\chapter{Simulations of Peptide Aggregation Near Surfaces}
\label{ch:surf}
Spontaneous adsorption of protein molecules at interfaces is generally observed as an aqueous solution of the protein comes into contact with an interface.\cite{Malmsten2003} Protein side chains vary widely in their hydrophobicity, which renders them amphiphilic. Thus, there is a general tendency of proteins to reside at the interface between two phases of different polarity. As the folded conformation of the proteins is only marginally stable against unfolding, an interface-induced conformational change of protein can lead to an optimized contact between the interface and the protein.\cite{Czeslik2004} The presence of surfaces can induce a concentration gradient in a protein solution, and in turn affect its aggregation. The two dimensional nature of the surfaces can also induce orientational ordering of peptides, and thus lead to the formation of two dimensional aggregates.

Since the effects of surfaces on peptide aggregation are multifacet, it would be useful to analyze them separately. The direct interaction between peptides and surfaces includes contributions from Coulombic interactions, dispersion interactions, hydrogen bonding, etc. The presence of the solvent causes the appearance of solvent-induced interactions between the peptides and the surface. Hence, the \emph{effective} peptide-surface interaction is determined by relative strengths and ranges of direct peptide-surface, solvent-surface and peptide-solvent interactions. 

In order to study the effects of solvent induced peptide-surface interactions on peptide aggregation, two types of peptide fragments were used, i.e., fragment A, with the amino acid sequence NFGAIL, having a molecular weight of 634 Dalton, and fragment B, with the amino acid sequence GNNQQNY and a molecular weight of 837 Dalton.
\begin{figure}[hbt]
	\begin{minipage}{7.5cm}
		\centering
		 \includegraphics[height=6.0cm]{p1o}
		 \caption{Fragment A, the hydrophobic fragment.}
		  \label{fig:p1o}
	\end{minipage}
	\hfill
	\begin{minipage}{7.5cm}
		\centering
		\includegraphics[height=6.0cm]{p1i}
		\caption{Fragment B, the hydrophilic fragment.}
		\label{fig:p1i}
	\end{minipage}
\end{figure}
Fragment A is part of hIAPP (residue number 25-29) and fragment B is part of the Sup35 yeast protein (residue number 7-13). Sup35 is a 685 residue yeast protein. It is a subunit of the translation termination factor and its normal cellular role is to terminate translation. Yeast harboring the $[PSI^+]$ determinant exhibits increased translation read-through of stop codons due to aggregation and subsequent loss of function of Sup35. The first 123 residues constitute the prion determining domain and are sufficient for fibril formation.\cite{Glover1997} The sequence GNNQQNY has been shown to display the same amyloid properties as Sup35.\cite{Gustavsson1991} Even though both fragments are capable of forming fibrils, they differ greatly in their amino acid composition. Fragment A  mainly consists of hydrophobic amino acids, whereas fragment B is made up of polar amino acids. Figures \ref{fig:p1o} and \ref{fig:p1i} show the molecular surfaces of both the fragments, colored with values according to the hydrophobicity scale of Kyte and Doolittle,\cite{Kyte1982} with colors ranging from dodger blue for the most hydrophilic amino acid residue (-4.5) to white at 0.0, and towards orange-red (4.5) for the most hydrophobic residue . In the following text, fragments A and B are referred to as the ``hydrophobic'' and the ``hydrophilic'' fragment, respectively.
%
\section{System setup}
All simulations were performed using the Gromacs software, OPLS force field\cite{Jorgensen1988,Kaminski2001} and SPCE water molecules.\cite{Berendsen1987} The short range cutoff of 1.2 nm was used for LJ and Coloumbic interactions. LINCS algorithm was used to constrain all the bonds to their equilibrium bond lengths, allowing an integration time step of 2 fs. The neighbor list was updated every 20 fs and the coordinates were recorded every 2 ps. The following systems were simulated:
\begin{enumerate}
\item Water in slit-like pores.
\item The liquid-vapor interface of water.
\item Peptides in bulk water.
\item Peptides in slit-like pores and at the liquid-vapor interface.
\end{enumerate}
%
\subsection{Water in slit-like pores}
The interaction of water molecules with the surfaces was represented using the 9-3 potential
%
\begin{eqnarray}
U(r) &=& \frac{4\pi\epsilon n_{wall}\sigma^3}{6}\left[ \frac{2}{15} \left(\frac{\sigma}{r}\right)^9 - \left(\frac{\sigma}{r}\right)^3 \right] \\ \nonumber 
&=& \pi n_{wall}\left( \frac{A_{ij}}{45r^9} -\frac{B_{ij}}{6r^3}\right), 
\end{eqnarray}
%
where $A_{ij}=\sqrt{A_i A_j}$ , $A_i=4\pi\epsilon_i\sigma_i^{12}$, $B_{ij}=\sqrt{B_i B_j}$,  $B_i=4\pi\epsilon_i\sigma_i^6$. The ``hydrophilic wall'' was represented by $\sigma$ and $\epsilon$ values of  0.373 nm and 0.030800 kJ/mol, respectively, whereas the ``hydrophobic wall'' interacts with $\sigma$ and $\epsilon$ values of  0.373 nm and 5.2 kJ/mol, respectively (see Figure \ref{fig:surfpot}).
%
\begin{figure}[hbt]
\centering
\includegraphics[width=12.0cm]{surfpot}
\caption{Potential energy as a function of distance, between an oxygen atom of water and the hydrophilic or the hydrophobic wall  are representated by black and red  lines, respectively. }
\label{fig:surfpot}
 \end{figure}
%
The number density of the wall $n_{wall}$ was set to 100 nm$^{-3}$. Simulations were carried out on SPCE water molecules  in a cubic box of length 6 nm with the model walls at 0 and 6 nm in the $z$ dimension.  PME with 3dc geometry was used  to calculate long range Coloumbic interactions. In 3dc geometry, the reciprocal sum is still performed in 3d, but a force and potential correction are applied in the $z$ dimension to produce a pseudo-2d summation.\cite{Yeh1999} For Ewald summation, a scaling factor of 3 was used to scale down the interaction in the $z$ direction. The system was simulated for 20 ps at 600 K and subsequently for 300 ps at 330 K for initial equilibration. Water molecules were added or removed from the system until the density at the center of the box matched that of the bulk density of SPCE water molecules (see Figure \ref{fig:surfdensp}).
\subsection{Liquid-vapor interface}
The cubic box of length 6 nm was filled with 6400 SPCE water molecules. The box was extended in the $z$ direction in such a way that an empty space of 6 nm exists between the two periodic images in $z$ direction. The box was equilibrated for the duration of 10 ns at 330 K in an NVT ensemble using  periodic boundary conditions in all three directions. 
\subsection{Peptides in bulk water}
%
Six copies  of  one fragment were added randomly into a cubic box of  6 nm length in such a way that the fragments were at least 0.8 nm  away from each other. The boxes were solvated with water molecules. Five simulations were started at different initial velocities for each of the systems, i.e., five simulations for the hydrophobic fragment and five simulations for the hydrophilic fragment. Each system was initially equilibrated for 1 ns with position restrains on the peptide fragments. After 1 ns, the position restrains were removed, and the system was further simulated for the duration of 70 ns. The number of water molecules for the systems with hydrophobic and hydrophilic fragments were 6439 and 6394, respectively. All the simulations were carried out in an NVT ensemble. An equilibration period of 30 ns was estimated from the time evolution of various system parameters and was excluded from the analysis of system properties.
%
\subsection{Peptides in slit-like pores and at liquid-vapor interface}
%
The interaction between wall and peptide atoms was represented by $A_{ij} = 0.0 $ kJ mol$^{-1}$ nm$^{12}$ and $B_{ij}$ = 0.0 kJ mol$^{-1}$ nm$^{6}$, which lead to no interaction between the peptide atoms and the pore walls. Six copies of the peptide fragment were added randomly into the box in such a way that each fragment is at least 0.8 nm  away from other fragments and at least 1.5 nm away from the walls. The water molecules overlapping with the peptides were removed. The system was equilibrated for 1 ns using Berendsen's temperature coupling\cite{Berendsen1984} with a coupling constant of 0.5 ps. In the case of peptides with a liquid-vapor interface, position restrains were used on peptides during the initial 100 ps. After 1 ns of equilibration using Berendsen's temperature coupling, the system was switched to Nose-Hoover temperature coupling\cite{Nose1984,Hoover1985} with a coupling constant of 1.5 ps and thereupon further simulated for 70 ns.  Five NVT simulations, each starting with a different distribution of initial velocities at 330 K and 1 atm pressure, were carried out for each of the following systems:
\begin{enumerate}
\item Fragment A with hydrophilic walls: 6689 water molecules and 6 peptide fragments.
\item Fragment A with hydrophobic walls: 6189 water molecules and 6 peptide fragments.
\item Fragment A with a liquid-vapor interface: 6184 water molecules and 6 peptide fragments.
\item Fragment B with hydrophilic walls: 6643 water molecules and 6 peptide fragments.
\item Fragment B with hydrophobic walls: 6145 water molecules and 6 peptide fragments.
\end{enumerate}
\section{Results}
The density profiles of liquid water in the two pores and the density profile in the slab of liquid water are shown in Figure \ref{fig:surfdensp}. There is a pronounced density depletion of liquid water near the hydrophobic surface caused by the domination of the effect of missing neighbors over the weak water-surface attractive potential. The depletion of water density is more gradual near the liquid-vapor interface. Conversely, there are two pronounced and highly ordered water layers near the hydrophilic surface. The high value of the local density at the maximum of the first peak does not correspond to an enhancement of water density in the first layer, but rather to a narrow localization of water molecules in a plane parallel to the surface.
\begin{figure}[hbt]
\centering
\includegraphics[trim= 0.5cm 0.5cm 0.7cm 0.5cm,clip,width=12cm]{surfdensp}
\caption{Density profile of liquid water in a hydrophobic pore, hydrophilic pore, and in a slab of liquid water.  Pore walls are located at 0 and 6 nm, respectively. The density profile of the liquid slab is shifted by 3.5 nm. The same graph is shown at an enlarged scale in the right panel.}
\label{fig:surfdensp}
 \end{figure}
%
\par
The radius of gyration $R_g$ of the peptides, calculated using the method described in Chapter \ref{ch:bulkphase}, decreases rapidly within 30 ns in all the simulations, while the number of inter-peptide hydrogen bonds increases during the same time period, indicating that the peptides aggregate in all the simulation runs. The equilibration period of 30 ns was estimated from the time evolution of various system parameters. The time dependence of the probability distribution of the center of mass of the peptides in pores and in the slab of liquid water is shown in Figure \ref{fig:p6com}.
\begin{figure}[hbt]
\centering
\includegraphics[trim= 1.4cm 0.5cm 0.6cm 0.5cm,clip,width=12.5cm]{surfcomtime}
\caption{Probability distribution of the center of mass (COM) of the peptides as a function of time. Blue lines depict the walls in the case of slit-like pores, and the midpoint of the liquid-vapor interface in the case of the liquid slab. The probabilites of COM are shown using the color range depicted above.}
\label{fig:p6com}
 \end{figure}
After the random insertion of hydrophobic peptides into the liquid water in the hydrophobic pore or in the slab of liquid water, system equilibration requires up to 30 ns. After equilibration, all peptides are adsorbed at the interfaces. Usually the peptides are adsorbed on the two opposing interfaces, but sometimes all the peptides are adsorbed at only one of them. The situation is quite different in hydrophilic pores, where both hydrophilic and hydrophobic peptides quickly  become localized in the center of the pore. A similar trend is also observed for hydrophilic peptides in the hydrophobic pore.
\par
The density profiles of peptide in pores, calculated after the equilibration period of 30 ns, are shown in Figures \ref{fig:surfdenspro1} and \ref{fig:surfdenspro2}. The degree of peptide localization in the pore center is quite similar for both the hydrophilic and hydrophobic peptides in the hydrophilic pore. Hydrophilic peptides show weaker localization in the center of the hydrophobic pore. An opposite behavior is observed for the hydrophobic peptides in the hydrophobic pore or in the slab of liquid water: in all simulation runs, peptides are strongly adsorbed at the interfaces (Figure \ref{fig:surfdenspro2}). The degree of peptide localization near interfaces essentially exceeds that in the pore center, as can be seen from the comparison of the width of the density distributions shown in Figures \ref{fig:surfdenspro1} and \ref{fig:surfdenspro2}, respectively.
\begin{figure}[hbt]
\centering
\begin{minipage}[t]{7.5cm}
\includegraphics[trim= 0.5cm 0.5cm 0.5cm 0.5cm,clip,width=7.0cm]{surfdenspro1}
\caption{Density profiles of peptides in aqueous solution in various pores: hydrophilic peptides in hydrophobic (PISO) and hydrophilic  pores (PISI and left vertical axis); hydrophobic peptides in the hydrophilic pore (POSI and right vertical axis). The scales of the left and right are proportional to the molecular weights of the hydrophilic and hydrophobic peptides. The green vertical lines represent the position of the pore walls.}
\label{fig:surfdenspro1}
\end{minipage}
\hfill
\begin{minipage}[t]{7.5cm}
\includegraphics[trim= 0.5cm 0.5cm 0.5cm 0.5cm,clip,width=7.0cm]{surfdenspro2}
\caption{Density profiles of hydrophobic peptides in the slab of aqueous solution (left panel, POSV) and in aqueous solution in the hydrophobic pore (right panel, POSO). The density profiles of the centers of mass of the peptides are shown by the blue lines (POSV-com and POSO-com). The midpoints of the liquid-vapor interface and the position of the pore walls are shown by the green dashed and solid lines, respectively.}
\label{fig:surfdenspro2}
\end{minipage}
 \end{figure}
The probability distribution of the angle $\alpha$ between the pore surface and the vector connecting two most distant peptide heavy atoms are shown in Figure \ref{fig:surfangl}. When peptides are repelled from the pore walls and localized in its center, the orientations of their longest axes are highly isotropic (left panel in Figure \ref{fig:surfangl}) and only a slight preferential orientation of these axes parallel to the wall can be noticed. The situation is quite different in the case of a strong adsorption of peptides at the interfaces (right panel in Figure \ref{fig:surfangl}). The localization near the interfaces essentially enhances the orientational ordering of peptides and makes their longest axes align parallel to the interfaces.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=14.00 cm]{surfangl}
\caption{Probability distribution of the angle $\alpha$  between the pore wall or the liquid-vapor interface and the vector connecting two most distant peptide heavy atoms. }
\label{fig:surfangl}
\end{center}
\end{figure}
\par
The droplet probability $R$ to have more than four peptides in a cluster based on the distances between the centers of mass of two pepties as a function of the distance $r_{crit}$ used as cutoff is shown in Figure \ref{fig:surfdrop}. The degree of aggregation of the hydrophobic and hydrophilic peptides in bulk solutions is quite similar (see upper panel in Figure \ref{fig:surfdrop}). It can be compared with that of the FLVHS peptides studied in Chapter \ref{ch:bulkphase}. The degree of aggregation is noticeably weaker in the latter case in a wide range of $r_{crit}$. This could be due to a weaker propensity of these peptides to aggregate, the presence of methyl caps in the FLVHS peptides, or due to the different force fields. The aggregation of hydrophilic peptides becomes weaker in both hydrophilic and hydrophobic pores (see middle panel in Figure \ref{fig:surfdrop}), as it can be inferred from a shift of the dependence  $R(r_{crit})$ to higher values of $r_{crit}$.  For hydrophobic peptides, the situation is opposite (lower panel in Figure \ref{fig:surfdrop}). The aggregation of hydrophobic peptides becomes slightly stronger upon confinement in hydrophilic pores, and it is enormously enhanced by confinement in the hydrophobic pore. 
\par
\begin{figure}[htb]
\centering
%\begin{minipage}[t]{7.5cm}
\includegraphics[width=9.00 cm]{surfdrop}
\caption{Dependence of the aggregation parameter $R$ on the distance $r_{crit}$ used as a criterion for inter-peptide connectivity. Upper panel: bulk aqueous solutions of peptides. Middle panel: aqueous solution of hydrophilic peptides in hydrophilic and hydrophobic pores. Lower panel: aqueous solution of hydrophobic peptides in pores. }
\label{fig:surfdrop}
%\end{minipage}
%\hfill
\end{figure}
%
The average peptide-water and inter-peptide hydrogen bonds, peptide-water contacts and the density of the hydration layer of the peptides for the various systems investigated are shown in Figure \ref{fig:surfavgprop}. The average number of peptide-water and inter-peptide hydrogen bonds is higher in the simulation runs with hydrophilic fragments, due to a higher number of polar atoms. Hydrophilic peptides also have fewer peptide-water hydrogen bonds upon bulk phase aggregation compared to the aggregation in pores. The number of water molecules in contact with the aggregate follows a similar trend: we observed a 12\% increase in the number of water molecules around the aggregate in pores as compared to the aggregate in the bulk. In the case of hydrophobic fragments, the number of peptide-water hydrogen bonds and water molecules in contact with the peptides decreases upon adsorption to the hydrophobic surface. As the peptides are aligned parallel to the surface, the formation of $\beta$-sheets is facilitated, thus compensating for the decreased peptide-water hydrogen bonding by formation of inter-peptide hydrogen bonds. As a result, an increase in the number of peptides adopting a  $\beta$-sheet conformation is observed upon adsorption. 
\par
The density of the hydration layer of the peptides for the various simulation runs (except for the hydrophobic peptides in hydrophobic pores) is shown in Figure \ref{fig:surfavgprop} (bottom left panel). In the case of hydrophilic peptides, the higher shell densities found for aggregates in pores as compared to aggregates in the bulk can be attributed to the suppression of aggregation of hydrophilic peptides in pores. The trend towards a more dense hydration shell upon increased aggregation has also been observed in the case of the hydrophobic fragment studied in Chapter \ref{ch:bulkphase} (see Figure \ref{fig:shelldens}). A similar trend is observed for the hydrophobic fragment in the hydrophilic pore, where aggregation is enhanced, as compared to the bulk simulations.
%
\begin{figure}[htb]
\includegraphics[trim= 0.5cm 0.5cm 0.5cm 0.7cm,clip,width=16.00 cm]{surfavgprop}
\caption{Average number of peptide-water hydrogen bonds (top left panel), water molecules in contact with peptides (top right panel), shell density (bottom left), and inter-peptide hydrogen bonds (bottom right) for each type of run labelled as follows: PI (hydrophilic fragment in bulk water), PISI (hydrophilic fragment in the hydrophilic pore), PISO (hydrophilic fragment in the hydrophobic pore), PO (hydrophobic fragment  in bulk water), POSO (hydrophobic fragment in the hydrophobic pore), POSI (hydrophobic fragment in the hydrophilic pore). Error bars represent RMSD from average values.}
\label{fig:surfavgprop}
\end{figure}
\clearpage
\section{Discussion}    
The effect of water mediated peptide-surface interactions is determined by the balance of  peptide-surface, peptide-water and  water-surface interactions. For simplicity, the direct peptide-surface interactions are switched off in these studies. The hydrophilic peptide is composed of residues having multiple polar atoms in its side chains, which makes water-peptide interactions much more favorable as compared to the case of the hydrophobic peptide. This prevents the adsorption of hydrophilic peptides at both hydrophilic and hydrophobic surfaces, as the adsorption would decrease the number of favorable peptide-water hydrogen bonds. On the contrary, the hydrophobic peptide has many apolar atoms, which makes direct peptide-water interactions much weaker as compared to water-water interactions. Hence, adsorption of hydrophobic peptides decreases the overall amount of hydrophobic surface exposed to water, and the formation of inter-peptide $\beta$-sheets leads to further stabilization. However, this trend can be overcome if the surface is hydrophilic enough. Near a hydrophilic silica-like surface, the tendency of the surface to be hydrated overcomes the tendency of the hydrophobic peptides to be dehydrated, which leads to their desorption from the surface. 
\par
Depending on the nature of the peptide and the surface, either an enhancement or a retardation of peptide aggregation can be observed. The adsorption of hydrophobic peptides on hydrophobic surfaces foster the aggregation drastically by facilitating the formation of inter-peptide $\beta$-sheets, which is a key structural element of amyloid fibril formation. A small increase in the aggregation propensity is also seen by confining hydrophobic peptides in hydrophilic pores, whereas confinement of hydrophilic peptides in either hydrophobic or hydrophilic pores decreases their aggregation propensity. 
