| 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} ,\\ |
| 278 |
|
caclculate $T(t + h)$ as well as $\chi(t + h)$, they indirectly |
| 279 |
|
depend on their own values at time $t + h$. {\tt moveB} is |
| 280 |
|
therefore done in an iterative fashion until $\chi(t + h)$ becomes |
| 281 |
< |
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. |
| 281 |
> |
self-consistent. |
| 282 |
|
|
| 283 |
|
The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for |
| 284 |
|
the extended system that is, to within a constant, identical to the |
| 285 |
< |
Helmholtz free energy,\cite{melchionna93} |
| 285 |
> |
Helmholtz free energy,\cite{Melchionna1993} |
| 286 |
|
\begin{equation} |
| 287 |
|
H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 288 |
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
| 293 |
|
last column of the {\tt .stat} file to allow checks on the quality |
| 294 |
|
of the integration. |
| 295 |
|
|
| 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 |
– |
|
| 296 |
|
\subsection{\label{methodSection:NPTi}Constant-pressure integration with |
| 297 |
|
isotropic box deformations (NPTi)} |
| 298 |
|
|
| 299 |
< |
To carry out isobaric-isothermal ensemble calculations {\sc oopse} |
| 300 |
< |
implements the Melchionna modifications to the |
| 301 |
< |
Nos\'e-Hoover-Andersen equations of motion,\cite{melchionna93} |
| 299 |
> |
Isobaric-isothermal ensemble integrator is implemented using the |
| 300 |
> |
Melchionna modifications to the Nos\'e-Hoover-Andersen equations of |
| 301 |
> |
motion,\cite{Melchionna1993} |
| 302 |
|
|
| 303 |
|
\begin{eqnarray} |
| 304 |
|
\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\ |
| 353 |
|
\end{equation} |
| 354 |
|
|
| 355 |
|
In eq.(\ref{eq:melchionna1}), $\tau_B$ is the time constant for |
| 356 |
< |
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 |
| 356 |
> |
relaxation of the pressure to the target value. Like in the NVT |
| 357 |
|
integrator, the integration of the equations of motion is carried |
| 358 |
|
out in a velocity-Verlet style 2 part algorithm: |
| 359 |
|
|
| 394 |
|
|
| 395 |
|
Most of these equations are identical to their counterparts in the |
| 396 |
|
NVT integrator, but the propagation of positions to time $t + h$ |
| 397 |
< |
depends on the positions at the same time. {\sc oopse} carries out |
| 398 |
< |
this step iteratively (with a limit of 5 passes through the |
| 399 |
< |
iterative loop). Also, the simulation box $\mathsf{H}$ is scaled |
| 400 |
< |
uniformly for one full time step by an exponential factor that |
| 401 |
< |
depends on the value of $\eta$ at time $t + h / 2$. Reshaping the |
| 412 |
< |
box uniformly also scales the volume of the box by |
| 397 |
> |
depends on the positions at the same time. The simulation box |
| 398 |
> |
$\mathsf{H}$ is scaled uniformly for one full time step by an |
| 399 |
> |
exponential factor that depends on the value of $\eta$ at time $t + |
| 400 |
> |
h / 2$. Reshaping the box uniformly also scales the volume of the |
| 401 |
> |
box by |
| 402 |
|
\begin{equation} |
| 403 |
|
\mathcal{V}(t + h) \leftarrow e^{ - 3 h \eta(t + h /2)}. |
| 404 |
|
\mathcal{V}(t) |
| 440 |
|
to caclculate $T(t + h)$, $P(t + h)$, $\chi(t + h)$, and $\eta(t + |
| 441 |
|
h)$, they indirectly depend on their own values at time $t + h$. |
| 442 |
|
{\tt moveB} is therefore done in an iterative fashion until $\chi(t |
| 443 |
< |
+ h)$ 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. |
| 443 |
> |
+ h)$ and $\eta(t + h)$ become self-consistent. |
| 444 |
|
|
| 445 |
|
The Melchionna modification of the Nos\'e-Hoover-Andersen algorithm |
| 446 |
|
is known to conserve a Hamiltonian for the extended system that is, |
| 462 |
|
P_{\mathrm{target}} \mathcal{V}(t). |
| 463 |
|
\end{equation} |
| 464 |
|
|
| 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 |
– |
|
| 465 |
|
\subsection{\label{methodSection:NPTf}Constant-pressure integration with a |
| 466 |
|
flexible box (NPTf)} |
| 467 |
|
|
| 535 |
|
\mathsf{H}(t + h) &\leftarrow \mathsf{H}(t) \cdot e^{-h |
| 536 |
|
\overleftrightarrow{\eta}(t + h / 2)} . |
| 537 |
|
\end{align*} |
| 538 |
< |
{\sc oopse} uses a power series expansion truncated at second order |
| 539 |
< |
for the exponential operation which scales the simulation box. |
| 538 |
> |
Here, a power series expansion truncated at second order for the |
| 539 |
> |
exponential operation is used to scale the simulation box. |
| 540 |
|
|
| 541 |
|
The {\tt moveB} portion of the algorithm is largely unchanged from |
| 542 |
|
the NPTi integrator: |
| 575 |
|
identical to those described for the NPTi integrator. |
| 576 |
|
|
| 577 |
|
The NPTf integrator is known to conserve the following Hamiltonian: |
| 578 |
< |
\begin{equation} |
| 579 |
< |
H_{\mathrm{NPTf}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 578 |
> |
\begin{eqnarray*} |
| 579 |
> |
H_{\mathrm{NPTf}} & = & V + K + f k_B T_{\mathrm{target}} \left( |
| 580 |
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
| 581 |
< |
dt^\prime \right) + P_{\mathrm{target}} \mathcal{V}(t) + \frac{f k_B |
| 582 |
< |
T_{\mathrm{target}}}{2} |
| 581 |
> |
dt^\prime \right) \\ |
| 582 |
> |
+ P_{\mathrm{target}} \mathcal{V}(t) + \frac{f |
| 583 |
> |
k_B T_{\mathrm{target}}}{2} |
| 584 |
|
\mathrm{Tr}\left[\overleftrightarrow{\eta}(t)\right]^2 \tau_B^2. |
| 585 |
< |
\end{equation} |
| 585 |
> |
\end{eqnarray*} |
| 586 |
|
|
| 587 |
|
This integrator must be used with care, particularly in liquid |
| 588 |
|
simulations. Liquids have very small restoring forces in the |
| 592 |
|
finds most use in simulating crystals or liquid crystals which |
| 593 |
|
assume non-orthorhombic geometries. |
| 594 |
|
|
| 595 |
< |
\subsection{\label{methodSection:NPAT}Constant Lateral Pressure and Constant Surface Area (NPAT)} |
| 595 |
> |
\subsection{\label{methodSection:NPAT}NPAT Ensemble} |
| 596 |
|
|
| 597 |
< |
\subsection{\label{methodSection:NPrT}Constant lateral Pressure and Constant Surface Tension (NP\gamma T) } |
| 597 |
> |
A comprehensive understanding of structure¨Cfunction relations of |
| 598 |
> |
biological membrane system ultimately relies on structure and |
| 599 |
> |
dynamics of lipid bilayer, which are strongly affected by the |
| 600 |
> |
interfacial interaction between lipid molecules and surrounding |
| 601 |
> |
media. One quantity to describe the interfacial interaction is so |
| 602 |
> |
called the average surface area per lipid. Constat area and constant |
| 603 |
> |
lateral pressure simulation can be achieved by extending the |
| 604 |
> |
standard NPT ensemble with a different pressure control strategy |
| 605 |
|
|
| 606 |
< |
\subsection{\label{methodSection:NPTxyz}Constant pressure in 3 axes (NPTxyz)} |
| 606 |
> |
\begin{equation} |
| 607 |
> |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
| 608 |
> |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau_{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }} |
| 609 |
> |
& \mbox{if $ \alpha = \beta = z$}\\ |
| 610 |
> |
0 & \mbox{otherwise}\\ |
| 611 |
> |
\end{array} |
| 612 |
> |
\right. |
| 613 |
> |
\end{equation} |
| 614 |
|
|
| 615 |
< |
There is one additional extended system integrator which is somewhat |
| 616 |
< |
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. |
| 615 |
> |
Note that the iterative schemes for NPAT are identical to those |
| 616 |
> |
described for the NPTi integrator. |
| 617 |
|
|
| 618 |
+ |
\subsection{\label{methodSection:NPrT}NP$\gamma$T |
| 619 |
+ |
Ensemble} |
| 620 |
|
|
| 621 |
< |
\section{\label{methodSection:constraintMethods}Constraint Methods} |
| 621 |
> |
Theoretically, the surface tension $\gamma$ of a stress free |
| 622 |
> |
membrane system should be zero since its surface free energy $G$ is |
| 623 |
> |
minimum with respect to surface area $A$ |
| 624 |
> |
\[ |
| 625 |
> |
\gamma = \frac{{\partial G}}{{\partial A}}. |
| 626 |
> |
\] |
| 627 |
> |
However, a surface tension of zero is not appropriate for relatively |
| 628 |
> |
small patches of membrane. In order to eliminate the edge effect of |
| 629 |
> |
the membrane simulation, a special ensemble, NP$\gamma$T, is |
| 630 |
> |
proposed to maintain the lateral surface tension and normal |
| 631 |
> |
pressure. The equation of motion for cell size control tensor, |
| 632 |
> |
$\eta$, in $NP\gamma T$ is |
| 633 |
> |
\begin{equation} |
| 634 |
> |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
| 635 |
> |
- A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ) & \mbox{$\alpha = \beta = x$ or $\alpha = \beta = y$}\\ |
| 636 |
> |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}}}} & \mbox{$\alpha = \beta = z$} \\ |
| 637 |
> |
0 & \mbox{$\alpha \ne \beta$} \\ |
| 638 |
> |
\end{array} |
| 639 |
> |
\right. |
| 640 |
> |
\end{equation} |
| 641 |
> |
where $ \gamma _{{\rm{target}}}$ is the external surface tension and |
| 642 |
> |
the instantaneous surface tensor $\gamma _\alpha$ is given by |
| 643 |
> |
\begin{equation} |
| 644 |
> |
\gamma _\alpha = - h_z( \overleftrightarrow{P} _{\alpha \alpha } |
| 645 |
> |
- P_{{\rm{target}}} ) |
| 646 |
> |
\label{methodEquation:instantaneousSurfaceTensor} |
| 647 |
> |
\end{equation} |
| 648 |
|
|
| 649 |
< |
\subsection{\label{methodSection:rattle}The {\sc rattle} Method for Bond |
| 650 |
< |
Constraints} |
| 649 |
> |
There is one additional extended system integrator (NPTxyz), in |
| 650 |
> |
which each attempt to preserve the target pressure along the box |
| 651 |
> |
walls perpendicular to that particular axis. The lengths of the box |
| 652 |
> |
axes are allowed to fluctuate independently, but the angle between |
| 653 |
> |
the box axes does not change. It should be noted that the NPTxyz |
| 654 |
> |
integrator is a special case of $NP\gamma T$ if the surface tension |
| 655 |
> |
$\gamma$ is set to zero. |
| 656 |
|
|
| 657 |
< |
\subsection{\label{methodSection:zcons}Z-Constraint Method} |
| 657 |
> |
\section{\label{methodSection:zcons}Z-Constraint Method} |
| 658 |
|
|
| 659 |
|
Based on the fluctuation-dissipation theorem, a force |
| 660 |
|
auto-correlation method was developed by Roux and Karplus to |
| 661 |
< |
investigate the dynamics of ions inside ion channels.\cite{Roux91} |
| 661 |
> |
investigate the dynamics of ions inside ion channels\cite{Roux1991}. |
| 662 |
|
The time-dependent friction coefficient can be calculated from the |
| 663 |
|
deviation of the instantaneous force from its mean force. |
| 664 |
|
\begin{equation} |
| 676 |
|
F(z,0)\rangle dt. |
| 677 |
|
\end{equation} |
| 678 |
|
Allowing diffusion constant to then be calculated through the |
| 679 |
< |
Einstein relation:\cite{Marrink94} |
| 679 |
> |
Einstein relation:\cite{Marrink1994} |
| 680 |
|
\begin{equation} |
| 681 |
|
D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
| 682 |
|
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.% |
| 685 |
|
The Z-Constraint method, which fixes the z coordinates of the |
| 686 |
|
molecules with respect to the center of the mass of the system, has |
| 687 |
|
been a method suggested to obtain the forces required for the force |
| 688 |
< |
auto-correlation calculation.\cite{Marrink94} However, simply |
| 688 |
> |
auto-correlation calculation.\cite{Marrink1994} However, simply |
| 689 |
|
resetting the coordinate will move the center of the mass of the |
| 690 |
< |
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 |
| 690 |
> |
whole system. To avoid this problem, we reset the forces of |
| 691 |
|
z-constrained molecules as well as subtract the total constraint |
| 692 |
|
forces from the rest of the system after the force calculation at |
| 693 |
< |
each time step. |
| 693 |
> |
each time step instead of resetting the coordinate. |
| 694 |
|
|
| 695 |
|
After the force calculation, define $G_\alpha$ as |
| 696 |
|
\begin{equation} |
| 743 |
|
F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= |
| 744 |
|
-k_{\text{Harmonic}}(z(t)-z_{\text{cons}}). |
| 745 |
|
\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} |
