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), \\ |
356 |
|
\end{equation} |
357 |
|
|
358 |
|
In eq.(\ref{eq:melchionna1}), $\tau_B$ is the time constant for |
359 |
< |
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 |
359 |
> |
relaxation of the pressure to the target value. Like in the NVT |
360 |
|
integrator, the integration of the equations of motion is carried |
361 |
|
out in a velocity-Verlet style 2 part algorithm: |
362 |
|
|
468 |
|
\left( \chi^2 \tau_T^2 + \eta^2 \tau_B^2 \right) + |
469 |
|
P_{\mathrm{target}} \mathcal{V}(t). |
470 |
|
\end{equation} |
478 |
– |
|
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}. |
471 |
|
|
472 |
|
\subsection{\label{methodSection:NPTf}Constant-pressure integration with a |
473 |
|
flexible box (NPTf)} |
600 |
|
|
601 |
|
\subsection{\label{methodSection:otherSpecialEnsembles}Other Special Ensembles} |
602 |
|
|
603 |
< |
\subsubsection{\label{methodSection:NPAT}Constant Normal Pressure, Constant Lateral Surface Area and Constant Temperature (NPAT) Ensemble} |
603 |
> |
\subsubsection{\label{methodSection:NPAT}\textbf{NPAT Ensemble}} |
604 |
|
|
605 |
|
A comprehensive understanding of structure¨Cfunction relations of |
606 |
|
biological membrane system ultimately relies on structure and |
610 |
|
called the average surface area per lipid. Constat area and constant |
611 |
|
lateral pressure simulation can be achieved by extending the |
612 |
|
standard NPT ensemble with a different pressure control strategy |
613 |
+ |
|
614 |
|
\begin{equation} |
615 |
< |
\dot |
616 |
< |
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} |
617 |
< |
\over \eta } _{\alpha \beta } = \left\{ \begin{array}{l} |
618 |
< |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }}{\rm{ }}(\alpha = \beta = z), \\ |
619 |
< |
0{\rm{ }}(\alpha \ne z{\rm{ }}or{\rm{ }}\beta \ne z) \\ |
620 |
< |
\end{array} \right. |
631 |
< |
\label{methodEquation:NPATeta} |
615 |
> |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
616 |
> |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau_{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }} |
617 |
> |
& \mbox{if $ \alpha = \beta = z$}\\ |
618 |
> |
0 & \mbox{otherwise}\\ |
619 |
> |
\end{array} |
620 |
> |
\right. |
621 |
|
\end{equation} |
622 |
+ |
|
623 |
|
Note that the iterative schemes for NPAT are identical to those |
624 |
|
described for the NPTi integrator. |
625 |
|
|
626 |
< |
\subsubsection{\label{methodSection:NPrT}Constant Normal Pressure, Constant Lateral Surface Tension and Constant Temperature (NP\gamma T) Ensemble } |
626 |
> |
\subsubsection{\label{methodSection:NPrT}\textbf{NP$\gamma$T Ensemble}} |
627 |
|
|
628 |
|
Theoretically, the surface tension $\gamma$ of a stress free |
629 |
|
membrane system should be zero since its surface free energy $G$ is |
633 |
|
\] |
634 |
|
However, a surface tension of zero is not appropriate for relatively |
635 |
|
small patches of membrane. In order to eliminate the edge effect of |
636 |
< |
the membrane simulation, a special ensemble, NP\gamma T, is proposed |
637 |
< |
to maintain the lateral surface tension and normal pressure. The |
638 |
< |
equation of motion for cell size control tensor, $\eta$, in NP\gamma |
639 |
< |
T is |
636 |
> |
the membrane simulation, a special ensemble, NP$\gamma$T, is |
637 |
> |
proposed to maintain the lateral surface tension and normal |
638 |
> |
pressure. The equation of motion for cell size control tensor, |
639 |
> |
$\eta$, in $NP\gamma T$ is |
640 |
|
\begin{equation} |
641 |
< |
\dot |
642 |
< |
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} |
643 |
< |
\over \eta } _{\alpha \beta } = \left\{ \begin{array}{l} |
644 |
< |
- A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ){\rm{ (}}\alpha = \beta = x{\rm{ or }} = y{\rm{)}} \\ |
645 |
< |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }}{\rm{ }}(\alpha = \beta = z) \\ |
646 |
< |
0{\rm{ }}(\alpha \ne \beta ) \\ |
657 |
< |
\end{array} \right. |
658 |
< |
\label{methodEquation:NPrTeta} |
641 |
> |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
642 |
> |
- A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ) & \mbox{$\alpha = \beta = x$ or $\alpha = \beta = y$}\\ |
643 |
> |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}}}} & \mbox{$\alpha = \beta = z$} \\ |
644 |
> |
0 & \mbox{$\alpha \ne \beta$} \\ |
645 |
> |
\end{array} |
646 |
> |
\right. |
647 |
|
\end{equation} |
648 |
|
where $ \gamma _{{\rm{target}}}$ is the external surface tension and |
649 |
|
the instantaneous surface tensor $\gamma _\alpha$ is given by |
650 |
|
\begin{equation} |
651 |
< |
\gamma _\alpha = - h_z |
652 |
< |
(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} |
665 |
< |
\over P} _{\alpha \alpha } - P_{{\rm{target}}} ) |
651 |
> |
\gamma _\alpha = - h_z( \overleftrightarrow{P} _{\alpha \alpha } |
652 |
> |
- P_{{\rm{target}}} ) |
653 |
|
\label{methodEquation:instantaneousSurfaceTensor} |
654 |
|
\end{equation} |
655 |
|
|
661 |
|
integrator is a special case of $NP\gamma T$ if the surface tension |
662 |
|
$\gamma$ is set to zero. |
663 |
|
|
664 |
< |
\section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies} |
664 |
> |
\section{\label{methodSection:zcons}Z-Constraint Method} |
665 |
|
|
666 |
< |
\subsection{\label{methodSection:temperature}Temperature Control} |
666 |
> |
Based on the fluctuation-dissipation theorem, a force |
667 |
> |
auto-correlation method was developed by Roux and Karplus to |
668 |
> |
investigate the dynamics of ions inside ion channels\cite{Roux1991}. |
669 |
> |
The time-dependent friction coefficient can be calculated from the |
670 |
> |
deviation of the instantaneous force from its mean force. |
671 |
> |
\begin{equation} |
672 |
> |
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T, |
673 |
> |
\end{equation} |
674 |
> |
where% |
675 |
> |
\begin{equation} |
676 |
> |
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle. |
677 |
> |
\end{equation} |
678 |
|
|
679 |
< |
\subsection{\label{methodSection:pressureControl}Pressure Control} |
679 |
> |
If the time-dependent friction decays rapidly, the static friction |
680 |
> |
coefficient can be approximated by |
681 |
> |
\begin{equation} |
682 |
> |
\xi_{\text{static}}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta |
683 |
> |
F(z,0)\rangle dt. |
684 |
> |
\end{equation} |
685 |
> |
Allowing diffusion constant to then be calculated through the |
686 |
> |
Einstein relation:\cite{Marrink1994} |
687 |
> |
\begin{equation} |
688 |
> |
D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
689 |
> |
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.% |
690 |
> |
\end{equation} |
691 |
|
|
692 |
< |
\section{\label{methodSection:hydrodynamics}Hydrodynamics} |
692 |
> |
The Z-Constraint method, which fixes the z coordinates of the |
693 |
> |
molecules with respect to the center of the mass of the system, has |
694 |
> |
been a method suggested to obtain the forces required for the force |
695 |
> |
auto-correlation calculation.\cite{Marrink1994} However, simply |
696 |
> |
resetting the coordinate will move the center of the mass of the |
697 |
> |
whole system. To avoid this problem, we reset the forces of |
698 |
> |
z-constrained molecules as well as subtract the total constraint |
699 |
> |
forces from the rest of the system after the force calculation at |
700 |
> |
each time step instead of resetting the coordinate. |
701 |
|
|
702 |
< |
%\section{\label{methodSection:coarseGrained}Coarse-Grained Modeling} |
702 |
> |
After the force calculation, define $G_\alpha$ as |
703 |
> |
\begin{equation} |
704 |
> |
G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1} |
705 |
> |
\end{equation} |
706 |
> |
where $F_{\alpha i}$ is the force in the z direction of atom $i$ in |
707 |
> |
z-constrained molecule $\alpha$. The forces of the z constrained |
708 |
> |
molecule are then set to: |
709 |
> |
\begin{equation} |
710 |
> |
F_{\alpha i} = F_{\alpha i} - |
711 |
> |
\frac{m_{\alpha i} G_{\alpha}}{\sum_i m_{\alpha i}}. |
712 |
> |
\end{equation} |
713 |
> |
Here, $m_{\alpha i}$ is the mass of atom $i$ in the z-constrained |
714 |
> |
molecule. Having rescaled the forces, the velocities must also be |
715 |
> |
rescaled to subtract out any center of mass velocity in the z |
716 |
> |
direction. |
717 |
> |
\begin{equation} |
718 |
> |
v_{\alpha i} = v_{\alpha i} - |
719 |
> |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}, |
720 |
> |
\end{equation} |
721 |
> |
where $v_{\alpha i}$ is the velocity of atom $i$ in the z direction. |
722 |
> |
Lastly, all of the accumulated z constrained forces must be |
723 |
> |
subtracted from the system to keep the system center of mass from |
724 |
> |
drifting. |
725 |
> |
\begin{equation} |
726 |
> |
F_{\beta i} = F_{\beta i} - \frac{m_{\beta i} \sum_{\alpha} |
727 |
> |
G_{\alpha}} |
728 |
> |
{\sum_{\beta}\sum_i m_{\beta i}}, |
729 |
> |
\end{equation} |
730 |
> |
where $\beta$ are all of the unconstrained molecules in the system. |
731 |
> |
Similarly, the velocities of the unconstrained molecules must also |
732 |
> |
be scaled. |
733 |
> |
\begin{equation} |
734 |
> |
v_{\beta i} = v_{\beta i} + \sum_{\alpha} |
735 |
> |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}. |
736 |
> |
\end{equation} |
737 |
|
|
738 |
< |
%\section{\label{methodSection:moleculeScale}Molecular-Scale Modeling} |
738 |
> |
At the very beginning of the simulation, the molecules may not be at |
739 |
> |
their constrained positions. To move a z-constrained molecule to its |
740 |
> |
specified position, a simple harmonic potential is used |
741 |
> |
\begin{equation} |
742 |
> |
U(t)=\frac{1}{2}k_{\text{Harmonic}}(z(t)-z_{\text{cons}})^{2},% |
743 |
> |
\end{equation} |
744 |
> |
where $k_{\text{Harmonic}}$ is the harmonic force constant, $z(t)$ |
745 |
> |
is the current $z$ coordinate of the center of mass of the |
746 |
> |
constrained molecule, and $z_{\text{cons}}$ is the constrained |
747 |
> |
position. The harmonic force operating on the z-constrained molecule |
748 |
> |
at time $t$ can be calculated by |
749 |
> |
\begin{equation} |
750 |
> |
F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= |
751 |
> |
-k_{\text{Harmonic}}(z(t)-z_{\text{cons}}). |
752 |
> |
\end{equation} |