4 |
|
|
5 |
|
In order to mimic the experiments, which are usually performed under |
6 |
|
constant temperature and/or pressure, extended Hamiltonian system |
7 |
< |
methods have been developed to generate statistical ensemble, such |
7 |
> |
methods have been developed to generate statistical ensembles, such |
8 |
|
as canonical ensemble and isobaric-isothermal ensemble \textit{etc}. |
9 |
|
In addition to the standard ensemble, specific ensembles have been |
10 |
|
developed to account for the anisotropy between the lateral and |
16 |
|
|
17 |
|
Integration schemes for rotational motion of the rigid molecules in |
18 |
|
microcanonical ensemble have been extensively studied in the last |
19 |
< |
two decades. Matubayasi and Nakahara developed a time-reversible |
20 |
< |
integrator for rigid bodies in quaternion representation. Although |
21 |
< |
it is not symplectic, this integrator still demonstrates a better |
22 |
< |
long-time energy conservation than traditional methods because of |
23 |
< |
the time-reversible nature. Extending Trotter-Suzuki to general |
24 |
< |
system with a flat phase space, Miller and his colleagues devised an |
25 |
< |
novel symplectic, time-reversible and volume-preserving integrator |
26 |
< |
in quaternion representation, which was shown to be superior to the |
27 |
< |
time-reversible integrator of Matubayasi and Nakahara. However, all |
28 |
< |
of the integrators in quaternion representation suffer from the |
19 |
> |
two decades. Matubayasi developed a time-reversible integrator for |
20 |
> |
rigid bodies in quaternion representation. Although it is not |
21 |
> |
symplectic, this integrator still demonstrates a better long-time |
22 |
> |
energy conservation than traditional methods because of the |
23 |
> |
time-reversible nature. Extending Trotter-Suzuki to general system |
24 |
> |
with a flat phase space, Miller and his colleagues devised an novel |
25 |
> |
symplectic, time-reversible and volume-preserving integrator in |
26 |
> |
quaternion representation, which was shown to be superior to the |
27 |
> |
Matubayasi's time-reversible integrator. However, all of the |
28 |
> |
integrators in quaternion representation suffer from the |
29 |
|
computational penalty of constructing a rotation matrix from |
30 |
|
quaternions to evolve coordinates and velocities at every time step. |
31 |
|
An alternative integration scheme utilizing rotation matrix directly |
139 |
|
average 7\% increase in computation time using the DLM method in |
140 |
|
place of quaternions. This cost is more than justified when |
141 |
|
comparing the energy conservation of the two methods as illustrated |
142 |
< |
in Fig.~\ref{timestep}. |
142 |
> |
in Fig.~\ref{methodFig:timestep}. |
143 |
|
|
144 |
|
\begin{figure} |
145 |
|
\centering |
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) |
296 |
|
last column of the {\tt .stat} file to allow checks on the quality |
297 |
|
of the integration. |
298 |
|
|
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}. |
301 |
– |
|
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), \\ |
474 |
|
\end{equation} |
475 |
|
|
476 |
|
Bond constraints are applied at the end of both the {\tt moveA} and |
477 |
< |
{\tt moveB} portions of the algorithm. Details on the constraint |
481 |
< |
algorithms are given in section \ref{oopseSec:rattle}. |
477 |
> |
{\tt moveB} portions of the algorithm. |
478 |
|
|
479 |
|
\subsection{\label{methodSection:NPTf}Constant-pressure integration with a |
480 |
|
flexible box (NPTf)} |
605 |
|
finds most use in simulating crystals or liquid crystals which |
606 |
|
assume non-orthorhombic geometries. |
607 |
|
|
608 |
< |
\subsection{\label{methodSection:NPAT}Constant Lateral Pressure and Constant Surface Area (NPAT)} |
608 |
> |
\subsection{\label{methodSection:otherSpecialEnsembles}Other Special Ensembles} |
609 |
|
|
610 |
< |
\subsection{\label{methodSection:NPrT}Constant lateral Pressure and Constant Surface Tension (NP\gamma T) } |
610 |
> |
\subsubsection{\label{methodSection:NPAT}\textbf{NPAT Ensemble}} |
611 |
|
|
612 |
< |
\subsection{\label{methodSection:NPTxyz}Constant pressure in 3 axes (NPTxyz)} |
612 |
> |
A comprehensive understanding of structure¨Cfunction relations of |
613 |
> |
biological membrane system ultimately relies on structure and |
614 |
> |
dynamics of lipid bilayer, which are strongly affected by the |
615 |
> |
interfacial interaction between lipid molecules and surrounding |
616 |
> |
media. One quantity to describe the interfacial interaction is so |
617 |
> |
called the average surface area per lipid. Constat area and constant |
618 |
> |
lateral pressure simulation can be achieved by extending the |
619 |
> |
standard NPT ensemble with a different pressure control strategy |
620 |
|
|
621 |
< |
There is one additional extended system integrator which is somewhat |
622 |
< |
simpler than the NPTf method described above. In this case, the |
623 |
< |
three axes have independent barostats which each attempt to preserve |
624 |
< |
the target pressure along the box walls perpendicular to that |
625 |
< |
particular axis. The lengths of the box axes are allowed to |
626 |
< |
fluctuate independently, but the angle between the box axes does not |
627 |
< |
change. The equations of motion are identical to those described |
628 |
< |
above, but only the {\it diagonal} elements of |
626 |
< |
$\overleftrightarrow{\eta}$ are computed. The off-diagonal elements |
627 |
< |
are set to zero (even when the pressure tensor has non-zero |
628 |
< |
off-diagonal elements). It should be noted that the NPTxyz |
629 |
< |
integrator is a special case of $NP\gamma T$ if the surface tension |
630 |
< |
$\gamma$ is set to zero. |
621 |
> |
\begin{equation} |
622 |
> |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
623 |
> |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau_{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }} |
624 |
> |
& \mbox{if $ \alpha = \beta = z$}\\ |
625 |
> |
0 & \mbox{otherwise}\\ |
626 |
> |
\end{array} |
627 |
> |
\right. |
628 |
> |
\end{equation} |
629 |
|
|
630 |
+ |
Note that the iterative schemes for NPAT are identical to those |
631 |
+ |
described for the NPTi integrator. |
632 |
|
|
633 |
< |
\section{\label{methodSection:constraintMethods}Constraint Methods} |
633 |
> |
\subsubsection{\label{methodSection:NPrT}\textbf{NP$\gamma$T Ensemble}} |
634 |
|
|
635 |
< |
\subsection{\label{methodSection:rattle}The {\sc rattle} Method for Bond |
636 |
< |
Constraints} |
635 |
> |
Theoretically, the surface tension $\gamma$ of a stress free |
636 |
> |
membrane system should be zero since its surface free energy $G$ is |
637 |
> |
minimum with respect to surface area $A$ |
638 |
> |
\[ |
639 |
> |
\gamma = \frac{{\partial G}}{{\partial A}}. |
640 |
> |
\] |
641 |
> |
However, a surface tension of zero is not appropriate for relatively |
642 |
> |
small patches of membrane. In order to eliminate the edge effect of |
643 |
> |
the membrane simulation, a special ensemble, NP$\gamma$T, is |
644 |
> |
proposed to maintain the lateral surface tension and normal |
645 |
> |
pressure. The equation of motion for cell size control tensor, |
646 |
> |
$\eta$, in $NP\gamma T$ is |
647 |
> |
\begin{equation} |
648 |
> |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
649 |
> |
- A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ) & \mbox{$\alpha = \beta = x$ or $\alpha = \beta = y$}\\ |
650 |
> |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}}}} & \mbox{$\alpha = \beta = z$} \\ |
651 |
> |
0 & \mbox{$\alpha \ne \beta$} \\ |
652 |
> |
\end{array} |
653 |
> |
\right. |
654 |
> |
\end{equation} |
655 |
> |
where $ \gamma _{{\rm{target}}}$ is the external surface tension and |
656 |
> |
the instantaneous surface tensor $\gamma _\alpha$ is given by |
657 |
> |
\begin{equation} |
658 |
> |
\gamma _\alpha = - h_z( \overleftrightarrow{P} _{\alpha \alpha } |
659 |
> |
- P_{{\rm{target}}} ) |
660 |
> |
\label{methodEquation:instantaneousSurfaceTensor} |
661 |
> |
\end{equation} |
662 |
|
|
663 |
< |
\subsection{\label{methodSection:zcons}Z-Constraint Method} |
663 |
> |
There is one additional extended system integrator (NPTxyz), in |
664 |
> |
which each attempt to preserve the target pressure along the box |
665 |
> |
walls perpendicular to that particular axis. The lengths of the box |
666 |
> |
axes are allowed to fluctuate independently, but the angle between |
667 |
> |
the box axes does not change. It should be noted that the NPTxyz |
668 |
> |
integrator is a special case of $NP\gamma T$ if the surface tension |
669 |
> |
$\gamma$ is set to zero. |
670 |
|
|
671 |
+ |
\section{\label{methodSection:zcons}Z-Constraint Method} |
672 |
+ |
|
673 |
|
Based on the fluctuation-dissipation theorem, a force |
674 |
|
auto-correlation method was developed by Roux and Karplus to |
675 |
< |
investigate the dynamics of ions inside ion channels.\cite{Roux91} |
675 |
> |
investigate the dynamics of ions inside ion channels\cite{Roux1991}. |
676 |
|
The time-dependent friction coefficient can be calculated from the |
677 |
|
deviation of the instantaneous force from its mean force. |
678 |
|
\begin{equation} |
690 |
|
F(z,0)\rangle dt. |
691 |
|
\end{equation} |
692 |
|
Allowing diffusion constant to then be calculated through the |
693 |
< |
Einstein relation:\cite{Marrink94} |
693 |
> |
Einstein relation:\cite{Marrink1994} |
694 |
|
\begin{equation} |
695 |
|
D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
696 |
|
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.% |
699 |
|
The Z-Constraint method, which fixes the z coordinates of the |
700 |
|
molecules with respect to the center of the mass of the system, has |
701 |
|
been a method suggested to obtain the forces required for the force |
702 |
< |
auto-correlation calculation.\cite{Marrink94} However, simply |
702 |
> |
auto-correlation calculation.\cite{Marrink1994} However, simply |
703 |
|
resetting the coordinate will move the center of the mass of the |
704 |
< |
whole system. To avoid this problem, a new method was used in {\sc |
672 |
< |
oopse}. Instead of resetting the coordinate, we reset the forces of |
704 |
> |
whole system. To avoid this problem, we reset the forces of |
705 |
|
z-constrained molecules as well as subtract the total constraint |
706 |
|
forces from the rest of the system after the force calculation at |
707 |
< |
each time step. |
707 |
> |
each time step instead of resetting the coordinate. |
708 |
|
|
709 |
|
After the force calculation, define $G_\alpha$ as |
710 |
|
\begin{equation} |
757 |
|
F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= |
758 |
|
-k_{\text{Harmonic}}(z(t)-z_{\text{cons}}). |
759 |
|
\end{equation} |
728 |
– |
|
729 |
– |
\section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies} |
730 |
– |
|
731 |
– |
\subsection{\label{methodSection:temperature}Temperature Control} |
732 |
– |
|
733 |
– |
\subsection{\label{methodSection:pressureControl}Pressure Control} |
734 |
– |
|
735 |
– |
\section{\label{methodSection:hydrodynamics}Hydrodynamics} |
736 |
– |
|
737 |
– |
%\section{\label{methodSection:coarseGrained}Coarse-Grained Modeling} |
738 |
– |
|
739 |
– |
%\section{\label{methodSection:moleculeScale}Molecular-Scale Modeling} |