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|
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\section{\label{methodSection:rigidBodyIntegrators}Integrators for Rigid Body Motion in Molecular Dynamics} |
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|
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In order to mimic the experiments, which are usually performed under |
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> |
In order to mimic experiments which are usually performed under |
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constant temperature and/or pressure, extended Hamiltonian system |
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methods have been developed to generate statistical ensembles, such |
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as canonical ensemble and isobaric-isothermal ensemble \textit{etc}. |
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In addition to the standard ensemble, specific ensembles have been |
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developed to account for the anisotropy between the lateral and |
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normal directions of membranes. The $NPAT$ ensemble, in which the |
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normal pressure and the lateral surface area of the membrane are |
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kept constant, and the $NP\gamma T$ ensemble, in which the normal |
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pressure and the lateral surface tension are kept constant were |
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proposed to address this issue. |
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as the canonical and isobaric-isothermal ensembles. In addition to |
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the standard ensemble, specific ensembles have been developed to |
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account for the anisotropy between the lateral and normal directions |
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of membranes. The $NPAT$ ensemble, in which the normal pressure and |
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the lateral surface area of the membrane are kept constant, and the |
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$NP\gamma T$ ensemble, in which the normal pressure and the lateral |
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surface tension are kept constant were proposed to address the |
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issues. |
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|
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Integration schemes for rotational motion of the rigid molecules in |
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microcanonical ensemble have been extensively studied in the last |
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two decades. Matubayasi developed a time-reversible integrator for |
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rigid bodies in quaternion representation. Although it is not |
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symplectic, this integrator still demonstrates a better long-time |
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energy conservation than traditional methods because of the |
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time-reversible nature. Extending Trotter-Suzuki to general system |
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with a flat phase space, Miller and his colleagues devised an novel |
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symplectic, time-reversible and volume-preserving integrator in |
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quaternion representation, which was shown to be superior to the |
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Matubayasi's time-reversible integrator. However, all of the |
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integrators in quaternion representation suffer from the |
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computational penalty of constructing a rotation matrix from |
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quaternions to evolve coordinates and velocities at every time step. |
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An alternative integration scheme utilizing rotation matrix directly |
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proposed by Dullweber, Leimkuhler and McLachlan (DLM) also preserved |
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the same structural properties of the Hamiltonian flow. In this |
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section, the integration scheme of DLM method will be reviewed and |
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extended to other ensembles. |
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Integration schemes for the rotational motion of the rigid molecules |
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in the microcanonical ensemble have been extensively studied over |
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> |
the last two decades. Matubayasi developed a time-reversible |
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> |
integrator for rigid bodies in quaternion representation. Although |
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it is not symplectic, this integrator still demonstrates a better |
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> |
long-time energy conservation than Euler angle methods because of |
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> |
the time-reversible nature. Extending the Trotter-Suzuki |
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factorization to general system with a flat phase space, Miller and |
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> |
his colleagues devised a novel symplectic, time-reversible and |
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> |
volume-preserving integrator in the quaternion representation, which |
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was shown to be superior to the Matubayasi's time-reversible |
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> |
integrator. However, all of the integrators in the quaternion |
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> |
representation suffer from the computational penalty of constructing |
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> |
a rotation matrix from quaternions to evolve coordinates and |
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> |
velocities at every time step. An alternative integration scheme |
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utilizing the rotation matrix directly proposed by Dullweber, |
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Leimkuhler and McLachlan (DLM) also preserved the same structural |
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properties of the Hamiltonian flow. In this section, the integration |
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scheme of DLM method will be reviewed and extended to other |
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ensembles. |
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|
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\subsection{\label{methodSection:DLM}Integrating the Equations of Motion: the |
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DLM method} |
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- \left(\frac{\partial V}{\partial {\bf r}}\right)_{{\bf r}(t + h)}, \\ |
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% |
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{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h) |
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< |
\times \frac{\partial V}{\partial {\bf u}}, \\ |
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> |
\times (\frac{\partial V}{\partial {\bf u}})_{u(t+h)}, \\ |
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% |
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{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) |
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\cdot {\bf \tau}^s(t + h). |
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\end{align*} |
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|
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${\bf u}$ will be automatically updated when the rotation matrix |
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> |
${\bf u}$ is automatically updated when the rotation matrix |
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$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
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|
torques have been obtained at the new time step, the velocities can |
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be advanced to the same time value. |
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\begin{equation} |
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f = 3 N + 3 N_{\mathrm{orient}} - N_{\mathrm{constraints}}, |
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\end{equation} |
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and $K$ is the total kinetic energy, |
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> |
where $N_{\mathrm{orient}}$ is the number of molecules with |
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> |
orientational degrees of freedom, and $K$ is the total kinetic |
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> |
energy, |
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\begin{equation} |
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K = \sum_{i=1}^{N} \frac{1}{2} m_i {\bf v}_i^T \cdot {\bf v}_i + |
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\sum_{i=1}^{N_{\mathrm{orient}}} \frac{1}{2} {\bf j}_i^T \cdot |
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\subsection{\label{methodSection:NPTi}Constant-pressure integration with |
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|
isotropic box deformations (NPTi)} |
| 297 |
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|
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< |
Isobaric-isothermal ensemble integrator is implemented using the |
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< |
Melchionna modifications to the Nos\'e-Hoover-Andersen equations of |
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< |
motion,\cite{Melchionna1993} |
| 298 |
> |
We can used an isobaric-isothermal ensemble integrator which is |
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> |
implemented using the Melchionna modifications to the |
| 300 |
> |
Nos\'e-Hoover-Andersen equations of motion,\cite{Melchionna1993} |
| 301 |
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|
| 302 |
|
\begin{eqnarray} |
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|
\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\ |
| 593 |
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|
| 594 |
|
\subsection{\label{methodSection:NPAT}NPAT Ensemble} |
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|
| 596 |
< |
A comprehensive understanding of structure¨Cfunction relations of |
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< |
biological membrane system ultimately relies on structure and |
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< |
dynamics of lipid bilayer, which are strongly affected by the |
| 599 |
< |
interfacial interaction between lipid molecules and surrounding |
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< |
media. One quantity to describe the interfacial interaction is so |
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< |
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 |
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|
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|
\begin{equation} |
| 607 |
|
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
| 626 |
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\] |
| 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} |
