--- trunk/tengDissertation/Methodology.tex 2006/04/24 18:49:04 2729 +++ trunk/tengDissertation/Methodology.tex 2006/04/26 01:27:56 2739 @@ -4,7 +4,7 @@ methods have been developed to generate statistical en In order to mimic the experiments, which are usually performed under constant temperature and/or pressure, extended Hamiltonian system -methods have been developed to generate statistical ensemble, such +methods have been developed to generate statistical ensembles, such as canonical ensemble and isobaric-isothermal ensemble \textit{etc}. In addition to the standard ensemble, specific ensembles have been developed to account for the anisotropy between the lateral and @@ -608,123 +608,71 @@ assume non-orthorhombic geometries. electrostatic or Lennard-Jones cutoff radii. The NPTf integrator finds most use in simulating crystals or liquid crystals which assume non-orthorhombic geometries. - -\subsection{\label{methodSection:NPAT}Constant Lateral Pressure and Constant Surface Area (NPAT)} - -\subsection{\label{methodSection:NPrT}Constant lateral Pressure and Constant Surface Tension (NP\gamma T) } -\subsection{\label{methodSection:NPTxyz}Constant pressure in 3 axes (NPTxyz)} - -There is one additional extended system integrator which is somewhat -simpler than the NPTf method described above. In this case, the -three axes have independent barostats which each attempt to preserve -the target pressure along the box walls perpendicular to that -particular axis. The lengths of the box axes are allowed to -fluctuate independently, but the angle between the box axes does not -change. The equations of motion are identical to those described -above, but only the {\it diagonal} elements of -$\overleftrightarrow{\eta}$ are computed. The off-diagonal elements -are set to zero (even when the pressure tensor has non-zero -off-diagonal elements). It should be noted that the NPTxyz -integrator is a special case of $NP\gamma T$ if the surface tension -$\gamma$ is set to zero. +\subsection{\label{methodSection:otherSpecialEnsembles}Other Special Ensembles} +\subsubsection{\label{methodSection:NPAT}Constant Normal Pressure, Constant Lateral Surface Area and Constant Temperature (NPAT) Ensemble} -\section{\label{methodSection:constraintMethods}Constraint Methods} - -\subsection{\label{methodSection:rattle}The {\sc rattle} Method for Bond - Constraints} - -\subsection{\label{methodSection:zcons}Z-Constraint Method} - -Based on the fluctuation-dissipation theorem, a force -auto-correlation method was developed by Roux and Karplus to -investigate the dynamics of ions inside ion channels.\cite{Roux91} -The time-dependent friction coefficient can be calculated from the -deviation of the instantaneous force from its mean force. -\begin{equation} -\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T, -\end{equation} -where% +A comprehensive understanding of structure¨Cfunction relations of +biological membrane system ultimately relies on structure and +dynamics of lipid bilayer, which are strongly affected by the +interfacial interaction between lipid molecules and surrounding +media. One quantity to describe the interfacial interaction is so +called the average surface area per lipid. Constat area and constant +lateral pressure simulation can be achieved by extending the +standard NPT ensemble with a different pressure control strategy \begin{equation} -\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle. +\dot +\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} +\over \eta } _{\alpha \beta } = \left\{ \begin{array}{l} + \frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }}{\rm{ }}(\alpha = \beta = z), \\ + 0{\rm{ }}(\alpha \ne z{\rm{ }}or{\rm{ }}\beta \ne z) \\ + \end{array} \right. +\label{methodEquation:NPATeta} \end{equation} +Note that the iterative schemes for NPAT are identical to those +described for the NPTi integrator. -If the time-dependent friction decays rapidly, the static friction -coefficient can be approximated by -\begin{equation} -\xi_{\text{static}}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta -F(z,0)\rangle dt. -\end{equation} -Allowing diffusion constant to then be calculated through the -Einstein relation:\cite{Marrink94} -\begin{equation} -D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty -}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.% -\end{equation} +\subsubsection{\label{methodSection:NPrT}Constant Normal Pressure, Constant Lateral Surface Tension and Constant Temperature (NP\gamma T) Ensemble } -The Z-Constraint method, which fixes the z coordinates of the -molecules with respect to the center of the mass of the system, has -been a method suggested to obtain the forces required for the force -auto-correlation calculation.\cite{Marrink94} However, simply -resetting the coordinate will move the center of the mass of the -whole system. To avoid this problem, a new method was used in {\sc -oopse}. Instead of resetting the coordinate, we reset the forces of -z-constrained molecules as well as subtract the total constraint -forces from the rest of the system after the force calculation at -each time step. - -After the force calculation, define $G_\alpha$ as +Theoretically, the surface tension $\gamma$ of a stress free +membrane system should be zero since its surface free energy $G$ is +minimum with respect to surface area $A$ +\[ +\gamma = \frac{{\partial G}}{{\partial A}}. +\] +However, a surface tension of zero is not appropriate for relatively +small patches of membrane. In order to eliminate the edge effect of +the membrane simulation, a special ensemble, NP\gamma T, is proposed +to maintain the lateral surface tension and normal pressure. The +equation of motion for cell size control tensor, $\eta$, in NP\gamma +T is \begin{equation} -G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1} +\dot +\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} +\over \eta } _{\alpha \beta } = \left\{ \begin{array}{l} + - A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ){\rm{ (}}\alpha = \beta = x{\rm{ or }} = y{\rm{)}} \\ + \frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }}{\rm{ }}(\alpha = \beta = z) \\ + 0{\rm{ }}(\alpha \ne \beta ) \\ + \end{array} \right. +\label{methodEquation:NPrTeta} \end{equation} -where $F_{\alpha i}$ is the force in the z direction of atom $i$ in -z-constrained molecule $\alpha$. The forces of the z constrained -molecule are then set to: +where $ \gamma _{{\rm{target}}}$ is the external surface tension and +the instantaneous surface tensor $\gamma _\alpha$ is given by \begin{equation} -F_{\alpha i} = F_{\alpha i} - - \frac{m_{\alpha i} G_{\alpha}}{\sum_i m_{\alpha i}}. +\gamma _\alpha = - h_z +(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} +\over P} _{\alpha \alpha } - P_{{\rm{target}}} ) +\label{methodEquation:instantaneousSurfaceTensor} \end{equation} -Here, $m_{\alpha i}$ is the mass of atom $i$ in the z-constrained -molecule. Having rescaled the forces, the velocities must also be -rescaled to subtract out any center of mass velocity in the z -direction. -\begin{equation} -v_{\alpha i} = v_{\alpha i} - - \frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}, -\end{equation} -where $v_{\alpha i}$ is the velocity of atom $i$ in the z direction. -Lastly, all of the accumulated z constrained forces must be -subtracted from the system to keep the system center of mass from -drifting. -\begin{equation} -F_{\beta i} = F_{\beta i} - \frac{m_{\beta i} \sum_{\alpha} -G_{\alpha}} - {\sum_{\beta}\sum_i m_{\beta i}}, -\end{equation} -where $\beta$ are all of the unconstrained molecules in the system. -Similarly, the velocities of the unconstrained molecules must also -be scaled. -\begin{equation} -v_{\beta i} = v_{\beta i} + \sum_{\alpha} - \frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}. -\end{equation} -At the very beginning of the simulation, the molecules may not be at -their constrained positions. To move a z-constrained molecule to its -specified position, a simple harmonic potential is used -\begin{equation} -U(t)=\frac{1}{2}k_{\text{Harmonic}}(z(t)-z_{\text{cons}})^{2},% -\end{equation} -where $k_{\text{Harmonic}}$ is the harmonic force constant, $z(t)$ -is the current $z$ coordinate of the center of mass of the -constrained molecule, and $z_{\text{cons}}$ is the constrained -position. The harmonic force operating on the z-constrained molecule -at time $t$ can be calculated by -\begin{equation} -F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= - -k_{\text{Harmonic}}(z(t)-z_{\text{cons}}). -\end{equation} +There is one additional extended system integrator (NPTxyz), in +which each attempt to preserve the target pressure along the box +walls perpendicular to that particular axis. The lengths of the box +axes are allowed to fluctuate independently, but the angle between +the box axes does not change. It should be noted that the NPTxyz +integrator is a special case of $NP\gamma T$ if the surface tension +$\gamma$ is set to zero. \section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies}