| 883 |
|
|
| 884 |
|
\section{\label{introSection:molecularDynamics}Molecular Dynamics} |
| 885 |
|
|
| 886 |
< |
As a special discipline of molecular modeling, Molecular dynamics |
| 887 |
< |
has proven to be a powerful tool for studying the functions of |
| 888 |
< |
biological systems, providing structural, thermodynamic and |
| 889 |
< |
dynamical information. |
| 886 |
> |
As one of the principal tools of molecular modeling, Molecular |
| 887 |
> |
dynamics has proven to be a powerful tool for studying the functions |
| 888 |
> |
of biological systems, providing structural, thermodynamic and |
| 889 |
> |
dynamical information. The basic idea of molecular dynamics is that |
| 890 |
> |
macroscopic properties are related to microscopic behavior and |
| 891 |
> |
microscopic behavior can be calculated from the trajectories in |
| 892 |
> |
simulations. For instance, instantaneous temperature of an |
| 893 |
> |
Hamiltonian system of $N$ particle can be measured by |
| 894 |
> |
\[ |
| 895 |
> |
T(t) = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
| 896 |
> |
\] |
| 897 |
> |
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle |
| 898 |
> |
respectively, $f$ is the number of degrees of freedom, and $k_B$ is |
| 899 |
> |
the boltzman constant. |
| 900 |
|
|
| 901 |
< |
One of the principal tools for modeling proteins, nucleic acids and |
| 902 |
< |
their complexes. Stability of proteins Folding of proteins. |
| 903 |
< |
Molecular recognition by:proteins, DNA, RNA, lipids, hormones STP, |
| 904 |
< |
etc. Enzyme reactions Rational design of biologically active |
| 905 |
< |
molecules (drug design) Small and large-scale conformational |
| 906 |
< |
changes. determination and construction of 3D structures (homology, |
| 907 |
< |
Xray diffraction, NMR) Dynamic processes such as ion transport in |
| 908 |
< |
biological systems. |
| 901 |
> |
A typical molecular dynamics run consists of three essential steps: |
| 902 |
> |
\begin{enumerate} |
| 903 |
> |
\item Initialization |
| 904 |
> |
\begin{enumerate} |
| 905 |
> |
\item Preliminary preparation |
| 906 |
> |
\item Minimization |
| 907 |
> |
\item Heating |
| 908 |
> |
\item Equilibration |
| 909 |
> |
\end{enumerate} |
| 910 |
> |
\item Production |
| 911 |
> |
\item Analysis |
| 912 |
> |
\end{enumerate} |
| 913 |
> |
These three individual steps will be covered in the following |
| 914 |
> |
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
| 915 |
> |
initialization of a simulation. Sec.~\ref{introSec:production} will |
| 916 |
> |
discusses issues in production run, including the force evaluation |
| 917 |
> |
and the numerical integration schemes of the equations of motion . |
| 918 |
> |
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
| 919 |
> |
trajectory analysis. |
| 920 |
|
|
| 921 |
< |
Macroscopic properties are related to microscopic behavior. |
| 921 |
> |
\subsection{\label{introSec:initialSystemSettings}Initialization} |
| 922 |
|
|
| 923 |
< |
Time dependent (and independent) microscopic behavior of a molecule |
| 903 |
< |
can be calculated by molecular dynamics simulations. |
| 923 |
> |
\subsubsection{Preliminary preparation} |
| 924 |
|
|
| 925 |
< |
\subsection{\label{introSec:mdInit}Initialization} |
| 925 |
> |
When selecting the starting structure of a molecule for molecular |
| 926 |
> |
simulation, one may retrieve its Cartesian coordinates from public |
| 927 |
> |
databases, such as RCSB Protein Data Bank \textit{etc}. Although |
| 928 |
> |
thousands of crystal structures of molecules are discovered every |
| 929 |
> |
year, many more remain unknown due to the difficulties of |
| 930 |
> |
purification and crystallization. Even for the molecule with known |
| 931 |
> |
structure, some important information is missing. For example, the |
| 932 |
> |
missing hydrogen atom which acts as donor in hydrogen bonding must |
| 933 |
> |
be added. Moreover, in order to include electrostatic interaction, |
| 934 |
> |
one may need to specify the partial charges for individual atoms. |
| 935 |
> |
Under some circumstances, we may even need to prepare the system in |
| 936 |
> |
a special setup. For instance, when studying transport phenomenon in |
| 937 |
> |
membrane system, we may prepare the lipids in bilayer structure |
| 938 |
> |
instead of placing lipids randomly in solvent, since we are not |
| 939 |
> |
interested in self-aggregation and it takes a long time to happen. |
| 940 |
|
|
| 941 |
< |
\subsection{\label{introSec:forceEvaluation}Force Evaluation} |
| 941 |
> |
\subsubsection{Minimization} |
| 942 |
|
|
| 943 |
< |
\subsection{\label{introSection:mdIntegration} Integration of the Equations of Motion} |
| 943 |
> |
It is quite possible that some of molecules in the system from |
| 944 |
> |
preliminary preparation may be overlapped with each other. This |
| 945 |
> |
close proximity leads to high potential energy which consequently |
| 946 |
> |
jeopardizes any molecular dynamics simulations. To remove these |
| 947 |
> |
steric overlaps, one typically performs energy minimization to find |
| 948 |
> |
a more reasonable conformation. Several energy minimization methods |
| 949 |
> |
have been developed to exploit the energy surface and to locate the |
| 950 |
> |
local minimum. While converging slowly near the minimum, steepest |
| 951 |
> |
descent method is extremely robust when systems are far from |
| 952 |
> |
harmonic. Thus, it is often used to refine structure from |
| 953 |
> |
crystallographic data. Relied on the gradient or hessian, advanced |
| 954 |
> |
methods like conjugate gradient and Newton-Raphson converge rapidly |
| 955 |
> |
to a local minimum, while become unstable if the energy surface is |
| 956 |
> |
far from quadratic. Another factor must be taken into account, when |
| 957 |
> |
choosing energy minimization method, is the size of the system. |
| 958 |
> |
Steepest descent and conjugate gradient can deal with models of any |
| 959 |
> |
size. Because of the limit of computation power to calculate hessian |
| 960 |
> |
matrix and insufficient storage capacity to store them, most |
| 961 |
> |
Newton-Raphson methods can not be used with very large models. |
| 962 |
|
|
| 963 |
+ |
\subsubsection{Heating} |
| 964 |
+ |
|
| 965 |
+ |
Typically, Heating is performed by assigning random velocities |
| 966 |
+ |
according to a Gaussian distribution for a temperature. Beginning at |
| 967 |
+ |
a lower temperature and gradually increasing the temperature by |
| 968 |
+ |
assigning greater random velocities, we end up with setting the |
| 969 |
+ |
temperature of the system to a final temperature at which the |
| 970 |
+ |
simulation will be conducted. In heating phase, we should also keep |
| 971 |
+ |
the system from drifting or rotating as a whole. Equivalently, the |
| 972 |
+ |
net linear momentum and angular momentum of the system should be |
| 973 |
+ |
shifted to zero. |
| 974 |
+ |
|
| 975 |
+ |
\subsubsection{Equilibration} |
| 976 |
+ |
|
| 977 |
+ |
The purpose of equilibration is to allow the system to evolve |
| 978 |
+ |
spontaneously for a period of time and reach equilibrium. The |
| 979 |
+ |
procedure is continued until various statistical properties, such as |
| 980 |
+ |
temperature, pressure, energy, volume and other structural |
| 981 |
+ |
properties \textit{etc}, become independent of time. Strictly |
| 982 |
+ |
speaking, minimization and heating are not necessary, provided the |
| 983 |
+ |
equilibration process is long enough. However, these steps can serve |
| 984 |
+ |
as a means to arrive at an equilibrated structure in an effective |
| 985 |
+ |
way. |
| 986 |
+ |
|
| 987 |
+ |
\subsection{\label{introSection:production}Production} |
| 988 |
+ |
|
| 989 |
+ |
\subsubsection{\label{introSec:forceCalculation}The Force Calculation} |
| 990 |
+ |
|
| 991 |
+ |
\subsubsection{\label{introSection:integrationSchemes} Integration |
| 992 |
+ |
Schemes} |
| 993 |
+ |
|
| 994 |
+ |
\subsection{\label{introSection:Analysis} Analysis} |
| 995 |
+ |
|
| 996 |
+ |
Recently, advanced visualization technique are widely applied to |
| 997 |
+ |
monitor the motions of molecules. Although the dynamics of the |
| 998 |
+ |
system can be described qualitatively from animation, quantitative |
| 999 |
+ |
trajectory analysis are more appreciable. According to the |
| 1000 |
+ |
principles of Statistical Mechanics, |
| 1001 |
+ |
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
| 1002 |
+ |
thermodynamics properties, analyze fluctuations of structural |
| 1003 |
+ |
parameters, and investigate time-dependent processes of the molecule |
| 1004 |
+ |
from the trajectories. |
| 1005 |
+ |
|
| 1006 |
+ |
\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties} |
| 1007 |
+ |
|
| 1008 |
+ |
\subsubsection{\label{introSection:structuralProperties}Structural Properties} |
| 1009 |
+ |
|
| 1010 |
+ |
Structural Properties of a simple fluid can be described by a set of |
| 1011 |
+ |
distribution functions. Among these functions,\emph{pair |
| 1012 |
+ |
distribution function}, also known as \emph{radial distribution |
| 1013 |
+ |
function}, are of most fundamental importance to liquid-state |
| 1014 |
+ |
theory. Pair distribution function can be gathered by Fourier |
| 1015 |
+ |
transforming raw data from a series of neutron diffraction |
| 1016 |
+ |
experiments and integrating over the surface factor \cite{Powles73}. |
| 1017 |
+ |
The experiment result can serve as a criterion to justify the |
| 1018 |
+ |
correctness of the theory. Moreover, various equilibrium |
| 1019 |
+ |
thermodynamic and structural properties can also be expressed in |
| 1020 |
+ |
terms of radial distribution function \cite{allen87:csl}. |
| 1021 |
+ |
|
| 1022 |
+ |
A pair distribution functions $g(r)$ gives the probability that a |
| 1023 |
+ |
particle $i$ will be located at a distance $r$ from a another |
| 1024 |
+ |
particle $j$ in the system |
| 1025 |
+ |
\[ |
| 1026 |
+ |
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
| 1027 |
+ |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle. |
| 1028 |
+ |
\] |
| 1029 |
+ |
Note that the delta function can be replaced by a histogram in |
| 1030 |
+ |
computer simulation. Figure |
| 1031 |
+ |
\ref{introFigure:pairDistributionFunction} shows a typical pair |
| 1032 |
+ |
distribution function for the liquid argon system. The occurrence of |
| 1033 |
+ |
several peaks in the plot of $g(r)$ suggests that it is more likely |
| 1034 |
+ |
to find particles at certain radial values than at others. This is a |
| 1035 |
+ |
result of the attractive interaction at such distances. Because of |
| 1036 |
+ |
the strong repulsive forces at short distance, the probability of |
| 1037 |
+ |
locating particles at distances less than about 2.5{\AA} from each |
| 1038 |
+ |
other is essentially zero. |
| 1039 |
+ |
|
| 1040 |
+ |
%\begin{figure} |
| 1041 |
+ |
%\centering |
| 1042 |
+ |
%\includegraphics[width=\linewidth]{pdf.eps} |
| 1043 |
+ |
%\caption[Pair distribution function for the liquid argon |
| 1044 |
+ |
%]{Pair distribution function for the liquid argon} |
| 1045 |
+ |
%\label{introFigure:pairDistributionFunction} |
| 1046 |
+ |
%\end{figure} |
| 1047 |
+ |
|
| 1048 |
+ |
\subsubsection{\label{introSection:timeDependentProperties}Time-dependent |
| 1049 |
+ |
Properties} |
| 1050 |
+ |
|
| 1051 |
+ |
Time-dependent properties are usually calculated using \emph{time |
| 1052 |
+ |
correlation function}, which correlates random variables $A$ and $B$ |
| 1053 |
+ |
at two different time |
| 1054 |
+ |
\begin{equation} |
| 1055 |
+ |
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
| 1056 |
+ |
\label{introEquation:timeCorrelationFunction} |
| 1057 |
+ |
\end{equation} |
| 1058 |
+ |
If $A$ and $B$ refer to same variable, this kind of correlation |
| 1059 |
+ |
function is called \emph{auto correlation function}. One example of |
| 1060 |
+ |
auto correlation function is velocity auto-correlation function |
| 1061 |
+ |
which is directly related to transport properties of molecular |
| 1062 |
+ |
liquids. Another example is the calculation of the IR spectrum |
| 1063 |
+ |
through a Fourier transform of the dipole autocorrelation function. |
| 1064 |
+ |
|
| 1065 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
| 1066 |
|
|
| 1067 |
|
Rigid bodies are frequently involved in the modeling of different |
| 1310 |
|
e^{\Delta tR_1 } \approx (1 - \Delta tR_1 )^{ - 1} (1 + \Delta tR_1 |
| 1311 |
|
) |
| 1312 |
|
\] |
| 1313 |
< |
|
| 1160 |
< |
The flow maps for $T_2^r$ and $T_2^r$ can be found in the same |
| 1313 |
> |
The flow maps for $T_2^r$ and $T_3^r$ can be found in the same |
| 1314 |
|
manner. |
| 1315 |
|
|
| 1316 |
|
In order to construct a second-order symplectic method, we split the |
| 1633 |
|
coefficient $\xi _0$ can either be calculated from spectral density |
| 1634 |
|
or be determined by Stokes' law for regular shaped particles.A |
| 1635 |
|
briefly review on calculating friction tensor for arbitrary shaped |
| 1636 |
< |
particles is given in section \ref{introSection:frictionTensor}. |
| 1636 |
> |
particles is given in Sec.~\ref{introSection:frictionTensor}. |
| 1637 |
|
|
| 1638 |
|
\subsubsection{\label{introSection:secondFluctuationDissipation}The Second Fluctuation Dissipation Theorem} |
| 1639 |
|
|
| 1923 |
|
\] |
| 1924 |
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 1925 |
|
joining center of resistance $R$ and origin $O$. |
| 1773 |
– |
|
| 1774 |
– |
%\section{\label{introSection:correlationFunctions}Correlation Functions} |
