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\section{\label{Sec:zcons}Z-Constraint Method}
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Based on fluctuatin-dissipation theorem,\bigskip\ force auto-correlation
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method was developed to investigate the dynamics of ions inside the ion
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channels.\cite{Roux91} Time-dependent friction coefficient can be calculated
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from the deviation of the instaneous force from its mean force.
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%
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\begin{equation}
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\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T
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\end{equation}
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where%
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\begin{equation}
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\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle
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\end{equation}
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If the time-dependent friction decay rapidly, static friction coefficient can
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be approximated by%
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\begin{equation}
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\xi^{static}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta F(z,0)\rangle dt
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\end{equation}
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Hence, diffusion constant can be estimated by
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\begin{equation}
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D(z)=\frac{k_{B}T}{\xi^{static}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty
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}\langle\delta F(z,t)\delta F(z,0)\rangle dt}%
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\end{equation}
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\bigskip Z-Constraint method, which fixed the z coordinates of the molecules
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with respect to the center of the mass of the system, was proposed to obtain
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the forces required in force auto-correlation method.\cite{Marrink94} However,
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simply resetting the coordinate will move the center of the mass of the whole
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system. To avoid this problem, a new method was used at OOPSE. Instead of
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resetting the coordinate, we reset the forces of z-constraint molecules as
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well as subtract the total constraint forces from the rest of the system after
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force calculation at each time step.
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\begin{verbatim}
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$F_{\alpha i}=0$
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$V_{\alpha i}=V_{\alpha i}-\frac{\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{i}M_{_{\alpha i}}}$
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$F_{\alpha i}=F_{\alpha i}-\frac{M_{_{\alpha i}}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}}\sum\limits_{\beta}F_{\beta}$
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$V_{\alpha i}=V_{\alpha i}-\frac{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}V_{\alpha i}}{\sum\limits_{\alpha}\sum\limits_{i}M_{_{\alpha i}}}$
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\end{verbatim}
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At the very beginning of the simulation, the molecules may not be at its
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constraint position. To move the z-constraint molecule to the specified
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position, a simple harmonic potential is used%
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\begin{equation}
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U(t)=\frac{1}{2}k_{Harmonic}(z(t)-z_{cons})^{2}%
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\end{equation}
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where $k_{Harmonic}$\bigskip\ is the harmonic force constant, $z(t)$ is
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current z coordinate of the center of mass of the z-constraint molecule, and
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$z_{cons}$ is the restraint position. Therefore, the harmonic force operated
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on the z-constraint molecule at time $t$ can be calculated by%
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\begin{equation}
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F_{z_{Harmonic}}(t)=-\frac{\partial U(t)}{\partial z(t)}=-k_{Harmonic}%
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(z(t)-z_{cons})
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\end{equation}
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Worthy of mention, other kinds of potential functions can also be used to
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drive the z-constraint molecule. |
