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 |
117 |
|
\cdot {\bf \tau}^s(t + h). |
118 |
|
\end{align*} |
119 |
|
|
120 |
< |
{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix |
120 |
> |
${\bf u}$ will be automatically updated when the rotation matrix |
121 |
|
$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
122 |
|
torques have been obtained at the new time step, the velocities can |
123 |
|
be advanced to the same time value. |
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} ,\\ |
206 |
|
\end{equation} |
207 |
|
|
208 |
|
In eq.(\ref{eq:nosehooverext}), $\tau_T$ is the time constant for |
209 |
< |
relaxation of the temperature to the target value. To set values |
210 |
< |
for $\tau_T$ or $T_{\mathrm{target}}$ in a simulation, one would use |
211 |
< |
the {\tt tauThermostat} and {\tt targetTemperature} keywords in the |
211 |
< |
{\tt .bass} file. The units for {\tt tauThermostat} are fs, and the |
212 |
< |
units for the {\tt targetTemperature} are degrees K. The |
213 |
< |
integration of the equations of motion is carried out in a |
214 |
< |
velocity-Verlet style 2 part algorithm: |
209 |
> |
relaxation of the temperature to the target value. The integration |
210 |
> |
of the equations of motion is carried out in a velocity-Verlet style |
211 |
> |
2 part algorithm: |
212 |
|
|
213 |
|
{\tt moveA:} |
214 |
|
\begin{align*} |
274 |
|
caclculate $T(t + h)$ as well as $\chi(t + h)$, they indirectly |
275 |
|
depend on their own values at time $t + h$. {\tt moveB} is |
276 |
|
therefore done in an iterative fashion until $\chi(t + h)$ becomes |
277 |
< |
self-consistent. The relative tolerance for the self-consistency |
281 |
< |
check defaults to a value of $\mbox{10}^{-6}$, but {\sc oopse} will |
282 |
< |
terminate the iteration after 4 loops even if the consistency check |
283 |
< |
has not been satisfied. |
277 |
> |
self-consistent. |
278 |
|
|
279 |
|
The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for |
280 |
|
the extended system that is, to within a constant, identical to the |
281 |
< |
Helmholtz free energy,\cite{melchionna93} |
281 |
> |
Helmholtz free energy,\cite{Melchionna1993} |
282 |
|
\begin{equation} |
283 |
|
H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left( |
284 |
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
289 |
|
last column of the {\tt .stat} file to allow checks on the quality |
290 |
|
of the integration. |
291 |
|
|
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 |
– |
|
292 |
|
\subsection{\label{methodSection:NPTi}Constant-pressure integration with |
293 |
|
isotropic box deformations (NPTi)} |
294 |
|
|
295 |
< |
To carry out isobaric-isothermal ensemble calculations {\sc oopse} |
296 |
< |
implements the Melchionna modifications to the |
297 |
< |
Nos\'e-Hoover-Andersen equations of motion,\cite{melchionna93} |
295 |
> |
Isobaric-isothermal ensemble integrator is implemented using the |
296 |
> |
Melchionna modifications to the Nos\'e-Hoover-Andersen equations of |
297 |
> |
motion,\cite{Melchionna1993} |
298 |
|
|
299 |
|
\begin{eqnarray} |
300 |
|
\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\ |
349 |
|
\end{equation} |
350 |
|
|
351 |
|
In eq.(\ref{eq:melchionna1}), $\tau_B$ is the time constant for |
352 |
< |
relaxation of the pressure to the target value. To set values for |
363 |
< |
$\tau_B$ or $P_{\mathrm{target}}$ in a simulation, one would use the |
364 |
< |
{\tt tauBarostat} and {\tt targetPressure} keywords in the {\tt |
365 |
< |
.bass} file. The units for {\tt tauBarostat} are fs, and the units |
366 |
< |
for the {\tt targetPressure} are atmospheres. Like in the NVT |
352 |
> |
relaxation of the pressure to the target value. Like in the NVT |
353 |
|
integrator, the integration of the equations of motion is carried |
354 |
|
out in a velocity-Verlet style 2 part algorithm: |
355 |
|
|
390 |
|
|
391 |
|
Most of these equations are identical to their counterparts in the |
392 |
|
NVT integrator, but the propagation of positions to time $t + h$ |
393 |
< |
depends on the positions at the same time. {\sc oopse} carries out |
394 |
< |
this step iteratively (with a limit of 5 passes through the |
395 |
< |
iterative loop). Also, the simulation box $\mathsf{H}$ is scaled |
396 |
< |
uniformly for one full time step by an exponential factor that |
397 |
< |
depends on the value of $\eta$ at time $t + h / 2$. Reshaping the |
412 |
< |
box uniformly also scales the volume of the box by |
393 |
> |
depends on the positions at the same time. The simulation box |
394 |
> |
$\mathsf{H}$ is scaled uniformly for one full time step by an |
395 |
> |
exponential factor that depends on the value of $\eta$ at time $t + |
396 |
> |
h / 2$. Reshaping the box uniformly also scales the volume of the |
397 |
> |
box by |
398 |
|
\begin{equation} |
399 |
|
\mathcal{V}(t + h) \leftarrow e^{ - 3 h \eta(t + h /2)}. |
400 |
|
\mathcal{V}(t) |
436 |
|
to caclculate $T(t + h)$, $P(t + h)$, $\chi(t + h)$, and $\eta(t + |
437 |
|
h)$, they indirectly depend on their own values at time $t + h$. |
438 |
|
{\tt moveB} is therefore done in an iterative fashion until $\chi(t |
439 |
< |
+ h)$ and $\eta(t + h)$ become self-consistent. The relative |
455 |
< |
tolerance for the self-consistency check defaults to a value of |
456 |
< |
$\mbox{10}^{-6}$, but {\sc oopse} will terminate the iteration after |
457 |
< |
4 loops even if the consistency check has not been satisfied. |
439 |
> |
+ h)$ and $\eta(t + h)$ become self-consistent. |
440 |
|
|
441 |
|
The Melchionna modification of the Nos\'e-Hoover-Andersen algorithm |
442 |
|
is known to conserve a Hamiltonian for the extended system that is, |
458 |
|
P_{\mathrm{target}} \mathcal{V}(t). |
459 |
|
\end{equation} |
460 |
|
|
479 |
– |
Bond constraints are applied at the end of both the {\tt moveA} and |
480 |
– |
{\tt moveB} portions of the algorithm. Details on the constraint |
481 |
– |
algorithms are given in section \ref{oopseSec:rattle}. |
482 |
– |
|
461 |
|
\subsection{\label{methodSection:NPTf}Constant-pressure integration with a |
462 |
|
flexible box (NPTf)} |
463 |
|
|
531 |
|
\mathsf{H}(t + h) &\leftarrow \mathsf{H}(t) \cdot e^{-h |
532 |
|
\overleftrightarrow{\eta}(t + h / 2)} . |
533 |
|
\end{align*} |
534 |
< |
{\sc oopse} uses a power series expansion truncated at second order |
535 |
< |
for the exponential operation which scales the simulation box. |
534 |
> |
Here, a power series expansion truncated at second order for the |
535 |
> |
exponential operation is used to scale the simulation box. |
536 |
|
|
537 |
|
The {\tt moveB} portion of the algorithm is largely unchanged from |
538 |
|
the NPTi integrator: |
571 |
|
identical to those described for the NPTi integrator. |
572 |
|
|
573 |
|
The NPTf integrator is known to conserve the following Hamiltonian: |
574 |
< |
\begin{equation} |
575 |
< |
H_{\mathrm{NPTf}} = V + K + f k_B T_{\mathrm{target}} \left( |
574 |
> |
\begin{eqnarray*} |
575 |
> |
H_{\mathrm{NPTf}} & = & V + K + f k_B T_{\mathrm{target}} \left( |
576 |
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
577 |
< |
dt^\prime \right) + P_{\mathrm{target}} \mathcal{V}(t) + \frac{f k_B |
577 |
> |
dt^\prime \right) \\ |
578 |
> |
& & + P_{\mathrm{target}} \mathcal{V}(t) + \frac{f k_B |
579 |
|
T_{\mathrm{target}}}{2} |
580 |
|
\mathrm{Tr}\left[\overleftrightarrow{\eta}(t)\right]^2 \tau_B^2. |
581 |
< |
\end{equation} |
581 |
> |
\end{eqnarray*} |
582 |
|
|
583 |
|
This integrator must be used with care, particularly in liquid |
584 |
|
simulations. Liquids have very small restoring forces in the |
588 |
|
finds most use in simulating crystals or liquid crystals which |
589 |
|
assume non-orthorhombic geometries. |
590 |
|
|
591 |
< |
\subsection{\label{methodSection:NPAT}Constant Lateral Pressure and Constant Surface Area (NPAT)} |
591 |
> |
\subsection{\label{methodSection:NPAT}NPAT Ensemble} |
592 |
|
|
593 |
< |
\subsection{\label{methodSection:NPrT}Constant lateral Pressure and Constant Surface Tension (NP\gamma T) } |
593 |
> |
A comprehensive understanding of structure¨Cfunction relations of |
594 |
> |
biological membrane system ultimately relies on structure and |
595 |
> |
dynamics of lipid bilayer, which are strongly affected by the |
596 |
> |
interfacial interaction between lipid molecules and surrounding |
597 |
> |
media. One quantity to describe the interfacial interaction is so |
598 |
> |
called the average surface area per lipid. Constat area and constant |
599 |
> |
lateral pressure simulation can be achieved by extending the |
600 |
> |
standard NPT ensemble with a different pressure control strategy |
601 |
|
|
602 |
< |
\subsection{\label{methodSection:NPTxyz}Constant pressure in 3 axes (NPTxyz)} |
602 |
> |
\begin{equation} |
603 |
> |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
604 |
> |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau_{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }} |
605 |
> |
& \mbox{if $ \alpha = \beta = z$}\\ |
606 |
> |
0 & \mbox{otherwise}\\ |
607 |
> |
\end{array} |
608 |
> |
\right. |
609 |
> |
\end{equation} |
610 |
|
|
611 |
< |
There is one additional extended system integrator which is somewhat |
612 |
< |
simpler than the NPTf method described above. In this case, the |
620 |
< |
three axes have independent barostats which each attempt to preserve |
621 |
< |
the target pressure along the box walls perpendicular to that |
622 |
< |
particular axis. The lengths of the box axes are allowed to |
623 |
< |
fluctuate independently, but the angle between the box axes does not |
624 |
< |
change. The equations of motion are identical to those described |
625 |
< |
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. |
611 |
> |
Note that the iterative schemes for NPAT are identical to those |
612 |
> |
described for the NPTi integrator. |
613 |
|
|
614 |
+ |
\subsection{\label{methodSection:NPrT}NP$\gamma$T |
615 |
+ |
Ensemble} |
616 |
|
|
617 |
< |
\section{\label{methodSection:constraintMethods}Constraint Methods} |
617 |
> |
Theoretically, the surface tension $\gamma$ of a stress free |
618 |
> |
membrane system should be zero since its surface free energy $G$ is |
619 |
> |
minimum with respect to surface area $A$ |
620 |
> |
\[ |
621 |
> |
\gamma = \frac{{\partial G}}{{\partial A}}. |
622 |
> |
\] |
623 |
> |
However, a surface tension of zero is not appropriate for relatively |
624 |
> |
small patches of membrane. In order to eliminate the edge effect of |
625 |
> |
the membrane simulation, a special ensemble, NP$\gamma$T, is |
626 |
> |
proposed to maintain the lateral surface tension and normal |
627 |
> |
pressure. The equation of motion for cell size control tensor, |
628 |
> |
$\eta$, in $NP\gamma T$ is |
629 |
> |
\begin{equation} |
630 |
> |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
631 |
> |
- A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ) & \mbox{$\alpha = \beta = x$ or $\alpha = \beta = y$}\\ |
632 |
> |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}}}} & \mbox{$\alpha = \beta = z$} \\ |
633 |
> |
0 & \mbox{$\alpha \ne \beta$} \\ |
634 |
> |
\end{array} |
635 |
> |
\right. |
636 |
> |
\end{equation} |
637 |
> |
where $ \gamma _{{\rm{target}}}$ is the external surface tension and |
638 |
> |
the instantaneous surface tensor $\gamma _\alpha$ is given by |
639 |
> |
\begin{equation} |
640 |
> |
\gamma _\alpha = - h_z( \overleftrightarrow{P} _{\alpha \alpha } |
641 |
> |
- P_{{\rm{target}}} ) |
642 |
> |
\label{methodEquation:instantaneousSurfaceTensor} |
643 |
> |
\end{equation} |
644 |
|
|
645 |
< |
\subsection{\label{methodSection:rattle}The {\sc rattle} Method for Bond |
646 |
< |
Constraints} |
645 |
> |
There is one additional extended system integrator (NPTxyz), in |
646 |
> |
which each attempt to preserve the target pressure along the box |
647 |
> |
walls perpendicular to that particular axis. The lengths of the box |
648 |
> |
axes are allowed to fluctuate independently, but the angle between |
649 |
> |
the box axes does not change. It should be noted that the NPTxyz |
650 |
> |
integrator is a special case of $NP\gamma T$ if the surface tension |
651 |
> |
$\gamma$ is set to zero. |
652 |
|
|
653 |
< |
\subsection{\label{methodSection:zcons}Z-Constraint Method} |
653 |
> |
\section{\label{methodSection:zcons}Z-Constraint Method} |
654 |
|
|
655 |
|
Based on the fluctuation-dissipation theorem, a force |
656 |
|
auto-correlation method was developed by Roux and Karplus to |
657 |
< |
investigate the dynamics of ions inside ion channels.\cite{Roux91} |
657 |
> |
investigate the dynamics of ions inside ion channels\cite{Roux1991}. |
658 |
|
The time-dependent friction coefficient can be calculated from the |
659 |
|
deviation of the instantaneous force from its mean force. |
660 |
|
\begin{equation} |
672 |
|
F(z,0)\rangle dt. |
673 |
|
\end{equation} |
674 |
|
Allowing diffusion constant to then be calculated through the |
675 |
< |
Einstein relation:\cite{Marrink94} |
675 |
> |
Einstein relation:\cite{Marrink1994} |
676 |
|
\begin{equation} |
677 |
|
D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
678 |
|
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.% |
681 |
|
The Z-Constraint method, which fixes the z coordinates of the |
682 |
|
molecules with respect to the center of the mass of the system, has |
683 |
|
been a method suggested to obtain the forces required for the force |
684 |
< |
auto-correlation calculation.\cite{Marrink94} However, simply |
684 |
> |
auto-correlation calculation.\cite{Marrink1994} However, simply |
685 |
|
resetting the coordinate will move the center of the mass of the |
686 |
< |
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 |
686 |
> |
whole system. To avoid this problem, we reset the forces of |
687 |
|
z-constrained molecules as well as subtract the total constraint |
688 |
|
forces from the rest of the system after the force calculation at |
689 |
< |
each time step. |
689 |
> |
each time step instead of resetting the coordinate. |
690 |
|
|
691 |
|
After the force calculation, define $G_\alpha$ as |
692 |
|
\begin{equation} |
739 |
|
F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= |
740 |
|
-k_{\text{Harmonic}}(z(t)-z_{\text{cons}}). |
741 |
|
\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} |