| 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 |
| 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), \\ |
| 606 |
|
finds most use in simulating crystals or liquid crystals which |
| 607 |
|
assume non-orthorhombic geometries. |
| 608 |
|
|
| 609 |
< |
\subsection{\label{methodSection:NPAT}Constant Lateral Pressure and Constant Surface Area (NPAT)} |
| 613 |
< |
|
| 614 |
< |
\subsection{\label{methodSection:NPrT}Constant lateral Pressure and Constant Surface Tension (NP\gamma T) } |
| 609 |
> |
\subsection{\label{methodSection:otherSpecialEnsembles}Other Special Ensembles} |
| 610 |
|
|
| 611 |
< |
\subsection{\label{methodSection:NPTxyz}Constant pressure in 3 axes (NPTxyz)} |
| 611 |
> |
\subsubsection{\label{methodSection:NPAT}NPAT Ensemble} |
| 612 |
|
|
| 613 |
< |
There is one additional extended system integrator which is somewhat |
| 614 |
< |
simpler than the NPTf method described above. In this case, the |
| 615 |
< |
three axes have independent barostats which each attempt to preserve |
| 616 |
< |
the target pressure along the box walls perpendicular to that |
| 617 |
< |
particular axis. The lengths of the box axes are allowed to |
| 618 |
< |
fluctuate independently, but the angle between the box axes does not |
| 619 |
< |
change. The equations of motion are identical to those described |
| 620 |
< |
above, but only the {\it diagonal} elements of |
| 621 |
< |
$\overleftrightarrow{\eta}$ are computed. The off-diagonal elements |
| 622 |
< |
are set to zero (even when the pressure tensor has non-zero |
| 623 |
< |
off-diagonal elements). It should be noted that the NPTxyz |
| 624 |
< |
integrator is a special case of $NP\gamma T$ if the surface tension |
| 625 |
< |
$\gamma$ is set to zero. |
| 613 |
> |
A comprehensive understanding of structure¨Cfunction relations of |
| 614 |
> |
biological membrane system ultimately relies on structure and |
| 615 |
> |
dynamics of lipid bilayer, which are strongly affected by the |
| 616 |
> |
interfacial interaction between lipid molecules and surrounding |
| 617 |
> |
media. One quantity to describe the interfacial interaction is so |
| 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 |
> |
|
| 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$}\\ |
| 626 |
> |
0 & \mbox{otherwise}\\ |
| 627 |
> |
\end{array} |
| 628 |
> |
\right. |
| 629 |
> |
\end{equation} |
| 630 |
|
|
| 631 |
+ |
Note that the iterative schemes for NPAT are identical to those |
| 632 |
+ |
described for the NPTi integrator. |
| 633 |
|
|
| 634 |
< |
\section{\label{methodSection:constraintMethods}Constraint Methods} |
| 634 |
> |
\subsubsection{\label{methodSection:NPrT}NP$\gamma$T Ensemble} |
| 635 |
|
|
| 636 |
< |
\subsection{\label{methodSection:rattle}The {\sc rattle} Method for Bond |
| 637 |
< |
Constraints} |
| 638 |
< |
|
| 639 |
< |
\subsection{\label{methodSection:zcons}Z-Constraint Method} |
| 640 |
< |
|
| 641 |
< |
Based on the fluctuation-dissipation theorem, a force |
| 642 |
< |
auto-correlation method was developed by Roux and Karplus to |
| 643 |
< |
investigate the dynamics of ions inside ion channels.\cite{Roux91} |
| 644 |
< |
The time-dependent friction coefficient can be calculated from the |
| 645 |
< |
deviation of the instantaneous force from its mean force. |
| 636 |
> |
Theoretically, the surface tension $\gamma$ of a stress free |
| 637 |
> |
membrane system should be zero since its surface free energy $G$ is |
| 638 |
> |
minimum with respect to surface area $A$ |
| 639 |
> |
\[ |
| 640 |
> |
\gamma = \frac{{\partial G}}{{\partial A}}. |
| 641 |
> |
\] |
| 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 |
| 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 |
< |
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T, |
| 650 |
< |
\end{equation} |
| 651 |
< |
where% |
| 652 |
< |
\begin{equation} |
| 653 |
< |
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle. |
| 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 |
< |
|
| 657 |
< |
If the time-dependent friction decays rapidly, the static friction |
| 654 |
< |
coefficient can be approximated by |
| 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 |
< |
\xi_{\text{static}}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta |
| 660 |
< |
F(z,0)\rangle dt. |
| 659 |
> |
\gamma _\alpha = - h_z( \overleftrightarrow{P} _{\alpha \alpha } |
| 660 |
> |
- P_{{\rm{target}}} ) |
| 661 |
> |
\label{methodEquation:instantaneousSurfaceTensor} |
| 662 |
|
\end{equation} |
| 659 |
– |
Allowing diffusion constant to then be calculated through the |
| 660 |
– |
Einstein relation:\cite{Marrink94} |
| 661 |
– |
\begin{equation} |
| 662 |
– |
D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
| 663 |
– |
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.% |
| 664 |
– |
\end{equation} |
| 663 |
|
|
| 664 |
< |
The Z-Constraint method, which fixes the z coordinates of the |
| 665 |
< |
molecules with respect to the center of the mass of the system, has |
| 666 |
< |
been a method suggested to obtain the forces required for the force |
| 667 |
< |
auto-correlation calculation.\cite{Marrink94} However, simply |
| 668 |
< |
resetting the coordinate will move the center of the mass of the |
| 669 |
< |
whole system. To avoid this problem, a new method was used in {\sc |
| 670 |
< |
oopse}. Instead of resetting the coordinate, we reset the forces of |
| 673 |
< |
z-constrained molecules as well as subtract the total constraint |
| 674 |
< |
forces from the rest of the system after the force calculation at |
| 675 |
< |
each time step. |
| 664 |
> |
There is one additional extended system integrator (NPTxyz), in |
| 665 |
> |
which each attempt to preserve the target pressure along the box |
| 666 |
> |
walls perpendicular to that particular axis. The lengths of the box |
| 667 |
> |
axes are allowed to fluctuate independently, but the angle between |
| 668 |
> |
the box axes does not change. It should be noted that the NPTxyz |
| 669 |
> |
integrator is a special case of $NP\gamma T$ if the surface tension |
| 670 |
> |
$\gamma$ is set to zero. |
| 671 |
|
|
| 672 |
< |
After the force calculation, define $G_\alpha$ as |
| 678 |
< |
\begin{equation} |
| 679 |
< |
G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1} |
| 680 |
< |
\end{equation} |
| 681 |
< |
where $F_{\alpha i}$ is the force in the z direction of atom $i$ in |
| 682 |
< |
z-constrained molecule $\alpha$. The forces of the z constrained |
| 683 |
< |
molecule are then set to: |
| 684 |
< |
\begin{equation} |
| 685 |
< |
F_{\alpha i} = F_{\alpha i} - |
| 686 |
< |
\frac{m_{\alpha i} G_{\alpha}}{\sum_i m_{\alpha i}}. |
| 687 |
< |
\end{equation} |
| 688 |
< |
Here, $m_{\alpha i}$ is the mass of atom $i$ in the z-constrained |
| 689 |
< |
molecule. Having rescaled the forces, the velocities must also be |
| 690 |
< |
rescaled to subtract out any center of mass velocity in the z |
| 691 |
< |
direction. |
| 692 |
< |
\begin{equation} |
| 693 |
< |
v_{\alpha i} = v_{\alpha i} - |
| 694 |
< |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}, |
| 695 |
< |
\end{equation} |
| 696 |
< |
where $v_{\alpha i}$ is the velocity of atom $i$ in the z direction. |
| 697 |
< |
Lastly, all of the accumulated z constrained forces must be |
| 698 |
< |
subtracted from the system to keep the system center of mass from |
| 699 |
< |
drifting. |
| 700 |
< |
\begin{equation} |
| 701 |
< |
F_{\beta i} = F_{\beta i} - \frac{m_{\beta i} \sum_{\alpha} |
| 702 |
< |
G_{\alpha}} |
| 703 |
< |
{\sum_{\beta}\sum_i m_{\beta i}}, |
| 704 |
< |
\end{equation} |
| 705 |
< |
where $\beta$ are all of the unconstrained molecules in the system. |
| 706 |
< |
Similarly, the velocities of the unconstrained molecules must also |
| 707 |
< |
be scaled. |
| 708 |
< |
\begin{equation} |
| 709 |
< |
v_{\beta i} = v_{\beta i} + \sum_{\alpha} |
| 710 |
< |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}. |
| 711 |
< |
\end{equation} |
| 672 |
> |
%\section{\label{methodSection:constraintMethod}Constraint Method} |
| 673 |
|
|
| 674 |
< |
At the very beginning of the simulation, the molecules may not be at |
| 714 |
< |
their constrained positions. To move a z-constrained molecule to its |
| 715 |
< |
specified position, a simple harmonic potential is used |
| 716 |
< |
\begin{equation} |
| 717 |
< |
U(t)=\frac{1}{2}k_{\text{Harmonic}}(z(t)-z_{\text{cons}})^{2},% |
| 718 |
< |
\end{equation} |
| 719 |
< |
where $k_{\text{Harmonic}}$ is the harmonic force constant, $z(t)$ |
| 720 |
< |
is the current $z$ coordinate of the center of mass of the |
| 721 |
< |
constrained molecule, and $z_{\text{cons}}$ is the constrained |
| 722 |
< |
position. The harmonic force operating on the z-constrained molecule |
| 723 |
< |
at time $t$ can be calculated by |
| 724 |
< |
\begin{equation} |
| 725 |
< |
F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= |
| 726 |
< |
-k_{\text{Harmonic}}(z(t)-z_{\text{cons}}). |
| 727 |
< |
\end{equation} |
| 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} |
