\chapter{Introduction}
\label{ch:intro}
Proteins were first described and named by the Swedish chemist J\"{o}ns Jakob Berzelius in 1838. The word protein comes from the Greek word `$\pi\rho\omega\tau\epsilon\iota o\varsigma$', which means `standing in the front', `in the lead'.\cite{Tanford2001} More than a century of research has shown that proteins are central to all living systems due to their involvement in all aspects of cell life, immune protection, cell-cell communication, physical support (hair and skin), and muscle movement. The solubility of proteins has been measured for more than a century, and the concept played a prominent role in the development of protein physical chemistry.\cite{Arakawa1985} The majority of proteins in the human body are biologically active in a narrow range of temperature and pressure, and a `folded' conformation is necessary for biological activity.
\par
Relatively recently, several diseases have been associated with protein aggregation. Some proteins or protein fragments have been found to undergo a transition from the normally soluble forms to insoluble fibrils or plaques, which accumulate in various organs such as liver, spleen and brain.  The mature aggregates often have well-defined fibrillar structures, and are generally known as amyloids, hence the term amyloidosis is used to describe many of the clinical conditions with which they are associated (see Ref. \citenum{Sipe2000} and references therein). Protein aggregation is also a relevant issue in pharmaceutical industry, as it also affects the shelf-life of a protein solution, which is directly related to the economic viability of a typical protein based pharmaceutical product.\cite{Chi2003}  Hence, the understanding of protein sequence-structure relationships and the phase behavior of proteins has become a major research area in the field of biophysics. 
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\section{Proteins}
Proteins and peptides are built from smaller, individual amino acid units, with each amino acid containing an $\alpha$-carbon bonded to an $\alpha$-amino group, an $\alpha$-carboxyl group, one hydrogen, and a side chain, R, of varying lengths and functionalities. There are twenty naturally occurring amino acids. Their names are commonly abbreviated with either a three-letter code or a one-letter code. Depending on the chemical structure of the side chain, the amino acids can be broadly divided into two different classes. The first class comprises those with hydrophobic side chains (non-polar) e.g. Ala (A), Val (V), Leu (L), Phe (F), etc. The second class comprises amino acids having hydrophilic side chains (polar) that are either charged (Lys (K), Glu (E), Asp (D) and Arg (R)) or uncharged (Gln (Q), His (H), etc). The amino acid glycine (G), the simplest of all the twenty amino acids, has only a hydrogen atom as the side chain. The growth of a protein occurs by a condensation reaction between the free amine and the free carboxylic group of two distinct amino acids, resulting in the formation of amide bonds. The secondary structure of the $\alpha$-helix is formed by the hydrogen bonding between the NH and CO groups of the same strand, whereas the $\beta$-strand forms hydrogen bonds with other $\beta$-strands. These $\beta$-strands stabilize each other through intermolecular hydrogen bonding, forming $\beta$-sheets. Tertiary structures are built on these secondary structures, with the corresponding formation of higher-order structures such as $\beta\alpha\beta$-units, $\beta$-hairpins, and $\alpha$-helix bundles. If a protein has two or more polypeptide subunits, their spatial arrangement is referred to as quaternary structure. On the basis of higher level structures, proteins can be classified into two major groups: fibrous proteins, having polypeptide chains arranged in long strands or sheets, and globular proteins, having polypeptide chains folded into a spherical or globular shape. The two groups are structurally distinct: fibrous proteins usually consist largely of a single type of secondary structure; globular proteins often contain several types of secondary structures. The two groups also differ functionally; the structures that provide support, shape, and external protection to vertebrates are made of fibrous proteins, whereas most enzymes and regulatory proteins are globular proteins. \cite{Stryer1996} 
\section{Protein folding}
``Protein folding'' is a term used to describe the process by which a polypeptide chain adopts its three-dimensional ``native'' conformation. To carry out their functions, proteins must fold rapidly and reliably. They must satisfy a kinetic requirement (folding must be completed within a reasonable time) and a thermodynamic requirement (the folded conformation must be stable under physiological conditions). Due to the vast amount of configurational space available to a protein chain,  predicting protein folding from its primary sequence still remains one of the most important challenges in science.\cite{2005a}  It has been shown experimentally that, for some proteins, the stable native state of the protein survives in a closed range in the pressure-temperature plane.\cite{Hawley1971} In these studies, it was assumed that there are only two possible states of the protein, the native and the denatured state, and that the transition occurs without any intermediate states. The Gibbs free energy difference ($\Delta G$) between the denatured and the native state was calculated. The transition line was defined by $\Delta G = 0$ and the curve was found to be elliptic-like in shape. In the framework of this theory, heat, pressure and cold denaturation are the three ways of protein unfolding.\cite{Smeller2002}  
\begin{figure}[hbt]
\centering
\includegraphics[width=9.5cm]{ptdiag}
\caption {Schematic pressure/temperature stability diagram of a typical monomeric protein. The different routes of unfolding of the native protein (green area) as well as corresponding thermodynamic properties are depicted. Heat denaturation is often accompanied by irreversible aggregation. The water/ice phase transition line is also given.\cite{Mishra2008}}
\label{fig:ptdiag}
\end{figure}
\par
Solvent plays a crucial role in determining the state of a polymer. In polymer physics, it is well known that the compactness of a polymer greatly depends on the type of liquid in which it is placed. If there is a strong attractive interaction between solvent molecules and the polymer chain, the solvent is called a ``good solvent''. Polymers swell or expand in a good solvent, in a way similar to the swelling caused by repulsion between various segments of the chain itself. On the other hand, if there is a much weaker attractive interaction between solvent molecules and the polymer than between segments of the polymer molecules themselves, the solvent is called a ``poor solvent''. In a poor solvent, the polymer chain will be forced to adopt a more compact conformation. This is equivalent to an attractive interaction between the segments of the chain.\cite{Doi1996,Bower2002} 
Thus, various interactions between constituent amino acid residues of the protein, and with the surrounding solvent, determines the preferable structure of protein in a given thermodynamic state. These interactions include electrostatic  and van der Waal interactions between various atoms, formation of hydrogen bonds within the polypeptide chain or with solvent molecules, and hydrophobic interactions.  The  term ``hydrophobic interaction''  refers to the total interaction between two or more interacting non-polar solutes in an aqueous solution.  The net attractive interaction between hydrophobic particles was initially explained using the ``iceberg'' model.  It was believed that water molecules near hydrophobic particles are ``ordered'',  i.e., in order to avoid the enthalpic penalty of losing hydrogen bonds, water molecules arrange themselves around a hydrophobe in a relatively ordered fashion, causing a loss of entropy, which is to some extent compensated by enhanced hydrogen bonding in the hydration layer, resulting in the formation of semicrystalline hydration shell. Thus, if the two hydrophobes comes together, the ``structured'' water in the region between them is returned to the bulk, leading to an entropy increase.\cite{Frank1945,Kauzmann1959} This view has been discredited as many evidences have shown that  water does not undergo any structural enhancement near hydrophobic solutes. The favorable enthalpy of hydrophobic hydration can be attributed to van der Waals interactions between water and an apolar solute. The water molecules simply orient themselves near the apolar solutes in such a way that the O--H bonds are tangential to the solute surface. The increase in either the number or the size of hydrophobic particles causes the loss of hydrogen bonds between water molecules,  an energetically unfavorable event, and results in aggregation of apolar particles (see References \citenum{Blokzijl1993} and \citenum{Ball2008} for detailed reviews).   
\par
One of the main ideas in the theory of protein folding is the energy landscape theory. The protein energy landscape is considered to resemble a rugged funnel, i.e. riddled with traps in which the protein can transiently reside. Native contacts and local conformation energies are considered more stabilizing than non-native contacts; hence there is an overall slope of the energy landscape towards the native structure. In the early stages of protein folding, the funnel guides the protein through many different sequences of traps towards the low-energy folded (native) structure. There are no pathways, but a multiplicity of folding routes, and discrete pathways only appear later in the folding process, when most of the protein has already achieved a correct configuration.\cite{Onuchic1997} 

\section{Protein aggregation}
In 1854, Rudolph Virchow introduced the term ``amyloid'' to denote a macroscopic tissue abnormality that exhibited a positive iodine staining reaction. However, it was later demonstrated that amyloids consist mainly of proteins rather than carbohydrates. The variety of pathological conditions were related to the inability of certain proteins to fold correctly or to remain correctly folded, resulting in their aggregation.\cite{Sipe2000,Chiti2006} The insolubility of amyloid fibrils has hindered their experimental evaluation for a long time. The X-ray diffraction examination of amyloid fibrils has led to the general conclusion that fibrils are made up of a cross-$\beta$ structure in which the polypeptide chains are organized as $\beta$-sheets arranged parallel to the fibril axis, having their constituents $\beta$-strands perpendicular to the fibril axis. The characteristic cross-$\beta$ diffraction pattern and the fibrillar morphology are accepted as a diagnostic hallmark of amyloids, suggesting that the fibrils formed from different protein precursors share a degree of similarity.\cite{Sunde1997} 
%
\begin{figure}[htb]
\begin{center}
\includegraphics[height=100mm,angle=90]{fibril}
\caption{{Model of the generic amyloid fibril structure depicting four $\beta$ sheets running parallel to the fibril axis with component $\beta$ strands perpendicular to the fibril axis.\cite{Sunde1997}}}
\end{center}
\end{figure}
%
%
\begin{figure}[htb]
\begin{minipage}[t]{7.0cm}
	\includegraphics[height=60mm]{kinetic}
	\caption{Experimentally observable formation of an aggregate for a nucleation dependent process, above its critical concentration. Aggregate formation at low  and high concentrations is indicated by a solid and dashed line, respectively. Addition of seed also results in rapid aggregation.\cite{Jarrett1993}}
	\label{fig:kin}
\end{minipage}
\hfill
\begin{minipage}[t]{7.5cm}
	\includegraphics[height=60mm]{protphase}
	\caption{Phase diagram for a Yukawa fluid with screening parameter $b$=7.5. The crosses represent the fluid-solid phase transition, the continuous line is the binodal, and the dashed one is the spinodal. The filled circle is the critical point. The glass line (open circles) as evaluated for mode coupling theory and the glass line shifted to obtain the experimental packing fraction $\phi=0.58$ (stars).\cite{Foffi2002}}
\label{fig:protphase}
\end{minipage}
\end{figure}
%

The process of amyloid fibril formation follows a nucleated growth mechanism, as depicted in Figure \ref{fig:kin}. In a typical nucleated growth mechanism, a threshold in protein concentration, called ``critical concentration'', is required for aggregation to be initiated. The critical concentration depends on several factors, such as protein structure, presence of co-solvents, pH, and presence of interface. If the protein concentration slightly exceeds the critical concentration, there is a lag time before polymerization occurs. During this lag time, the addition of preformed fibrillar species or ``seeds'' causes a shortening of the lag phase. The length of the lag time drastically depends on the concentration of protein solution. A protein solution above its critical concentration could remain kinetically soluble for a considerable amount of time depending on the level of supersaturation.\cite{Jarrett1993} Once the critical nucleus is formed, the monomer concentration decreases exponentially. After aggregation is complete and the equilibrium is established, the solution contains predominantly monomers and large aggregates. Nucleation dependent protein aggregation is also a characteristic of many processes such as protein crystallization, microtubule assembly, and sickle cell hemoglobin formation.\cite{Harper1997} 

Experimental studies have shown that the temperature-concentration phase diagram for a variety of globular proteins follows the pattern depicted in Figure \ref{fig:protphase}.\cite{Dumetz2008} This phase diagram can be obtained by using a very short range interaction between protein molecules in comparison to their diameter. If the range of attraction is long, the phase diagram resembles the elementary phase diagram of atoms and molecules where van der Waals interactions predominate. As illustrated in Figure \ref{fig:protphase}, the binodal and spinodal lines lie completely within the region of fluid-solid phase separation and the critical point is also below the coexistence curve between the low-density fluid and the solid. At high temperatures, the fluid to solid phase transition is not interrupted by the glass transition. As the temperature is lowered below solubility limits, the solution becomes metastable; however, the fluid-fluid coexistence curve persists in the metastable regime and for particular temperatures and concentrations, the protein solution separates into  a liquid, dilute in protein, and  a liquid, rich in protein. \cite{Foffi2002,Dumetz2008}  It can also be speculated that the phase diagram of amyloid forming proteins that do not have any native structure should be similar to the above mentioned phase diagram, with the exception that the solid phase would not be a crystalline phase, but a solid fibril. In this case, the morphology of the resulting fibrils could depend on the anisotropy of the protein structure and the kinetics of the system. 
\par
Aggregation and precipitation of proteins also occur upon heating (see Figure \ref{fig:ptdiag}). The aggregation behavior of proteins in liquid water is similar to the demixing phase transition of binary mixtures, in general, and to aqueous solutions of organic molecules, in particular. Many aqueous solutions of simple organic molecules (pyridines, tetrahydrofuran, etc.) show demixing upon heating \cite{Narayanan:94a,Brovchenko:97a}. Aqueous solutions of large polymeric macromolecules (poly-N-isopropylacrylamide, polyoxyethylene, etc.) also show demixing upon heating, accompanied by drastic changes in  polymer conformation \cite{Winnik:90a,Luna-Barcenas:97a,Maeda:01a}.  The occurrence of a first-order demixing phase transition in aqueous solutions of proteins is therefore not surprising, being a common phenomenon in binary systems. Upon demixing, aqueous solutions of organic molecules separate into a water-rich phase and an organic-rich phase. When the thermodynamic conditions are close to ambient conditions, the water-rich phase is a liquid phase. The state of the organic-rich phase mainly depends on the phase state of the corresponding organic substance at ambient conditions. It may be a vapor phase (for example, in water-methane mixture), a liquid phase (for example, in aqueous solutions of pyridines) or an amorphous (solid-like) phase (aqueous solutions of macromolecules). Dry biomolecular substances are typically in an amorphous state at ambient conditions. Therefore, upon demixing, their aqueous solutions separate into a liquid water-rich phase having the critical concentration of biomolecules, and a solid-like organic-rich phase, which could be amorphous or ordered (crystalline or fibrillar). 
\section{Islet amyloid polypeptide}
Human islet amyloid polypeptide (hIAPP), also known as amylin, is a pancreatic hormone secreted by islet $\beta$-cells. It is a 37 amino acid residue polypeptide, having a disulfide bond between the two cysteine residues at position 2 and 7 and an amidated C-terminus, which is important for its hormonal functions. As hormone, it modulates carbohydrate metabolism in muscle and liver by stimulating glycogen breakdown and acting as a non-competitive antagonist of insulin; additionally, IAPP is involved in processes associated with feeding and maintenance of bone cells, renal proximal tubular cells, and islet $\beta$-cells.\cite{Cooper1994}  Fibrillar deposits containing hIAPP have been found in over 95\% of type 2 diabetic subjects postmortem and their abundance correlates with the severity of the disease.\cite{Jaikaran2001} Studies indicate that islet amyloid is usually juxtaposed with $\beta$ cell membranes. Sequence comparison between amylin from different species suggested that amylin (25-29) could serve as a core recognition motif and lack of proline residues at one or more of positions 25, 28, and 29 attributes to fibril formation in hIAPP.\cite{Cooper1994} A sequence comparison between human and rat amylin is depicted in Figure \ref{fig:iapp_seq}. The amino acid sequences of all known amylin molecules are highly conserved at their N- and C-termini, but vary among themselves to a greater extent in the central segment.  In contrast to human amylin, rodent amylin does not form fibrils. It was further demonstrated that penta- and hexapeptide sequences, hIAPP(23-27) and hIAPP(22-27), are sufficient for formation of $\beta$-fibrils.\cite{Tenidis2000} However, studies have also shown that proline substitutions outside the core domain diminishes fibril formation and single residue substitutions in rat IAPP from corresponding positions of human IAPP render rat IAPP competent for fibril formation, pointing towards the contribution of other regions in fibril formation.\cite{Abedini2006,Green2003} Several other fragments of hIAPP have also been shown to form fibrils.\cite{Jaikaran2001,Mazor2002,Nilsson1999} Amyloid formation by short peptides derived from  region 10-19 of  hIAPP are strongly influenced by pH. His-18 is considered to play a crucial role in modulating the pH dependent effect on aggregation.\cite{Abedini2005}

There are several reports  indicating that hIAPP is able to interact with membranes,\cite{Engel2006,Quist2005,Green2004,Porat2003,Lopes2007} and it seems that the interaction results in the formation of fibrillar amyloid deposits in the extracellular matrix of the $\beta$-cells, leading to membrane damage. Details of the nature of this interaction are still not fully understood. Some studies have suggested that the binding of hIAPP to the cell membrane is followed by loss of lipids from the membrane, \cite{Sparr2004} whereas other reports propose insertion of hIAPP into the membranes \cite{Harroun2001}. It has also been observed that phospholipids are able to catalyze hIAPP amyloid formation, leaving the lipid bilayer intact.\cite{Knight2004} Also, it is still unclear which of the hIAPP species, monomers, oligomers, protofibrils, or mature fibrils are mainly involved in these membrane interactions. Several studies have indicated that hIAPP oligomers, and not the fibrils, may be involved in the interaction with membranes.\cite{Knight2004,Mirzabekov1996,Kayed2003} According to Porat et al. \cite{Porat2003}, oligomers are believed to be intermediate species in the formation of hIAPP amyloid fibrils. Another report pointed to the possible role of hIAPP monomers in membrane interactions.\cite{Green2004a} It has also been suggested that the process of hIAPP amyloid formation and not the presence of a particular hIAPP species is related to its cytotoxicity.\cite{Harroun2001,Green2004a,Knight2004} These various results indicate that the interaction between hIAPP and lipid membranes is still far from being fully understood.
\section{Computer simulations of protein aggregation}
There have been several lattice studies on protein aggregation depicting the relationship between the stability of the native state and the propensity of aggregation suggesting distinct mechanisms for polymerization and its dependence on protein sequence .\cite{Broglia1998,Harrison1999,Dima2002} MD simulation methods have also been used to predict the possible atomic scale amyloid organization for many peptide fragments. In these studies, several models of $\beta$-clusters were built and MD simulations were used to investigate the structure and stability of such models.\cite{Zanuy2003,Zanuy2003a} Simulations starting from random orientations of peptides have also been carried out to investigate the initial steps in aggregation, either using implicit solvent and discrete molecular dynamics\cite{Nguyen2004} or explicit solvent molecular dynamics simulations.\cite{Colombo2005} Wu \textit{et al.} have reported the formation and elongation of ordered peptide aggregates using MD simulations with explicit solvent.\cite{Wu2004,Wu2005} The aggregation of small peptide fragments has also been studied using replica exchange molecular dynamics simulations with implicit solvent,\cite{Baumketner2005,Cecchini2004} and also with explicit solvent.\cite{Tsai2005} 
\section{Thesis objectives}
Protein aggregation and fibril formation seem to be generic in origin as they can be induced in proteins of diverse sequences and structures.  The demixing of a protein aqueous solution is similar to the demixing observed in two component systems. Thus,  protein aggregation should be studied as a function of concentration.  In molecular simulation studies, the system size is an additional factor that can have a drastic effect on the simulation results. Even though studies of peptide aggregation using molecular simulation techniques are reported in the literature, the effect of concentration and system size on aggregation is generally neglected. Even the parameters needed to quantify aggregation in simulation studies have not yet been fully established.  
\par
Therefore, the first objective of this thesis is to establish parameters that can be used to quantify the aggregation process in simulation studies of peptide aggregation. Once the parameters have been established, systematic studies on the effect of concentration and system size on aggregation can be studied. These studies will also reveal the extent to which the finite size effect can distort simulation results meant to reproduce the properties of macroscopic systems.  The finite size effects have been well understood for Ising magnets and Lennard-Jones fluids, but have never been studied for complex systems like aqueous solutions of peptides in full atomistic details.  The second objective of this thesis is to study the effect of temperature on peptide aggregation. The change in aggregation parameters as a function of temperature would provide information about the transition temperature and the nature of the transition involved. As various experimental studies have shown that protein aggregation is generally  affected by the presence of interfaces, the third and final aim of this thesis is to study the effect of model surfaces on the peptide aggregation process. 
\par
This thesis has been divided into five chapters: this chapter provides a general overview of protein aggregation; the second chapter gives a brief introduction of molecular simulation techniques and associated issues; the selection of appropriate parameters for the quantification of peptide aggregation, effect of concentration and system size on peptide aggregation in the bulk solution are described in the third chapter; the effect of temperature on peptide aggregation is described in chapter four; the last chapter describes the effect of surfaces on peptide aggregation by using two different types of amyloidogenic peptide fragments, in both hydrophobic and hydrophilic pores. 
