the nullcline (see Sec. 8.2.5), which is not verified by the semi-stochastic
approach for large forces, we find small deviations of ν+(Fmax) = 0 and ν+(F )stoc = 0.
To conclude: Semi-stochastic calculations and mean-field approximations agree well for
both rates and forces F/F0 . 2. Results derived from the pure dynamical mean-field
126 Microtubule ensemble dynamics
theory, which is based on the final approximation step, are in agreement with semi-
stochastic calculations (see Fig. 8.22) and the mean-field approximation remain valid
for forces up to F/F0 . 2. At larger forces F/F0 & 2, semi-stochastic and mean-field
approximation exhibit major differences. Results for the average polymerization force
Fs,N , obtained from the pure dynamical mean field theory are only valid in the context
of the mean-field theory and are not confirmed by semi-stochastic calculations. Also
the improved mean field theory is not fully confirmed by the semi-stochastic approach
for forces F/F0 > 2. However, results for the maximal ensemble force Fmax and thus
for the average polymerization force Fs,N , are in good agreement with results from the
semi-stochastic approach.
Chapter 9
Final conclusion 1
We extend the single MT model to an ensemble of N MTs, growing against an elastic
barrier (see Fig. 6.1).
In cooperative force generation by an ensemble of N MTs, the interplay between force-
sharing and MT dynamic instability gives rise to a complex dynamics, which can be
described in terms of collective catastrophe and rescue events.
We developed a dynamical mean-field theory [see eqs. (8.1) and (8.12)] which gives a
quantitative description of the cooperative MT dynamics in terms of two parameters, the
mean force 〈F 〉 and the mean number of pushing MTs 〈n+〉, in both the absence and
presence of rescue events.
The dynamical mean-field theory is validated by stochastic simulations of the MT ensem-
ble dynamics.
In the absence of rescue events, the maximal polymerization force before collective catas-
trophes grows linearly with N for small N or a stiff elastic barrier, in agreement with
existing experimental data [53], whereas it crosses over to a logarithmic dependence for
larger N or soft elastic barrier [see eq. (8.5) and Fig. 8.1]. This crossover should be acces-
sible in experiments by varying the stiffness of optical traps. The maximal polymerization
force increases approximately linear in the tubulin on-rate [see eqs. (8.4) and (8.5)]. Again
this should be accessible in experiments by varying the tubulin concentration.
In the presence of rescue events and for a soft elastic barrier, the dynamics becomes
strongly modified: Collective catastrophes and rescues lead to an oscillatory stable limit
cycle dynamics far from a Hopf bifurcation. These oscillations should be observable in
vitro in experiments such as in Ref. [53] if the MT lengths are sufficient to observe rescue
events and if the stiffness of optical traps is reduced. Moreover, in vivo the behavior of
polarized MT ensembles can be explored, as has been shown in Ref. [92], and our model
predicts synchronized growth and shrinkage in oscillations if a polarized MT ensemble is
growing against an elastic barrier such as the cell cortex.
1Parts of the text have been published in reference [55] and is ©2012 American Physical Society.
http://publish.aps.org/copyrightFAQ.html
128 Final conclusion
In the presence of oscillations, we have quantified the maximal polymerization force Fc =
Nfc in eq. (8.19) and the time-averaged polymerization force Fs,N in eq. (8.22) based on
the nullclines of the dynamics mean field theory. Both forces are linear in N (see Figs.
8.6), and the relevant force scale is the force scale F0, above which the MT growth velocity
decreases exponentially and the MT catastrophe rate increases exponentially. The time-
averaged polymerization force strongly increases with increasing tubulin on-rate (and,
thus, decreasing catastrophe rate) or increasing MT rescue rate (see Figs. 8.7).
Including stochastic fluctuations, which reduce the maximal polymerization force, we find
a linear N dependence of forces for small MT numbers [see eq. (8.34)].
The linear N dependence of forces in the presence of rescue events is remarkable because
we find an only logarithmic increase with N in the absence of rescue events for soft barriers
(see Fig. 8.1).
Nevertheless, even the maximal polymerization force is significantly smaller than the N -
fold single-MT stall force. This shows that MTs are not optimized with respect to force
generation because of their dynamic instability, even if they cooperate in an ensemble.
On the other hand, our analysis also shows that force generation in MT ensembles is very
sensitive to changes of system parameters related to the dynamic instability of MTs. The
combination of both results suggests that a MT ensemble is not efficient to generate high
forces but that the dynamic instability in connection with the ensemble dynamics allows
us to efficiently regulate force generation through several system parameters.
We show that our results are robustness against variations of the catastrophe model In
terms of the relevant force scale F/F0, the catastrophe models by Flyvbjerg et al. and
Janson et al. yield similar results. The linear catastrophe model, lacking an exponentially
increase for forces larger than F0, does not exhibits a oscillatory limit cycle. We find a
stable fixed point of the cooperative dynamics,
Finally we present a semi-stochastic approach to the dynamics of the n+ leading MTs
and investigate the limits of the dynamical mean field theory. For small forces, results
obtained from the mean field theory are confirmed by semi-stochastic calculations, while
for larger forces results exhibit major differences. In addition, we illustrate a possible
connection of MT ensembles growing under constant force and growing against an elastic
barrier.
Chapter 10
Outlook
We discuss some open issues, which can be addressed by the theoretical framework pre-
sented in this thesis:
• Regulation by feedback mechanisms
Within a cell, MTs and regulating proteins form feedback loops. Recent work on
the regulation of single MT growth dynamics by a positive feedback loop, revealed
a complex interplay of single MT and protein dynamics [93, 94]. Simple feedback
mechanisms can easily be incorporated into the MT ensemble model, to investigate
their influence on the force generation by MT ensembles.
• MT ensembles under constant force
Within this thesis, we discussed the polymerization dynamics of MT ensembles
growing against an elastic obstacle. But, as in the case of a single MT, it is also
possible to study MT ensemble dynamics under a constant force. Simulation re-
sults, not shown here, of MT ensemble growing under a constant force indicate two
different growth regimes, resembling the regime of bounded and unbounded growth
for a single MT. For small forces the average length of the MT ensemble increases
linearly in time, whereas for large forces the average length approaches zero. There
is a sharp transition between the two regimes. This transition can also be seen in
the probability distributions pn+ to find n+ leading MTs, as discussed in Sec. 8.2.6.
It should be possible, as in the case of a single MT, to find a critical force, which
provides the transition between the two regimes. For a single MT we found that the
critical force ηc approximately equals the average steady state force 〈η〉 for growth
against an elastic obstacle (see Sec. 4.3.2). A similar connection may exist for MT
ensembles growing under a constant force and growing against an elastic obstacle.
• No force sharing in actin filaments
Recent optical trap experiments on the force generation of actin filaments showed
that polymerizing bundles of parallelly arranged actin filaments do not share a
130 Outlook
resiting force [51]. Existing polymerization models for parallel actin filaments do
not address the force generation [26]. The MT ensemble model, presented in this
thesis, can be modified in order to describe the polymerization dynamics of parallel
actin filaments. Based on the modified model, the force generation of actin bundles
can be investigated in a similar way, as the force generation of MT bundles.
• Protofilament dynamics
The MT ensemble model can be modified to describe the dynamics of coupled
protofilaments within a single MT. A single MT now acts as a single protofilament,
which can either be in a state of growth or in a state of stall, resembling the state
of shrinkage. In a state of growth, new GTP-tubulin monomers are incorporated
into the tip of a single protofilament, whereas in a state of stall no GTP-tubulin
monomers are attached. The growth of the protofilament is stalled and the entire
protofilament consists of GDP-tubulin. Protofilaments can be coupled to neighbor-
ing filaments by lateral bonds, resembling the coupling of MTs via force-sharing.
The number of lateral bonds can be increased by the addition of new GTP-tubulin
monomers and decreased by hydrolysis. If a single protofilament switches into the
state of stall, all lateral bonds are lost. If a critical amount of protofilaments is in
the state of stall, too many lateral bonds are lost and the protofilament lattice is
destabilized, corresponding to a collective catastrophe in the case of MT ensemble
dynamics. The dynamics of the coupled protofilaments can than be analyzed in a
similar way as the cooperative dynamics of MT ensembles.
Appendix A
Literature values for MT growth
parameters and simulation
parameters
132 Literature values for MT growth parameters and simulation parameters
R
ef.
v+ (0)
[m
/s]
ω
on
[1/s]
v−
[m
/s]
ω
r
[1/s]
D
rechsel[95]
(0.7
...2)×
10 −
8
(11
...32)
∼
1.8×
10 −
7
-
G
ildersleeve
[96]
∼
4.2×
10 −
8
∼
68
∼
4.2×
10 −
7
-
W
alker
[15]
(4
...8)×
10 −
8
(63
...130)
∼
5×
10 −
7
(0.05
...0.08)(T
U
B
)
Laan
[53]
∼
4.2×
10 −
8
68.25
-
-
Janson
[60]
(3
...4.3)×
10 −
8
(53
...74)
-
-
Pryer
[16]
-
-
-
...0.5
(T
U
B
)
..0.15
(M
A
PS)
D
ham
odharan
[97]
-
-
-
...0.07
(C
ell)
...0.085
(M
A
PS)
N
akao
[98]
-
-
-
...0.1
(T
U
B
)
Shelden
[99]
-
-
-
(0.03
...0.2)
(C
ell)
T
able
A
.1:
Literature
values
for
param
eters.
T
U
B
:
in
vitro
results
for
tubulin
solutions,C
ell:
in
vivo
results,M
A
PS:
effect
from
M
T
associated
proteins.
Valuesforω
on
are
estim
ated
from
m
easured
grow
th
velocitiesvia
ω
on ≈
v+ (0)N
/d
neglecting
ω
off
=
6s −
1
[60].
H
ere
N
=
13
denotes
the
num
ber
ofprotofilam
ents
w
ithin
a
single
M
T
.
Param
eter
v−
[m
/s](see
Table
A
.1)
ω
off
[s −
1]
d
[m
]
r
[m
−
1s −
1]
vh
[m
/s]
∆
t[s]
Value
3×
10 −
7
6
0.6×
10 −
9
3.7×
10 6
4.2×
10 −
9
0.1
T
able
A
.2:
Fixed
param
eter
values
for
calculations
and
sim
ulations.
Bibliography
[1] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter. Molec-
ular Biology of the Cell. Garland Science, New York (2008). 1, 1.1, 1.1, 1.2, 1.1, 1.1,
1.4, 1.1
[2] J. Howard. Mechanics of Motor Proteins and the Cytoskeleton. Sinauer Associates,
Inc., Sunderland, Massachusetts (2001). 1, 1.1, 1.1, 1.1, 2.2, 2.2, 2.2
[3] J. Howard, A. A. Hyman. Dynamics and mechanics of the microtubule plus end.
Nature 422(6933), 753 (2003). doi:10.1038/nature01600.
URL http://www.ncbi.nlm.nih.gov/pubmed/12700769 1, 1.1
[4] L. A. Amos, T. S. Baker. The three-dimensional structure of tubulin protofila-
ments. Nature 279(5714), 607 (1979). doi:10.1038/279607a0.
URL http://www.nature.com/nature/journal/v279/n5714/abs/279607a0.html 1.1
[5] E. Mandelkow, E. M. Mandelkow. Microtubular structure and tubulin
polymerization. Current Opinion in Cell Biology 1(1), 5 (1989). doi:10.1016/
S0955-0674(89)80029-8. 1.1
[6] E. Nogales, S. G. Wolf, K. H. Downing. Structure of the αβ tubulin dimer
by electron crystallography. Nature 391(6663), 199 (1998). doi:10.1038/34465.
URL http://www.nature.com/nature/journal/v391/n6663/full/391199a0.html 1.1
[7] V. VanBuren, D. J. Odde, L. Cassimeris. Estimates of lateral and longitu-
dinal bond energies within the microtubule lattice. Proceedings of the National
Academy of Sciences of the United States of America 99(9), 6035 (2002). doi:
10.1073/pnas.092504999.
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=122897&tool=
pmcentrez&rendertype=abstract 1.1, 1.1
[8] A. Grafmüller, E. G. Noya, G. A. Voth. Nucleotide-Dependent Lateral and
Longitudinal Interactions in Microtubules. Journal of Molecular Biology 425(12),
2232 (2013). doi:10.1016/j.jmb.2013.03.029.
URL http://www.ncbi.nlm.nih.gov/pubmed/23541590 1.1
134 BIBLIOGRAPHY
[9] T. Mitchison, M. Kirschner. Dynamic instability of microtubule growth. Nature
312(5991), 237 (1984). doi:10.1038/312237a0.
URL http://www.ncbi.nlm.nih.gov/pubmed/6504138 1.1, 1.1
[10] J. Luders, T. Stearns. Opinion - Microtubule-organizing centres: a re-evaluation.
Nature Reviews Molecular Cell Biology 8(2), 161 (2007). doi:10.1038/nrm2100.
URL http://www.nature.com/nrm/journal/v8/n2/full/nrm2100.html 1.1
[11] D. K. Fygenson, E. Braun, A. Libchaber. Phase diagram of microtubules.
Physical Review E 50(2), 1579 (1994). doi:10.1103/PhysRevE.50.1579.
URL http://pre.aps.org/abstract/PRE/v50/i2/p1579_1 1.1
[12] D. K. Fygenson, H. Flyvbjerg, K. Sneppen, A. Libchaber, S. Leibler.
Spontaneous nucleation of microtubules. Physical Review E 51(5), 5058 (1995). doi:
10.1103/PhysRevE.51.5058.
URL http://pre.aps.org/abstract/PRE/v51/i5/p5058_1 1.1
[13] T. D. Pollard. Rate Constants for the Reaction of ATP-Actin and ADP-Actin
with the End of Actin filaments. The Journal of Cell Biology 103(6), 2747 (1986).
doi:10.1083/jcb.103.6.2747.
URL http://jcb.rupress.org/content/103/6/2747 1.1
[14] H. P. Erickson, E. T. O’Brien. Microtubule dynamic instability and GTP
hydrolysis. Annual Review of Biophysics and Biomolecular Structure 21, 145 (1992).
doi:10.1146/annurev.bb.21.060192.001045.
URL http://www.ncbi.nlm.nih.gov/pubmed/1525467 1.1, 1.5
[15] R. A. Walker, E. T. O’Brien, N. K. Pryer, M. F. Soboeiro, W. A. Voter,
H. P. Erickson, E. D. Salmon. Dynamic instability of individual microtubules
analyzed by video light microscopy: rate constants and transition frequencies. The
Journal of Cell Biology 107(4), 1437 (1988). doi:10.1083/jcb.107.4.1437.
URL http://www.ncbi.nlm.nih.gov/pubmed/3170635 1.1, 2.3, A
[16] N. K. Pryer, R. A. Walker, V. P. Skeen, B. D. Bourns, M. F. Soboeiro,
E. D. Salmon. Brain microtubule-associated proteins modulate microtubule dy-
namic instability in vitro. Real-time observations using video microscopy. Journal of
Cell Science 103(4), 965 (1992).
URL http://www.ncbi.nlm.nih.gov/pubmed/1487507 1.1, A
[17] H. T. Schek, M. K. Gardner, J. Cheng, D. J. Odde, A. J. Hunt. Micro-
tubule assembly dynamics at the nanoscale. Current Biology 17(17), 1445 (2007).
doi:10.1016/j.cub.2007.07.011.
BIBLIOGRAPHY 135
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2094715&tool=
pmcentrez&rendertype=abstract 1.1, 1.4, 1.9, 2.2, 4.1, 4.3, 4.3.1
[18] H. Flyvbjerg, T. E. Holy, S. Leibler. Microtubule dynamics: Caps, catas-
trophes, and coupled hydrolysis. Physical Review E 54(5), 5538 (1996). doi:
10.1103/PhysRevE.54.5538.
URL http://www.ncbi.nlm.nih.gov/pubmed/9965740 1.1, 2.3, 2.3, 2.3, 4.2, 4.3.1, 8.2.4
[19] D. N. Drechsel, M. W. Kirschner. The minimum GTP cap required to stabilize
microtubules. Current Biology 4(12), 1053 (1994). doi:http://dx.doi.org/10.1016/
S0960-9822(00)00243-8.
URL http://www.ncbi.nlm.nih.gov/pubmed/7704569 1.1
[20] H. Y. Kueh, T. J. Mitchison. Structural plasticity in actin and tubulin polymer
dynamics. Science 325(5943), 960 (2009). doi:10.1126/science.1168823.
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2864651&tool=
pmcentrez&rendertype=abstract 1.1, 2.3
[21] G. Margolin, H. V. Goodson, M. S. Alber. Mean-field study of the role of
lateral cracks in microtubule dynamics. Physical Review E 83(4), 041905 (2011).
doi:10.1103/PhysRevE.83.041905.
URL http://link.aps.org/doi/10.1103/PhysRevE.83.041905 1.1
[22] A. Akhmanova, M. Dogterom. Kinesins lead aging microtubules to catastrophe.
Cell 147(5), 966 (2011). doi:10.1016/j.cell.2011.11.011.
URL http://www.ncbi.nlm.nih.gov/pubmed/22118452 1.1, 1.1
[23] A. Dimitrov, M. Quesnoit, S. Moutel, I. Cantaloube, C. Poüs, F. Perez.
Detection of GTP-tubulin conformation in vivo reveals a role for GTP remnants in
microtubule rescues. Science 322(5906), 1353 (2008). doi:10.1126/science.1165401.
URL http://www.ncbi.nlm.nih.gov/pubmed/18927356 1.1
[24] C. Tropini, E. A. Roth, M. Zanic, M. K. Gardner, J. Howard. Islands
containing slowly hydrolyzable GTP analogs promote microtubule rescues. PloS
One 7(1), e30103 (2012). doi:10.1371/journal.pone.0030103.
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=3260198&tool=
pmcentrez&rendertype=abstract 1.1
[25] L. Cassimeris, N. K. Pryer, E. Salmon. Real-Time Observations of Microtubule
Dynamic Instability in Living Cells. The Journal of Cell Biology 107(6), 2223 (198).
doi:10.1083/jcb.107.6.2223.
URL http://jcb.rupress.org/content/107/6/2223 1.5
136 BIBLIOGRAPHY
[26] K. Tsekouras, D. Lacoste, K. Mallick, J. F. Joanny. Condensation of actin
filaments pushing against a barrier. New Journal of Physics 13(10), 103032 (2011).
doi:10.1088/1367-2630/13/10/103032.
URL http://stacks.iop.org/1367-2630/13/i=10/a=103032?key=crossref.
4e3dd120185e7373ff9a6810fea08277 1.1, 10
[27] X. Li, J. Kierfeld, R. Lipowsky. Actin Polymerization and Depolymerization
Coupled to Cooperative Hydrolysis. Physical Review Letters 103(4), 1 (2009). doi:
10.1103/PhysRevLett.103.048102.
URL http://link.aps.org/doi/10.1103/PhysRevLett.103.048102 1.1, 2.3
[28] P. A. Curmi, S. S. L. Andersen, S. Lachkar, O. Gavet, E. Karsenti,
M. Knossow, A. Sobel. The stathmin/tubulin interaction in vitro. The Journal
of Biological Chemistry 272(40), 25029 (1997). doi:10.1074/jbc.272.40.25029.
URL http://www.ncbi.nlm.nih.gov/pubmed/9312110 1.1, 2.1, 4.3.2.3
[29] M. K. Gardner, M. Zanic, C. Gell, V. Bormuth, J. Howard. Depolymeriz-
ing kinesins Kip3 and MCAK shape cellular microtubule architecture by differential
control of catastrophe. Cell 147(5), 1092 (2011). doi:10.1016/j.cell.2011.10.037.
URL http://www.ncbi.nlm.nih.gov/pubmed/22118464 1.1, 2.3
[30] P. T. Tran, L. Marsh, V. Doye, S. Inoué, F. Chang. A mechanism for
nuclear positioning in fission yeast based on microtubule pushing. The Journal of
Cell Biology 153(2), 397 (2001). doi:10.1083/jcb.153.2.397.
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2169469&tool=
pmcentrez&rendertype=abstract 1.2
[31] D. R. Drummond, R. A. Cross. Dynamics of interphase microtubules in
Schizosaccharomyces pombe. Current Biology 10(13), 766 (2000). doi:10.1016/
S0960-9822(00)00570-4.
URL http://www.sciencedirect.com/science/article/pii/S0960982200005704 1.2
[32] R. R. Daga, K. G. Lee, S. Bratman, S. Salas-Pino, F. Chang. Self-
organization of microtubule bundles in anucleate fission yeast cells. Nature Cell
Biology 8(10), 1108 (2006). doi:10.1038/ncb1480.
URL http://www.ncbi.nlm.nih.gov/pubmed/16998476 1.2
[33] M. E. Janson, R. Loughlin, I. Loïodice, F. Chuanhai, D. Brunner, F. J.
Nédélec, P. T. Tran. Crosslinkers and Motors Organize Dynamic Microtubules
to Form Stable Bipolar Arrays in Fission Yeast. Cell 128(2), 357 (2007). doi:
10.1016/j.cell.2006.12.030.
URL http://www.sciencedirect.com/science/article/pii/S0092867407000487 1.2
BIBLIOGRAPHY 137
[34] R. R. Daga, A. Yonetani, F. Chang. Asymmetric microtubule pushing forces
in nuclear centering. Current Biology 16(15), 1544 (2006). doi:10.1016/j.cub.2006.
06.026.
URL http://www.ncbi.nlm.nih.gov/pubmed/16890530 1.2, 4.1
[35] L. Sacconi, I. M. Tolić-Nørrelykke, C. Stringari, R. Antolini, F. S.
Pavone. Optical micromanipulations inside yeast cells. Applied Optics 44(11), 2001
(2005). doi:10.1364/AO.44.002001.
URL http://www.opticsinfobase.org/ao/abstract.cfm?uri=AO-44-11-2001 1.2
[36] C. Tischer, D. Brunner, M. Dogterom. Force- and kinesin-8-dependent effects
in the spatial regulation of fission yeast microtubule dynamics. Molecular Systems
Biology 5(250), 250 (2009). doi:10.1038/msb.2009.5.
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2671921&tool=
pmcentrez&rendertype=abstract 1.2
[37] D. Foethke, T. Makushok, D. Brunner, F. J. Nédélec. Force- and length-
dependent catastrophe activities explain interphase microtubule organization in fis-
sion yeast. Molecular Systems Biology 5(241), 241 (2009). doi:10.1038/msb.2008.76.
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2671915&tool=
pmcentrez&rendertype=abstract 1.2, 1.7
[38] D. K. Fygenson, J. F. Marko, A. Libchaber. Mechanics of Microtubule-
Based Membrane Extension. Physical Review Letters 79(22), 4497 (1997). doi:
10.1103/PhysRevLett.79.4497.
URL http://link.aps.org/doi/10.1103/PhysRevLett.79.4497 1.3, 1.8
[39] L. Dehmelt, F. M. Smart, R. S. Ozer, S. Halpain. The role of microtubule-
associated protein 2c in the reorganization of microtubules and lamellipodia during
neurite initiation. The Journal of Neuroscience 23(29), 9479 (2003).
URL http://www.ncbi.nlm.nih.gov/pubmed/14573527 1.3
[40] K. Franze, J. Guck. The biophysics of neuronal growth. Reports on Progress in
Physics 73(9), 094601 (2010). doi:10.1088/0034-4885/73/9/094601.
URL http://stacks.iop.org/0034-4885/73/i=9/a=094601?key=crossref.
1e6c2d7988b5dab945ccce0f5a6ef772 1.3
[41] J. R. Moffitt, Y. R. Chemla, S. B. Smith, C. Bustamante. Recent advances
in optical tweezers. Annual Review of Biochemistry 77, 205 (2008). doi:10.1146/
annurev.biochem.77.043007.090225.
URL http://www.ncbi.nlm.nih.gov/pubmed/18307407 1.4
138 BIBLIOGRAPHY
[42] K. C. Neuman, S. M. Block. Optical trapping. The Review of Scientific
Instruments 75(9), 2787 (2004). doi:10.1063/1.1785844.
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1523313&tool=
pmcentrez&rendertype=abstract 1.4
[43] K. C. Neuman, A. Nagy. Single-molecule force spectroscopy: optical tweezers,
magnetic tweezers and atomic force microscopy. Nature Methods 5(6), 491 (2008).
doi:10.1038/NMETH.1218.
URL http://www.nature.com/nmeth/journal/v5/n6/full/nmeth.1218.html 1.4
[44] T. T. Perkins. Optical traps for single molecule biophysics: a primer. Laser &
Phontonics Reviews 3(1-2), 203 (2009). doi:10.1002/lpor.200810014.
URL http://jila.colorado.edu/perkins/publications/
optical-traps-single-molecule-biophysics-primer 1.4
[45] A. Ashkin. Obervation of a single-beam gradient force optical trap for dielectric
particles. Optics Letters 11(5), 288 (19862). doi:10.1364/OL.11.000288.
URL http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-11-5-288 1.4
[46] A. Ashkin. Forces of a single-beam gradient laser trap on a dielectric sphere
in the ray optics regime. Biophysical Journal 61(2), 569 (1992). doi:10.1016/
S0006-3495(92)81860-X.
URL http://linkinghub.elsevier.com/retrieve/pii/S000634959281860X 1.4
[47] G. J. Brouhard, H. T. Schek, A. J. Hunt. Advanced optical tweezers for the
study of cellular and molecular biomechanics. Transactions on Biomedical Engineer-
ing 50(1), 121 (2003). doi:0.1109/TBME.2002.805463.
URL http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1179141 1.4, 2.2
[48] K. Visscher, S. P. Gross, S. M. Block. Construction of multiple-beam opti-
cal traps with nanometer-resolution position sensing. Journal of Selected Topics in
Quantum Electronics 2(4), 1066 (1996). doi:10.1109/2944.577338.
URL http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=577338 1.4, 2.2
[49] S. C. Kuo, M. P. Sheetz. Force of single kinesin molecules measured with optical
tweezers. Science 260(5105), 232 (1993). doi:10.1126/science.8469975.
URL http://www.ncbi.nlm.nih.gov/pubmed/20151444 1.4
[50] J. D. Wen, M. Manosas, P. T. X. Li, S. B. Smith, C. Bustamante,
F. Ritort, I. Tinoco. Force unfolding kinetics of RNA using optical tweezers. I.
Effects of experimental variables on measured results. Biophysical Journal 92(9),
2996 (2007). doi:10.1529/biophysj.106.094052.
BIBLIOGRAPHY 139
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1852336&tool=
pmcentrez&rendertype=abstract 1.4
[51] M. J. Footer, J. W. J. Kerssemakers, J. A. Theriot, M. Dogterom.
Direct measurement of force generation by actin filament polymerization using an
optical trap. Proceedings of the National Academy of Sciences of the United States
of America 104(7), 2181 (2007). doi:10.1073/pnas.0607052104.
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1892916&tool=
pmcentrez&rendertype=abstract 1.4, 10
[52] J. W. J. Kerssemakers, E. L. Munteanu, L. Laan, T. L. Noetzel, M. E.
Janson, M. Dogterom. Assembly dynamics of microtubules at molecular resolu-
tion. Nature 442(7103), 709 (2006). doi:10.1038/nature04928.
URL http://www.ncbi.nlm.nih.gov/pubmed/16799566 1.4, 2.2
[53] L. Laan, J. Husson, E. L. Munteanu, J. W. J. Kerssemakers,
M. Dogterom. Force-generation and dynamic instability of microtubule bundles.
Proceedings of the National Academy of Sciences of the United States of America
105(26), 8920 (2008). doi:10.1073/pnas.0710311105.
URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2449340/ 1.4, 1.9, 2.2, 2.2, 4.3,
4.3.1, 6.1, 8, 8.1, 8.1, 8.2, 8.2.1, 9, A
[54] B. Zelinski, J. Kierfeld. Cooperative dynamics of microtubule ensembles: Poly-
merization forces and rescue-induced oscillations. Physical Review E 87, 012703
(2013). doi:10.1103/PhysRevE.87.012703.
URL http://link.aps.org/doi/10.1103/PhysRevE.87.012703 1.4, 1, 1, 1
[55] B. Zelinski, N. Müller, J. Kierfeld. Dynamics and length distribution of
microtubules under force and confinement. Physical Review E 86, 041918 (2012).
doi:10.1103/PhysRevE.86.041918.
URL http://link.aps.org/doi/10.1103/PhysRevE.86.041918 1, 1, 1, 1, 8.2.3, 1
[56] F. Verde, M. Dogterom, E. Stelzer, E. Karsenti, S. Leibler. Control
of microtubule dynamics and length by cyclin A- and cyclin B-dependent kinases
in Xenopus egg extracts. The Journal of Cell Biology 118(5), 1097 (1992). doi:
10.1083/jcb.118.5.1097.
URL http://www.jstor.org/stable/1615232 2.1, 2.1, 2.1, 2.1, 4.1, 4.2
[57] M. Dogterom, S. Leibler. Physical aspects of the growth and regulation of
microtubule structures. Physical Review Letters 70(9), 1347 (1993). doi:10.1103/
PhysRevLett.70.1347.
URL http://link.aps.org/doi/10.1103/PhysRevLett.70.1347 2.1, 2.1, 2.1, 4.2
140 BIBLIOGRAPHY
[58] B. Mulder. Microtubules interacting with a boundary: Mean length and mean first-
passage times. Physical Review E 86(1), 1 (2012). doi:10.1103/PhysRevE.86.011902.
URL http://link.aps.org/doi/10.1103/PhysRevE.86.011902 2.1, 4.1
[59] K. Ookata, S. Hisanaga, J. C. Bulinski, H. Murofushi, H. Aizawa, T. J.
Itoh, H. Hotani, E. Okumura, K. Tachibana, T. Kishimoto. Cyclin B
Interaction with Microtubule-associated Protein 4 (MAP4) Targets p34cdc2 Kinase
to Micrombules and Is a Potential Regulator of M-phase Microtubule Dynamics.
The Journal of Cell Biology 128(5), 849 (1995). doi:10.1083/jcb.128.5.849.
URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2120387/?tool=
pmcentrez&rendertype=abstract 2.1
[60] M. E. Janson, M. Dogterom. Scaling of Microtubule Force-Velocity Curves Ob-
tained at Different Tubulin Concentrations. Physical Review Letters 92(24), 248101
(2004). doi:10.1103/PhysRevLett.92.248101.
URL http://link.aps.org/doi/10.1103/PhysRevLett.92.248101 2.2, A, A.1
[61] C. S. Peskin, G. M. Odell, G. F. Oster. Cellular motions and thermal
fluctuations: the Brownian ratchet. Biophysical Journal 65(1), 316 (1993). doi:
10.1016/S0006-3495(93)81035-X.
URL http://linkinghub.elsevier.com/retrieve/pii/S000634959381035X 2.2, 2.2
[62] N. G. van Kampen. Stochastic Processes in Physics and Chemistry. Elsevier,
Amsterdam (2007). 2.2, 8.2.5, 8.2.6, 8.2.6, 8.2.6
[63] C. Gardiner. Stochastic Methods. Springer, Heidelberg (2009). 2.2, 8.2.6
[64] A. Mogilner, G. Oster. The polymerization ratchet model explains the force-
velocity relation for growing microtubules. European Biophysics Journal 28(3), 235
(1999). doi:10.1007/s002490050204.
URL http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/
s002490050204 2.2
[65] A. E. Carlsson. Force-velocity relation for growing biopolymers. Physical Review
E 62(5), 7082 (2000). doi:10.1103/PhysRevE.62.7082.
URL http://www.ncbi.nlm.nih.gov/pubmed/11102065 2.2
[66] J. Krawczyk, J. Kierfeld. Stall force of polymerizing microtubules and filament
bundles. Europhysics Letters 93(2), 28006 (2011). doi:10.1209/0295-5075/93/28006.
URL http://stacks.iop.org/0295-5075/93/i=2/a=28006 2.2, 8.2.3
[67] A. B. Kolomeisky, M. E. Fisher. Force-Velocity Relation for Growing Micro-
tubules. Biophysical Journal 80(1), 149 (2001).
URL http://dx.doi.org/10.1016/S0006-3495(01)76002-X 2.2
BIBLIOGRAPHY 141
[68] H. Flyvbjerg, T. E. Holy, S. Leibler. Stochastic Dynamics of Microtubules:
A Model for Caps and Catastrophes. Physical Review Letters 73(17), 2372 (1994).
doi:10.1103/PhysRevLett.73.2372.
URL http://link.aps.org/doi/10.1103/PhysRevLett.73.2372 2.3, 2.3, 2.3, 8.2.4
[69] L. Brun, B. Rupp, J. J. Ward, F. Nédélec. A theory of microtubule catastro-
phes and their regulation. Proceedings of the National Academy of Sciences of the
United States of America 106(50), 21173 (2009). doi:10.1073/pnas.0910774106.
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2795527&tool=
pmcentrez&rendertype=abstract 2.3
[70] H. Bowne-Anderson, M. Zanicy, M. Kauer, J. Howard. Microtubule dy-
namic instability: A new model with coupled GTP hydrolysis and multistep catas-
trophe. BioEssays 35(6), 579 (2013). doi:10.1002/bies.201370066.
URL http://www.ncbi.nlm.nih.gov/pubmed/23532586 2.3
[71] X. Li, R. Lipowsky, J. Kierfeld. Coupling of actin hydrolysis and polymer-
ization: Reduced description with two nucleotide states. Europhysics Letters 89(3)
(2010). doi:10.1209/0295-5075/89/38010.
URL http://stacks.iop.org/0295-5075/89/i=3/a=38010 2.3
[72] F. Schwabl. Quantenmechanik. Springer, Heidelberg (2005). 2.3
[73] M. Abramowitz, A. I. Stegun. Handbook of Mathematical Functions. Dover
Publications, Inc., New York (1965). 2.3
[74] O. Vallée, M. Soares. Airy Functions and Applications to Physics. Imperial
College Press, London (2004). 2.3, 2.3
[75] M. E. Janson, M. E. de Dood, M. Dogterom. Dynamic instability of
microtubules is regulated by force. The Journal of Cell Biology 161(6), 1029 (2003).
doi:10.1083/jcb.200301147.
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2173003&tool=
pmcentrez&rendertype=abstract 2.3, 2.3, 2.3, 2.3, 8.2.4
[76] W. H. Press, S. A. Teukolsky, S. A. Teukolsky, B. Flannery. Numerical
Recipes 3rd Edition. Cambridge University Press, Cambridge (2007). 3.1
[77] T. E. Holy, S. Leibler. Dynamic instability of microtubules as an efficient way
to search in space. Proceedings of the National Academy of Sciences of the United
States of America 91(12), 5682 (1994). doi:10.1073/pnas.91.12.5682.
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=44060&tool=
pmcentrez&rendertype=abstract 4.1
142 BIBLIOGRAPHY
[78] M. Dogterom, B. Yurke. Microtubule Dynamics and the Positioning of Mi-
crotubule Organizing Centers. Physical Review Letters 81(2), 485 (1998). doi:
10.1103/PhysRevLett.81.485.
URL http://link.aps.org/doi/10.1103/PhysRevLett.81.485 4.1
[79] B. Govindan, W. Spillman. Steady states of a microtubule assembly in a confined
geometry. Physical Review E 70(3), 1 (2004). doi:10.1103/PhysRevE.70.032901.
URL http://link.aps.org/doi/10.1103/PhysRevE.70.032901 4.1, 4.1
[80] Y. A. Komarova, I. A. Vorobjev, G. G. Borisy. Life cycle of MTs: persistent
growth in the cell interior, asymmetric transition frequencies and effects of the cell
boundary. Journal of Cell Science 115(Pt 17), 3527 (2002). doi:10.3410/f.1008394.
105507.
URL http://www.ncbi.nlm.nih.gov/pubmed/12154083 4.1
[81] C. Faivre-Moskalenko, M. Dogterom. Dynamics of microtubule asters in
microfabricated chambers: the role of catastrophes. Proceedings of the National
Academy of Sciences of the United States of America 99(26), 16788 (2002). doi:
10.1073/pnas.252407099.
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=139222&tool=
pmcentrez&rendertype=abstract 4.1
[82] N. Müller. Polymerisationskinetik von Mikrotubuli unter Kraft. Bachelor thesis,
Technische Universität Dortmund (2011). 2
[83] C. Tischer, P. R. Ten Wolde, M. Dogterom. Providing positional informa-
tion with active transport on dynamic microtubules. Biophysical Journal 99(3), 726
(2010). doi:10.1016/j.bpj.2010.05.026.
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2913178&tool=
pmcentrez&rendertype=abstract 4.3.2.1
[84] R. A. Walker, N. K. Pryer, E. D. Salmon. Dilution of Individual Microtubules
Observed in Real Time. In Vitro : Evidence That Cap Size Is Small and Independent
of Elongation Rate. The Journal of Cell Biology 114(1), 73 (1991). doi:10.1083/jcb.
114.1.73.
URL http://jcb.rupress.org/content/114/1/73 4.3.2.3
[85] R. M. Corless, G. H. G. D. E. G. Hare, D. J. Jeffrey, D. E. Knuth. On
the Lambert W function. Advances in Computational Mathematics 5(4), 329 (1996).
doi:10.1007/BF02124750.
URL https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf 8.1
BIBLIOGRAPHY 143
[86] M. Dogterom, B. Yurke. Measurement of the Force-Velocity Relation for Grow-
ing Microtubules. Science 278(5339), 856 (1997). doi:10.1126/science.278.5339.856.
URL http://www.sciencemag.org/cgi/doi/10.1126/science.278.5339.856 8.1
[87] T. Erdmann, U. S. Schwarz. Stability of Adhesion Clusters under Constant Force.
Physical Review Letters 92(10), 12 (2004). doi:10.1103/PhysRevLett.92.108102.
URL http://link.aps.org/doi/10.1103/PhysRevLett.92.108102 8.2.1
[88] T. Erdmann, U. S. Schwarz. Stochastic dynamics of adhesion clusters under
shared constant force and with rebinding. The Journal of Chemical Physics 121(18),
8997 (2004). doi:10.1063/1.1805496.
URL http://www.ncbi.nlm.nih.gov/pubmed/15527366 8.2.1
[89] M. F. Carlier. Synchronous Oscillations in Microtubule Polymerization. Proceed-
ings of the National Academy of Sciences of the United States of America 84(15),
5257 (1987). doi:10.1073/pnas.84.15.5257.
URL http://www.pnas.org/cgi/doi/10.1073/pnas.84.15.5257 8.2.3
[90] A. Marx, E. Mandelkow. A model of microtubule oszillation. European Biophysis
Journal 22(6), 405 (1994). doi:10.1007/BF00180162.
URL http://www.ncbi.nlm.nih.gov/pubmed/8149923 8.2.3
[91] S. H. Strogatz. Nonlinear Dynamics and Chaos. Westview Press, Boulder, Col-
orado (2000). 8.2.4
[92] R. Picone, X. Ren, K. D. Ivanovitch, J. D. W. Clarke, R. A. McK-
endry, B. Baum. A polarised population of dynamic microtubules mediates
homeostatic length control in animal cells. PLoS Biology 8(11), e1000542 (2010).
doi:10.1371/journal.pbio.1000542.
URL http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=2982804&tool=
pmcentrez&rendertype=abstract 9
[93] S. E. Kassar. Regulation and filament growth in the cytoskeleton. Diploma thesis,
Technische Universität Dortmund (2013). 10
[94] M. Zeitz. Feedback mechanism for microtubule polymerization regulation. Diploma
thesis, Technische Universität Dortmund (2013). 10
[95] D. N. Drechsel, A. A. Hyman, M. H. Cobb, M. W. Kirschner. Modulation
of the dynamic instability of tubulin assembly by the microtubule-associated protein
tau. Molecular Biology of the Cell 3(10), 1141 (1992).
URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC275678/ A
144 BIBLIOGRAPHY
[96] R. F. Gildersleeve, A. R. Cross, K. E. Cullen, A. P. Fagen, R. C.
Williams. Microtubules grow and shorten at intrinsically variable rates. The Jour-
nal of Biological Chemistry 267(12), 7995 (1992).
URL http://www.ncbi.nlm.nih.gov/pubmed/1569058 A
[97] R. Dhamodharan, P. Wadsworth. Modulation of microtubule dynamic instabil-
ity in vivo by brain microtubule associated proteins. Journal of Cell Science 108(4),
1679 (1995).
URL http://www.ncbi.nlm.nih.gov/pubmed/7615685 A
[98] C. Nakao, T. J. Itoh, H. Hotani, N. Mori. Modulation of the stathmin-like
microtubule destabilizing activity of RB3, a neuron-specific member of the SCG10
family, by its N-terminal domain. The Journal of Biological Chemistry 279(22),
23014 (2004). doi:10.1074/jbc.M313693200.
URL http://www.ncbi.nlm.nih.gov/pubmed/15039434 A
[99] E. Shelden, P. Wadsworth. Observation and quantification of individual micro-
tubule behavior in vivo: microtubule dynamics are cell-type specific. The Journal of
Cell Biology 120(4), 935 (1993). doi:10.1083/jcb.120.4.935.
URL http://jcb.rupress.org/content/120/4/935 A
Teilpublikationen
• B. Zelinski, N. Müller and J. Kierfeld (2012).
Dynamics and length distribution of microtubules under force and confinement.
Physical Review E 86, 041918, arXiv:1212.3583
• B. Zelinski and J. Kierfeld (2013).
Cooperative dynamics of microtubule ensembles: Polymerization forces and rescue-
induced oscillations.
Physical Review E 87, 012703, arXiv:1301.1155
146 BIBLIOGRAPHY
Danksagung
An dieser Stelle möchte ich mich herzlich bei all denen bedanken, die zum Gelingen
dieser Arbeit beigetragen haben.
An erster Stelle möchte ich mich bei Prof. Jan Kierfeld für die Vergabe des
interessanten Themas und für die hervorragende Betreuung bedanken. Die vielen offen
geführten Diskussionen und Anregungen haben maßgeblich zum Gelingen dieser Arbeit
beigetragen. Ich habe mich in seiner Arbeitsgruppe stets wohlgefühlt und sehr viel gelernt.
Des Weiteren bedanke ich mich bei allen ehemaligen und aktuellen Mitgliedern der
Arbeitsgruppe. Sie hatten immer ein offenes Ohr für Fragen und Probleme, forderten
aber auch Ihren Teil meiner Zeit und Aufmerksamkeit.
Dr. Kai Schmidt, Prof. Dr. Ulrich Schwarz und Dr. Carsten Raas danke ich für die
Begutachtung meiner Arbeit. Großer Dank gebührt auch allen Mitarbeitern von T1 und
T2 für die angenehme und entspannte Arbeitsatmosphäre.
Vielen Dank auch an meine WG-Mitbewohner Tim Fischer, Felix Gorschlüter und Stefan
Schmidt. Es ist immer schön nach Hause zu kommen.
Im Besonderen möchte ich mich bei meinen Eltern und meinem Bruder bedanken, die
mich zu jeder Zeit und in jeglicher Hinsicht unterstüzt haben.