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Revision 2725 by tim, Fri Apr 21 05:45:14 2006 UTC

+\]
+\section{\label{introSection:molecularDynamics}Molecular Dynamics}
-–
-–
+As a special discipline of molecular modeling, Molecular dynamics
-–
+has proven to be a powerful tool for studying the functions of
-–
+biological systems, providing structural, thermodynamic and
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+dynamical information.
-<
+One of the principal tools for modeling proteins, nucleic acids and
-<
+their complexes. Stability of proteins Folding of proteins.
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+Molecular recognition by:proteins, DNA, RNA, lipids, hormones STP,
-<
+etc. Enzyme reactions Rational design of biologically active
-<
+molecules (drug design) Small and large-scale conformational
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+changes. determination and construction of 3D structures (homology,
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+Xray diffraction, NMR) Dynamic processes such as ion transport in
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+biological systems.
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+Macroscopic properties are related to microscopic behavior.
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-<
+Time dependent (and independent) microscopic behavior of a molecule
-<
+can be calculated by molecular dynamics simulations.
-<
-<
+\subsection{\label{introSec:mdInit}Initialization}
->
+As one of the principal tools of molecular modeling, Molecular
->
+dynamics has proven to be a powerful tool for studying the functions
->
+of biological systems, providing structural, thermodynamic and
->
+dynamical information. The basic idea of molecular dynamics is that
->
+macroscopic properties are related to microscopic behavior and
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+microscopic behavior can be calculated from the trajectories in
->
+simulations. For instance, instantaneous temperature of an
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+Hamiltonian system of $N$ particle can be measured by
->
+\[
->
+T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}}
->
+\]
->
+where $m_i$ and $v_i$ are the mass and velocity of $i$th particle
->
+respectively, $f$ is the number of degrees of freedom, and $k_B$ is
->
+the boltzman constant.
-<
+\subsection{\label{introSec:forceEvaluation}Force Evaluation}
->
+A typical molecular dynamics run consists of three essential steps:
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+\begin{enumerate}
->
+  \item Initialization
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+    \begin{enumerate}
->
+    \item Preliminary preparation
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+    \item Minimization
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+    \item Heating
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+    \item Equilibration
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+    \end{enumerate}
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+  \item Production
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+  \item Analysis
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+\end{enumerate}
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+These three individual steps will be covered in the following
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+sections. Sec.~\ref{introSec:initialSystemSettings} deals with the
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+initialization of a simulation. Sec.~\ref{introSec:production} will
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+discusses issues in production run. Sec.~\ref{introSection:Analysis}
->
+provides the theoretical tools for trajectory analysis.
-<
+\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion}
->
+\subsection{\label{introSec:initialSystemSettings}Initialization}
->
->
+\subsubsection{Preliminary preparation}
->
->
+When selecting the starting structure of a molecule for molecular
->
+simulation, one may retrieve its Cartesian coordinates from public
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+databases, such as RCSB Protein Data Bank \textit{etc}. Although
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+thousands of crystal structures of molecules are discovered every
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+year, many more remain unknown due to the difficulties of
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+purification and crystallization. Even for the molecule with known
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+structure, some important information is missing. For example, the
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+missing hydrogen atom which acts as donor in hydrogen bonding must
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+be added. Moreover, in order to include electrostatic interaction,
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+one may need to specify the partial charges for individual atoms.
->
+Under some circumstances, we may even need to prepare the system in
->
+a special setup. For instance, when studying transport phenomenon in
->
+membrane system, we may prepare the lipids in bilayer structure
->
+instead of placing lipids randomly in solvent, since we are not
->
+interested in self-aggregation and it takes a long time to happen.
->
->
+\subsubsection{Minimization}
->
->
+It is quite possible that some of molecules in the system from
->
+preliminary preparation may be overlapped with each other. This
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+close proximity leads to high potential energy which consequently
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+jeopardizes any molecular dynamics simulations. To remove these
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+steric overlaps, one typically performs energy minimization to find
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+a more reasonable conformation. Several energy minimization methods
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+have been developed to exploit the energy surface and to locate the
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+local minimum. While converging slowly near the minimum, steepest
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+descent method is extremely robust when systems are far from
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+harmonic. Thus, it is often used to refine structure from
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+crystallographic data. Relied on the gradient or hessian, advanced
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+methods like conjugate gradient and Newton-Raphson converge rapidly
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+to a local minimum, while become unstable if the energy surface is
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+far from quadratic. Another factor must be taken into account, when
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+choosing energy minimization method, is the size of the system.
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+Steepest descent and conjugate gradient can deal with models of any
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+size. Because of the limit of computation power to calculate hessian
->
+matrix and insufficient storage capacity to store them, most
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+Newton-Raphson methods can not be used with very large models.
-+
+\subsubsection{Heating}
-+
-+
+Typically, Heating is performed by assigning random velocities
-+
+according to a Gaussian distribution for a temperature. Beginning at
-+
+a lower temperature and gradually increasing the temperature by
-+
+assigning greater random velocities, we end up with setting the
-+
+temperature of the system to a final temperature at which the
-+
+simulation will be conducted. In heating phase, we should also keep
-+
+the system from drifting or rotating as a whole. Equivalently, the
-+
+net linear momentum and angular momentum of the system should be
-+
+shifted to zero.
-+
-+
+\subsubsection{Equilibration}
-+
-+
+The purpose of equilibration is to allow the system to evolve
-+
+spontaneously for a period of time and reach equilibrium. The
-+
+procedure is continued until various statistical properties, such as
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+temperature, pressure, energy, volume and other structural
-+
+properties \textit{etc}, become independent of time. Strictly
-+
+speaking, minimization and heating are not necessary, provided the
-+
+equilibration process is long enough. However, these steps can serve
-+
+as a means to arrive at an equilibrated structure in an effective
-+
+way.
-+
-+
+\subsection{\label{introSection:production}Production}
-+
-+
+Production run is the most important steps of the simulation, in
-+
+which the equilibrated structure is used as a starting point and the
-+
+motions of the molecules are collected for later analysis. In order
-+
+to capture the macroscopic properties of the system, the molecular
-+
+dynamics simulation must be performed in correct and efficient way.
-+
-+
+The most expensive part of a molecular dynamics simulation is the
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+calculation of non-bonded forces, such as van der Waals force and
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+Coulombic forces \textit{etc}. For a system of $N$ particles, the
-+
+complexity of the algorithm for pair-wise interactions is $O(N^2 )$,
-+
+which making large simulations prohibitive in the absence of any
-+
+computation saving techniques.
-+
-+
+A natural approach to avoid system size issue is to represent the
-+
+bulk behavior by a finite number of the particles. However, this
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+approach will suffer from the surface effect. To offset this,
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+\textit{Periodic boundary condition} is developed to simulate bulk
-+
+properties with a relatively small number of particles. In this
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+method, the simulation box is replicated throughout space to form an
-+
+infinite lattice. During the simulation, when a particle moves in
-+
+the primary cell, its image in other cells move in exactly the same
-+
+direction with exactly the same orientation. Thus, as a particle
-+
+leaves the primary cell, one of its images will enter through the
-+
+opposite face.
-+
+%\begin{figure}
-+
+%\centering
-+
+%\includegraphics[width=\linewidth]{pbcFig.eps}
-+
+%\caption[An illustration of periodic boundary conditions]{A 2-D
-+
+%illustration of periodic boundary conditions. As one particle leaves
-+
+%the right of the simulation box, an image of it enters the left.}
-+
+%\label{introFig:pbc}
-+
+%\end{figure}
-+
-+
+%cutoff and minimum image convention
-+
+Another important technique to improve the efficiency of force
-+
+evaluation is to apply cutoff where particles farther than a
-+
+predetermined distance, are not included in the calculation
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+\cite{Frenkel1996}. The use of a cutoff radius will cause a
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+discontinuity in the potential energy curve
-+
+(Fig.~\ref{introFig:shiftPot}). Fortunately, one can shift the
-+
+potential to ensure the potential curve go smoothly to zero at the
-+
+cutoff radius. Cutoff strategy works pretty well for Lennard-Jones
-+
+interaction because of its short range nature. However, simply
-+
+truncating the electrostatic interaction with the use of cutoff has
-+
+been shown to lead to severe artifacts in simulations. Ewald
-+
+summation, in which the slowly conditionally convergent Coulomb
-+
+potential is transformed into direct and reciprocal sums with rapid
-+
+and absolute convergence, has proved to minimize the periodicity
-+
+artifacts in liquid simulations. Taking the advantages of the fast
-+
+Fourier transform (FFT) for calculating discrete Fourier transforms,
-+
+the particle mesh-based methods are accelerated from $O(N^{3/2})$ to
-+
+$O(N logN)$. An alternative approach is \emph{fast multipole
-+
+method}, which treats Coulombic interaction exactly at short range,
-+
+and approximate the potential at long range through multipolar
-+
+expansion. In spite of their wide acceptances at the molecular
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+simulation community, these two methods are hard to be implemented
-+
+correctly and efficiently. Instead, we use a damped and
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+charge-neutralized Coulomb potential method developed by Wolf and
-+
+his coworkers. The shifted Coulomb potential for particle $i$ and
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+particle $j$ at distance $r_{rj}$ is given by:
-+
+\begin{equation}
-+
+V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha
-+
+r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow
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+R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha
-+
+r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb}
-+
+\end{equation}
-+
+where $\alpha$ is the convergence parameter. Due to the lack of
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+inherent periodicity and rapid convergence,this method is extremely
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+efficient and easy to implement.
-+
+%\begin{figure}
-+
+%\centering
-+
+%\includegraphics[width=\linewidth]{pbcFig.eps}
-+
+%\caption[An illustration of shifted Coulomb potential]{An illustration of shifted Coulomb potential.}
-+
+%\label{introFigure:shiftedCoulomb}
-+
+%\end{figure}
-+
-+
+%multiple time step
-+
-+
+\subsection{\label{introSection:Analysis} Analysis}
-+
-+
+Recently, advanced visualization technique are widely applied to
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+monitor the motions of molecules. Although the dynamics of the
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+system can be described qualitatively from animation, quantitative
-+
+trajectory analysis are more appreciable. According to the
-+
+principles of Statistical Mechanics,
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+Sec.~\ref{introSection:statisticalMechanics}, one can compute
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+thermodynamics properties, analyze fluctuations of structural
-+
+parameters, and investigate time-dependent processes of the molecule
-+
+from the trajectories.
-+
-+
+\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties}
-+
-+
+Thermodynamics properties, which can be expressed in terms of some
-+
+function of the coordinates and momenta of all particles in the
-+
+system, can be directly computed from molecular dynamics. The usual
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+way to measure the pressure is based on virial theorem of Clausius
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+which states that the virial is equal to $-3Nk_BT$. For a system
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+with forces between particles, the total virial, $W$, contains the
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+contribution from external pressure and interaction between the
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+particles:
-+
+\[
-+
+W =  - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot
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+f_{ij} } } \right\rangle
-+
+\]
-+
+where $f_{ij}$ is the force between particle $i$ and $j$ at a
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+distance $r_{ij}$. Thus, the expression for the pressure is given
-+
+by:
-+
+\begin{equation}
-+
+P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i
-+
+< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle
-+
+\end{equation}
-+
-+
+\subsubsection{\label{introSection:structuralProperties}Structural Properties}
-+
-+
+Structural Properties of a simple fluid can be described by a set of
-+
+distribution functions. Among these functions,\emph{pair
-+
+distribution function}, also known as \emph{radial distribution
-+
+function}, is of most fundamental importance to liquid-state theory.
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+Pair distribution function can be gathered by Fourier transforming
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+raw data from a series of neutron diffraction experiments and
-+
+integrating over the surface factor \cite{Powles73}. The experiment
-+
+result can serve as a criterion to justify the correctness of the
-+
+theory. Moreover, various equilibrium thermodynamic and structural
-+
+properties can also be expressed in terms of radial distribution
-+
+function \cite{allen87:csl}.
-+
-+
+A pair distribution functions $g(r)$ gives the probability that a
-+
+particle $i$ will be located at a distance $r$ from a another
-+
+particle $j$ in the system
-+
+\[
-+
+g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j
-+
+\ne i} {\delta (r - r_{ij} )} } } \right\rangle.
-+
+\]
-+
+Note that the delta function can be replaced by a histogram in
-+
+computer simulation. Figure
-+
+\ref{introFigure:pairDistributionFunction} shows a typical pair
-+
+distribution function for the liquid argon system. The occurrence of
-+
+several peaks in the plot of $g(r)$ suggests that it is more likely
-+
+to find particles at certain radial values than at others. This is a
-+
+result of the attractive interaction at such distances. Because of
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+the strong repulsive forces at short distance, the probability of
-+
+locating particles at distances less than about 2.5{\AA} from each
-+
+other is essentially zero.
-+
-+
+%\begin{figure}
-+
+%\centering
-+
+%\includegraphics[width=\linewidth]{pdf.eps}
-+
+%\caption[Pair distribution function for the liquid argon
-+
+%]{Pair distribution function for the liquid argon}
-+
+%\label{introFigure:pairDistributionFunction}
-+
+%\end{figure}
-+
-+
+\subsubsection{\label{introSection:timeDependentProperties}Time-dependent
-+
+Properties}
-+
-+
+Time-dependent properties are usually calculated using \emph{time
-+
+correlation function}, which correlates random variables $A$ and $B$
-+
+at two different time
-+
+\begin{equation}
-+
+C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle.
-+
+\label{introEquation:timeCorrelationFunction}
-+
+\end{equation}
-+
+If $A$ and $B$ refer to same variable, this kind of correlation
-+
+function is called \emph{auto correlation function}. One example of
-+
+auto correlation function is velocity auto-correlation function
-+
+which is directly related to transport properties of molecular
-+
+liquids:
-+
+\[
-+
+D = \frac{1}{3}\int\limits_0^\infty  {\left\langle {v(t) \cdot v(0)}
-+
+\right\rangle } dt
-+
+\]
-+
+where $D$ is diffusion constant. Unlike velocity autocorrelation
-+
+function which is averaging over time origins and over all the
-+
+atoms, dipole autocorrelation are calculated for the entire system.
-+
+The dipole autocorrelation function is given by:
-+
+\[
-+
+c_{dipole}  = \left\langle {u_{tot} (t) \cdot u_{tot} (t)}
-+
+\right\rangle
-+
+\]
-+
+Here $u_{tot}$ is the net dipole of the entire system and is given
-+
+by
-+
+\[
-+
+u_{tot} (t) = \sum\limits_i {u_i (t)}
-+
+\]
-+
+In principle, many time correlation functions can be related with
-+
+Fourier transforms of the infrared, Raman, and inelastic neutron
-+
+scattering spectra of molecular liquids. In practice, one can
-+
+extract the IR spectrum from the intensity of dipole fluctuation at
-+
+each frequency using the following relationship:
-+
+\[
-+
+\hat c_{dipole} (v) = \int_{ - \infty }^\infty  {c_{dipole} (t)e^{ -
-+
+i2\pi vt} dt}
-+
+\]
-+
+\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies}
+Rigid bodies are frequently involved in the modeling of different
+e^{\Delta tR_1 }  \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1
+\]
-<
-<
+The flow maps for $T_2^r$ and $T_2^r$ can be found in the same
->
+The flow maps for $T_2^r$ and $T_3^r$ can be found in the same
+manner.
+In order to construct a second-order symplectic method, we split the
+coefficient $\xi _0$ can either be calculated from spectral density
+or be determined by Stokes' law for regular shaped particles.A
+briefly review on calculating friction tensor for arbitrary shaped
-<
+particles is given in section \ref{introSection:frictionTensor}.
->
+particles is given in Sec.~\ref{introSection:frictionTensor}.
+\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem}
+\]
+where $x_OR$, $y_OR$, $z_OR$ are the components of the vector
+joining center of resistance $R$ and origin $O$.
-–
-–
+%\section{\label{introSection:correlationFunctions}Correlation Functions}

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Comparing trunk/tengDissertation/Introduction.tex (file contents): Revision 2719 by tim, Tue Apr 18 22:38:19 2006 UTC vs. Revision 2725 by tim, Fri Apr 21 05:45:14 2006 UTC

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Comparing trunk/tengDissertation/Introduction.tex (file contents):
Revision 2719 by tim, Tue Apr 18 22:38:19 2006 UTC vs.
Revision 2725 by tim, Fri Apr 21 05:45:14 2006 UTC