| 3 |
|
\section{Background on the Problem\label{In:sec:pro}} |
| 4 |
|
Phospholipid molecules are the primary topic of this dissertation |
| 5 |
|
because of their critical role as the foundation of biological |
| 6 |
< |
membranes. The chemical structure of phospholipids includes the polar |
| 7 |
< |
head group which is due to a large charge separation between phosphate |
| 8 |
< |
and amino alcohol, and the nonpolar tails that contains fatty acid |
| 9 |
< |
chains. Depending on the alcohol which phosphate and fatty acid chains |
| 10 |
< |
are esterified to, the phospholipids are divided into |
| 11 |
< |
glycerophospholipids and sphingophospholipids.~\cite{Cevc80} The |
| 12 |
< |
chemical structures are shown in figure~\ref{Infig:lipid}. |
| 6 |
> |
membranes. The chemical structure of phospholipids includes a head |
| 7 |
> |
group with a large dipole moment which is due to the large charge |
| 8 |
> |
separation between phosphate and amino alcohol, and a nonpolar tail |
| 9 |
> |
that contains fatty acid chains. Depending on the specific alcohol |
| 10 |
> |
which the phosphate and fatty acid chains are esterified to, the |
| 11 |
> |
phospholipids are divided into glycerophospholipids and |
| 12 |
> |
sphingophospholipids.~\cite{Cevc80} The chemical structures are shown |
| 13 |
> |
in figure~\ref{Infig:lipid}. |
| 14 |
|
\begin{figure} |
| 15 |
|
\centering |
| 16 |
|
\includegraphics[width=\linewidth]{./figures/inLipid.pdf} |
| 17 |
< |
\caption{The chemical structure of glycerophospholipids (left) and |
| 18 |
< |
sphingophospholipids (right).\cite{Cevc80}} |
| 17 |
> |
\caption[The chemical structure of lipids]{The chemical structure of |
| 18 |
> |
glycerophospholipids (left) and sphingophospholipids |
| 19 |
> |
(right).\cite{Cevc80}} |
| 20 |
|
\label{Infig:lipid} |
| 21 |
|
\end{figure} |
| 22 |
< |
The glycerophospholipid is the dominant phospholipid in membranes. The |
| 23 |
< |
types of glycerophospholipids depend on the X group, and the |
| 24 |
< |
chains. For example, if X is choline, the lipids are known as |
| 25 |
< |
phosphatidylcholine (PC), or if X is ethanolamine, the lipids are |
| 26 |
< |
known as phosphatidyethanolamine (PE). Table~\ref{Intab:pc} listed a |
| 27 |
< |
number types of phosphatidycholine with different fatty acids as the |
| 28 |
< |
lipid chains. |
| 22 |
> |
Glycerophospholipids are the dominant phospholipids in biological |
| 23 |
> |
membranes. The type of glycerophospholipid depends on the identity of |
| 24 |
> |
the X group, and the chains. For example, if X is choline |
| 25 |
> |
[(CH$_3$)$_3$N$^+$CH$_2$CH$_2$OH], the lipids are known as |
| 26 |
> |
phosphatidylcholine (PC), or if X is ethanolamine |
| 27 |
> |
[H$_3$N$^+$CH$_2$CH$_2$OH], the lipids are known as |
| 28 |
> |
phosphatidyethanolamine (PE). Table~\ref{Intab:pc} listed a number |
| 29 |
> |
types of phosphatidycholine with different fatty acids as the lipid |
| 30 |
> |
chains. |
| 31 |
|
\begin{table*} |
| 32 |
|
\begin{minipage}{\linewidth} |
| 33 |
|
\begin{center} |
| 34 |
< |
\caption{A number types of phosphatidycholine.} |
| 34 |
> |
\caption{A NUMBER TYPES OF PHOSPHATIDYCHOLINE} |
| 35 |
|
\begin{tabular}{lll} |
| 36 |
|
\hline |
| 37 |
|
& Fatty acid & Generic Name \\ |
| 46 |
|
\end{center} |
| 47 |
|
\end{minipage} |
| 48 |
|
\end{table*} |
| 49 |
< |
When dispersed in water, lipids self assemble into a mumber of |
| 49 |
> |
When dispersed in water, lipids self assemble into a number of |
| 50 |
|
topologically distinct bilayer structures. The phase behavior of lipid |
| 51 |
|
bilayers has been explored experimentally~\cite{Cevc80}, however, a |
| 52 |
|
complete understanding of the mechanism and driving forces behind the |
| 53 |
|
various phases has not been achieved. |
| 54 |
|
|
| 55 |
< |
\subsection{Ripple Phase\label{In:ssec:ripple}} |
| 55 |
> |
\subsection{The Ripple Phase\label{In:ssec:ripple}} |
| 56 |
|
The $P_{\beta'}$ {\it ripple phase} of lipid bilayers, named from the |
| 57 |
|
periodic buckling of the membrane, is an intermediate phase which is |
| 58 |
|
developed either from heating the gel phase $L_{\beta'}$ or cooling |
| 59 |
< |
the fluid phase $L_\alpha$. A Sketch is shown in |
| 59 |
> |
the fluid phase $L_\alpha$. A sketch of the phases is shown in |
| 60 |
|
figure~\ref{Infig:phaseDiagram}.~\cite{Cevc80} |
| 61 |
|
\begin{figure} |
| 62 |
|
\centering |
| 63 |
|
\includegraphics[width=\linewidth]{./figures/inPhaseDiagram.pdf} |
| 64 |
< |
\caption{A phase diagram of lipid bilayer. With increasing the |
| 65 |
< |
temperature, the bilayer can go through a gel ($L_{\beta'}$), ripple |
| 66 |
< |
($P_{\beta'}$) to fluid ($L_\alpha$) phase transition.~\cite{Cevc80}} |
| 64 |
> |
\caption[Phases of PC lipid bilayers]{Phases of PC lipid |
| 65 |
> |
bilayers. With increasing temperature, phosphotidylcholine (PC) |
| 66 |
> |
bilayers can go through $L_{\beta'} \rightarrow P_{\beta'}$ (gel |
| 67 |
> |
$\rightarrow$ ripple) and $P_{\beta'} \rightarrow L_\alpha$ (ripple |
| 68 |
> |
$\rightarrow$ fluid) phase transitions.~\cite{Cevc80}} |
| 69 |
|
\label{Infig:phaseDiagram} |
| 70 |
|
\end{figure} |
| 71 |
< |
Most structural information of the ripple phase has been obtained by |
| 72 |
< |
the X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture |
| 71 |
> |
Most structural information about the ripple phase has been obtained |
| 72 |
> |
by X-ray diffraction~\cite{Sun96,Katsaras00} and freeze-fracture |
| 73 |
|
electron microscopy (FFEM).~\cite{Copeland80,Meyer96} The X-ray |
| 74 |
|
diffraction work by Katsaras {\it et al.} showed that a rich phase |
| 75 |
|
diagram exhibiting both {\it asymmetric} and {\it symmetric} ripples |
| 80 |
|
\begin{figure} |
| 81 |
|
\centering |
| 82 |
|
\includegraphics[width=\linewidth]{./figures/inRipple.pdf} |
| 83 |
< |
\caption{The experimental observed ripple phase. The top image is |
| 84 |
< |
obtained by Sun {\it et al.} using X-ray diffraction~\cite{Sun96}, |
| 85 |
< |
and the bottom one is observed by Kaasgaard {\it et al.} using |
| 86 |
< |
AFM.~\cite{Kaasgaard03}} |
| 83 |
> |
\caption[Experimental observations of the riple phase]{Experimental |
| 84 |
> |
observations of the riple phase. The top image is an electrostatic |
| 85 |
> |
density map obtained by Sun {\it et al.} using X-ray |
| 86 |
> |
diffraction~\cite{Sun96}. The lower figures are the surface topology |
| 87 |
> |
of various ripple domains in bilayers supported in mica. The AFM |
| 88 |
> |
images were observed by Kaasgaard {\it et al.}.~\cite{Kaasgaard03}} |
| 89 |
|
\label{Infig:ripple} |
| 90 |
|
\end{figure} |
| 91 |
< |
Figure~\ref{Infig:ripple} shows the ripple phase oberved by X-ray |
| 91 |
> |
Figure~\ref{Infig:ripple} shows the ripple phase oberved by both X-ray |
| 92 |
|
diffraction and AFM. The experimental results provide strong support |
| 93 |
|
for a 2-dimensional triangular packing lattice of the lipid molecules |
| 94 |
|
within the ripple phase. This is a notable change from the observed |
| 95 |
|
lipid packing within the gel phase,~\cite{Cevc80} although Tenchov |
| 96 |
< |
{\it et al.} have recently observed near-hexagonal packing in some |
| 96 |
> |
{\it et al.} have recently observed near-triangular packing in some |
| 97 |
|
phosphatidylcholine (PC) gel phases.~\cite{Tenchov2001} However, the |
| 98 |
|
physical mechanism for the formation of the ripple phase has never |
| 99 |
|
been explained and the microscopic structure of the ripple phase has |
| 100 |
|
never been elucidated by experiments. Computational simulation is a |
| 101 |
< |
perfect tool to study the microscopic properties for a |
| 102 |
< |
system. However, the large length scale the ripple structure and the |
| 103 |
< |
long time scale of the formation of the ripples are crucial obstacles |
| 104 |
< |
to performing the actual work. The principal ideas explored in this |
| 105 |
< |
dissertation are attempts to break the computational task up by |
| 101 |
> |
very good tool to study the microscopic properties for a |
| 102 |
> |
system. However, the large length scale of the ripple structures and |
| 103 |
> |
the long time required for the formation of the ripples are crucial |
| 104 |
> |
obstacles to performing the actual work. The principal ideas explored |
| 105 |
> |
in this dissertation are attempts to break the computational task up |
| 106 |
> |
by |
| 107 |
|
\begin{itemize} |
| 108 |
|
\item Simplifying the lipid model. |
| 109 |
< |
\item Improving algorithm for integrating the equations of motion. |
| 109 |
> |
\item Improving the algorithm for integrating the equations of motion. |
| 110 |
|
\end{itemize} |
| 111 |
|
In chapters~\ref{chap:mc} and~\ref{chap:md}, we use a simple point |
| 112 |
< |
dipole spin model and a coarse-grained molecualr scale model to |
| 112 |
> |
dipole spin model and a coarse-grained molecular scale model to |
| 113 |
|
perform the Monte Carlo and Molecular Dynamics simulations |
| 114 |
|
respectively, and in chapter~\ref{chap:ld}, we develop a Langevin |
| 115 |
|
Dynamics algorithm which excludes the explicit solvent to improve the |
| 116 |
|
efficiency of the simulations. |
| 117 |
|
|
| 118 |
< |
\subsection{Lattice Model\label{In:ssec:model}} |
| 119 |
< |
The gel-like characteristic (relatively immobile molecules) exhibited |
| 118 |
> |
\subsection{Lattice Models\label{In:ssec:model}} |
| 119 |
> |
The gel-like characteristic (laterally immobile molecules) exhibited |
| 120 |
|
by the ripple phase makes it feasible to apply a lattice model to |
| 121 |
|
study the system. The popular $2$ dimensional lattice models, {\it |
| 122 |
|
i.e.}, the Ising, $X-Y$, and Heisenberg models, show {\it frustration} |
| 137 |
|
\begin{figure} |
| 138 |
|
\centering |
| 139 |
|
\includegraphics[width=3in]{./figures/inFrustration.pdf} |
| 140 |
< |
\caption{Frustration on triangular lattice, the spins and dipoles are |
| 141 |
< |
represented by arrows. The multiple local minima of energy states |
| 142 |
< |
induce the frustration for spins and dipoles picking the directions.} |
| 140 |
> |
\caption[Frustration on triangular lattice]{Frustration on triangular |
| 141 |
> |
lattice, the spins and dipoles are represented by arrows. The multiple |
| 142 |
> |
local minima of energy states induce frustration for spins and dipoles |
| 143 |
> |
resulting in disordered low-temperature phases.} |
| 144 |
|
\label{Infig:frustration} |
| 145 |
|
\end{figure} |
| 146 |
< |
The spins in figure~\ref{Infig:frustration} shows an illustration of |
| 147 |
< |
the frustration for $J < 0$ on a triangular lattice. There are |
| 148 |
< |
multiple local minima energy states which are independent of the |
| 149 |
< |
direction of the spin on top of the triangle, therefore infinite |
| 150 |
< |
possibilities for the packing of spins which induces what is known as |
| 151 |
< |
``complete regular frustration'' which leads to disordered low |
| 152 |
< |
temperature phases. The similarity goes to the dipoles on a hexagonal |
| 153 |
< |
lattice, which are shown by the dipoles in |
| 154 |
< |
figure~\ref{Infig:frustration}. In this circumstance, the dipoles want |
| 155 |
< |
to be aligned, however, due to the long wave fluctuation, at low |
| 156 |
< |
temperature, the aligned state becomes unstable, vortex is formed and |
| 157 |
< |
results in multiple local minima of energy states. The dipole on the |
| 148 |
< |
center of the hexagonal lattice is frustrated. |
| 146 |
> |
The spins in figure~\ref{Infig:frustration} illustrate frustration for |
| 147 |
> |
$J < 0$ on a triangular lattice. There are multiple local minima |
| 148 |
> |
energy states which are independent of the direction of the spin on |
| 149 |
> |
top of the triangle, therefore infinite possibilities for orienting |
| 150 |
> |
large numbers spins. This induces what is known as ``complete regular |
| 151 |
> |
frustration'' which leads to disordered low temperature phases. This |
| 152 |
> |
behavior extends to dipoles on a triangular lattices, which are shown |
| 153 |
> |
in the lower portion of figure~\ref{Infig:frustration}. In this case, |
| 154 |
> |
dipole-aligned structures are energetically favorable, however, at low |
| 155 |
> |
temperature, vortices are easily formed, and, this results in multiple |
| 156 |
> |
local minima of energy states for a central dipole. The dipole on the |
| 157 |
> |
center of the hexagonal lattice is therefore frustrated. |
| 158 |
|
|
| 159 |
|
The lack of translational degrees of freedom in lattice models |
| 160 |
|
prevents their utilization in models for surface buckling. In |
| 168 |
|
to key concepts of classical statistical mechanics that we used in |
| 169 |
|
this dissertation. Tolman gives an excellent introduction to the |
| 170 |
|
principles of statistical mechanics.~\cite{Tolman1979} A large part of |
| 171 |
< |
section~\ref{In:sec:SM} will follow Tolman's notation. |
| 171 |
> |
section~\ref{In:sec:SM} follows Tolman's notation. |
| 172 |
|
|
| 173 |
|
\subsection{Ensembles\label{In:ssec:ensemble}} |
| 174 |
|
In classical mechanics, the state of the system is completely |
| 278 |
|
and the rate of density change is zero in the neighborhood of any |
| 279 |
|
selected moving representative points in the phase space. |
| 280 |
|
|
| 281 |
< |
The condition of the ensemble is determined by the density |
| 281 |
> |
The type of thermodynamic ensemble is determined by the density |
| 282 |
|
distribution. If we consider the density distribution as only a |
| 283 |
|
function of $q$ and $p$, which means the rate of change of the phase |
| 284 |
|
space density in the neighborhood of all representative points in the |
| 307 |
|
\rho = \mathrm{const.} |
| 308 |
|
\label{Ineq:uniformEnsemble} |
| 309 |
|
\end{equation} |
| 310 |
< |
the ensemble is called {\it uniform ensemble}. |
| 310 |
> |
the ensemble is called {\it uniform ensemble}, but this ensemble has |
| 311 |
> |
little relevance for physical chemistry. It is an ensemble with |
| 312 |
> |
essentially infinite temperature. |
| 313 |
|
|
| 314 |
|
\subsubsection{The Microcanonical Ensemble\label{In:sssec:microcanonical}} |
| 315 |
< |
Another useful ensemble is the {\it microcanonical ensemble}, for |
| 316 |
< |
which: |
| 315 |
> |
The most useful ensemble for Molecular Dynamics is the {\it |
| 316 |
> |
microcanonical ensemble}, for which: |
| 317 |
|
\begin{equation} |
| 318 |
|
\rho = \delta \left( H(q^N, p^N) - E \right) \frac{1}{\Sigma (N, V, E)} |
| 319 |
|
\label{Ineq:microcanonicalEnsemble} |
| 330 |
|
\end{equation} |
| 331 |
|
where $k_B$ is the Boltzmann constant and $C^N$ is a number which |
| 332 |
|
makes the argument of $\ln$ dimensionless. In this case, $C^N$ is the |
| 333 |
< |
total phase space volume of one state. The entropy of a microcanonical |
| 334 |
< |
ensemble is given by |
| 333 |
> |
total phase space volume of one state which has the same units as |
| 334 |
> |
$\Sigma(N, V, E)$. The entropy of a microcanonical ensemble is given |
| 335 |
> |
by |
| 336 |
|
\begin{equation} |
| 337 |
|
S = k_B \ln \left(\frac{\Sigma(N, V, E)}{C^N}\right). |
| 338 |
|
\label{Ineq:entropy} |
| 349 |
|
Z_N = \int d \vec q~^N \int_\Gamma d \vec p~^N e^{-H(q^N, p^N) / k_B T}, |
| 350 |
|
\label{Ineq:partitionFunction} |
| 351 |
|
\end{equation} |
| 352 |
< |
which is also known as the canonical{\it partition function}. $\Gamma$ |
| 353 |
< |
indicates that the integral is over all phase space. In the canonical |
| 354 |
< |
ensemble, $N$, the total number of particles, $V$, total volume, and |
| 355 |
< |
$T$, the temperature, are constants. The systems with the lowest |
| 356 |
< |
energies hold the largest population. According to maximum principle, |
| 357 |
< |
thermodynamics maximizes the entropy $S$, implying that |
| 352 |
> |
which is also known as the canonical {\it partition |
| 353 |
> |
function}. $\Gamma$ indicates that the integral is over all phase |
| 354 |
> |
space. In the canonical ensemble, $N$, the total number of particles, |
| 355 |
> |
$V$, total volume, and $T$, the temperature, are constants. The |
| 356 |
> |
systems with the lowest energies hold the largest |
| 357 |
> |
population. Thermodynamics maximizes the entropy, $S$, implying that |
| 358 |
|
\begin{equation} |
| 359 |
|
\begin{array}{ccc} |
| 360 |
|
\delta S = 0 & \mathrm{and} & \delta^2 S < 0. |
| 375 |
|
system and the distribution of microscopic states. |
| 376 |
|
|
| 377 |
|
There is an implicit assumption that our arguments are based on so |
| 378 |
< |
far. A representative point in the phase space is equally likely to be |
| 379 |
< |
found in any energetically allowed region. In other words, all |
| 380 |
< |
energetically accessible states are represented equally, the |
| 381 |
< |
probabilities to find the system in any of the accessible states is |
| 382 |
< |
equal. This is called the principle of equal a {\it priori} |
| 378 |
> |
far. Tow representative points in phase space are equally likely to be |
| 379 |
> |
found if they have the same energy. In other words, all energetically |
| 380 |
> |
accessible states are represented , and the probabilities to find the |
| 381 |
> |
system in any of the accessible states is equal to that states |
| 382 |
> |
Boltzmann weight. This is called the principle of equal a {\it priori} |
| 383 |
|
probabilities. |
| 384 |
|
|
| 385 |
|
\subsection{Statistical Averages\label{In:ssec:average}} |
| 414 |
|
\frac{1}{T} \int_{0}^{T} F[q^N(t), p^N(t)] dt |
| 415 |
|
\label{Ineq:timeAverage2} |
| 416 |
|
\end{equation} |
| 417 |
< |
for an infinite time interval. |
| 417 |
> |
for an finite time interval, $T$. |
| 418 |
|
|
| 419 |
|
\subsubsection{Ergodicity\label{In:sssec:ergodicity}} |
| 420 |
|
The {\it ergodic hypothesis}, an important hypothesis governing modern |
| 457 |
|
\frac{1}{T} \int_{0}^{T} dt A(t) B(t + \tau), |
| 458 |
|
\label{Ineq:crosscorrelationFunction} |
| 459 |
|
\end{equation} |
| 460 |
< |
and called {\it cross correlation function}. |
| 460 |
> |
and is called a {\it cross correlation function}. |
| 461 |
|
|
| 462 |
|
We know from the ergodic hypothesis that there is a relationship |
| 463 |
|
between time average and ensemble average. We can put the correlation |
| 464 |
< |
function in a classical mechanics form, |
| 464 |
> |
function in a classical mechanical form, |
| 465 |
|
\begin{equation} |
| 466 |
< |
C_{AA}(\tau) = \int d \vec q~^N \int d \vec p~^N A[(q^N(t), p^N(t)] |
| 467 |
< |
A[q^N(t+\tau), p^N(t+\tau)] \rho(q, p) |
| 466 |
> |
C_{AA}(\tau) = \int d \vec q~^N \int d \vec p~^N A[(q^N, p^N] |
| 467 |
> |
A[q^N(\tau), p^N(\tau)] \rho(q^N, p^N) |
| 468 |
|
\label{Ineq:autocorrelationFunctionCM} |
| 469 |
|
\end{equation} |
| 470 |
< |
and |
| 470 |
> |
where $q^N(\tau)$, $p^N(\tau)$ is the phase space point that follows |
| 471 |
> |
classical evolution of the point $q^N$, $p^N$ after a tme $\tau$ has |
| 472 |
> |
elapsed, and |
| 473 |
|
\begin{equation} |
| 474 |
< |
C_{AB}(\tau) = \int d \vec q~^N \int d \vec p~^N A[(q^N(t), p^N(t)] |
| 475 |
< |
B[q^N(t+\tau), p^N(t+\tau)] \rho(q, p) |
| 474 |
> |
C_{AB}(\tau) = \int d \vec q~^N \int d \vec p~^N A[(q^N, p^N] |
| 475 |
> |
B[q^N(\tau), p^N(\tau)] \rho(q^N, p^N) |
| 476 |
|
\label{Ineq:crosscorrelationFunctionCM} |
| 477 |
|
\end{equation} |
| 478 |
|
as the autocorrelation function and cross correlation functions |
| 493 |
|
simulations. One is that the Monte Carlo simulations are time |
| 494 |
|
independent methods of sampling structural features of an ensemble, |
| 495 |
|
while Molecular Dynamics simulations provide dynamic |
| 496 |
< |
information. Additionally, Monte Carlo methods are a stochastic |
| 497 |
< |
processes, the future configurations of the system are not determined |
| 496 |
> |
information. Additionally, Monte Carlo methods are stochastic |
| 497 |
> |
processes; the future configurations of the system are not determined |
| 498 |
|
by its past. However, in Molecular Dynamics, the system is propagated |
| 499 |
< |
by Newton's equation of motion, and the trajectory of the system |
| 499 |
> |
by Hamilton's equations of motion, and the trajectory of the system |
| 500 |
|
evolving in phase space is deterministic. Brief introductions of the |
| 501 |
|
two algorithms are given in section~\ref{In:ssec:mc} |
| 502 |
|
and~\ref{In:ssec:md}. Langevin Dynamics, an extension of the Molecular |
| 507 |
|
A Monte Carlo integration algorithm was first introduced by Metropolis |
| 508 |
|
{\it et al.}~\cite{Metropolis53} The basic Metropolis Monte Carlo |
| 509 |
|
algorithm is usually applied to the canonical ensemble, a |
| 510 |
< |
Boltzmann-weighted ensemble, in which the $N$, the total number of |
| 511 |
< |
particles, $V$, total volume, $T$, temperature are constants. An |
| 512 |
< |
average in this ensemble is given |
| 510 |
> |
Boltzmann-weighted ensemble, in which $N$, the total number of |
| 511 |
> |
particles, $V$, the total volume, and $T$, the temperature are |
| 512 |
> |
constants. An average in this ensemble is given |
| 513 |
|
\begin{equation} |
| 514 |
|
\langle A \rangle = \frac{1}{Z_N} \int d \vec q~^N \int d \vec p~^N |
| 515 |
|
A(q^N, p^N) e^{-H(q^N, p^N) / k_B T}. |
| 533 |
|
\rangle$ is now a configuration integral, and |
| 534 |
|
Eq.~\ref{Ineq:configurationIntegral} is equivalent to |
| 535 |
|
\begin{equation} |
| 536 |
< |
\langle A \rangle = \int d \vec q~^N A \rho(q^N). |
| 536 |
> |
\langle A \rangle = \int d \vec q~^N A \rho(q^N), |
| 537 |
|
\label{Ineq:configurationAve} |
| 538 |
|
\end{equation} |
| 539 |
+ |
where $\rho(q^N)$ is a configurational probability |
| 540 |
+ |
\begin{equation} |
| 541 |
+ |
\rho(q^N) = \frac{e^{-U(q^N) / k_B T}}{\int d \vec q~^N e^{-U(q^N) / k_B T}}. |
| 542 |
+ |
\label{Ineq:configurationProb} |
| 543 |
+ |
\end{equation} |
| 544 |
|
|
| 545 |
|
In a Monte Carlo simulation of a system in the canonical ensemble, the |
| 546 |
|
probability of the system being in a state $s$ is $\rho_s$, the change |
| 555 |
|
\frac{d \rho_{s}^{equilibrium}}{dt} = 0, |
| 556 |
|
\label{Ineq:equiProb} |
| 557 |
|
\end{equation} |
| 558 |
< |
which means $\sum_{s'} [ -w_{ss'}\rho_s + w_{s's}\rho_{s'} ]$ is $0$ |
| 559 |
< |
for all $s'$. So |
| 558 |
> |
the sum of transition probabilities $\sum_{s'} [ -w_{ss'}\rho_s + |
| 559 |
> |
w_{s's}\rho_{s'} ]$ is $0$ for all $s'$. So |
| 560 |
|
\begin{equation} |
| 561 |
|
\frac{\rho_s^{equilibrium}}{\rho_{s'}^{equilibrium}} = \frac{w_{s's}}{w_{ss'}}. |
| 562 |
|
\label{Ineq:relationshipofRhoandW} |
| 571 |
|
\frac{w_{s's}}{w_{ss'}} = e^{-(U_s - U_{s'}) / k_B T}, |
| 572 |
|
\label{Ineq:conditionforBoltzmannStatistics} |
| 573 |
|
\end{equation} |
| 574 |
< |
An algorithm that shows how Monte Carlo simulation generates a |
| 574 |
> |
An algorithm that indicates how a Monte Carlo simulation generates a |
| 575 |
|
transition probability governed by |
| 576 |
|
\ref{Ineq:conditionforBoltzmannStatistics}, is given schematically as, |
| 577 |
|
\begin{enumerate} |
| 578 |
|
\item\label{Initm:oldEnergy} Choose a particle randomly, and calculate |
| 579 |
< |
the energy of the rest of the system due to the location of the particle. |
| 579 |
> |
the energy of the rest of the system due to the current location of |
| 580 |
> |
the particle. |
| 581 |
|
\item\label{Initm:newEnergy} Make a random displacement of the particle, |
| 582 |
|
calculate the new energy taking the movement of the particle into account. |
| 583 |
|
\begin{itemize} |
| 584 |
< |
\item If the energy goes down, keep the new configuration and return to |
| 565 |
< |
step~\ref{Initm:oldEnergy}. |
| 584 |
> |
\item If the energy goes down, keep the new configuration. |
| 585 |
|
\item If the energy goes up, pick a random number between $[0,1]$. |
| 586 |
|
\begin{itemize} |
| 587 |
|
\item If the random number smaller than |
| 588 |
< |
$e^{-(U_{new} - U_{old})} / k_B T$, keep the new configuration and return to |
| 570 |
< |
step~\ref{Initm:oldEnergy}. |
| 588 |
> |
$e^{-(U_{new} - U_{old})} / k_B T$, keep the new configuration. |
| 589 |
|
\item If the random number is larger than |
| 590 |
< |
$e^{-(U_{new} - U_{old})} / k_B T$, keep the old configuration and return to |
| 573 |
< |
step~\ref{Initm:oldEnergy}. |
| 590 |
> |
$e^{-(U_{new} - U_{old})} / k_B T$, keep the old configuration. |
| 591 |
|
\end{itemize} |
| 592 |
|
\end{itemize} |
| 593 |
|
\item\label{Initm:accumulateAvg} Accumulate the averages based on the |
| 594 |
< |
current configuartion. |
| 594 |
> |
current configuration. |
| 595 |
|
\item Go to step~\ref{Initm:oldEnergy}. |
| 596 |
|
\end{enumerate} |
| 597 |
< |
It is important for sampling accurately that the old configuration is |
| 597 |
> |
It is important for sampling accuracy that the old configuration is |
| 598 |
|
sampled again if it is kept. |
| 599 |
|
|
| 600 |
|
\subsection{Molecular Dynamics\label{In:ssec:md}} |
| 606 |
|
evolution of the system obeys the laws of classical mechanics, and in |
| 607 |
|
most situations, there is no need to account for quantum effects. In a |
| 608 |
|
real experiment, the instantaneous positions and momenta of the |
| 609 |
< |
particles in the system are neither important nor measurable, the |
| 610 |
< |
observable quantities are usually an average value for a finite time |
| 611 |
< |
interval. These quantities are expressed as a function of positions |
| 612 |
< |
and momenta in Molecular Dynamics simulations. For example, |
| 609 |
> |
particles in the system are ofter neither important nor measurable, |
| 610 |
> |
the observable quantities are usually an average value for a finite |
| 611 |
> |
time interval. These quantities are expressed as a function of |
| 612 |
> |
positions and momenta in Molecular Dynamics simulations. For example, |
| 613 |
|
temperature of the system is defined by |
| 614 |
|
\begin{equation} |
| 615 |
|
\frac{3}{2} N k_B T = \langle \sum_{i=1}^N \frac{1}{2} m_i v_i \rangle, |
| 622 |
|
The initial positions of the particles are chosen so that there is no |
| 623 |
|
overlap of the particles. The initial velocities at first are |
| 624 |
|
distributed randomly to the particles using a Maxwell-Boltzmann |
| 625 |
< |
ditribution, and then shifted to make the total linear momentum of the |
| 626 |
< |
system $0$. |
| 625 |
> |
distribution, and then shifted to make the total linear momentum of |
| 626 |
> |
the system $0$. |
| 627 |
|
|
| 628 |
|
The core of Molecular Dynamics simulations is the calculation of |
| 629 |
|
forces and the integration algorithm. Calculation of the forces often |
| 649 |
|
equations of motion. The Taylor expansion of the position at time $t$ |
| 650 |
|
is |
| 651 |
|
\begin{equation} |
| 652 |
< |
q(t+\Delta t)= q(t) + v(t) \Delta t + \frac{f(t)}{2m}\Delta t^2 + |
| 652 |
> |
q(t+\Delta t)= q(t) + \frac{p(t)}{m} \Delta t + \frac{f(t)}{2m}\Delta t^2 + |
| 653 |
|
\frac{\Delta t^3}{3!}\frac{\partial^3 q(t)}{\partial t^3} + |
| 654 |
|
\mathcal{O}(\Delta t^4) |
| 655 |
|
\label{Ineq:verletFuture} |
| 656 |
|
\end{equation} |
| 657 |
|
for a later time $t+\Delta t$, and |
| 658 |
|
\begin{equation} |
| 659 |
< |
q(t-\Delta t)= q(t) - v(t) \Delta t + \frac{f(t)}{2m}\Delta t^2 - |
| 659 |
> |
q(t-\Delta t)= q(t) - \frac{p(t)}{m} \Delta t + \frac{f(t)}{2m}\Delta t^2 - |
| 660 |
|
\frac{\Delta t^3}{3!}\frac{\partial^3 q(t)}{\partial t^3} + |
| 661 |
|
\mathcal{O}(\Delta t^4) , |
| 662 |
|
\label{Ineq:verletPrevious} |
| 683 |
|
code can be found in Ref.~\cite{Meineke2005}. |
| 684 |
|
|
| 685 |
|
\subsection{Langevin Dynamics\label{In:ssec:ld}} |
| 686 |
< |
In many cases, the properites of the solvent in a system, like the |
| 687 |
< |
water in the lipid-water system studied in this dissertation, are less |
| 688 |
< |
interesting to the researchers than the behavior of the |
| 689 |
< |
solute. However, the major computational expense is ofter the |
| 690 |
< |
solvent-solvent interation, this is due to the large number of the |
| 691 |
< |
solvent molecules when compared to the number of solute molecules, the |
| 692 |
< |
ratio of the number of lipid molecules to the number of water |
| 693 |
< |
molecules is $1:25$ in our lipid-water system. The efficiency of the |
| 694 |
< |
Molecular Dynamics simulations is greatly reduced, with up to 85\% of |
| 695 |
< |
CPU time spent calculating only water-water interactions. |
| 686 |
> |
In many cases, the properites of the solvent (like the water in the |
| 687 |
> |
lipid-water system studied in this dissertation) are less interesting |
| 688 |
> |
to the researchers than the behavior of the solute. However, the major |
| 689 |
> |
computational expense is ofter the solvent-solvent interactions, this |
| 690 |
> |
is due to the large number of the solvent molecules when compared to |
| 691 |
> |
the number of solute molecules. The ratio of the number of lipid |
| 692 |
> |
molecules to the number of water molecules is $1:25$ in our |
| 693 |
> |
lipid-water system. The efficiency of the Molecular Dynamics |
| 694 |
> |
simulations is greatly reduced, with up to 85\% of CPU time spent |
| 695 |
> |
calculating only water-water interactions. |
| 696 |
|
|
| 697 |
|
As an extension of the Molecular Dynamics methodologies, Langevin |
| 698 |
|
Dynamics seeks a way to avoid integrating the equations of motion for |
| 745 |
|
q_\alpha(0)}{\omega_\alpha} \sin(\omega_\alpha t), |
| 746 |
|
\label{Ineq:randomForceforGLE} |
| 747 |
|
\end{equation} |
| 748 |
< |
depends only on the initial conditions. The relationship of friction |
| 749 |
< |
kernel $\xi(t)$ and random force $R(t)$ is given by the second |
| 750 |
< |
fluctuation dissipation theorem, |
| 748 |
> |
that depends only on the initial conditions. The relationship of |
| 749 |
> |
friction kernel $\xi(t)$ and random force $R(t)$ is given by the |
| 750 |
> |
second fluctuation dissipation theorem, |
| 751 |
|
\begin{equation} |
| 752 |
|
\xi(t) = \frac{1}{k_B T} \langle R(t)R(0) \rangle. |
| 753 |
|
\label{Ineq:relationshipofXiandR} |
