2 |
|
|
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 |
< |
${\bf u}$ will be automatically updated 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 |
295 |
|
\subsection{\label{methodSection:NPTi}Constant-pressure integration with |
296 |
|
isotropic box deformations (NPTi)} |
297 |
|
|
298 |
< |
Isobaric-isothermal ensemble integrator is implemented using the |
299 |
< |
Melchionna modifications to the Nos\'e-Hoover-Andersen equations of |
300 |
< |
motion,\cite{Melchionna1993} |
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} |
303 |
|
\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\ |
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) \\ |
581 |
< |
& & + P_{\mathrm{target}} \mathcal{V}(t) + \frac{f k_B |
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{eqnarray*} |
593 |
|
|
594 |
|
\subsection{\label{methodSection:NPAT}NPAT Ensemble} |
595 |
|
|
596 |
< |
A comprehensive understanding of structure¨Cfunction relations of |
597 |
< |
biological membrane system ultimately relies on structure and |
598 |
< |
dynamics of lipid bilayer, which are strongly affected by the |
599 |
< |
interfacial interaction between lipid molecules and surrounding |
600 |
< |
media. One quantity to describe the interfacial interaction is so |
601 |
< |
called the average surface area per lipid. Constat area and constant |
602 |
< |
lateral pressure simulation can be achieved by extending the |
603 |
< |
standard NPT ensemble with a different pressure control strategy |
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 |
|
|
606 |
|
\begin{equation} |
607 |
|
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
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} |