\chapter{Simulations of Peptide Aggregation in Water}
\label{ch:bulkphase}
Molecular simulations can be an efficient tool to characterize  aggregation and conformational changes of biomolecules in water at a molecular level. However, any simulation study is unavoidably affected by the finite size of the simulated system. Therefore, the evolution of system properties towards those encountered in the macroscopic limit should be properly understood and taken into account. The possible occurrence of phase transitions in the system studied must also be taken into account by choosing an appropriate ensemble. Nowadays, simulation studies of aggregation of biomolecules can be performed in simple ensembles, such as those with constant volume or constant pressure. Typically, these simulations are performed under conditions of strong oversaturation, i.e., deeply inside the two-phase region, where the system undergoes demixing. When a finite size system is in the two-phase region, the behavior of the minor phase noticeably differs from that encountered in the corresponding macroscopic system. \cite{Furukawa:82a,Binder:03a,Macdowell:04a,Macdowell:06a,Nussbaumer:06a,Nussbaumer:08a} For example, in a macroscopic one-component fluid, the oversaturated vapor separates into two coexisting phases, the liquid and the vapor phase, such as a liquid droplet surrounded by saturated vapor. However, when the fluid volume is finite, a new stable state of the system emerges at the same density: an evaporated (at least partially) droplet surrounded by oversaturated vapor. The state with the droplet and the state without the droplet are both stable and replace each other with time. The occurrence probability of these states is determined by the level of oversaturation and by the system size. An increase of the system size eventually stabilizes the droplet state, which is the only stable state in the macroscopic limit. A state without droplet is just an artificial state, whose properties have no relation to those expected in the macroscopic limit. A similar behavior may be expected for an aqueous solution of peptides in small volumes at concentrations exceeding  solubility limits.
\par 
To study the effect of the finite system size on the aggregation of peptides, we chose aqueous solutions of amyloidogenic peptides, which are highly insoluble in water. Aggregation and subsequent amyloid formation is a central phenomenon in a number of diseases, such as Alzheimer's, Parkinson's and type II Diabetes Mellitus, and seems to be a key factor in the development of the symptoms of these diseases.\cite{Chiti:06a} Upon formation of amyloid fibrils, protein molecules adopt ordered, stacked cross-$\beta$-sheet structures.\cite{Harper1997,Jansen:05a,Chiti:06a,Grudzielanek:06a} It has been discovered that certain short sequence fragments contained within a respective protein can form amyloids even in isolated forms. Many fragments of A$\beta$ and hIAPP,\cite{Balbach2000,Westermark1990,Tenidis2000,Azriel2001,Mazor2002} fragments consisting of residues 10-20 and 105-115 of the protein transthyretin,\cite{Gustavsson1991} residues 7-13 of the yeast protein, Sup35, are few such examples.\cite{Balbirnie2001} Most of these amyloidogenic peptides have also been shown to be cytotoxic. This makes them particularly useful for studies of amyloid formation and elongation. However, the fibrils formed by short peptide fragments may or may not have the same fibril morphology and the same cytotoxicity as those of the parent protein. Nevertheless, studies of such short fragments may lead to understanding of fundamental processes taking place in amyloid formation in large systems. Short peptide sequences also provide a major technical advantage in MD simulations, as the size of the system containing small fragments will be much smaller than if the whole protein were to be simulated. Aqueous solutions of  the amyloidogenic fragment of hIAPP corresponding to residues 15 to 19 were chosen for the present studies. It is one of the shortest peptide fragments capable of self-assembly. Experimental studies have reported that the aqueous solution of the fragment forms broad ribbon-like fibrils, with a short lag time.\cite{Mazor2002} All the aspects mentioned above make this fragment an ideal candidate for studying protein aggregation by using molecular simulations.
\begin{figure}[hbt]
\begin{center}
\includegraphics[scale=0.72]{seq_iapp}
\caption{Sequences of human and rat IAPP. Conserved mutations are colored in blue and nonconserved mutations in red. The sequence of the simulated fragment is depicted by the yellow box. }
\label{fig:iapp_seq}
\end{center}
\end{figure}
%*******************************************
\section{System setup}
\label{bulksetup}
All atomic MD simulations were performed using Amber99s (C. Simmerling’s modifications of $\phi$ and $\psi$ dihedral parameters of the original Amber99).\cite{Simmerling2002} The N- and C-termini of the peptide fragments were capped with acetyl and N-methylamine groups, respectively. All starting configurations were solvated with the TIP3P water model and the simulations were carried out in an isothermic-isobaric ensemble and under periodic boundary conditions, at 330 K and 1 atm pressure, using GROMACS software package.\cite{Lindahl2001} The Particle Mesh Ewald (PME) method was used to treat long range electrostatic interactions.\cite{Essmann1995} SHAKE was applied to constrain all the bonds to their equilibrium bond lengths allowing a time step of 2.0 fs. Initial equilibration was carried out using Berendsen's thermostat for 1 ns, followed by the production run of various lengths using Nose-Hoover temperature coupling and Parrinello-Rahman pressure coupling. The number $N_p$ of peptides in the simulation box varied from 1 to 56 and the number $N_w$ of water molecules varied from 591 to 29702. A total number of 12 peptide-water systems were studied (see \ref{tab:conc}). The peptide weight concentration $C$ is effectively equal to zero in the case of a single peptide in a simulation box, representing an infinite dilution, and varied from $\sim$ 2.5 to 42 $\%$ in systems with $N_p>$ 1. Accordingly, the lateral size $L$ of the cubic simulation box was in the range of 2.7 to 9.9 nm.
\par
Initial configurations of these systems were prepared by random insertion of peptides in a cubic box such that the shortest peptide-peptide distance exceeds a certain minimal value, which depends on concentration. 1 to 10 different initial configurations were used for different systems. Equilibration periods, estimated from the time evolution of various system parameters, varied from 10 to 25 ns, depending on the system size and peptide concentration. Configurations were saved every 2 to 5 ps and equilibration periods were excluded from the analysis.
%
\begin{table}[hbt]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$N_p$ & $N(sim)$ & $box(\textmd{\AA})$ & $N_w$ & $C_w\%$ & $Time(ns)$ \\ \hline 
1 & 10 & 26.9 & 591 & 5.8 & 300 \\ \hline
3 & 5 & 37.4 & 1590 & 6.4 & 250 \\ \hline
3 & 5 & 29.7 & 750 & 12.7 & 250 \\ \hline
6 & 1 & 65.0 & 8600 & 2.5 & 100 \\ \hline
6 & 3 & 48.8 & 3536 & 5.8 & 300 \\ \hline
6 & 3 & 37.2 & 1480 & 12.9 & 600 \\ \hline
6 & 3 & 29.2 & 626 & 25.9 & 300 \\ \hline
12 & 3 & 59.7 & 6448 & 6.4 & 450 \\ \hline
12 & 6 & 47.5 & 3079 & 12.4 & 750 \\ \hline
12 & 3 & 36.7 & 1232 & 26.2 & 150 \\ \hline
12 & 1 & 30.8 & 586 & 42.7 & 160 \\ \hline
56 & 1 & 99.3 & 29702 & 6.4 & 145 \\  \hline
\end{tabular}
\end{center}
\caption{\label{tab:conc}Table summarizing the simulations carried out depicting the following columns: number of peptides in the simulation box $N_p$, number of simulations $N(sim)$, averaged box size $box$,  average number of water molecules in the simulations $N_w$, concentration of protein in weight percent $C_w\%$, total duration of simulation runs in nanoseconds $Time(ns)$.} 
\end{table}
%
\par
For the analysis of peptide clustering based on the number of hydrophobic contacts as a criterion for the connectivity between two peptides, a hydrophobic contact was defined as follows: one hydrophobic contact between two peptides exists, when the distance between two sidechain carbon atoms involved (sidechains of F, L and V) does not exceed the sum of their van der Waals radii plus 2.8 $\mbox{\AA}$. The radius of gyration $R_g$ of the largest peptide cluster, as well as its maximal extension $L_{max}$, measured as the maximal distance between two heavy atoms of peptides in the cluster were also calculated. The secondary structures were determined using the corresponding distributions of  $\phi$ and $\psi$ dihedral angles in the Ramachandran plot. A residue was considered to contribute to $\alpha$-helices, when -110$^{\circ}\le\phi\le-30^{\circ}$ and -90$^{\circ}\le\psi\le-10^{\circ}$; to $\beta$-sheets, when -180$^{\circ}\le\phi\le-100^{\circ}$ and 60$^{\circ}\le\psi\le180^{\circ}$; to polyproline II structures, when -100$^{\circ}\le\phi\le-30^{\circ}$ and 60$^{\circ}\le\psi\le180^{\circ}$. The solvent accessible surface area (SASA) of all peptides was obtained with a probe radius of 1.4 $\mbox{\AA}$. Water molecules were considered as belonging to the hydration shell of the peptides, if the shortest distance between a water oxygen atom and at least one of the heavy atoms of the peptides, did not exceed 4.5 $\mbox{\AA}$. The volume of the surface layer of water can be approximated to be $V_h=SASA \cdot D$. The width $D$ of the water layer is considered to be 0.3 nm. Thus, the density of the hydration shell $\rho_h$ is calculated as $\rho_h =N_h \cdot m_{wat}/V_h$. The peptide-peptide and peptide-water hydrogen bonds (H-bonds) were identified using the following criteria: a H-bond exists if the distance between donor and acceptor did not exceed 0.35 nm and if the donor-hydrogen-acceptor angle exceeds 120$^{\circ}$.
%\subsection{Fragment B and C}
%
\section{Characterization of aggregation}
The snapshots of the largest simulated system ($N_p=56$) are shown in Figure \ref{fig:p56snap}. Within the first 0.05 ns, the randomly oriented peptides in the initial configuration form well distinguished clusters. These clusters grow and merge with time (see snapshots at $t$ = 1 ns and $t$ = 2.5 ns). Two large droplets can be seen at $t$ = 10 ns, which later merge to form a single droplet at $t$ = 25 ns. The simulation was carried out at a concentration manyfold higher than the expected critical concentration. It is expected that all the peptides should belong to the peptide aggregate. However,  the system exhibits two stable states, which interchange with time: state A, where all the peptides are in one large cluster (lower-middle snapshot in Figure \ref{fig:p56snap}) and state B, where few peptides break up from the large cluster (lower-right snapshot in Figure \ref{fig:p56snap}). The behavior of the system can be considered as analogous to that of finite Lennard-Jones fluids at a constant density in the two phase region or finite Ising magnet at constant magnetization, where only the state A is relevant in the macroscopic systems and state B is an artifact of the finite size of the simulation box and is not seen in macroscopic systems. As the system size increases, the effect of state B on the system properties diminishes and vice versa. The snapshots of the simulation at the same peptide concentration, $C_w \approx 6\%$, but with only 6 peptides, are shown in Figure \ref{fig:rgyr6}. Visual comparison with Figure \ref{fig:p56snap} shows that the degree of peptide aggregation is lower in small systems, as a relatively large fraction of the total aggregate repeatedly breaks up.  However, the systematic and quantitative study of the effects of concentration and system size on peptide aggregation can only be performed by introducing parameters reflecting the degree of clustering in simulations. 
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=10cm]{p56snap}
\caption{Time evolution of peptide clustering in the system with $N_p$ = 56 and $C \approx$ 6$\%$. After the nucleation process ($t <$ 25 ns), the system exhibits two stable states (lower-middle and lower-right snapshots): all peptides are in one cluster (lower-middle snapshot); a few small clusters are separated from the large peptide cluster (lower-right snapshot). Backbones and side chains are shown in white and black, respectively.}
\label{fig:p56snap}
\end{center}
\end{figure}
%
\subsection{Radius of gyration}
The radius of gyration ($R_g$) of all peptides can be used to characterize aggregation; however, in MD simulations of several peptide fragments with periodic boundary conditions, the radius of gyration is not well defined (see Figure \ref{fig:rgdef}). 
%
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=8cm]{rg_2d}
\caption{Figure depicting three particles in a box surrounded by its images. Depending on the selection of the images of particles, as depicted by colored ellipses, distinct radii of gyration can be obtained.}
\label{fig:rgdef}
\end{center}
\end{figure}
%
The following approach was used to calculate the radius of gyration: one of the peptides was selected as a reference and the box was translated in such a way that the center of mass of the reference peptide coincides with the center of the box. The radius of gyration of all the peptides based on the $i^{th}$ reference peptide can be calculated as:
\begin{equation}
\label{eq:rg1}
Rg_{i}=\sqrt{\frac{\sum\limits_{j=1}^Nm_j\left(\mathbf{Ra}_{i}-\mathbf{r}_j\right)^2}{\sum\limits_{j=1}^N m_j}} ,
 \end{equation}
  where $N$ is the total number of atoms, $m_j$ is the mass of atom $j$, $\mathbf{Ra}_{i}$ is the center of mass of all peptides based on the $i^{th}$ reference peptide, calculated form the following equation:
\begin{equation}
\label{eq:rg2}
\mathbf{Ra}_{i}=\frac{1}{J}\sum\limits_{j=1}^J\mathbf{R}_{j}.
 \end{equation}
$J$ is the total number of peptide fragments and $\mathbf{R}_j$ is the center of mass of a peptide fragment $j$. Note that the center of mass of reference peptide $i$ is also the center of the box. $Rg_{i}$ is calculated using each peptide fragment as reference and the minimum is reported: $R_g = min(Rg_{i})$, where $i$ is iterated sequentially over all the peptides.

Figure \ref{fig:rgyr6} depicts the time evolution of the radius of gyration for the two systems at the same concentration, calculated by using the above mentioned method for one of the simulations. Two states of the system are clearly visible for the small system, i.e.,  the ``droplet'' state where the total $R_g$ of the peptides is low and all the peptides are together forming one large cluster, and the ``dissolved'' state where the peptides split into smaller clusters with a high total $R_g$ of peptides.  The system switches between these two states throughout the course of the simulation, whereas for the large system  $R_g$ varies  in a relatively narrow range. 
%  
\begin{figure}[htb]
\begin{center}
\includegraphics[width=15cm]{rgyr6}
\caption{Time evolution of peptide clustering in the system with $N_p$ = 6 and $C\approx 6\%$. Upper panel: Snapshots of peptides at various time steps, as indicated in the figure. Backbones and side chains are shown in light grey and black, respectively. Lower panel: Time evolution of the radius of gyration $R_g$ of all peptides.}
\label{fig:rgyr6}
\end{center}
\end{figure}
%
The probability distribution $P(R_g)$ of $R_g$ for the large system is highly symmetric and can be well fitted to a Gaussian function (dashed area in the lower right panel of Figure \ref{fig:rgyr6gaus}). In contrast, $P(R_g)$ for the smaller system is asymmetric (upper right panel of Figure \ref{fig:rgyr6gaus}), with the narrow peak of $\sim$ 1 nm corresponding to a state where all the peptides form a compact aggregate, whereas the wide tail of $P(R_g)$ extending towards large $R_g$ values corresponds to a state without a compact aggregate. To estimate the existence probabilities of these states, the distribution should be decomposed into two constituents. For the large system where the peptide aggregate dominates and the distribution is symmetric, the area under the Gaussian can be taken as the existence probability $R_1$ of a state with peptide aggregate. In the smaller system with 6 peptides, $R_1$ can be approximated by fitting the Gaussian function to the first maxima of $R_g$. 
\par
The characterization of the degree of peptide aggregation by use of the distribution $P(R_g)$ does not require any criterion for the connectivity between peptides. This is the main advantage of the proposed method, as the choice of an adequate connectivity criterion is not  trivial. On the other hand, this approach suffers from a rather
ambiguous extraction of the Gaussian contribution to $P(R_g)$. In particular, it is not applicable in very small systems (with $N_p$ = 3 in our studies), where this contribution to the $P(R_g)$ distribution is indistinguishable.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=15cm]{rgyr6gaus}
\caption{Left panels: Time evolution of the radius of gyration $R_g$ of all peptides for two systems with the same peptide concentration, but a different number of peptides. For the system with $N_p = 6$, three time intervals of 75 ns, each from different simulation runs, are placed successively. Right panels: Probability distribution of $R_g$ for the dependences shown in the left panels (solid points). Fit of the symmetrical part of the probability distribution to a Gaussian is shown by the dashed area.}
\label{fig:rgyr6gaus}
\end{center}
\end{figure}
%
\subsection{Cluster analysis: search for connectivity criteria}
Cluster analysis is the most basic approach for characterizing the arrangement of particles in a system. It is widely used in various fields of statistical physics. A system of particles can be described as an ensemble of clusters. Typically, the definition of clusters is based on a certain criterion for the connectivity between two particles. This criterion can be based on interparticle distance, potential and relative kinetic energies of two particles, the existence of H-bonds, etc. The particular choice of a connectivity criterion depends on the problem considered. In studies of condensation phenomena, two particles are usually considered as belonging to the same cluster, if the distance between them does not exceed a certain critical value $r_{crit}$, comparable to the distance between two particles in the condensed phase. 
\par
When considering aggregation of amyloidogenic peptides, the condensed organic-rich phase appears as fibrillar aggregates. Therefore, the connectivity criterion should be related to the structure of fibrils. There are two characteristic interpeptide distances in the case of fibrils: the interstrand distance between the neighboring peptides in the $\beta$-sheets is $\sim$ 0.5 nm and the intersheet distance is $\sim$ 1.0--1.1 nm. All peptides in the fibril belong to one cluster, if $r_{crit}$ is close to the latter distance. Hence, in the cluster analysis of peptide aggregate formation in water, two peptides can be considered as belonging to the same cluster, when the distance between their centers of mass is $\lesssim$ 1.0--1.1 nm. Additionally, we can use a qualitatively different connectivity criterion, based on the number $n_c$ of hydrophobic contacts between two peptides. However, typical values of $n_c$ in fibrils are not known. The choice of a particular value of $r_{crit}$ or $n_c$ should satisfy well-established aspects of peptide aggregation; in particular, the degree of peptide aggregation must increase with increasing peptide concentration.
\par
The clustering analysis allows distinguishing the largest peptide cluster, which mimics an organic-rich phase of the studied system. Accordingly, various properties of the largest peptide cluster (shape, fractal dimension, density, etc.) can be studied. Additionally, the fraction $f$ of the peptides in the largest cluster can be used as a parameter reflecting the degree of aggregation.
%
\subsubsection{Distance between centers of mass}
To choose the value $r_{crit}$ for the distance between the center of mass of two peptides, which adequately characterizes peptide aggregation, we consider the largest system studied with $N_p$ = 56. In this system, almost all peptides form one large aggregate permanently (see Figure \ref{fig:p56snap}), hence, the existence probability $R$ of the aggregate state is equal to 1. The dependence of the existence probability $R$ to find at least $fN_p$ peptides in the largest cluster of this system for various choices of $r_{crit}$ is shown in Figure \ref{fig:critcom}.
When $r_{crit}$ $<$ 0.9 nm, the fraction $f$ of the peptides in the largest cluster never exceeds $\sim$ 0.2. This  contradicts visual observation and should be considered as unreasonable. When $r_{crit}>$ 1.2 nm, the fraction $f$ is always close to 1. At $R$ = 1, a drastic increase of the fraction $f$ occurs, when $r_{crit}$ changes from 0.9 to 1.1 nm, in agreement with literature values of intersheet distances in fibrils. \cite{Serpell:00a} The presence of the majority of peptides (0.5 $< f <$ 1.0) in the largest cluster can be considered as a signature of the state with peptide aggregate. Even in the largest system studied, $f$ is never equal to 1, since a small fraction of peptides periodically splits from the main aggregate (see Figure \ref{fig:p56snap}). Therefore, any value of $f$, which exceeds 0.5 and is not very close to 1, seems to be reasonable for the definition of the existence probability $R$ of a state with peptide aggregate. In Ising magnets, a fraction of 2/3 of the excess magnetization forms a droplet at the transition point.\cite{Nussbaumer:06a,Nussbaumer:08a} Therefore, we use $f$ = 2/3 for the definition of $R$.
\par
The dependence of the existence probability $R$ defined in such a way on $r_{crit}$ is shown in the upper and middle panels of Figure \ref{fig:r2dist} for systems with $N_p$ = 6 and $N_p$ = 12, respectively, and with several peptide concentrations $C$.
\begin{figure}[hbt]
	\begin{minipage}[t]{8.0cm}
		 \includegraphics[trim= 1.2cm 0.5cm 0cm 0.5cm,clip,height=9cm]{critcom}
		 \caption{Probability $R$ to find $>fN_p$ peptides in the largest peptide cluster at various choices of the distance $r_{crit}$ between the center of mass of two peptides, which is used as a connectivity criterion}
		  \label{fig:critcom}
	\end{minipage}
	\hfill
	\begin{minipage}[t]{6.0cm}
		\includegraphics[trim=0.2cm 0.2cm 0.6cm 0.1cm,clip,height=9cm]{r2dist}
		\caption{Dependence of the aggregation parameter $R_2$ (with $f=2/3$) on the connectivity criterion $r_{crit}$, which is the distance between the center of mass of two peptides.}
		\label{fig:r2dist}
	\end{minipage}
\end{figure}
As expected, $R$ increases, when the connectivity criterion weakens, i.e., ${r}_{crit}$ increases. A physically reasonable choice of the value $r_{crit}$ should provide an increase of $R$ with increasing peptide concentration at   constant number of peptides. This condition is satisfied for $r_{crit}$ $>$ 1.0 nm, which can be considered as the lower limit for $r_{crit}$. The existence of such a lower limit for $r_{crit}$ is not expected for systems of simple isotropic molecules and evidences a multilevel structural organization of peptide aggregates. This can be seen, when systems with approximately the same peptide concentration ($C \approx 6\%$) but with a different number of peptides are compared (lower panel in Figure \ref{fig:r2dist}). When the connectivity criterion applied is stricter ($r_{crit} <$ 1.0 nm), the systems with smaller peptide numbers $N_p$ exhibit increased aggregation in comparison with larger systems. This is due to the fact that in the systems with just a few peptides, the formation of a single $\beta$-sheet with interstrand distance of about 0.5 nm is the main form of aggregation. In contrast, in larger systems, more then one $\beta$-sheet can be formed, and the use of $r_{crit}<$ 1.0 nm artificially breaks the peptide aggregate into separate $\beta$-sheets even in the case of an ideal fibril. For the largest system studied with $N_p$ = 56, the aggregation parameter $R$ shows a pronounced sigmoid-like dependence on $r_{crit}$ with an inflection point at $\sim$ 1.0 nm. Such sigmoid-like dependence, although less steep, is still seen in the system with $N_p$ = 12, but it disappears in the systems  $N_p$ = 6 and 3 (see lower panel in Figure \ref{fig:r2dist}).
\par
The choice of the connectivity criterion in studies of peptide aggregation should be meaningful also in the limit $N_p\rightarrow\infty$, since simulation studies are typically aimed to reproduce the properties of macroscopic systems. Therefore, a connectivity criterion of $r_{crit}>$ 1.0 should be used, keeping in mind that their use overestimates peptide aggregation in small systems. In the largest system studied with $N_p$ = 56, the state with peptide aggregate exists almost permanently with a probability of $R$ = 1, when $r_{crit}$ exceeds 1.1 nm (see lower panel in Figure \ref{fig:r2dist}). Below, we use the connectivity criterion $r_{crit}$ = 1.1 nm for the distance between the centers of mass of two peptides and the minimal fraction $f$ = 2/3 of peptides in the largest cluster, to estimate the existence probability of the state with aggregate which is denoted as the aggregation parameter $R_2$.
%
\subsubsection{Number of hydrophobic contacts}
The connectivity criterion based on the distance between the centers of mass does not take into account  specific interpeptide interactions, such as interpeptide H-bonds or direct contact of the atomic groups of two peptides. For the description of the formation of ordered peptide aggregates, the connectivity criteria which deal with specific interpeptide interactions might be important. As a first step in this direction, we have also analyzed peptide aggregation using the number $n_c$ of hydrophobic contacts between two peptides as connectivity criterion. 
\par
Thus, if we consider the largest system with $N_p = 56$, the dependence of the existence probability $R$ to find at least $fN_p$ peptides in the largest cluster of this system for various choices of number of contacts $n_c$ is shown in Figure \ref{fig:crithd}. The existence probability rapidly approaches 1 as  $n_c$ is lowered from 30 to 20 for the $f$ between 0.4--0.7. The dependence of the existence probability $R$ (with $f$ = 2/3) on peptide concentration is shown in the upper and middle panels of Figure \ref{fig:r3cont} for systems with $N_p = 6$ and $N_p=12$, respectively. With decreasing the number of contacts $n_c$, the probability $R$ approaches 1, as expected. The physically justified enhancement of aggregation with increasing the peptide concentration is observed, when $n_c\leq$ 10. On the other hand, in the system with $N_p$ = 56, the requirement $R$ = 1 is satisfied for $n_c\leq$ 15. We use $\geq$10 hydrophobic contacts between two peptides as a connectivity criterion and the corresponding existence probability of the state with aggregate is denoted as aggregation parameter $R_3$.
\begin{figure}[hbt]
	\begin{minipage}[t]{8.0cm}
		 \includegraphics[trim= 0cm 0.5cm 0cm 0.5cm,clip,height=8cm]{crithd}
		 \caption{Probability $R$ to find $>fN_p$ peptides in the largest peptide cluster at various choices of the number of hydrophobic contacts $n_c$ between the two peptides, used as a connectivity criterion.}
		  \label{fig:crithd}
	\end{minipage}
	\hfill
	\begin{minipage}[t]{6.0cm}
		\includegraphics[trim= 0cm 0.2cm 0.7cm 0.2cm,clip,height=9cm]{r3cont}
		\caption{Dependence of the aggregation parameter $R$ on the number of hydrophobic contacts between two peptides, used as a connectivity criterion.}
		\label{fig:r3cont}
	\end{minipage}
\end{figure}
%
\subsection{Peptide aggregation: concentration and system size}
The dependences of the aggregation parameters $R_1$, $R_2$ and $R_3$ on peptide concentration $C$ are shown in Figure \ref{fig:allRs}. All three aggregation parameters used show a qualitatively similar behavior: the aggregation is fostered with increasing $C$, when the number of peptides is fixed. Although such a dependence was, in fact, imposed by the choice of the connectivity criteria for parameters $R_2$ and $R_3$, this is not the case for the aggregation parameter $R_1$, which characterizes the degree of aggregation without imposing any connectivity criterion. The enhancement of aggregation with increasing peptide concentration is physically obvious and we are not aware of any mechanism, which can decrease the aggregation propensity when the number of peptide is fixed but the amount of solvent decreases. This aspect of peptide aggregation did not get proper attention in simulation studies of aggregation of peptides or other particles in liquid water so far. The effect of concentration on peptide aggregation is illustrated by the dependence of the size $S_{max}$ of the largest peptide cluster on $C$ for the systems with $N_p$ = 6 (Figure \ref{fig:R2C6}). Obviously, $S_{max}$ = 1 upon infinite dilution ($C = 0\%$) and $S_{max}$ = 6 in the absence of solvent ($C = 100\%$). The steepness of the dependence $S_{max}(C)$ is determined by the degree of solubility of the peptides considered. For more soluble peptides, we may expect a more gradual increase of $S_{max}$ with $C$.
%
\begin{figure}[hbt]
\begin{minipage}[t]{7.5cm}
\includegraphics[height=11cm]{allRs}
\caption{Dependence of the aggregation parameters $R_1$, $R_2$ and $R_3$ (upper, middle and lower panel, respectively) on the peptide concentration C at fixed numbers of peptides (indicated in the figure).}
\label{fig:allRs}
\end{minipage}
\hfill
\begin{minipage}[t]{7.5cm}
\includegraphics[height=11cm]{allRswithL}
\caption{Dependence of the aggregation parameters $R_1$, $R_2$ and $R_3$ (upper, middle and lower panel, respectively) on the system size $L$ at fixed peptide concentrations (indicated in the figure).}
\label{fig:allRswithL}
\end{minipage}
\end{figure}
%
The data shown in Figure \ref{fig:allRs} also demonstrate that the degree of aggregation increases strongly when the concentration is fixed but the number of peptides in the system increases. This is shown explicitly in Figure \ref{fig:allRswithL}, where the aggregation parameters $R_1$, $R_2$ and $R_3$ are given as a function of the box size $L$. All three different aggregation parameters depend on the system size in a drastic way. In fact, the effect of system size on aggregation (Figure \ref{fig:allRswithL}) is as strong as the effect of peptide concentration (Figure \ref{fig:allRs}). Decreasing the system size has the same effect as decreasing the peptide concentration: the peptides become apparently more ``dissolved'' in water. The physical origin of this phenomenon is the effect of the finite system size on the minority phase in the two-phase region. The same behavior is seen in simple fluid and magnet systems, when they are at constant density or magnetization, respectively. \cite{Furukawa:82a,Binder:03a,Macdowell:04a,Macdowell:06a,Nussbaumer:06a,Nussbaumer:08a} 
 \begin{figure}[hbt]
 \begin{center}
 \includegraphics[height=8cm]{R2C6}
 \caption{Dependence of the size $S_{max}$ of the largest peptide cluster on the peptide concentration $C$ at fixed peptide number $N_p$.}
 \label{fig:R2C6}
 \end{center}
 \end{figure}
% %
Since the concentration and the system size strongly affect the degree of peptide aggregation as measured by different parameters, various other properties of the peptide-water system should also be strongly sensitive to these two factors. The dependence of the average number  of interpeptide hydrogen bonds per peptide, $n^{pp}_H$, on the system size is shown in the lower panel of Figure \ref{fig:Hpepwat} for various peptide concentrations.
These dependences are qualitatively similar to those of the aggregation parameters shown in Figure \ref{fig:R2C6}, i.e., $n^{pp}_H$ increases with increasing concentration $C$ and with increasing system size $L$. The average number of peptide-water H-bonds, $n^{pw}_H$, is also strongly sensitive to $C$ and $L$, although this dependence is opposite to that of $n^{pp}_H$. The clear correlation between the aggregation parameters and other properties of the peptide-water system evidences the reasonable choice of the aggregation parameters. 
\par
The secondary structure content of peptides is an important property, which is used in the characterization of peptide aggregation both in simulations and experiments. We have found that the $\beta$-sheet and $\alpha$-helical contents are strongly sensitive to the system size and concentration, and vary from 0.28 to 0.42 and from 0.11 to 0.33, respectively. Nevertheless, there is a clear anti-correlation between the $\alpha$-helical and $\beta$-sheet contents of the peptide-water system (see upper panel in Figure \ref{fig:secstr}). In turn, the $\beta$-sheet content is proportional to the average number of interpeptide H-bonds per peptide $n^{pp}_H$ (see lower panel in Figure \ref{fig:secstr}).
%
\begin{figure}[hbt]
\begin{minipage}[t]{7.50cm}
\includegraphics[height=9cm]{Hpepwat}
\caption{ Dependence of the average number $n_H^{pw}$ of peptide-water (upper panel) and the average number $n_H^{pp}$ of peptide-peptide (lower panel) H-bonds on the system size $L$ at several peptide concentrations.}
\label{fig:Hpepwat}
\end{minipage}
\hfill
\begin{minipage}[t]{7.50cm}
\includegraphics[height=9cm]{secstr}
\caption{Correlation between the $\beta$-sheet content and the average number  of  peptide-peptide H-bonds $n_H^{pp}$ (lower panel) and correlation between the $\beta$-sheet  and $\alpha$-helical contents (upper panel) in all systems studied.}
\label{fig:secstr}
\end{minipage}
\end{figure}
%
\par
Important information about the driving forces of peptide aggregation can be obtained from the analysis of the properties of hydration water. The water-mediated attraction between hydrophobic groups of peptides as well as Coulombic and H-bonding interactions between peptides are the main driving forces of peptide aggregation. An analysis of the effective hydrophobicity/hydrophilicity of the surface of peptide aggregates may give insight into the mechanism of aggregation. If the peptide surface exposed to water becomes more hydrophilic upon aggregation, the attraction between hydrophobic groups may be considered as the main driving force in aggregation. Otherwise, aggregation should be attributed mainly to the interpeptide Coulombic and H-bonding interactions. The strength of water-peptide interaction can be characterized by the number  of water-peptide H-bonds $n^{pw}_H$ per unit area of solvent accessible surface. The dependence of this number on the system size \textit{L} at peptide concentration $C \approx$ 6 $\%$ is shown in the lower panel of Figure \ref{fig:shelldens}, where the value for a single peptide is shown by a horizontal line. With increasing system size (and, accordingly with increasing aggregation), the surface of peptides exposed to water becomes more hydrophilic. We can conclude that the hydrophilicity of the peptide surface exposed to water is enhanced upon aggregation and that the hydrophobic attraction between hydrophobic groups of the peptides studied is the main driving force of their aggregation. This could be attributed to the presence of hydrophobic amino acid residues and hydrophobic caps in the peptide. 
\par
The density $\rho_h$ of water in the hydration shell of peptides can also be used as a measure of the effective strength of peptide-water interaction: more hydrophobic surfaces cause a decrease in the density of hydration water. The density $\rho_h$ of hydration water can be simply estimated, if the solvent accessible peptide surface as well as the number of water molecules in the shell of a certain width near the surface are known. The dependence of the density $\rho_h$ of hydration water on the system size $L$ at a peptide concentration $C$ $\approx$ 6$\%$ is shown in the upper panel of Figure \ref{fig:shelldens}. The trend towards more dense hydration water upon aggregation is clearly seen, corroborating our conclusion about the leading role of hydrophobic attraction in the aggregation of the studied fragments, as derived from the analysis of water-peptide H-bonding (see lower panel in Figure \ref{fig:shelldens}). 
%
\begin{figure}[hbt]
\begin{center}
\includegraphics[trim= 0.2cm 0.5cm 0.5cm 0.2cm,clip,height=9cm]{shelldens}
\caption{Dependence of the average number of peptide-water H-bonds, $n_H^{pw}$, normalized by the peptide SASA, on the system size $L$ (lower panel) and dependence of the average density $\rho_h$ of hydration water (upper panel) on the system size $L$ at peptide concentration $C$ $\approx$ 6 $\%$. The dashed lines represent corresponding values observed in the simulations of single peptides. }
\label{fig:shelldens}
\end{center}
\end{figure}
%
\subsection{Properties of the largest peptide cluster}
A clustering analysis provides the possibility to explore various properties of the largest peptide cluster, which can be considered as an embryo of the organic-rich fibrillar phase. The time evolution of the size $S_{max}$ of the largest peptide cluster (lower panel in Figure \ref{fig:P56rg}) as well as the time evolution of its radius of gyration $R_g$ (upper panel in Figure \ref{fig:P56rg}) can be used to characterize the equilibration process. Similar to the radius of gyration $R_g$ of all peptides, the properties of the largest peptide cluster achieve saturation in about 25 ns. Note that the $R_g$ of all peptides and of the peptides in the largest cluster are very close due to the strong degree of aggregation in the largest system studied (upper panel in Figure \ref{fig:P56rg}).
%
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=9.0cm]{P56rg}
\caption{Upper panel: Time evolution of the radius of gyration, $R_g$, of all peptides (black line) and of the peptides in the largest cluster (red line). Lower panel: Time evolution of the size $S_{max}$ of the largest peptide cluster.}
\label{fig:P56rg}
\end{center}
\end{figure}
%
\par
The structural properties of the largest peptide cluster may be characterized by the relation between its average mass $M$ and volume $V$. In a first approximation, $V$ can be estimated from the radius of gyration $R_g$ of the largest peptide cluster, assuming that it is a spherical body: $V = (4/3)\pi$ $(R_g\sqrt{5/3})^3$. The obtained dependence $M(V)$ in the various systems studied is shown in Figure \ref{fig:lclustmv}. If the peptide cluster is a three-dimensional object, its density is equal to the slope of the dependence $M(V)$, which is $\rho$ = 0.66 g/cm$^3$.
Since a spherical shape of the peptide cluster was imposed, the estimation given above is a lower limit for the aggregate density, as the sphere provides the maximal volume at a fixed radius of gyration. Information about the shape of the largest cluster can be obtained from the analysis of the maximal extension $L_{max}$ of the largest cluster. The dependence of the average value of $L_{max}$ on the average value of $R_{g}$ in the various systems is shown in the upper panel of Figure \ref{fig:lcluslmax}.For comparison, the dependence expected for a spherical object, whose $L_{max} = 2R_g\sqrt{3/5}$, is also shown. The dependence $L_{max}(R_{g})$ for the peptide cluster deviates from the one expected for spherical objects, indicating an elongated shape of the peptide clusters. 
\par
To gain further insight into the mass distribution within the largest cluster, we plotted the dependence of $M$ on the radius of gyration, $R_g$, in a double logarithmic scale (lower panel in Figure \ref{fig:lcluslmax}). The slope of this dependence is equal to the fractal dimensionality of the object. The fit of the dependence $M(R_g)$, shown in the lower panel in Figure \ref{fig:lcluslmax}, to a power law yields the fractal dimension of the largest peptide cluster equal to 2.8 (solid line), which notably differs from 3 (dashed line), corresponding to compact three-dimensional objects. A fractal-like structure of the peptide aggregate is also supported by the dependence $L_{max}(R_g)$, which is not linear, but $L_{max}\sim R_{g}^{1.1}$, indicating that the fractal dimension of the peptide clusters is $<$ 3. In fact, the fractal dimensionality of proteins is always $<$ 3 and, therefore, a low fractal dimension of peptide clusters is not surprising.\cite{Enright:05a}  
\begin{figure}[hbt]
\begin{minipage}[t]{7.5cm}
\includegraphics[width=8.0cm]{lclustmv}
\caption {Dependence of the average mass $M$ of the largest peptide cluster in various systems on its volume, estimated as the volume of a sphere with the corresponding radius of gyration (circles). The lower limit for the density of the largest peptide cluster is estimated from the slope of the linear fit (solid line).}
\label{fig:lclustmv}
\end{minipage}
\hfill
\begin{minipage}[t]{7.5cm}
\includegraphics[width=7.0cm]{lclustlmax}
\caption {Dependence of the maximal extension $L_{max}$ of the largest peptide cluster on its radius of gyration, $R_{g}$, in various systems. Lower panel: Dependence of the mass $M$ of the largest peptide cluster on its radius of gyration, $R_{g}$, in the double logarithmic scale.
}
\label{fig:lcluslmax}
\end{minipage}
\end{figure}
\subsection{Time evolution of secondary structure}
In order to evaluate the time evolution of the secondary structure of the peptides, each residue is assigned to a particular secondary structure using SEGNO,\cite{Cubellis2005} and a peptide is assigned to a particular secondary structure, if three consecutive residues have the same secondary structure and no other consecutive secondary structure is present, else, the peptide is assigned to a coil. Two peptides are considered to be forming $\beta$-sheets if both peptides are assigned to $\beta$-strands by SEGNO and at least two polar backbone hydrogen bonds are present between them. Figure \ref{fig:segnoc12} shows the time evolution of two representative simulations, Run1 and Run2, at $C= 12\%$. In the initial configuration of the system, all the peptides were in an isolated strand conformation.
\begin{figure}[htb]
\begin{center}
\begin{minipage}[t]{14.5cm}
\includegraphics[width=14.5cm]{segnoc12}
\caption {Plot of the fraction of peptides in a particular secondary structure for Run1 and Run2 as a function of time. Data was smoothed using Savitzky-Golay filtering with a fourth degree polynomial and 25 points on each side.}
\label{fig:segnoc12}
\end{minipage}
\hfill
\begin{minipage}[t]{14.5cm}
% \end{center}
% \end{figure}
% \begin{figure}[htb]
% \begin{center}
\includegraphics[width=14.5cm]{clustd}
\caption {Log-log plot of $n_S$, the number of peptide molecules in a cluster of size $S$. Clusters were defined using a hydrophobic contact cutoff of $ \ge 20 $ between two peptides.}
\label{fig:clustd}
\end{minipage}
\end{center}
\end{figure}
A rapid increase in random coil conformation, accompanied by a corresponding decrease in isolated strands, is observed within the first 5 ns  in both simulations. In Run1, $\beta$-sheets increase steadily up to 100 ns, which correlates well with the decrease in random coils. There seems to be no other secondary structure surge during this transformation, indicating the direct transition from coil to $\beta$-sheets. Run2 has a very low fraction of $\beta$-sheets and most of the peptides have no distinct secondary structure; however, the relative amount of peptides in the helix conformation is higher. The cluster size distribution for both simulations at various time intervals is shown in Figure \ref{fig:clustd}. The power law $n_S \approx S^{-2.2}$ represents the behavior of $n_S$ at the percolation threshold.\cite{Brovchenko:04a} A hump of $n_S$ at a large $S$ and its drop-off at intermediate $S$ values indicate the presence of a large aggregate in the case of Run1, which is missing in Run2. The increase in the amount of $\beta$-sheets with time is only seen in simulations where the droplet survives for multiple nanoseconds.
%\clearpage
%
%******************************************************************
%
\section {Discussion}
The results of the simulation studies presented have two main implications. The first implication is related to the necessity to take into account the finite size of the simulated system, when one intends to reproduce properties of a real macroscopic system. The second implication is related to natural systems, where the finite system size is an intrinsic property. 
\subsection {Finite system size \textit{in silico}: simulation studies of binary systems}
Generally, the simulation studies are aimed at reproducing the properties of macroscopic systems. There are various factors, which complicate the realization of this goal, and it is important to know to what extent these factors make the properties of the simulated system different from those of its macroscopic analog. For example, the ability of available force fields to reproduce various system properties is approximate even for simple fluids and their mixtures. Another complication arises from the necessity to reproduce correctly the phase behavior of a system of interacting particles that requires application of sophisticated simulation methods (such as Monte Carlo simulations in the grand canonical ensemble). Besides, there is another general problem, which accompanies \textit{a priori} any simulation studies aimed to reproduce the properties of macroscopic systems; this is the finite size of the simulated systems, which even in the distant future will not noticeably approach the size of a macroscopic system.
\par
The effect of the finite system size on its properties is well known in statistical and computational physics. Finite size scaling allows approaching the properties of the macroscopic system. This can be achieved by simulations of several systems of different sizes (measured, for example, by their linear extensions $L$) with subsequent extrapolation of the results to the macroscopic limit $L\rightarrow \infty$. The effect of the finite size depends on the thermodynamic state of the system. For example, it is especially strong in the vicinity of the critical point due to the suppression of  fluctuations by the finite system size. On the other hand, when the system is in a thermodynamic state that is distant from the phase transition, system properties are not strongly affected by its finite size.
\par
Typically, simulation studies of aqueous solutions are performed in constant-volume or constant-pressure ensembles. As an aqueous solution is a mixture, the concentration of solutes is a key parameter of such a system, which is also kept constant. Simulations of a single solute in water can be used for reproducing the properties of a macroscopic system only at infinite dilution. In this case, the finite size does not affect the system properties noticeably when the box size essentially exceeds the size of the solute. The addition of just another solute molecule to the system changes the status of simulations in a drastic way. Now the clustering (aggregation) of solutes is possible and strongly depends on the concentration of the solutes and the thermodynamic state of the system.\cite{Fisher:67a,Oleinikova:02a,Brovchenko:04a} If this state is distant from the demixing phase transition, the probability to find a cluster containing $S$ molecules drops in a drastic way with increasing $S$ and the vast majority of solute molecules exists as monomers or belongs to small clusters. Of course, the finite system with only a few solutes fails to reproduce the cluster size distribution of a macroscopic system. However, this affects mainly the large clusters whose population in solution is low, whereas the clustering of the majority of solute molecules is not strongly perturbed. 
\par
The distorting effect of the finite size on the system properties becomes enormous as the system enters the two-phase region.\cite{Furukawa:82a,Binder:03a,Macdowell:04a,Macdowell:06a,Nussbaumer:06a,Nussbaumer:08a} This situation is typically encountered in the simulation studies of  strongly aggregating solutes in liquid water, which are usually performed in one simulation box with the solute concentration deeply inside the two-phase region. However, the effect of a finite system size is generally neglected not only in simulations of biomolecules, but also in simulation studies of simple small hydrophobic solutes in liquid water (see, for example, Refs. \citenum{Pangali:79a,Smith:93a,Li:07a}). Moreover, the effect of concentration on the aggregation of solute molecules was considered in only few studies. \cite{Wallqvist:91a,Raschke:01a} In large enough systems, the solution may separate into two coexisting phases with an explicit interface between them. However, this is not the case in the majority of simulation studies of solute aggregation in water, where the number of solutes is usually small. The finite system size suppresses the minority (organic-rich) phase and produces an artificial stable state of the system in which the organic-rich phase is dissolved. The existence probability of this state increases with decreasing the system size and with decreasing solute concentration (within the two-phase region). To make the simulation studies of such systems relevant to real systems, it is necessary to consider the type of phase transition, the character of the coexisting phases, and to perform a correct extrapolation of the simulation results to the macroscopic limit. 
\subsection{Finite system size \textit{in silico}: simulation studies of peptide aggregation}
In the case of amyloidogenic peptides in water, the organic-rich phase appears as a solid-like ordered aggregate. Typically, the organic-rich phase is a minority phase and its properties are strongly affected by the finite system size. When the peptide concentration exceeds the critical one, the peptides should form highly ordered fibrils in the macroscopic limit. However, the number of peptides is relatively small in simulations. Just a few peptides in a simulation box can only to some extent exhibit properties of the peptide aggregate expected in the macroscopic limit. Therefore, simulation studies of aqueous solutions of peptides must include the analysis of the effect of the system size on the degree of aggregation and on all other system properties, and the artificial system properties, produced solely by the finite size effect, should be excluded from the consideration. In a finite system of an aqueous solution of peptides whose concentration is within the two-phase region, there are two stable (equilibrium) states. One state is a state with the peptide aggregate, which represents an organic-rich phase. Of course, this phase, reproduced by only a few peptides, differs strongly from the ordered peptide aggregate seen in macroscopic systems. However, this state will evolve towards its analog in the macroscopic limit upon increasing the system size (the number of peptides in the simulation box at fixed concentration). In the other state, the organic-rich phase is dissolved, either partially or completely. This state is a pure artifact of the finite system size and has no analog in the macroscopic limit, as it disappears with increasing the system size. The time intervals of the molecular dynamics trajectory, where the system exists in such an artificial state, can be determined using, for example, the time evolution of the total radius of gyration of all peptides (see Figure \ref{fig:rgyr6}). The exclusion of these time intervals from the analysis should help approaching the properties of the macroscopic system. Note that an equilibrium between the previously mentioned states was also observed in simulation studies of three amyloidogenic peptides in implicit water,\cite{Cecchini2004} although the origin of such a behavior was not discussed.
\par
The most important finding of our studies is the drastic effect of the system size on peptide aggregation. This effect seems to be responsible for the instability of aggregates consisting from just a few peptides, as seen in simulations.\cite{Zheng:06a,Rohrig:06a,Nguyen:07a} This effect of the finite size of a system which is in the two-phase state, is known and well understood for Ising magnets and LJ fluids, has not been studied so far in more complex systems. To the best of our knowledge, this is the first study showing the manifestation of this effect in binary mixtures of complex peptide-water systems with low solubility of the peptide. This system was chosen due to the importance of simulation studies of the aggregation of biomolecules in water. Our simulation studies of peptide aggregation with various peptide concentrations clearly show that the concentration affects all system properties in a drastic way. This effect is well understood and seems to be obvious. The degree of peptide aggregation (Figure \ref{fig:allRs}), H-bonding (Figure \ref{fig:Hpepwat}), secondary structure content and other system properties strongly depend on peptide concentration. The dependence of the size of the largest peptide cluster on concentration at a fixed  number of peptide (Figure \ref{fig:allRswithL}) clearly illustrates the necessity to account for the concentration effect in the simulation studies of aggregation phenomena. This effect is unavoidable, when the number of aggregating particles exceeds 1. In particular, any property of a system with just two aggregating particles are concentration dependent. 
\par
When studying the effect of concentration and system size on aggregation, it is important to introduce an adequate parameter characterizing the degree of aggregation. Although the use of the radius of gyration of all peptides for this purpose does not require a criterion for connectivity between peptides, it does not allow obtaining of the cluster size distribution and an analysis of the properties of the largest peptide cluster. A more detailed analysis of aggregation requires the introduction of a connectivity criterion between two peptides. Such a choice is not unambiguous, even for simple molecules, and it becomes even more difficult for biomolecules. In a first approximation, we have used the distance between the center of mass of two peptides as a measure of connectivity. The disadvantage of this measure is the inability to distinguish the ordered character of the peptide aggregate. Clearly, the search for more adequate connectivity criteria in studies of formation of ordered peptide aggregates is necessary. These criteria should be derived from the structure of the macroscopic ordered peptide aggregates (fibrils) and include the interatomic distances between two peptides and H-bonds in particular. It is to be noted that the effect of the finite system size on the ordered character of a peptide aggregate is so far unknown. We may assume that a more ordered aggregate will be more affected by this effect. However, this assumption should be tested in simulations.
\subsection{Finite system size \textit{in vivo}: aggregation of proteins in cells}
The effect of the system size on peptide aggregation complicates the attempts to reproduce the properties of macroscopic systems by simulations of finite systems. However, this is an intrinsic property of the finite system. Therefore, if the real system of interest is not macroscopic and contains a relatively small number of peptides, their aggregation will be suppressed by the finite system size as well. This situation may be relevant in the case of peptides in small volumes, such as biological cells or their compartments. \textit{In vivo}, amyloidogenic peptides can be found both in intracellular and extracellular fluids and these two pools of peptides often seem to be mutually related. \cite{Wilson:99a,Oddo:06a} The extracellular fluid may be regarded as an essentially macroscopic system and formation of the peptide-rich phase via the phase transition should largely follow the regularities normally encountered in \textit{in vitro} experiments with bulk aqueous solutions of the amyloidogenic peptides. The picomolar concentrations of islet amyloid polypeptide in the plasma \cite{Hanabusa:92a} and of \textit{A}$\beta$ in the cerebrospinal fluid \cite{Ida:96a,Lue:99a} should be considered as critical for their aggregation, because the extracellular fluid is in direct contact with amyloid plaques. The production of amyloidogenic peptides in cells and their accumulation with time \cite{Tienari:97a,Wilson:99a,Echeverria:02a} evidence that the intracellular concentration noticeably exceeds the critical one. The absence (or at least the slowing down) of extensive fibrillation in cells might be due to the effect of the relatively small volume of cells or their compartments. Due to the high insolubility of amyloidogenic peptides, the presence of dozens or hundreds of peptides in a cell may already provide conditions of strong oversaturation. Hence, a small cell volume should suppress peptide aggregation in general, and formation of ordered aggregates, in particular. The escape of these peptides into the extracellular fluid (for example, due to the destruction of the cell membrane upon its death), makes fibrillation unavoidable at the same peptide concentration. If oversaturation is not too strong, the lag time of fibril formation may still take many years.
\par
The suppression of aggregation by a small system size has a general physical origin and, therefore, is unavoidable. However, it is rather difficult to estimate the magnitude of this effect, which depends on the system and detailed solution conditions considered. Additionally, other factors (first of all, surface effects) may be equally or even more important. We may expect that peptide adsorption on some surfaces is favorable for the ordered character of their aggregation, whereas confinement in a small volume should suppress the formation of ordered aggregates. Further studies are necessary to identify those cases  where the finite size of biological cells noticeably affect intracellular peptide aggregation.
