| 150 |
|
increasing time step. For each time step, the dotted line is total |
| 151 |
|
energy using the DLM integrator, and the solid line comes from the |
| 152 |
|
quaternion integrator. The larger time step plots are shifted up |
| 153 |
< |
from the true energy baseline for clarity.} \label{timestep} |
| 153 |
> |
from the true energy baseline for clarity.} |
| 154 |
> |
\label{methodFig:timestep} |
| 155 |
|
\end{figure} |
| 156 |
|
|
| 157 |
< |
In Fig.~\ref{timestep}, the resulting energy drift at various time |
| 158 |
< |
steps for both the DLM and quaternion integration schemes is |
| 159 |
< |
compared. All of the 1000 molecule water simulations started with |
| 160 |
< |
the same configuration, and the only difference was the method for |
| 161 |
< |
handling rotational motion. At time steps of 0.1 and 0.5 fs, both |
| 162 |
< |
methods for propagating molecule rotation conserve energy fairly |
| 163 |
< |
well, with the quaternion method showing a slight energy drift over |
| 164 |
< |
time in the 0.5 fs time step simulation. At time steps of 1 and 2 |
| 165 |
< |
fs, the energy conservation benefits of the DLM method are clearly |
| 166 |
< |
demonstrated. Thus, while maintaining the same degree of energy |
| 167 |
< |
conservation, one can take considerably longer time steps, leading |
| 168 |
< |
to an overall reduction in computation time. |
| 157 |
> |
In Fig.~\ref{methodFig:timestep}, the resulting energy drift at |
| 158 |
> |
various time steps for both the DLM and quaternion integration |
| 159 |
> |
schemes is compared. All of the 1000 molecule water simulations |
| 160 |
> |
started with the same configuration, and the only difference was the |
| 161 |
> |
method for handling rotational motion. At time steps of 0.1 and 0.5 |
| 162 |
> |
fs, both methods for propagating molecule rotation conserve energy |
| 163 |
> |
fairly well, with the quaternion method showing a slight energy |
| 164 |
> |
drift over time in the 0.5 fs time step simulation. At time steps of |
| 165 |
> |
1 and 2 fs, the energy conservation benefits of the DLM method are |
| 166 |
> |
clearly demonstrated. Thus, while maintaining the same degree of |
| 167 |
> |
energy conservation, one can take considerably longer time steps, |
| 168 |
> |
leading to an overall reduction in computation time. |
| 169 |
|
|
| 170 |
|
\subsection{\label{methodSection:NVT}Nos\'{e}-Hoover Thermostatting} |
| 171 |
|
|
| 172 |
< |
The Nos\'e-Hoover equations of motion are given by\cite{Hoover85} |
| 172 |
> |
The Nos\'e-Hoover equations of motion are given by\cite{Hoover1985} |
| 173 |
|
\begin{eqnarray} |
| 174 |
|
\dot{{\bf r}} & = & {\bf v}, \\ |
| 175 |
|
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} ,\\ |
| 285 |
|
|
| 286 |
|
The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for |
| 287 |
|
the extended system that is, to within a constant, identical to the |
| 288 |
< |
Helmholtz free energy,\cite{melchionna93} |
| 288 |
> |
Helmholtz free energy,\cite{Melchionna1993} |
| 289 |
|
\begin{equation} |
| 290 |
|
H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 291 |
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
| 295 |
|
$H_{\mathrm{NVT}}$, so the conserved quantity is maintained in the |
| 296 |
|
last column of the {\tt .stat} file to allow checks on the quality |
| 297 |
|
of the integration. |
| 297 |
– |
|
| 298 |
– |
Bond constraints are applied at the end of both the {\tt moveA} and |
| 299 |
– |
{\tt moveB} portions of the algorithm. Details on the constraint |
| 300 |
– |
algorithms are given in section \ref{oopseSec:rattle}. |
| 298 |
|
|
| 299 |
|
\subsection{\label{methodSection:NPTi}Constant-pressure integration with |
| 300 |
|
isotropic box deformations (NPTi)} |
| 301 |
|
|
| 302 |
|
To carry out isobaric-isothermal ensemble calculations {\sc oopse} |
| 303 |
|
implements the Melchionna modifications to the |
| 304 |
< |
Nos\'e-Hoover-Andersen equations of motion,\cite{melchionna93} |
| 304 |
> |
Nos\'e-Hoover-Andersen equations of motion,\cite{Melchionna1993} |
| 305 |
|
|
| 306 |
|
\begin{eqnarray} |
| 307 |
|
\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\ |
| 608 |
|
|
| 609 |
|
\subsection{\label{methodSection:otherSpecialEnsembles}Other Special Ensembles} |
| 610 |
|
|
| 611 |
< |
\subsubsection{\label{methodSection:NPAT}Constant Normal Pressure, Constant Lateral Surface Area and Constant Temperature (NPAT) Ensemble} |
| 611 |
> |
\subsubsection{\label{methodSection:NPAT}NPAT Ensemble} |
| 612 |
|
|
| 613 |
|
A comprehensive understanding of structure¨Cfunction relations of |
| 614 |
|
biological membrane system ultimately relies on structure and |
| 618 |
|
called the average surface area per lipid. Constat area and constant |
| 619 |
|
lateral pressure simulation can be achieved by extending the |
| 620 |
|
standard NPT ensemble with a different pressure control strategy |
| 621 |
< |
\begin{equation} |
| 622 |
< |
\dot |
| 623 |
< |
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} |
| 624 |
< |
\over \eta } _{\alpha \beta } = \left\{ \begin{array}{l} |
| 625 |
< |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }}{\rm{ }}(\alpha = \beta = z), \\ |
| 626 |
< |
0{\rm{ }}(\alpha \ne z{\rm{ }}or{\rm{ }}\beta \ne z) \\ |
| 627 |
< |
\end{array} \right. |
| 628 |
< |
\label{methodEquation:NPATeta} |
| 621 |
> |
|
| 622 |
> |
\begin{equation} |
| 623 |
> |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
| 624 |
> |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau_{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }} |
| 625 |
> |
& \mbox{if $ \alpha = \beta = z$}\\ |
| 626 |
> |
0 & \mbox{otherwise}\\ |
| 627 |
> |
\end{array} |
| 628 |
> |
\right. |
| 629 |
|
\end{equation} |
| 630 |
+ |
|
| 631 |
|
Note that the iterative schemes for NPAT are identical to those |
| 632 |
|
described for the NPTi integrator. |
| 633 |
|
|
| 634 |
< |
\subsubsection{\label{methodSection:NPrT}Constant Normal Pressure, Constant Lateral Surface Tension and Constant Temperature (NP\gamma T) Ensemble } |
| 634 |
> |
\subsubsection{\label{methodSection:NPrT}NP$\gamma$T Ensemble} |
| 635 |
|
|
| 636 |
|
Theoretically, the surface tension $\gamma$ of a stress free |
| 637 |
|
membrane system should be zero since its surface free energy $G$ is |
| 641 |
|
\] |
| 642 |
|
However, a surface tension of zero is not appropriate for relatively |
| 643 |
|
small patches of membrane. In order to eliminate the edge effect of |
| 644 |
< |
the membrane simulation, a special ensemble, NP\gamma T, is proposed |
| 645 |
< |
to maintain the lateral surface tension and normal pressure. The |
| 646 |
< |
equation of motion for cell size control tensor, $\eta$, in NP\gamma |
| 647 |
< |
T is |
| 644 |
> |
the membrane simulation, a special ensemble, NP$\gamma$T, is |
| 645 |
> |
proposed to maintain the lateral surface tension and normal |
| 646 |
> |
pressure. The equation of motion for cell size control tensor, |
| 647 |
> |
$\eta$, in $NP\gamma T$ is |
| 648 |
|
\begin{equation} |
| 649 |
< |
\dot |
| 650 |
< |
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} |
| 651 |
< |
\over \eta } _{\alpha \beta } = \left\{ \begin{array}{l} |
| 652 |
< |
- A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ){\rm{ (}}\alpha = \beta = x{\rm{ or }} = y{\rm{)}} \\ |
| 653 |
< |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }}{\rm{ }}(\alpha = \beta = z) \\ |
| 654 |
< |
0{\rm{ }}(\alpha \ne \beta ) \\ |
| 657 |
< |
\end{array} \right. |
| 658 |
< |
\label{methodEquation:NPrTeta} |
| 649 |
> |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
| 650 |
> |
- A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ) & \mbox{$\alpha = \beta = x$ or $\alpha = \beta = y$}\\ |
| 651 |
> |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}}}} & \mbox{$\alpha = \beta = z$} \\ |
| 652 |
> |
0 & \mbox{$\alpha \ne \beta$} \\ |
| 653 |
> |
\end{array} |
| 654 |
> |
\right. |
| 655 |
|
\end{equation} |
| 656 |
|
where $ \gamma _{{\rm{target}}}$ is the external surface tension and |
| 657 |
|
the instantaneous surface tensor $\gamma _\alpha$ is given by |
| 658 |
|
\begin{equation} |
| 659 |
< |
\gamma _\alpha = - h_z |
| 660 |
< |
(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} |
| 665 |
< |
\over P} _{\alpha \alpha } - P_{{\rm{target}}} ) |
| 659 |
> |
\gamma _\alpha = - h_z( \overleftrightarrow{P} _{\alpha \alpha } |
| 660 |
> |
- P_{{\rm{target}}} ) |
| 661 |
|
\label{methodEquation:instantaneousSurfaceTensor} |
| 662 |
|
\end{equation} |
| 663 |
|
|
| 669 |
|
integrator is a special case of $NP\gamma T$ if the surface tension |
| 670 |
|
$\gamma$ is set to zero. |
| 671 |
|
|
| 672 |
+ |
%\section{\label{methodSection:constraintMethod}Constraint Method} |
| 673 |
+ |
|
| 674 |
+ |
%\subsection{\label{methodSection:bondConstraint}Bond Constraint for Rigid Body} |
| 675 |
+ |
|
| 676 |
+ |
%\subsection{\label{methodSection:zcons}Z-constraint Method} |
| 677 |
+ |
|
| 678 |
|
\section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies} |
| 679 |
|
|
| 680 |
|
\subsection{\label{methodSection:temperature}Temperature Control} |
