| 2 |
|
|
| 3 |
|
Recent examples of the usefulness of Langevin simulations include a |
| 4 |
|
study of met-enkephalin in which Langevin simulations predicted |
| 5 |
< |
dynamical properties that were largely in agreement with explicit |
| 5 |
> |
dynamical properties that were large\-ly in agreement with explicit |
| 6 |
|
solvent simulations.\cite{Shen2002} By applying Langevin dynamics with |
| 7 |
|
the UNRES model, Liwo and his coworkers suggest that protein folding |
| 8 |
|
pathways can be explored within a reasonable amount of |
| 119 |
|
our Langevin integrator. |
| 120 |
|
|
| 121 |
|
The goal of the present work is to develop a Langevin dynamics |
| 122 |
< |
algorithm for arbitrary-shaped rigid particles by integrating an |
| 122 |
> |
algorithm for ar\-bi\-trary-shaped rigid particles by integrating an |
| 123 |
|
accurate estimate of the friction tensor from hydrodynamics theory |
| 124 |
|
into a stable and efficient rigid body dynamics propagator. In the |
| 125 |
|
sections below, we review some of the theory of hydrodynamic tensors |
| 709 |
|
\begin{table*} |
| 710 |
|
\begin{minipage}{\linewidth} |
| 711 |
|
\begin{center} |
| 712 |
< |
\caption{Parameters for the primary particles in use by the rigid body |
| 713 |
< |
models in figure \ref{ldfig:models}.} |
| 712 |
> |
\caption{PARAMETERS FOR THE PRIMARY PARTICLES IN USE BY THE RIGID BODY |
| 713 |
> |
MODELS IN FIGURE \ref{ldfig:models}} |
| 714 |
|
\begin{tabular}{lrcccccccc} |
| 715 |
|
\hline |
| 716 |
|
& & & & & & \multicolumn{3}c{$\overleftrightarrow{\mathsf I}$ (amu \AA$^2$)} \\ |
| 861 |
|
\tau = \ell (\ell + 1) \int_0^{\infty} C_{\ell}(t) dt. |
| 862 |
|
\end{equation} |
| 863 |
|
In lower-friction solvents, the Legendre correlation functions often |
| 864 |
< |
exhibit non-exponential decay, and may not be characterized by a |
| 864 |
> |
exhibit non-ex\-po\-nen\-tial decay, and may not be characterized by a |
| 865 |
|
single decay constant. |
| 866 |
|
|
| 867 |
|
In table \ref{ldtab:rotation} we show the characteristic rotational |
| 924 |
|
\label{ldGPerrin} |
| 925 |
|
\end{equation} |
| 926 |
|
Again, there is some uncertainty about the correct boundary conditions |
| 927 |
< |
to use for molecular-scale ellipsoids in a sea of similarly-sized |
| 927 |
> |
to use for molecular scale ellipsoids in a sea of similarly-sized |
| 928 |
|
solvent particles. Ravichandran and Bagchi found that {\it slip} |
| 929 |
|
boundary conditions most closely resembled the simulation |
| 930 |
|
results,\cite{Ravichandran:1999fk} in agreement with earlier work of |
| 1118 |
|
\centering |
| 1119 |
|
\includegraphics[width=\linewidth]{./figures/ldGraph} |
| 1120 |
|
\caption[Mean squared displacements and orientational |
| 1121 |
< |
correlation functions for each of the model rigid bodies.]{The |
| 1121 |
> |
correlation functions for each of the model rigid bodies]{The |
| 1122 |
|
mean-squared displacements ($\langle r^2(t) \rangle$) and |
| 1123 |
|
orientational correlation functions ($C_2(t)$) for each of the model |
| 1124 |
|
rigid bodies studied. The circles are the results for microcanonical |
| 1133 |
|
\end{figure} |
| 1134 |
|
|
| 1135 |
|
\begin{table*} |
| 1136 |
– |
\begin{minipage}{\linewidth} |
| 1136 |
|
\begin{center} |
| 1137 |
< |
\caption{Translational diffusion constants (D) for the model systems |
| 1138 |
< |
calculated using microcanonical simulations (with explicit solvent), |
| 1139 |
< |
theoretical predictions, and Langevin simulations (with implicit solvent). |
| 1140 |
< |
Analytical solutions for the exactly-solved hydrodynamics models are obtained |
| 1142 |
< |
from: Stokes' law (sphere), and Refs. \citen{Perrin1934} and \citen{Perrin1936} |
| 1143 |
< |
(ellipsoid), \citen{Stimson:1926qy} and \citen{Davis:1969uq} |
| 1144 |
< |
(dumbbell). The other model systems have no known analytic solution. |
| 1145 |
< |
All diffusion constants are reported in units of $10^{-3}$ cm$^2$ / ps (= |
| 1146 |
< |
$10^{-4}$ \AA$^2$ / fs). } |
| 1137 |
> |
\caption{TRANSLATIONAL DIFFUSION CONSTANTS (D) FOR THE MODEL SYSTEMS |
| 1138 |
> |
CALCULATED USING MICROCANONICAL SIM\-U\-LA\-TIONS (WITH EXPLICIT |
| 1139 |
> |
SOLVENT), THEORETICAL PREDICTIONS, AND LANGEVIN SIMULATIONS (WITH |
| 1140 |
> |
IMPLICIT SOLVENT)} |
| 1141 |
|
\begin{tabular}{lccccccc} |
| 1142 |
|
\hline |
| 1143 |
|
& \multicolumn{2}c{microcanonical} & & \multicolumn{3}c{Theoretical} & Langevin \\ |
| 1151 |
|
& 0.308 & 2.06 & & 1.64 & rough shell & 1.59 & 1.62 \\ |
| 1152 |
|
banana & 0.298 & 1.53 & & & rough shell & 1.56 & 1.55 \\ |
| 1153 |
|
lipid & 0.349 & 1.41 & & & rough shell & 1.33 & 1.32 \\ |
| 1154 |
+ |
\hline |
| 1155 |
|
\end{tabular} |
| 1156 |
+ |
\begin{minipage}{\linewidth} |
| 1157 |
+ |
%\centering |
| 1158 |
+ |
\vspace{2mm} |
| 1159 |
+ |
Analytical solutions for the exactly-solved hydrodynamics models are |
| 1160 |
+ |
obtained from: Stokes' law (sphere), and Refs. \citen{Perrin1934} and |
| 1161 |
+ |
\citen{Perrin1936} (ellipsoid), \citen{Stimson:1926qy} and |
| 1162 |
+ |
\citen{Davis:1969uq} (dumbbell). The other model systems have no known |
| 1163 |
+ |
analytic solution. All diffusion constants are reported in units of |
| 1164 |
+ |
$10^{-3}$ cm$^2$ / ps (= $10^{-4}$ \AA$^2$ / fs). |
| 1165 |
|
\label{ldtab:translation} |
| 1162 |
– |
\end{center} |
| 1166 |
|
\end{minipage} |
| 1167 |
+ |
\end{center} |
| 1168 |
|
\end{table*} |
| 1169 |
|
|
| 1170 |
|
\begin{table*} |
| 1167 |
– |
\begin{minipage}{\linewidth} |
| 1171 |
|
\begin{center} |
| 1172 |
< |
\caption{Orientational relaxation times ($\tau$) for the model systems using |
| 1173 |
< |
microcanonical simulation (with explicit solvent), theoretical |
| 1174 |
< |
predictions, and Langevin simulations (with implicit solvent). All |
| 1172 |
< |
relaxation times are for the rotational correlation function with |
| 1173 |
< |
$\ell = 2$ and are reported in units of ps. The ellipsoidal model has |
| 1174 |
< |
an exact solution for the orientational correlation time due to |
| 1175 |
< |
Perrin, but the other model systems have no known analytic solution.} |
| 1172 |
> |
\caption{ORIENTATIONAL RELAXATION TIMES ($\tau$) FOR THE MODEL SYSTEMS USING |
| 1173 |
> |
MICROCANONICAL SIMULATION (WITH EXPLICIT SOLVENT), THEORETICAL |
| 1174 |
> |
PREDICTIONS, AND LANGEVIN SIMULATIONS (WITH IMPLICIT SOLVENT)} |
| 1175 |
|
\begin{tabular}{lccccccc} |
| 1176 |
|
\hline |
| 1177 |
|
& \multicolumn{2}c{microcanonical} & & \multicolumn{3}c{Theoretical} & Langevin \\ |
| 1187 |
|
lipid & 0.349 & 78.0 & & & rough shell & 76.9 & 77.9 \\ |
| 1188 |
|
\hline |
| 1189 |
|
\end{tabular} |
| 1190 |
+ |
\begin{minipage}{\linewidth} |
| 1191 |
+ |
%\centering |
| 1192 |
+ |
\vspace{2mm} |
| 1193 |
+ |
All relaxation times are for the rotational correlation function with |
| 1194 |
+ |
$\ell = 2$ and are reported in units of ps. The ellipsoidal model has |
| 1195 |
+ |
an exact solution for the orientational correlation time due to |
| 1196 |
+ |
Perrin, but the other model systems have no known analytic solution. |
| 1197 |
|
\label{ldtab:rotation} |
| 1192 |
– |
\end{center} |
| 1198 |
|
\end{minipage} |
| 1199 |
+ |
\end{center} |
| 1200 |
|
\end{table*} |
| 1201 |
|
|
| 1202 |
|
\section{Application: A rigid-body lipid bilayer} |
