| 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 |
| 608 |
|
electrostatic or Lennard-Jones cutoff radii. The NPTf integrator |
| 609 |
|
finds most use in simulating crystals or liquid crystals which |
| 610 |
|
assume non-orthorhombic geometries. |
| 611 |
– |
|
| 612 |
– |
\subsection{\label{methodSection:NPAT}Constant Lateral Pressure and Constant Surface Area (NPAT)} |
| 613 |
– |
|
| 614 |
– |
\subsection{\label{methodSection:NPrT}Constant lateral Pressure and Constant Surface Tension (NP\gamma T) } |
| 611 |
|
|
| 612 |
< |
\subsection{\label{methodSection:NPTxyz}Constant pressure in 3 axes (NPTxyz)} |
| 617 |
< |
|
| 618 |
< |
There is one additional extended system integrator which is somewhat |
| 619 |
< |
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. |
| 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} |
| 615 |
|
|
| 616 |
< |
\section{\label{methodSection:constraintMethods}Constraint Methods} |
| 617 |
< |
|
| 618 |
< |
\subsection{\label{methodSection:rattle}The {\sc rattle} Method for Bond |
| 619 |
< |
Constraints} |
| 620 |
< |
|
| 621 |
< |
\subsection{\label{methodSection:zcons}Z-Constraint Method} |
| 622 |
< |
|
| 623 |
< |
Based on the fluctuation-dissipation theorem, a force |
| 641 |
< |
auto-correlation method was developed by Roux and Karplus to |
| 642 |
< |
investigate the dynamics of ions inside ion channels.\cite{Roux91} |
| 643 |
< |
The time-dependent friction coefficient can be calculated from the |
| 644 |
< |
deviation of the instantaneous force from its mean force. |
| 645 |
< |
\begin{equation} |
| 646 |
< |
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T, |
| 647 |
< |
\end{equation} |
| 648 |
< |
where% |
| 616 |
> |
A comprehensive understanding of structure¨Cfunction relations of |
| 617 |
> |
biological membrane system ultimately relies on structure and |
| 618 |
> |
dynamics of lipid bilayer, which are strongly affected by the |
| 619 |
> |
interfacial interaction between lipid molecules and surrounding |
| 620 |
> |
media. One quantity to describe the interfacial interaction is so |
| 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 |
|
\begin{equation} |
| 625 |
< |
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle. |
| 625 |
> |
\dot |
| 626 |
> |
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} |
| 627 |
> |
\over \eta } _{\alpha \beta } = \left\{ \begin{array}{l} |
| 628 |
> |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }}{\rm{ }}(\alpha = \beta = z), \\ |
| 629 |
> |
0{\rm{ }}(\alpha \ne z{\rm{ }}or{\rm{ }}\beta \ne z) \\ |
| 630 |
> |
\end{array} \right. |
| 631 |
> |
\label{methodEquation:NPATeta} |
| 632 |
|
\end{equation} |
| 633 |
+ |
Note that the iterative schemes for NPAT are identical to those |
| 634 |
+ |
described for the NPTi integrator. |
| 635 |
|
|
| 636 |
< |
If the time-dependent friction decays rapidly, the static friction |
| 654 |
< |
coefficient can be approximated by |
| 655 |
< |
\begin{equation} |
| 656 |
< |
\xi_{\text{static}}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta |
| 657 |
< |
F(z,0)\rangle dt. |
| 658 |
< |
\end{equation} |
| 659 |
< |
Allowing diffusion constant to then be calculated through the |
| 660 |
< |
Einstein relation:\cite{Marrink94} |
| 661 |
< |
\begin{equation} |
| 662 |
< |
D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
| 663 |
< |
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.% |
| 664 |
< |
\end{equation} |
| 636 |
> |
\subsubsection{\label{methodSection:NPrT}Constant Normal Pressure, Constant Lateral Surface Tension and Constant Temperature (NP\gamma T) Ensemble } |
| 637 |
|
|
| 638 |
< |
The Z-Constraint method, which fixes the z coordinates of the |
| 639 |
< |
molecules with respect to the center of the mass of the system, has |
| 640 |
< |
been a method suggested to obtain the forces required for the force |
| 641 |
< |
auto-correlation calculation.\cite{Marrink94} However, simply |
| 642 |
< |
resetting the coordinate will move the center of the mass of the |
| 643 |
< |
whole system. To avoid this problem, a new method was used in {\sc |
| 644 |
< |
oopse}. Instead of resetting the coordinate, we reset the forces of |
| 645 |
< |
z-constrained molecules as well as subtract the total constraint |
| 646 |
< |
forces from the rest of the system after the force calculation at |
| 647 |
< |
each time step. |
| 648 |
< |
|
| 649 |
< |
After the force calculation, define $G_\alpha$ as |
| 638 |
> |
Theoretically, the surface tension $\gamma$ of a stress free |
| 639 |
> |
membrane system should be zero since its surface free energy $G$ is |
| 640 |
> |
minimum with respect to surface area $A$ |
| 641 |
> |
\[ |
| 642 |
> |
\gamma = \frac{{\partial G}}{{\partial A}}. |
| 643 |
> |
\] |
| 644 |
> |
However, a surface tension of zero is not appropriate for relatively |
| 645 |
> |
small patches of membrane. In order to eliminate the edge effect of |
| 646 |
> |
the membrane simulation, a special ensemble, NP\gamma T, is proposed |
| 647 |
> |
to maintain the lateral surface tension and normal pressure. The |
| 648 |
> |
equation of motion for cell size control tensor, $\eta$, in NP\gamma |
| 649 |
> |
T is |
| 650 |
|
\begin{equation} |
| 651 |
< |
G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1} |
| 651 |
> |
\dot |
| 652 |
> |
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} |
| 653 |
> |
\over \eta } _{\alpha \beta } = \left\{ \begin{array}{l} |
| 654 |
> |
- A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ){\rm{ (}}\alpha = \beta = x{\rm{ or }} = y{\rm{)}} \\ |
| 655 |
> |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }}{\rm{ }}(\alpha = \beta = z) \\ |
| 656 |
> |
0{\rm{ }}(\alpha \ne \beta ) \\ |
| 657 |
> |
\end{array} \right. |
| 658 |
> |
\label{methodEquation:NPrTeta} |
| 659 |
|
\end{equation} |
| 660 |
< |
where $F_{\alpha i}$ is the force in the z direction of atom $i$ in |
| 661 |
< |
z-constrained molecule $\alpha$. The forces of the z constrained |
| 683 |
< |
molecule are then set to: |
| 660 |
> |
where $ \gamma _{{\rm{target}}}$ is the external surface tension and |
| 661 |
> |
the instantaneous surface tensor $\gamma _\alpha$ is given by |
| 662 |
|
\begin{equation} |
| 663 |
< |
F_{\alpha i} = F_{\alpha i} - |
| 664 |
< |
\frac{m_{\alpha i} G_{\alpha}}{\sum_i m_{\alpha i}}. |
| 663 |
> |
\gamma _\alpha = - h_z |
| 664 |
> |
(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} |
| 665 |
> |
\over P} _{\alpha \alpha } - P_{{\rm{target}}} ) |
| 666 |
> |
\label{methodEquation:instantaneousSurfaceTensor} |
| 667 |
|
\end{equation} |
| 688 |
– |
Here, $m_{\alpha i}$ is the mass of atom $i$ in the z-constrained |
| 689 |
– |
molecule. Having rescaled the forces, the velocities must also be |
| 690 |
– |
rescaled to subtract out any center of mass velocity in the z |
| 691 |
– |
direction. |
| 692 |
– |
\begin{equation} |
| 693 |
– |
v_{\alpha i} = v_{\alpha i} - |
| 694 |
– |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}, |
| 695 |
– |
\end{equation} |
| 696 |
– |
where $v_{\alpha i}$ is the velocity of atom $i$ in the z direction. |
| 697 |
– |
Lastly, all of the accumulated z constrained forces must be |
| 698 |
– |
subtracted from the system to keep the system center of mass from |
| 699 |
– |
drifting. |
| 700 |
– |
\begin{equation} |
| 701 |
– |
F_{\beta i} = F_{\beta i} - \frac{m_{\beta i} \sum_{\alpha} |
| 702 |
– |
G_{\alpha}} |
| 703 |
– |
{\sum_{\beta}\sum_i m_{\beta i}}, |
| 704 |
– |
\end{equation} |
| 705 |
– |
where $\beta$ are all of the unconstrained molecules in the system. |
| 706 |
– |
Similarly, the velocities of the unconstrained molecules must also |
| 707 |
– |
be scaled. |
| 708 |
– |
\begin{equation} |
| 709 |
– |
v_{\beta i} = v_{\beta i} + \sum_{\alpha} |
| 710 |
– |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}. |
| 711 |
– |
\end{equation} |
| 668 |
|
|
| 669 |
< |
At the very beginning of the simulation, the molecules may not be at |
| 670 |
< |
their constrained positions. To move a z-constrained molecule to its |
| 671 |
< |
specified position, a simple harmonic potential is used |
| 672 |
< |
\begin{equation} |
| 673 |
< |
U(t)=\frac{1}{2}k_{\text{Harmonic}}(z(t)-z_{\text{cons}})^{2},% |
| 674 |
< |
\end{equation} |
| 675 |
< |
where $k_{\text{Harmonic}}$ is the harmonic force constant, $z(t)$ |
| 720 |
< |
is the current $z$ coordinate of the center of mass of the |
| 721 |
< |
constrained molecule, and $z_{\text{cons}}$ is the constrained |
| 722 |
< |
position. The harmonic force operating on the z-constrained molecule |
| 723 |
< |
at time $t$ can be calculated by |
| 724 |
< |
\begin{equation} |
| 725 |
< |
F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= |
| 726 |
< |
-k_{\text{Harmonic}}(z(t)-z_{\text{cons}}). |
| 727 |
< |
\end{equation} |
| 669 |
> |
There is one additional extended system integrator (NPTxyz), in |
| 670 |
> |
which each attempt to preserve the target pressure along the box |
| 671 |
> |
walls perpendicular to that particular axis. The lengths of the box |
| 672 |
> |
axes are allowed to fluctuate independently, but the angle between |
| 673 |
> |
the box axes does not change. It should be noted that the NPTxyz |
| 674 |
> |
integrator is a special case of $NP\gamma T$ if the surface tension |
| 675 |
> |
$\gamma$ is set to zero. |
| 676 |
|
|
| 677 |
|
\section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies} |
| 678 |
|
|
