| 611 |
|
|
| 612 |
|
\subsection{\label{methodSection:otherSpecialEnsembles}Other Special Ensembles} |
| 613 |
|
|
| 614 |
< |
\subsubsection{\label{methodSection:NPAT}Constant Normal Pressure, Constant Lateral Surface Area and Constant Temperature (NPAT) Ensemble} |
| 614 |
> |
\subsubsection{\label{methodSection:NPAT}NPAT Ensemble} |
| 615 |
|
|
| 616 |
|
A comprehensive understanding of structure¨Cfunction relations of |
| 617 |
|
biological membrane system ultimately relies on structure and |
| 621 |
|
called the average surface area per lipid. Constat area and constant |
| 622 |
|
lateral pressure simulation can be achieved by extending the |
| 623 |
|
standard NPT ensemble with a different pressure control strategy |
| 624 |
+ |
|
| 625 |
|
\begin{equation} |
| 626 |
< |
\dot |
| 627 |
< |
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} |
| 628 |
< |
\over \eta } _{\alpha \beta } = \left\{ \begin{array}{l} |
| 629 |
< |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }}{\rm{ }}(\alpha = \beta = z), \\ |
| 630 |
< |
0{\rm{ }}(\alpha \ne z{\rm{ }}or{\rm{ }}\beta \ne z) \\ |
| 631 |
< |
\end{array} \right. |
| 631 |
< |
\label{methodEquation:NPATeta} |
| 626 |
> |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
| 627 |
> |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau_{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }} |
| 628 |
> |
& \mbox{if $ \alpha = \beta = z$}\\ |
| 629 |
> |
0 & \mbox{otherwise}\\ |
| 630 |
> |
\end{array} |
| 631 |
> |
\right. |
| 632 |
|
\end{equation} |
| 633 |
+ |
|
| 634 |
|
Note that the iterative schemes for NPAT are identical to those |
| 635 |
|
described for the NPTi integrator. |
| 636 |
|
|
| 637 |
< |
\subsubsection{\label{methodSection:NPrT}Constant Normal Pressure, Constant Lateral Surface Tension and Constant Temperature (NP\gamma T) Ensemble } |
| 637 |
> |
\subsubsection{\label{methodSection:NPrT}NP$\gamma$T Ensemble} |
| 638 |
|
|
| 639 |
|
Theoretically, the surface tension $\gamma$ of a stress free |
| 640 |
|
membrane system should be zero since its surface free energy $G$ is |
| 644 |
|
\] |
| 645 |
|
However, a surface tension of zero is not appropriate for relatively |
| 646 |
|
small patches of membrane. In order to eliminate the edge effect of |
| 647 |
< |
the membrane simulation, a special ensemble, NP\gamma T, is proposed |
| 648 |
< |
to maintain the lateral surface tension and normal pressure. The |
| 649 |
< |
equation of motion for cell size control tensor, $\eta$, in NP\gamma |
| 650 |
< |
T is |
| 647 |
> |
the membrane simulation, a special ensemble, NP$\gamma$T, is |
| 648 |
> |
proposed to maintain the lateral surface tension and normal |
| 649 |
> |
pressure. The equation of motion for cell size control tensor, |
| 650 |
> |
$\eta$, in $NP\gamma T$ is |
| 651 |
|
\begin{equation} |
| 652 |
< |
\dot |
| 653 |
< |
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} |
| 654 |
< |
\over \eta } _{\alpha \beta } = \left\{ \begin{array}{l} |
| 655 |
< |
- A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ){\rm{ (}}\alpha = \beta = x{\rm{ or }} = y{\rm{)}} \\ |
| 656 |
< |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }}{\rm{ }}(\alpha = \beta = z) \\ |
| 657 |
< |
0{\rm{ }}(\alpha \ne \beta ) \\ |
| 657 |
< |
\end{array} \right. |
| 658 |
< |
\label{methodEquation:NPrTeta} |
| 652 |
> |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
| 653 |
> |
- A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ) & \mbox{$\alpha = \beta = x$ or $\alpha = \beta = y$}\\ |
| 654 |
> |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}}}} & \mbox{$\alpha = \beta = z$} \\ |
| 655 |
> |
0 & \mbox{$\alpha \ne \beta$} \\ |
| 656 |
> |
\end{array} |
| 657 |
> |
\right. |
| 658 |
|
\end{equation} |
| 659 |
|
where $ \gamma _{{\rm{target}}}$ is the external surface tension and |
| 660 |
|
the instantaneous surface tensor $\gamma _\alpha$ is given by |
| 673 |
|
integrator is a special case of $NP\gamma T$ if the surface tension |
| 674 |
|
$\gamma$ is set to zero. |
| 675 |
|
|
| 676 |
+ |
%\section{\label{methodSection:constraintMethod}Constraint Method} |
| 677 |
+ |
|
| 678 |
+ |
%\subsection{\label{methodSection:bondConstraint}Bond Constraint for Rigid Body} |
| 679 |
+ |
|
| 680 |
+ |
%\subsection{\label{methodSection:zcons}Z-constraint Method} |
| 681 |
+ |
|
| 682 |
|
\section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies} |
| 683 |
|
|
| 684 |
|
\subsection{\label{methodSection:temperature}Temperature Control} |
