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The Z-Constraint Method

A force auto-correlation method based on the fluctuation-dissipation theorem was developed by Roux and Karplus to investigate the dynamics of ions inside ion channels.[49] The time-dependent friction coefficient can be calculated from the deviation of the instantaneous force from its mean value:

$\displaystyle \xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T,$ (4.44)

where

$\displaystyle \delta F(z,t)=F(z,t)-\langle F(z,t)\rangle.$ (4.45)

If the time-dependent friction decays rapidly, the static friction coefficient can be approximated by

$\displaystyle \xi_{\text{static}}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta F(z,0)\rangle dt.$ (4.46)

This allows the diffusion constant to then be calculated through the Einstein relation:[50]

$\displaystyle 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}.% $ (4.47)

The Z-Constraint method, which fixes the $ z$ coordinates of a few ``tagged'' molecules with respect to the center of the mass of the system is a technique that was proposed to obtain the forces required for the force auto-correlation calculation.[50] However, simply resetting the coordinate will move the center of the mass of the whole system. To avoid this problem, we have developed a new method that is utilized in OOPSE. Instead of resetting the coordinates, we reset the forces of $ z$ -constrained molecules and subtract the total constraint forces from the rest of the system after the force calculation at each time step.

After the force calculation, the total force on molecule $ \alpha$ is:

$\displaystyle G_{\alpha} = \sum_i F_{\alpha i},$ (4.48)

where $ F_{\alpha i}$ is the force in the $ z$ direction on atom $ i$ in $ z$ -constrained molecule $ \alpha$ . The forces on the atoms in the $ z$ -constrained molecule are then adjusted to remove the total force on molecule $ \alpha$ :

$\displaystyle F_{\alpha i} = F_{\alpha i} - \frac{m_{\alpha i} G_{\alpha}}{\sum_i m_{\alpha i}}.$ (4.49)

Here, $ m_{\alpha i}$ is the mass of atom $ i$ in the $ z$ -constrained molecule. After the forces have been adjusted, the velocities must also be modified to subtract out molecule $ \alpha$ 's center-of-mass velocity in the $ z$ direction.

$\displaystyle v_{\alpha i} = v_{\alpha i} - \frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}},$ (4.50)

where $ v_{\alpha i}$ is the velocity of atom $ i$ in the $ z$ direction. Lastly, all of the accumulated constraint forces must be subtracted from the rest of the unconstrained system to keep the system center of mass of the entire system from drifting.

$\displaystyle F_{\beta i} = F_{\beta i} - \frac{m_{\beta i} \sum_{\alpha} G_{\alpha}} {\sum_{\beta}\sum_i m_{\beta i}},$ (4.51)

where $ \beta$ denotes all unconstrained molecules in the system. Similarly, the velocities of the unconstrained molecules must also be scaled:

$\displaystyle v_{\beta i} = v_{\beta i} + \sum_{\alpha} \frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}.$ (4.52)

This method will pin down the centers-of-mass of all of the $ z$ -constrained molecules, and will also keep the entire system fixed at the original system center-of-mass location.

At the very beginning of the simulation, the molecules may not be at their desired positions. To steer a $ z$ -constrained molecule to its specified position, a simple harmonic potential is used:

$\displaystyle U(t)=\frac{1}{2}k_{\text{Harmonic}}(z(t)-z_{\text{cons}})^{2},% $ (4.53)

where $ k_{\text{Harmonic}}$ is an 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 desired constraint position. The harmonic force operating on the $ z$ -constrained molecule at time $ t$ can be calculated by

$\displaystyle F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= -k_{\text{Harmonic}}(z(t)-z_{\text{cons}}).$ (4.54)

The user may also specify the use of a constant velocity method (steered molecular dynamics) to move the molecules to their desired initial positions. Based on concepts from atomic force microscopy, SMD has been used to study many processes which occur via rare events on the time scale of a few hundreds of picoseconds. For example,SMD has been used to observe the dissociation of Streptavidin-biotin Complex.[51]

To use of the $ z$ -constraint method in an OOPSE simulation, the molecules must be specified using the nZconstraints keyword in the meta-data file. The other parameters for modifying the behavior of the $ z$ -constraint method are listed in table 4.1.


Table 4.1: Meta-data Keywords: Z-Constraint Parameters
\begin{table} \begin{center} \begin{tabularx}{\linewidth}% {>{ \setlength{\hsiz... ...armonic Forces are used by default\\ \par \end{tabularx}\end{center}\end{table}


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Updated on January 16, 2006