2 |
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|
3 |
|
\section{\label{methodSection:rigidBodyIntegrators}Integrators for Rigid Body Motion in Molecular Dynamics} |
4 |
|
|
5 |
< |
In order to mimic the experiments, which are usually performed under |
5 |
> |
In order to mimic experiments which are usually performed under |
6 |
|
constant temperature and/or pressure, extended Hamiltonian system |
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 |
11 |
< |
normal directions of membranes. The $NPAT$ ensemble, in which the |
12 |
< |
normal pressure and the lateral surface area of the membrane are |
13 |
< |
kept constant, and the $NP\gamma T$ ensemble, in which the normal |
14 |
< |
pressure and the lateral surface tension are kept constant were |
15 |
< |
proposed to address this issue. |
8 |
> |
as the canonical and isobaric-isothermal ensembles. In addition to |
9 |
> |
the standard ensemble, specific ensembles have been developed to |
10 |
> |
account for the anisotropy between the lateral and normal directions |
11 |
> |
of membranes. The $NPAT$ ensemble, in which the normal pressure and |
12 |
> |
the lateral surface area of the membrane are kept constant, and the |
13 |
> |
$NP\gamma T$ ensemble, in which the normal pressure and the lateral |
14 |
> |
surface tension are kept constant were proposed to address the |
15 |
> |
issues. |
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 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 |
32 |
< |
proposed by Dullweber, Leimkuhler and McLachlan (DLM) also preserved |
33 |
< |
the same structural properties of the Hamiltonian flow. In this |
34 |
< |
section, the integration scheme of DLM method will be reviewed and |
35 |
< |
extended to other ensembles. |
17 |
> |
Integration schemes for the rotational motion of the rigid molecules |
18 |
> |
in the microcanonical ensemble have been extensively studied over |
19 |
> |
the last two decades. Matubayasi 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 Euler angle methods because of |
23 |
> |
the time-reversible nature. Extending the Trotter-Suzuki |
24 |
> |
factorization to general system with a flat phase space, Miller and |
25 |
> |
his colleagues devised a novel symplectic, time-reversible and |
26 |
> |
volume-preserving integrator in the quaternion representation, which |
27 |
> |
was shown to be superior to the Matubayasi's time-reversible |
28 |
> |
integrator. However, all of the integrators in the quaternion |
29 |
> |
representation suffer from the computational penalty of constructing |
30 |
> |
a rotation matrix from quaternions to evolve coordinates and |
31 |
> |
velocities at every time step. An alternative integration scheme |
32 |
> |
utilizing the rotation matrix directly proposed by Dullweber, |
33 |
> |
Leimkuhler and McLachlan (DLM) also preserved the same structural |
34 |
> |
properties of the Hamiltonian flow. In this section, the integration |
35 |
> |
scheme of DLM method will be reviewed and extended to other |
36 |
> |
ensembles. |
37 |
|
|
38 |
|
\subsection{\label{methodSection:DLM}Integrating the Equations of Motion: the |
39 |
|
DLM method} |
112 |
|
- \left(\frac{\partial V}{\partial {\bf r}}\right)_{{\bf r}(t + h)}, \\ |
113 |
|
% |
114 |
|
{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h) |
115 |
< |
\times \frac{\partial V}{\partial {\bf u}}, \\ |
115 |
> |
\times (\frac{\partial V}{\partial {\bf u}})_{u(t+h)}, \\ |
116 |
|
% |
117 |
|
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) |
118 |
|
\cdot {\bf \tau}^s(t + h). |
119 |
|
\end{align*} |
120 |
|
|
121 |
< |
{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix |
121 |
> |
${\bf u}$ is automatically updated when the rotation matrix |
122 |
|
$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
123 |
|
torques have been obtained at the new time step, the velocities can |
124 |
|
be advanced to the same time value. |
199 |
|
\begin{equation} |
200 |
|
f = 3 N + 3 N_{\mathrm{orient}} - N_{\mathrm{constraints}}, |
201 |
|
\end{equation} |
202 |
< |
and $K$ is the total kinetic energy, |
202 |
> |
where $N_{\mathrm{orient}}$ is the number of molecules with |
203 |
> |
orientational degrees of freedom, and $K$ is the total kinetic |
204 |
> |
energy, |
205 |
|
\begin{equation} |
206 |
|
K = \sum_{i=1}^{N} \frac{1}{2} m_i {\bf v}_i^T \cdot {\bf v}_i + |
207 |
|
\sum_{i=1}^{N_{\mathrm{orient}}} \frac{1}{2} {\bf j}_i^T \cdot |
209 |
|
\end{equation} |
210 |
|
|
211 |
|
In eq.(\ref{eq:nosehooverext}), $\tau_T$ is the time constant for |
212 |
< |
relaxation of the temperature to the target value. To set values |
213 |
< |
for $\tau_T$ or $T_{\mathrm{target}}$ in a simulation, one would use |
214 |
< |
the {\tt tauThermostat} and {\tt targetTemperature} keywords in the |
212 |
< |
{\tt .bass} file. The units for {\tt tauThermostat} are fs, and the |
213 |
< |
units for the {\tt targetTemperature} are degrees K. The |
214 |
< |
integration of the equations of motion is carried out in a |
215 |
< |
velocity-Verlet style 2 part algorithm: |
212 |
> |
relaxation of the temperature to the target value. The integration |
213 |
> |
of the equations of motion is carried out in a velocity-Verlet style |
214 |
> |
2 part algorithm: |
215 |
|
|
216 |
|
{\tt moveA:} |
217 |
|
\begin{align*} |
277 |
|
caclculate $T(t + h)$ as well as $\chi(t + h)$, they indirectly |
278 |
|
depend on their own values at time $t + h$. {\tt moveB} is |
279 |
|
therefore done in an iterative fashion until $\chi(t + h)$ becomes |
280 |
< |
self-consistent. The relative tolerance for the self-consistency |
282 |
< |
check defaults to a value of $\mbox{10}^{-6}$, but {\sc oopse} will |
283 |
< |
terminate the iteration after 4 loops even if the consistency check |
284 |
< |
has not been satisfied. |
280 |
> |
self-consistent. |
281 |
|
|
282 |
|
The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for |
283 |
|
the extended system that is, to within a constant, identical to the |
295 |
|
\subsection{\label{methodSection:NPTi}Constant-pressure integration with |
296 |
|
isotropic box deformations (NPTi)} |
297 |
|
|
298 |
< |
To carry out isobaric-isothermal ensemble calculations {\sc oopse} |
299 |
< |
implements the Melchionna modifications to the |
298 |
> |
We can used an isobaric-isothermal ensemble integrator which is |
299 |
> |
implemented using the Melchionna modifications to the |
300 |
|
Nos\'e-Hoover-Andersen equations of motion,\cite{Melchionna1993} |
301 |
|
|
302 |
|
\begin{eqnarray} |
352 |
|
\end{equation} |
353 |
|
|
354 |
|
In eq.(\ref{eq:melchionna1}), $\tau_B$ is the time constant for |
355 |
< |
relaxation of the pressure to the target value. To set values for |
360 |
< |
$\tau_B$ or $P_{\mathrm{target}}$ in a simulation, one would use the |
361 |
< |
{\tt tauBarostat} and {\tt targetPressure} keywords in the {\tt |
362 |
< |
.bass} file. The units for {\tt tauBarostat} are fs, and the units |
363 |
< |
for the {\tt targetPressure} are atmospheres. Like in the NVT |
355 |
> |
relaxation of the pressure to the target value. Like in the NVT |
356 |
|
integrator, the integration of the equations of motion is carried |
357 |
|
out in a velocity-Verlet style 2 part algorithm: |
358 |
|
|
393 |
|
|
394 |
|
Most of these equations are identical to their counterparts in the |
395 |
|
NVT integrator, but the propagation of positions to time $t + h$ |
396 |
< |
depends on the positions at the same time. {\sc oopse} carries out |
397 |
< |
this step iteratively (with a limit of 5 passes through the |
398 |
< |
iterative loop). Also, the simulation box $\mathsf{H}$ is scaled |
399 |
< |
uniformly for one full time step by an exponential factor that |
400 |
< |
depends on the value of $\eta$ at time $t + h / 2$. Reshaping the |
409 |
< |
box uniformly also scales the volume of the box by |
396 |
> |
depends on the positions at the same time. The simulation box |
397 |
> |
$\mathsf{H}$ is scaled uniformly for one full time step by an |
398 |
> |
exponential factor that depends on the value of $\eta$ at time $t + |
399 |
> |
h / 2$. Reshaping the box uniformly also scales the volume of the |
400 |
> |
box by |
401 |
|
\begin{equation} |
402 |
|
\mathcal{V}(t + h) \leftarrow e^{ - 3 h \eta(t + h /2)}. |
403 |
|
\mathcal{V}(t) |
439 |
|
to caclculate $T(t + h)$, $P(t + h)$, $\chi(t + h)$, and $\eta(t + |
440 |
|
h)$, they indirectly depend on their own values at time $t + h$. |
441 |
|
{\tt moveB} is therefore done in an iterative fashion until $\chi(t |
442 |
< |
+ h)$ and $\eta(t + h)$ become self-consistent. The relative |
452 |
< |
tolerance for the self-consistency check defaults to a value of |
453 |
< |
$\mbox{10}^{-6}$, but {\sc oopse} will terminate the iteration after |
454 |
< |
4 loops even if the consistency check has not been satisfied. |
442 |
> |
+ h)$ and $\eta(t + h)$ become self-consistent. |
443 |
|
|
444 |
|
The Melchionna modification of the Nos\'e-Hoover-Andersen algorithm |
445 |
|
is known to conserve a Hamiltonian for the extended system that is, |
461 |
|
P_{\mathrm{target}} \mathcal{V}(t). |
462 |
|
\end{equation} |
463 |
|
|
476 |
– |
Bond constraints are applied at the end of both the {\tt moveA} and |
477 |
– |
{\tt moveB} portions of the algorithm. |
478 |
– |
|
464 |
|
\subsection{\label{methodSection:NPTf}Constant-pressure integration with a |
465 |
|
flexible box (NPTf)} |
466 |
|
|
534 |
|
\mathsf{H}(t + h) &\leftarrow \mathsf{H}(t) \cdot e^{-h |
535 |
|
\overleftrightarrow{\eta}(t + h / 2)} . |
536 |
|
\end{align*} |
537 |
< |
{\sc oopse} uses a power series expansion truncated at second order |
538 |
< |
for the exponential operation which scales the simulation box. |
537 |
> |
Here, a power series expansion truncated at second order for the |
538 |
> |
exponential operation is used to scale the simulation box. |
539 |
|
|
540 |
|
The {\tt moveB} portion of the algorithm is largely unchanged from |
541 |
|
the NPTi integrator: |
574 |
|
identical to those described for the NPTi integrator. |
575 |
|
|
576 |
|
The NPTf integrator is known to conserve the following Hamiltonian: |
577 |
< |
\begin{equation} |
578 |
< |
H_{\mathrm{NPTf}} = V + K + f k_B T_{\mathrm{target}} \left( |
577 |
> |
\begin{eqnarray*} |
578 |
> |
H_{\mathrm{NPTf}} & = & V + K + f k_B T_{\mathrm{target}} \left( |
579 |
|
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
580 |
< |
dt^\prime \right) + P_{\mathrm{target}} \mathcal{V}(t) + \frac{f k_B |
580 |
> |
dt^\prime \right) \\ |
581 |
> |
& & + P_{\mathrm{target}} \mathcal{V}(t) + \frac{f k_B |
582 |
|
T_{\mathrm{target}}}{2} |
583 |
|
\mathrm{Tr}\left[\overleftrightarrow{\eta}(t)\right]^2 \tau_B^2. |
584 |
< |
\end{equation} |
584 |
> |
\end{eqnarray*} |
585 |
|
|
586 |
|
This integrator must be used with care, particularly in liquid |
587 |
|
simulations. Liquids have very small restoring forces in the |
591 |
|
finds most use in simulating crystals or liquid crystals which |
592 |
|
assume non-orthorhombic geometries. |
593 |
|
|
594 |
< |
\subsection{\label{methodSection:otherSpecialEnsembles}Other Special Ensembles} |
594 |
> |
\subsection{\label{methodSection:NPAT}NPAT Ensemble} |
595 |
|
|
596 |
< |
\subsubsection{\label{methodSection:NPAT}\textbf{NPAT Ensemble}} |
596 |
> |
A comprehensive understanding of relations between structures and |
597 |
> |
functions in biological membrane system ultimately relies on |
598 |
> |
structure and dynamics of lipid bilayers, which are strongly |
599 |
> |
affected by the interfacial interaction between lipid molecules and |
600 |
> |
surrounding media. One quantity to describe the interfacial |
601 |
> |
interaction is so called the average surface area per lipid. |
602 |
> |
Constant area and constant lateral pressure simulations can be |
603 |
> |
achieved by extending the standard NPT ensemble with a different |
604 |
> |
pressure control strategy |
605 |
|
|
612 |
– |
A comprehensive understanding of structure¨Cfunction relations of |
613 |
– |
biological membrane system ultimately relies on structure and |
614 |
– |
dynamics of lipid bilayer, which are strongly affected by the |
615 |
– |
interfacial interaction between lipid molecules and surrounding |
616 |
– |
media. One quantity to describe the interfacial interaction is so |
617 |
– |
called the average surface area per lipid. Constat area and constant |
618 |
– |
lateral pressure simulation can be achieved by extending the |
619 |
– |
standard NPT ensemble with a different pressure control strategy |
620 |
– |
|
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}}} }} |
615 |
|
Note that the iterative schemes for NPAT are identical to those |
616 |
|
described for the NPTi integrator. |
617 |
|
|
618 |
< |
\subsubsection{\label{methodSection:NPrT}\textbf{NP$\gamma$T Ensemble}} |
618 |
> |
\subsection{\label{methodSection:NPrT}NP$\gamma$T |
619 |
> |
Ensemble} |
620 |
|
|
621 |
|
Theoretically, the surface tension $\gamma$ of a stress free |
622 |
|
membrane system should be zero since its surface free energy $G$ is |
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 |
629 |
> |
the membrane simulation, a special ensemble, NP$\gamma$T, has been |
630 |
|
proposed to maintain the lateral surface tension and normal |
631 |
< |
pressure. The equation of motion for cell size control tensor, |
631 |
> |
pressure. The equation of motion for the cell size control tensor, |
632 |
|
$\eta$, in $NP\gamma T$ is |
633 |
|
\begin{equation} |
634 |
|
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
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. |
655 |
> |
$\gamma$ is set to zero, and if $x$ and $y$ can move independently. |
656 |
|
|
657 |
< |
\section{\label{methodSection:zcons}Z-Constraint Method} |
657 |
> |
\section{\label{methodSection:zcons}The Z-Constraint Method} |
658 |
|
|
659 |
|
Based on the fluctuation-dissipation theorem, a force |
660 |
|
auto-correlation method was developed by Roux and Karplus to |
692 |
|
forces from the rest of the system after the force calculation at |
693 |
|
each time step instead of resetting the coordinate. |
694 |
|
|
695 |
< |
After the force calculation, define $G_\alpha$ as |
695 |
> |
After the force calculation, we define $G_\alpha$ as |
696 |
|
\begin{equation} |
697 |
|
G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1} |
698 |
|
\end{equation} |