| 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) |
| 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), \\ |
| 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)} |
| 607 |
|
|
| 608 |
|
\subsection{\label{methodSection:otherSpecialEnsembles}Other Special Ensembles} |
| 609 |
|
|
| 610 |
< |
\subsubsection{\label{methodSection:NPAT}NPAT Ensemble} |
| 610 |
> |
\subsubsection{\label{methodSection:NPAT}\textbf{NPAT Ensemble}} |
| 611 |
|
|
| 612 |
|
A comprehensive understanding of structure¨Cfunction relations of |
| 613 |
|
biological membrane system ultimately relies on structure and |
| 619 |
|
standard NPT ensemble with a different pressure control strategy |
| 620 |
|
|
| 621 |
|
\begin{equation} |
| 622 |
< |
\.{\overleftrightarrow{{\eta _{\alpha \beta}}}}=\left\{\begin{array}{ll} |
| 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)$}\\ |
| 624 |
> |
& \mbox{if $ \alpha = \beta = z$}\\ |
| 625 |
|
0 & \mbox{otherwise}\\ |
| 626 |
|
\end{array} |
| 627 |
|
\right. |
| 630 |
|
Note that the iterative schemes for NPAT are identical to those |
| 631 |
|
described for the NPTi integrator. |
| 632 |
|
|
| 633 |
< |
\subsubsection{\label{methodSection:NPrT}NP$\gamma$T Ensemble} |
| 633 |
> |
\subsubsection{\label{methodSection:NPrT}\textbf{NP$\gamma$T Ensemble}} |
| 634 |
|
|
| 635 |
|
Theoretically, the surface tension $\gamma$ of a stress free |
| 636 |
|
membrane system should be zero since its surface free energy $G$ is |
| 645 |
|
pressure. The equation of motion for cell size control tensor, |
| 646 |
|
$\eta$, in $NP\gamma T$ is |
| 647 |
|
\begin{equation} |
| 648 |
< |
\.{\overleftrightarrow{{\eta _{\alpha \beta}}}}=\left\{\begin{array}{ll} |
| 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 |
| 659 |
< |
(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} |
| 663 |
< |
\over P} _{\alpha \alpha } - P_{{\rm{target}}} ) |
| 658 |
> |
\gamma _\alpha = - h_z( \overleftrightarrow{P} _{\alpha \alpha } |
| 659 |
> |
- P_{{\rm{target}}} ) |
| 660 |
|
\label{methodEquation:instantaneousSurfaceTensor} |
| 661 |
|
\end{equation} |
| 662 |
|
|
| 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:constraintMethod}Constraint Method} |
| 671 |
> |
\section{\label{methodSection:zcons}Z-Constraint Method} |
| 672 |
|
|
| 673 |
< |
%\subsection{\label{methodSection:bondConstraint}Bond Constraint for Rigid Body} |
| 674 |
< |
|
| 675 |
< |
%\subsection{\label{methodSection:zcons}Z-constraint Method} |
| 676 |
< |
|
| 677 |
< |
\section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies} |
| 678 |
< |
|
| 679 |
< |
\subsection{\label{methodSection:temperature}Temperature Control} |
| 680 |
< |
|
| 681 |
< |
\subsection{\label{methodSection:pressureControl}Pressure Control} |
| 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{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} |
| 679 |
> |
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T, |
| 680 |
> |
\end{equation} |
| 681 |
> |
where% |
| 682 |
> |
\begin{equation} |
| 683 |
> |
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle. |
| 684 |
> |
\end{equation} |
| 685 |
|
|
| 686 |
< |
\section{\label{methodSection:hydrodynamics}Hydrodynamics} |
| 686 |
> |
If the time-dependent friction decays rapidly, the static friction |
| 687 |
> |
coefficient can be approximated by |
| 688 |
> |
\begin{equation} |
| 689 |
> |
\xi_{\text{static}}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta |
| 690 |
> |
F(z,0)\rangle dt. |
| 691 |
> |
\end{equation} |
| 692 |
> |
Allowing diffusion constant to then be calculated through the |
| 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}.% |
| 697 |
> |
\end{equation} |
| 698 |
|
|
| 699 |
< |
%\section{\label{methodSection:coarseGrained}Coarse-Grained Modeling} |
| 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{Marrink1994} However, simply |
| 703 |
> |
resetting the coordinate will move the center of the mass of the |
| 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 instead of resetting the coordinate. |
| 708 |
|
|
| 709 |
< |
%\section{\label{methodSection:moleculeScale}Molecular-Scale Modeling} |
| 709 |
> |
After the force calculation, define $G_\alpha$ as |
| 710 |
> |
\begin{equation} |
| 711 |
> |
G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1} |
| 712 |
> |
\end{equation} |
| 713 |
> |
where $F_{\alpha i}$ is the force in the z direction of atom $i$ in |
| 714 |
> |
z-constrained molecule $\alpha$. The forces of the z constrained |
| 715 |
> |
molecule are then set to: |
| 716 |
> |
\begin{equation} |
| 717 |
> |
F_{\alpha i} = F_{\alpha i} - |
| 718 |
> |
\frac{m_{\alpha i} G_{\alpha}}{\sum_i m_{\alpha i}}. |
| 719 |
> |
\end{equation} |
| 720 |
> |
Here, $m_{\alpha i}$ is the mass of atom $i$ in the z-constrained |
| 721 |
> |
molecule. Having rescaled the forces, the velocities must also be |
| 722 |
> |
rescaled to subtract out any center of mass velocity in the z |
| 723 |
> |
direction. |
| 724 |
> |
\begin{equation} |
| 725 |
> |
v_{\alpha i} = v_{\alpha i} - |
| 726 |
> |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}, |
| 727 |
> |
\end{equation} |
| 728 |
> |
where $v_{\alpha i}$ is the velocity of atom $i$ in the z direction. |
| 729 |
> |
Lastly, all of the accumulated z constrained forces must be |
| 730 |
> |
subtracted from the system to keep the system center of mass from |
| 731 |
> |
drifting. |
| 732 |
> |
\begin{equation} |
| 733 |
> |
F_{\beta i} = F_{\beta i} - \frac{m_{\beta i} \sum_{\alpha} |
| 734 |
> |
G_{\alpha}} |
| 735 |
> |
{\sum_{\beta}\sum_i m_{\beta i}}, |
| 736 |
> |
\end{equation} |
| 737 |
> |
where $\beta$ are all of the unconstrained molecules in the system. |
| 738 |
> |
Similarly, the velocities of the unconstrained molecules must also |
| 739 |
> |
be scaled. |
| 740 |
> |
\begin{equation} |
| 741 |
> |
v_{\beta i} = v_{\beta i} + \sum_{\alpha} |
| 742 |
> |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}. |
| 743 |
> |
\end{equation} |
| 744 |
> |
|
| 745 |
> |
At the very beginning of the simulation, the molecules may not be at |
| 746 |
> |
their constrained positions. To move a z-constrained molecule to its |
| 747 |
> |
specified position, a simple harmonic potential is used |
| 748 |
> |
\begin{equation} |
| 749 |
> |
U(t)=\frac{1}{2}k_{\text{Harmonic}}(z(t)-z_{\text{cons}})^{2},% |
| 750 |
> |
\end{equation} |
| 751 |
> |
where $k_{\text{Harmonic}}$ is the harmonic force constant, $z(t)$ |
| 752 |
> |
is the current $z$ coordinate of the center of mass of the |
| 753 |
> |
constrained molecule, and $z_{\text{cons}}$ is the constrained |
| 754 |
> |
position. The harmonic force operating on the z-constrained molecule |
| 755 |
> |
at time $t$ can be calculated by |
| 756 |
> |
\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} |
