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\chapter{\label{chapt:methodology}METHODOLOGY} |
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|
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\section{\label{methodSection:rigidBodyIntegrators}Integrators for Rigid Body Motion in Molecular Dynamics} |
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|
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In order to mimic experiments which are usually performed under |
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constant temperature and/or pressure, extended Hamiltonian system |
| 7 |
methods have been developed to generate statistical ensembles, such |
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as the canonical and isobaric-isothermal ensembles. In addition to |
| 9 |
the standard ensemble, specific ensembles have been developed to |
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account for the anisotropy between the lateral and normal directions |
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of membranes. The $NPAT$ ensemble, in which the normal pressure and |
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the lateral surface area of the membrane are kept constant, and the |
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$NP\gamma T$ ensemble, in which the normal pressure and the lateral |
| 14 |
surface tension are kept constant were proposed to address the |
| 15 |
issues. |
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|
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Integration schemes for the rotational motion of the rigid molecules |
| 18 |
in the microcanonical ensemble have been extensively studied over |
| 19 |
the last two decades. Matubayasi developed a |
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time-reversible integrator for rigid bodies in quaternion |
| 21 |
representation\cite{Matubayasi1999}. Although it is not symplectic, this integrator still |
| 22 |
demonstrates a better long-time energy conservation than Euler angle |
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methods because of the time-reversible nature. Extending the |
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Trotter-Suzuki factorization to general system with a flat phase |
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space, Miller\cite{Miller2002} and his colleagues devised a novel |
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symplectic, time-reversible and volume-preserving integrator in the |
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quaternion representation, which was shown to be superior to the |
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Matubayasi's time-reversible integrator. However, all of the |
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integrators in the quaternion representation suffer from the |
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computational penalty of constructing a rotation matrix from |
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quaternions to evolve coordinates and velocities at every time step. |
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An alternative integration scheme utilizing the rotation matrix |
| 33 |
directly proposed by Dullweber, Leimkuhler and McLachlan (DLM) also |
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preserved the same structural properties of the Hamiltonian |
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propagator\cite{Dullweber1997}. In this section, the integration |
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scheme of DLM method will be reviewed and extended to other |
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ensembles. |
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|
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\subsection{\label{methodSection:DLM}Integrating the Equations of Motion: the |
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DLM method} |
| 41 |
|
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The DLM method uses a Trotter factorization of the orientational |
| 43 |
propagator. This has three effects: |
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\begin{enumerate} |
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\item the integrator is area-preserving in phase space (i.e. it is |
| 46 |
{\it symplectic}), |
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\item the integrator is time-{\it reversible}, making it suitable for Hybrid |
| 48 |
Monte Carlo applications, and |
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\item the error for a single time step is of order $\mathcal{O}\left(h^3\right)$ |
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for timesteps of length $h$. |
| 51 |
\end{enumerate} |
| 52 |
The integration of the equations of motion is carried out in a |
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velocity-Verlet style 2-part algorithm, where $h= \delta t$: |
| 54 |
|
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{\tt moveA:} |
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\begin{align*} |
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{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
| 58 |
+ \frac{h}{2} \left( {\bf f}(t) / m \right), \\ |
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% |
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{\bf r}(t + h) &\leftarrow {\bf r}(t) |
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+ h {\bf v}\left(t + h / 2 \right), \\ |
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% |
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{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
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+ \frac{h}{2} {\bf \tau}^b(t), \\ |
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% |
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\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
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(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
| 68 |
\end{align*} |
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In this context, the $\mathrm{rotate}$ function is the reversible |
| 70 |
product of the three body-fixed rotations, |
| 71 |
\begin{equation} |
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\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot |
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\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y |
| 74 |
/ 2) \cdot \mathsf{G}_x(a_x /2), |
| 75 |
\end{equation} |
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where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
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rotates both the rotation matrix $\mathsf{Q}$ and the body-fixed |
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angular momentum ${\bf j}$ by an angle $\theta$ around body-fixed |
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axis $\alpha$, |
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\begin{equation} |
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\mathsf{G}_\alpha( \theta ) = \left\{ |
| 82 |
\begin{array}{lcl} |
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\mathsf{Q}(t) & \leftarrow & \mathsf{Q}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
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{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
| 85 |
j}(0). |
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\end{array} |
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\right. |
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\end{equation} |
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$\mathsf{R}_\alpha$ is a quadratic approximation to the single-axis |
| 90 |
rotation matrix. For example, in the small-angle limit, the |
| 91 |
rotation matrix around the body-fixed x-axis can be approximated as |
| 92 |
\begin{equation} |
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\mathsf{R}_x(\theta) \approx \left( |
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\begin{array}{ccc} |
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1 & 0 & 0 \\ |
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0 & \frac{1-\theta^2 / 4}{1 + \theta^2 / 4} & -\frac{\theta}{1+ |
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\theta^2 / 4} \\ |
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0 & \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 + |
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\theta^2 / 4} |
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\end{array} |
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\right). |
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\end{equation} |
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All other rotations follow in a straightforward manner. After the |
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first part of the propagation, the forces and body-fixed torques are |
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calculated at the new positions and orientations |
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|
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{\tt doForces:} |
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\begin{align*} |
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{\bf f}(t + h) &\leftarrow |
| 110 |
- \left(\frac{\partial V}{\partial {\bf r}}\right)_{{\bf r}(t + h)}, \\ |
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% |
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{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h) |
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\times (\frac{\partial V}{\partial {\bf u}})_{u(t+h)}, \\ |
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% |
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{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{Q}(t + h) |
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\cdot {\bf \tau}^s(t + h). |
| 117 |
\end{align*} |
| 118 |
${\bf u}$ is automatically updated when the rotation matrix |
| 119 |
$\mathsf{Q}$ is calculated in {\tt moveA}. Once the forces and |
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torques have been obtained at the new time step, the velocities can |
| 121 |
be advanced to the same time value. |
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|
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{\tt moveB:} |
| 124 |
\begin{align*} |
| 125 |
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t + h / 2 |
| 126 |
\right) |
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+ \frac{h}{2} \left( {\bf f}(t + h) / m \right), \\ |
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% |
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{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t + h / 2 |
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\right) |
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+ \frac{h}{2} {\bf \tau}^b(t + h) . |
| 132 |
\end{align*} |
| 133 |
The matrix rotations used in the DLM method end up being more costly |
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computationally than the simpler arithmetic quaternion propagation. |
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With the same time step, a 1000-molecule water simulation shows an |
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average 7\% increase in computation time using the DLM method in |
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place of quaternions. This cost is more than justified when |
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comparing the energy conservation of the two methods as illustrated |
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in Fig.~\ref{methodFig:timestep} where the resulting energy drifts at |
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various time steps for both the DLM and quaternion integration |
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schemes are compared. All of the 1000 molecule water simulations |
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started with the same configuration, and the only difference was the |
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method for handling rotational motion. At time steps of 0.1 and 0.5 |
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fs, both methods for propagating molecule rotation conserve energy |
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fairly well, with the quaternion method showing a slight energy |
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drift over time in the 0.5 fs time step simulation. At time steps of |
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1 and 2 fs, the energy conservation benefits of the DLM method are |
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clearly demonstrated. Thus, while maintaining the same degree of |
| 149 |
energy conservation, one can take considerably longer time steps, |
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leading to an overall reduction in computation time. |
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|
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{timeStep.eps} |
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\caption[Energy conservation for quaternion versus DLM |
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dynamics]{Energy conservation using quaternion based integration |
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versus the method proposed by Dullweber \emph{et al.} with |
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increasing time step. For each time step, the dotted line is total |
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energy using the DLM integrator, and the solid line comes from the |
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quaternion integrator. The larger time step plots are shifted up |
| 161 |
from the true energy baseline for clarity.} |
| 162 |
\label{methodFig:timestep} |
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\end{figure} |
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|
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\subsection{\label{methodSection:NVT}Nos\'{e}-Hoover Thermostatting} |
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|
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The Nos\'e-Hoover equations of motion are given by\cite{Hoover1985} |
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\begin{eqnarray} |
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\dot{{\bf r}} & = & {\bf v}, \\ |
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\dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} ,\\ |
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\dot{\mathsf{Q}} & = & \mathsf{Q} \cdot |
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\mbox{ skew}\left(\overleftrightarrow{\mathsf{I}}^{-1} \cdot {\bf j}\right), \\ |
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\dot{{\bf j}} & = & {\bf j} \times \left( |
| 174 |
\overleftrightarrow{\mathsf{I}}^{-1} \cdot {\bf j} \right) - \mbox{ |
| 175 |
rot}\left(\mathsf{Q}^{T} \cdot \frac{\partial V}{\partial |
| 176 |
\mathsf{Q}} \right) - \chi {\bf j}. \label{eq:nosehoovereom} |
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\end{eqnarray} |
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$\chi$ is an ``extra'' variable included in the extended system, and |
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it is propagated using the first order equation of motion |
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\begin{equation} |
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\dot{\chi} = \frac{1}{\tau_{T}^2} \left( |
| 182 |
\frac{T}{T_{\mathrm{target}}} - 1 \right). \label{eq:nosehooverext} |
| 183 |
\end{equation} |
| 184 |
where $\tau_T$ is the time constant for relaxation of the |
| 185 |
temperature to the target value, and the instantaneous temperature |
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$T$ is given by |
| 187 |
\begin{equation} |
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T = \frac{2 K}{f k_B}. |
| 189 |
\end{equation} |
| 190 |
Here, $f$ is the total number of degrees of freedom in the system, |
| 191 |
\begin{equation} |
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f = 3 N + 3 N_{\mathrm{orient}} - N_{\mathrm{constraints}}, |
| 193 |
\end{equation} |
| 194 |
where $N_{\mathrm{orient}}$ is the number of molecules with |
| 195 |
orientational degrees of freedom. The integration of the equations of motion |
| 196 |
is carried out in a velocity-Verlet style 2 part algorithm: |
| 197 |
|
| 198 |
{\tt moveA:} |
| 199 |
\begin{align*} |
| 200 |
T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
| 201 |
% |
| 202 |
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
| 203 |
+ \frac{h}{2} \left( \frac{{\bf f}(t)}{m} - {\bf v}(t) |
| 204 |
\chi(t)\right), \\ |
| 205 |
% |
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{\bf r}(t + h) &\leftarrow {\bf r}(t) |
| 207 |
+ h {\bf v}\left(t + h / 2 \right) ,\\ |
| 208 |
% |
| 209 |
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
| 210 |
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 211 |
\chi(t) \right) ,\\ |
| 212 |
% |
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\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate} |
| 214 |
\left(h * {\bf j}(t + h / 2) |
| 215 |
\overleftrightarrow{\mathsf{I}}^{-1} \right) ,\\ |
| 216 |
% |
| 217 |
\chi\left(t + h / 2 \right) &\leftarrow \chi(t) |
| 218 |
+ \frac{h}{2 \tau_T^2} \left( \frac{T(t)} |
| 219 |
{T_{\mathrm{target}}} - 1 \right) . |
| 220 |
\end{align*} |
| 221 |
Here $\mathrm{rotate}(h * {\bf j} |
| 222 |
\overleftrightarrow{\mathsf{I}}^{-1})$ is the same symplectic Strang |
| 223 |
factorization of the three rotation operations that was discussed in |
| 224 |
the section on the DLM integrator. Note that this operation |
| 225 |
modifies both the rotation matrix $\mathsf{Q}$ and the angular |
| 226 |
momentum ${\bf j}$. {\tt moveA} propagates velocities by a half |
| 227 |
time step, and positional degrees of freedom by a full time step. |
| 228 |
The new positions (and orientations) are then used to calculate a |
| 229 |
new set of forces and torques in exactly the same way they are |
| 230 |
calculated in the {\tt doForces} portion of the DLM integrator. Once |
| 231 |
the forces and torques have been obtained at the new time step, the |
| 232 |
temperature, velocities, and the extended system variable can be |
| 233 |
advanced to the same time value. |
| 234 |
|
| 235 |
{\tt moveB:} |
| 236 |
\begin{align*} |
| 237 |
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
| 238 |
\left\{{\bf j}(t + h)\right\}, \\ |
| 239 |
% |
| 240 |
\chi\left(t + h \right) &\leftarrow \chi\left(t + h / |
| 241 |
2 \right) + \frac{h}{2 \tau_T^2} \left( \frac{T(t+h)} |
| 242 |
{T_{\mathrm{target}}} - 1 \right), \\ |
| 243 |
% |
| 244 |
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t |
| 245 |
+ h / 2 \right) + \frac{h}{2} \left( |
| 246 |
\frac{{\bf f}(t + h)}{m} - {\bf v}(t + h) |
| 247 |
\chi(t h)\right) ,\\ |
| 248 |
% |
| 249 |
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t |
| 250 |
+ h / 2 \right) + \frac{h}{2} |
| 251 |
\left( {\bf \tau}^b(t + h) - {\bf j}(t + h) |
| 252 |
\chi(t + h) \right) . |
| 253 |
\end{align*} |
| 254 |
Since ${\bf v}(t + h)$ and ${\bf j}(t + h)$ are required to |
| 255 |
caclculate $T(t + h)$ as well as $\chi(t + h)$, they indirectly |
| 256 |
depend on their own values at time $t + h$. {\tt moveB} is |
| 257 |
therefore done in an iterative fashion until $\chi(t + h)$ becomes |
| 258 |
self-consistent. The Nos\'e-Hoover algorithm is known to conserve a |
| 259 |
Hamiltonian for the extended system that is, to within a constant, |
| 260 |
identical to the Helmholtz free energy,\cite{Melchionna1993} |
| 261 |
\begin{equation} |
| 262 |
H_{\mathrm{NVT}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 263 |
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
| 264 |
dt^\prime \right). |
| 265 |
\end{equation} |
| 266 |
Poor choices of $h$ or $\tau_T$ can result in non-conservation of |
| 267 |
$H_{\mathrm{NVT}}$, so the conserved quantity should be checked |
| 268 |
periodically to verify the quality of the integration. |
| 269 |
|
| 270 |
\subsection{\label{methodSection:NPTi}Constant-pressure integration with |
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isotropic box (NPTi)} |
| 272 |
|
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We can used an isobaric-isothermal ensemble integrator which is |
| 274 |
implemented using the Melchionna modifications to the |
| 275 |
Nos\'e-Hoover-Andersen equations of motion\cite{Melchionna1993} |
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\begin{eqnarray} |
| 277 |
\dot{{\bf r}} & = & {\bf v} + \eta \left( {\bf r} - {\bf R}_0 \right), \\ |
| 278 |
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - (\eta + \chi) {\bf v}, \\ |
| 279 |
\dot{\mathsf{Q}} & = & \mathsf{Q} \cdot |
| 280 |
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right),\\ |
| 281 |
\dot{{\bf j}} & = & {\bf j} \times \left( |
| 282 |
\overleftrightarrow{I}^{-1} \cdot {\bf j} \right) - \mbox{ |
| 283 |
rot}\left(\mathsf{Q}^{T} \cdot \frac{\partial |
| 284 |
V}{\partial \mathsf{Q}} \right) - \chi {\bf j}, \\ |
| 285 |
\dot{\chi} & = & \frac{1}{\tau_{T}^2} \left( |
| 286 |
\frac{T}{T_{\mathrm{target}}} - 1 \right) ,\\ |
| 287 |
\dot{\eta} & = & \frac{1}{\tau_{B}^2 f k_B T_{\mathrm{target}}} V |
| 288 |
\left( P - |
| 289 |
P_{\mathrm{target}} \right), \\ |
| 290 |
\dot{\mathcal{V}} & = & 3 \mathcal{V} \eta . \label{eq:melchionna1} |
| 291 |
\end{eqnarray} |
| 292 |
$\chi$ and $\eta$ are the ``extra'' degrees of freedom in the |
| 293 |
extended system. $\chi$ is a thermostat, and it has the same |
| 294 |
function as it does in the Nos\'e-Hoover NVT integrator. $\eta$ is |
| 295 |
a barostat which controls changes to the volume of the simulation |
| 296 |
box. ${\bf R}_0$ is the location of the center of mass for the |
| 297 |
entire system, and $\mathcal{V}$ is the volume of the simulation |
| 298 |
box. At any time, the volume can be calculated from the determinant |
| 299 |
of the matrix which describes the box shape: |
| 300 |
\begin{equation} |
| 301 |
\mathcal{V} = \det(\mathsf{H}). |
| 302 |
\end{equation} |
| 303 |
|
| 304 |
The NPTi integrator requires an instantaneous pressure. This |
| 305 |
quantity is calculated via the pressure tensor, |
| 306 |
\begin{equation} |
| 307 |
\overleftrightarrow{\mathsf{P}}(t) = \frac{1}{\mathcal{V}(t)} \left( |
| 308 |
\sum_{i=1}^{N} m_i {\bf v}_i(t) \otimes {\bf v}_i(t) \right) + |
| 309 |
\overleftrightarrow{\mathsf{W}}(t). |
| 310 |
\end{equation} |
| 311 |
The kinetic contribution to the pressure tensor utilizes the {\it |
| 312 |
outer} product of the velocities denoted by the $\otimes$ symbol. |
| 313 |
The stress tensor is calculated from another outer product of the |
| 314 |
inter-atomic separation vectors (${\bf r}_{ij} = {\bf r}_j - {\bf |
| 315 |
r}_i$) with the forces between the same two atoms, |
| 316 |
\begin{equation} |
| 317 |
\overleftrightarrow{\mathsf{W}}(t) = \sum_{i} \sum_{j>i} {\bf |
| 318 |
r}_{ij}(t) \otimes {\bf f}_{ij}(t). |
| 319 |
\end{equation} |
| 320 |
The instantaneous pressure is then simply obtained from the trace of |
| 321 |
the Pressure tensor, |
| 322 |
\begin{equation} |
| 323 |
P(t) = \frac{1}{3} \mathrm{Tr} \left( |
| 324 |
\overleftrightarrow{\mathsf{P}}(t) \right) . |
| 325 |
\end{equation} |
| 326 |
In Eq.~\ref{eq:melchionna1}, $\tau_B$ is the time constant for |
| 327 |
relaxation of the pressure to the target value. Like in the NVT |
| 328 |
integrator, the integration of the equations of motion is carried |
| 329 |
out in a velocity-Verlet style 2 part algorithm: |
| 330 |
|
| 331 |
{\tt moveA:} |
| 332 |
\begin{align*} |
| 333 |
T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
| 334 |
% |
| 335 |
P(t) &\leftarrow \left\{{\bf r}(t)\right\}, \left\{{\bf v}(t)\right\} ,\\ |
| 336 |
% |
| 337 |
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
| 338 |
+ \frac{h}{2} \left( \frac{{\bf f}(t)}{m} - {\bf v}(t) |
| 339 |
\left(\chi(t) + \eta(t) \right) \right), \\ |
| 340 |
% |
| 341 |
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
| 342 |
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 343 |
\chi(t) \right), \\ |
| 344 |
% |
| 345 |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
| 346 |
{\bf j}(t + h / 2) \overleftrightarrow{\mathsf{I}}^{-1} |
| 347 |
\right) ,\\ |
| 348 |
% |
| 349 |
\chi\left(t + h / 2 \right) &\leftarrow \chi(t) + |
| 350 |
\frac{h}{2 \tau_T^2} \left( \frac{T(t)}{T_{\mathrm{target}}} - 1 |
| 351 |
\right) ,\\ |
| 352 |
% |
| 353 |
\eta(t + h / 2) &\leftarrow \eta(t) + \frac{h |
| 354 |
\mathcal{V}(t)}{2 N k_B T(t) \tau_B^2} \left( P(t) |
| 355 |
- P_{\mathrm{target}} \right), \\ |
| 356 |
% |
| 357 |
{\bf r}(t + h) &\leftarrow {\bf r}(t) + h |
| 358 |
\left\{ {\bf v}\left(t + h / 2 \right) |
| 359 |
+ \eta(t + h / 2)\left[ {\bf r}(t + h) |
| 360 |
- {\bf R}_0 \right] \right\} ,\\ |
| 361 |
% |
| 362 |
\mathsf{H}(t + h) &\leftarrow e^{-h \eta(t + h / 2)} |
| 363 |
\mathsf{H}(t). |
| 364 |
\end{align*} |
| 365 |
Most of these equations are identical to their counterparts in the |
| 366 |
NVT integrator, but the propagation of positions to time $t + h$ |
| 367 |
depends on the positions at the same time. The simulation box |
| 368 |
$\mathsf{H}$ is scaled uniformly for one full time step by an |
| 369 |
exponential factor that depends on the value of $\eta$ at time $t + |
| 370 |
h / 2$. Reshaping the box uniformly also scales the volume of the |
| 371 |
box by |
| 372 |
\begin{equation} |
| 373 |
\mathcal{V}(t + h) \leftarrow e^{ - 3 h \eta(t + h /2)}. |
| 374 |
\mathcal{V}(t) |
| 375 |
\end{equation} |
| 376 |
The {\tt doForces} step for the NPTi integrator is exactly the same |
| 377 |
as in both the DLM and NVT integrators. Once the forces and torques |
| 378 |
have been obtained at the new time step, the velocities can be |
| 379 |
advanced to the same time value. |
| 380 |
|
| 381 |
{\tt moveB:} |
| 382 |
\begin{align*} |
| 383 |
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
| 384 |
\left\{{\bf j}(t + h)\right\} ,\\ |
| 385 |
% |
| 386 |
P(t + h) &\leftarrow \left\{{\bf r}(t + h)\right\}, |
| 387 |
\left\{{\bf v}(t + h)\right\}, \\ |
| 388 |
% |
| 389 |
\chi\left(t + h \right) &\leftarrow \chi\left(t + h / |
| 390 |
2 \right) + \frac{h}{2 \tau_T^2} \left( \frac{T(t+h)} |
| 391 |
{T_{\mathrm{target}}} - 1 \right), \\ |
| 392 |
% |
| 393 |
\eta(t + h) &\leftarrow \eta(t + h / 2) + |
| 394 |
\frac{h \mathcal{V}(t + h)}{2 N k_B T(t + h) |
| 395 |
\tau_B^2} \left( P(t + h) - P_{\mathrm{target}} \right), \\ |
| 396 |
% |
| 397 |
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t |
| 398 |
+ h / 2 \right) + \frac{h}{2} \left( |
| 399 |
\frac{{\bf f}(t + h)}{m} - {\bf v}(t + h) |
| 400 |
(\chi(t + h) + \eta(t + h)) \right) ,\\ |
| 401 |
% |
| 402 |
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t |
| 403 |
+ h / 2 \right) + \frac{h}{2} \left( {\bf |
| 404 |
\tau}^b(t + h) - {\bf j}(t + h) |
| 405 |
\chi(t + h) \right) . |
| 406 |
\end{align*} |
| 407 |
Once again, since ${\bf v}(t + h)$ and ${\bf j}(t + h)$ are required |
| 408 |
to caclculate $T(t + h)$, $P(t + h)$, $\chi(t + h)$, and $\eta(t + |
| 409 |
h)$, they indirectly depend on their own values at time $t + h$. |
| 410 |
{\tt moveB} is therefore done in an iterative fashion until $\chi(t |
| 411 |
+ h)$ and $\eta(t + h)$ become self-consistent. |
| 412 |
|
| 413 |
The Melchionna modification of the Nos\'e-Hoover-Andersen algorithm |
| 414 |
is known to conserve a Hamiltonian for the extended system that is, |
| 415 |
to within a constant, identical to the Gibbs free energy, |
| 416 |
\begin{equation} |
| 417 |
H_{\mathrm{NPTi}} = V + K + f k_B T_{\mathrm{target}} \left( |
| 418 |
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
| 419 |
dt^\prime \right) + P_{\mathrm{target}} \mathcal{V}(t). |
| 420 |
\end{equation} |
| 421 |
It is also known that this algorithm samples the equilibrium |
| 422 |
distribution for the enthalpy (including contributions for the |
| 423 |
thermostat and barostat), |
| 424 |
\begin{equation} |
| 425 |
H_{\mathrm{NPTi}} = V + K + \frac{f k_B T_{\mathrm{target}}}{2} |
| 426 |
\left( \chi^2 \tau_T^2 + \eta^2 \tau_B^2 \right) + |
| 427 |
P_{\mathrm{target}} \mathcal{V}(t). |
| 428 |
\end{equation} |
| 429 |
|
| 430 |
\subsection{\label{methodSection:NPTf}Constant-pressure integration with a |
| 431 |
flexible box (NPTf)} |
| 432 |
|
| 433 |
There is a relatively simple generalization of the |
| 434 |
Nos\'e-Hoover-Andersen method to include changes in the simulation |
| 435 |
box {\it shape} as well as in the volume of the box. This method |
| 436 |
utilizes the full $3 \times 3$ pressure tensor and introduces a |
| 437 |
tensor of extended variables ($\overleftrightarrow{\eta}$) to |
| 438 |
control changes to the box shape. The equations of motion for this |
| 439 |
method are |
| 440 |
\begin{eqnarray} |
| 441 |
\dot{{\bf r}} & = & {\bf v} + \overleftrightarrow{\eta} \cdot \left( {\bf r} - {\bf R}_0 \right), \\ |
| 442 |
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - (\overleftrightarrow{\eta} + |
| 443 |
\chi \cdot \mathsf{1}) {\bf v}, \\ |
| 444 |
\dot{\mathsf{Q}} & = & \mathsf{Q} \cdot |
| 445 |
\mbox{ skew}\left(\overleftrightarrow{I}^{-1} \cdot {\bf j}\right) ,\\ |
| 446 |
\dot{{\bf j}} & = & {\bf j} \times \left( |
| 447 |
\overleftrightarrow{I}^{-1} \cdot {\bf j} \right) - \mbox{ |
| 448 |
rot}\left(\mathsf{Q}^{T} \cdot \frac{\partial |
| 449 |
V}{\partial \mathsf{Q}} \right) - \chi {\bf j} ,\\ |
| 450 |
\dot{\chi} & = & \frac{1}{\tau_{T}^2} \left( |
| 451 |
\frac{T}{T_{\mathrm{target}}} - 1 \right) ,\\ |
| 452 |
\dot{\overleftrightarrow{\eta}} & = & \frac{1}{\tau_{B}^2 f k_B |
| 453 |
T_{\mathrm{target}}} V \left( \overleftrightarrow{\mathsf{P}} - P_{\mathrm{target}}\mathsf{1} \right) ,\\ |
| 454 |
\dot{\mathsf{H}} & = & \overleftrightarrow{\eta} \cdot \mathsf{H} . |
| 455 |
\label{eq:melchionna2} |
| 456 |
\end{eqnarray} |
| 457 |
|
| 458 |
Here, $\mathsf{1}$ is the unit matrix and |
| 459 |
$\overleftrightarrow{\mathsf{P}}$ is the pressure tensor. Again, |
| 460 |
the volume, $\mathcal{V} = \det \mathsf{H}$. |
| 461 |
|
| 462 |
The propagation of the equations of motion is nearly identical to |
| 463 |
the NPTi integration: |
| 464 |
|
| 465 |
{\tt moveA:} |
| 466 |
\begin{align*} |
| 467 |
T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
| 468 |
% |
| 469 |
\overleftrightarrow{\mathsf{P}}(t) &\leftarrow \left\{{\bf |
| 470 |
r}(t)\right\}, |
| 471 |
\left\{{\bf v}(t)\right\} ,\\ |
| 472 |
% |
| 473 |
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
| 474 |
+ \frac{h}{2} \left( \frac{{\bf f}(t)}{m} - |
| 475 |
\left(\chi(t)\mathsf{1} + \overleftrightarrow{\eta}(t) \right) \cdot |
| 476 |
{\bf v}(t) \right), \\ |
| 477 |
% |
| 478 |
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
| 479 |
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 480 |
\chi(t) \right), \\ |
| 481 |
% |
| 482 |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
| 483 |
{\bf j}(t + h / 2) \overleftrightarrow{\mathsf{I}}^{-1} |
| 484 |
\right), \\ |
| 485 |
% |
| 486 |
\chi\left(t + h / 2 \right) &\leftarrow \chi(t) + |
| 487 |
\frac{h}{2 \tau_T^2} \left( \frac{T(t)}{T_{\mathrm{target}}} |
| 488 |
- 1 \right), \\ |
| 489 |
% |
| 490 |
\overleftrightarrow{\eta}(t + h / 2) &\leftarrow |
| 491 |
\overleftrightarrow{\eta}(t) + \frac{h \mathcal{V}(t)}{2 N k_B |
| 492 |
T(t) \tau_B^2} \left( \overleftrightarrow{\mathsf{P}}(t) |
| 493 |
- P_{\mathrm{target}}\mathsf{1} \right), \\ |
| 494 |
% |
| 495 |
{\bf r}(t + h) &\leftarrow {\bf r}(t) + h \left\{ {\bf v} |
| 496 |
\left(t + h / 2 \right) + \overleftrightarrow{\eta}(t + |
| 497 |
h / 2) \cdot \left[ {\bf r}(t + h) |
| 498 |
- {\bf R}_0 \right] \right\}, \\ |
| 499 |
% |
| 500 |
\mathsf{H}(t + h) &\leftarrow \mathsf{H}(t) \cdot e^{-h |
| 501 |
\overleftrightarrow{\eta}(t + h / 2)} . |
| 502 |
\end{align*} |
| 503 |
Here, a power series expansion truncated at second order for the |
| 504 |
exponential operation is used to scale the simulation box. The {\tt |
| 505 |
moveB} portion of the algorithm is largely unchanged from the NPTi |
| 506 |
integrator: |
| 507 |
|
| 508 |
{\tt moveB:} |
| 509 |
\begin{align*} |
| 510 |
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
| 511 |
\left\{{\bf j}(t + h)\right\}, \\ |
| 512 |
% |
| 513 |
\overleftrightarrow{\mathsf{P}}(t + h) &\leftarrow \left\{{\bf r} |
| 514 |
(t + h)\right\}, \left\{{\bf v}(t |
| 515 |
+ h)\right\}, \left\{{\bf f}(t + h)\right\} ,\\ |
| 516 |
% |
| 517 |
\chi\left(t + h \right) &\leftarrow \chi\left(t + h / |
| 518 |
2 \right) + \frac{h}{2 \tau_T^2} \left( \frac{T(t+ |
| 519 |
h)}{T_{\mathrm{target}}} - 1 \right), \\ |
| 520 |
% |
| 521 |
\overleftrightarrow{\eta}(t + h) &\leftarrow |
| 522 |
\overleftrightarrow{\eta}(t + h / 2) + |
| 523 |
\frac{h \mathcal{V}(t + h)}{2 N k_B T(t + h) |
| 524 |
\tau_B^2} \left( \overleftrightarrow{P}(t + h) |
| 525 |
- P_{\mathrm{target}}\mathsf{1} \right) ,\\ |
| 526 |
% |
| 527 |
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t |
| 528 |
+ h / 2 \right) + \frac{h}{2} \left( |
| 529 |
\frac{{\bf f}(t + h)}{m} - |
| 530 |
(\chi(t + h)\mathsf{1} + \overleftrightarrow{\eta}(t |
| 531 |
+ h)) \right) \cdot {\bf v}(t + h), \\ |
| 532 |
% |
| 533 |
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t |
| 534 |
+ h / 2 \right) + \frac{h}{2} \left( {\bf \tau}^b(t |
| 535 |
+ h) - {\bf j}(t + h) \chi(t + h) \right) . |
| 536 |
\end{align*} |
| 537 |
The iterative schemes for both {\tt moveA} and {\tt moveB} are |
| 538 |
identical to those described for the NPTi integrator. The NPTf |
| 539 |
integrator is known to conserve the following Hamiltonian: |
| 540 |
\begin{eqnarray*} |
| 541 |
H_{\mathrm{NPTf}} & = & V + K + f k_B T_{\mathrm{target}} \left( |
| 542 |
\frac{\tau_{T}^2 \chi^2(t)}{2} + \int_{0}^{t} \chi(t^\prime) |
| 543 |
dt^\prime \right) \\ |
| 544 |
& & + P_{\mathrm{target}} \mathcal{V}(t) + \frac{f k_B |
| 545 |
T_{\mathrm{target}}}{2} |
| 546 |
\mathrm{Tr}\left[\overleftrightarrow{\eta}(t)\right]^2 \tau_B^2. |
| 547 |
\end{eqnarray*} |
| 548 |
This integrator must be used with care, particularly in liquid |
| 549 |
simulations. Liquids have very small restoring forces in the |
| 550 |
off-diagonal directions, and the simulation box can very quickly |
| 551 |
form elongated and sheared geometries which become smaller than the |
| 552 |
electrostatic or Lennard-Jones cutoff radii. The NPTf integrator |
| 553 |
finds most use in simulating crystals or liquid crystals which |
| 554 |
assume non-orthorhombic geometries. |
| 555 |
|
| 556 |
\subsection{\label{methodSection:NPAT}NPAT Ensemble} |
| 557 |
|
| 558 |
A comprehensive understanding of relations between structures and |
| 559 |
functions in biological membrane system ultimately relies on |
| 560 |
structure and dynamics of lipid bilayers, which are strongly |
| 561 |
affected by the interfacial interaction between lipid molecules and |
| 562 |
surrounding media. One quantity used to describe the interfacial |
| 563 |
interaction is the average surface area per lipid. |
| 564 |
Constant area and constant lateral pressure simulations can be |
| 565 |
achieved by extending the standard NPT ensemble with a different |
| 566 |
pressure control strategy |
| 567 |
|
| 568 |
\begin{equation} |
| 569 |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
| 570 |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau_{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}} }} |
| 571 |
& \mbox{if $ \alpha = \beta = z$}\\ |
| 572 |
0 & \mbox{otherwise}\\ |
| 573 |
\end{array} |
| 574 |
\right. |
| 575 |
\end{equation} |
| 576 |
Note that the iterative schemes for NPAT are identical to those |
| 577 |
described for the NPTi integrator. |
| 578 |
|
| 579 |
\subsection{\label{methodSection:NPrT}NP$\gamma$T |
| 580 |
Ensemble} |
| 581 |
|
| 582 |
Theoretically, the surface tension $\gamma$ of a stress free |
| 583 |
membrane system should be zero since its surface free energy $G$ is |
| 584 |
minimum with respect to surface area $A$, |
| 585 |
\begin{equation} |
| 586 |
\gamma = \frac{{\partial G}}{{\partial A}}. |
| 587 |
\end{equation}0 |
| 588 |
However, a surface tension of zero is not |
| 589 |
appropriate for relatively small patches of membrane. In order to |
| 590 |
eliminate the edge effect of membrane simulations, a special |
| 591 |
ensemble, NP$\gamma$T, has been proposed to maintain the lateral |
| 592 |
surface tension and normal pressure. The equation of motion for the |
| 593 |
cell size control tensor, $\eta$, in $NP\gamma T$ is |
| 594 |
\begin{equation} |
| 595 |
\dot {\overleftrightarrow{\eta}} _{\alpha \beta}=\left\{\begin{array}{ll} |
| 596 |
- A_{xy} (\gamma _\alpha - \gamma _{{\rm{target}}} ) & \mbox{$\alpha = \beta = x$ or $\alpha = \beta = y$}\\ |
| 597 |
\frac{{V(P_{\alpha \beta } - P_{{\rm{target}}} )}}{{\tau _{\rm{B}}^{\rm{2}} fk_B T_{{\rm{target}}}}} & \mbox{$\alpha = \beta = z$} \\ |
| 598 |
0 & \mbox{$\alpha \ne \beta$} \\ |
| 599 |
\end{array} |
| 600 |
\right. |
| 601 |
\end{equation} |
| 602 |
where $ \gamma _{{\rm{target}}}$ is the external surface tension and |
| 603 |
the instantaneous surface tensor $\gamma _\alpha$ is given by |
| 604 |
\begin{equation} |
| 605 |
\gamma _\alpha = - h_z( \overleftrightarrow{P} _{\alpha \alpha } |
| 606 |
- P_{{\rm{target}}} ) |
| 607 |
\label{methodEquation:instantaneousSurfaceTensor} |
| 608 |
\end{equation} |
| 609 |
There is one additional extended system integrator (NPTxyz), in |
| 610 |
which each attempt to preserve the target pressure along the box |
| 611 |
walls perpendicular to that particular axis. The lengths of the box |
| 612 |
axes are allowed to fluctuate independently, but the angle between |
| 613 |
the box axes does not change. It should be noted that the NPTxyz |
| 614 |
integrator is a special case of $NP\gamma T$ if the surface tension |
| 615 |
$\gamma$ is set to zero, and if $x$ and $y$ can move independently. |
| 616 |
|
| 617 |
\section{\label{methodSection:zcons}The Z-Constraint Method} |
| 618 |
|
| 619 |
Based on the fluctuation-dissipation theorem, a force |
| 620 |
auto-correlation method was developed by Roux and Karplus to |
| 621 |
investigate the dynamics of ions inside ion channels\cite{Roux1991}. |
| 622 |
The time-dependent friction coefficient can be calculated from the |
| 623 |
deviation of the instantaneous force from its mean force. |
| 624 |
\begin{equation} |
| 625 |
\xi(z,t)=\langle\delta F(z,t)\delta F(z,0)\rangle/k_{B}T, |
| 626 |
\end{equation} |
| 627 |
where% |
| 628 |
\begin{equation} |
| 629 |
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle. |
| 630 |
\end{equation} |
| 631 |
If the time-dependent friction decays rapidly, the static friction |
| 632 |
coefficient can be approximated by |
| 633 |
\begin{equation} |
| 634 |
\xi_{\text{static}}(z)=\int_{0}^{\infty}\langle\delta F(z,t)\delta |
| 635 |
F(z,0)\rangle dt. |
| 636 |
\end{equation} |
| 637 |
Allowing diffusion constant to then be calculated through the |
| 638 |
Einstein relation:\cite{Marrink1994} |
| 639 |
\begin{equation} |
| 640 |
D(z)=\frac{k_{B}T}{\xi_{\text{static}}(z)}=\frac{(k_{B}T)^{2}}{\int_{0}^{\infty |
| 641 |
}\langle\delta F(z,t)\delta F(z,0)\rangle dt}.% |
| 642 |
\end{equation} |
| 643 |
The Z-Constraint method, which fixes the z coordinates of the |
| 644 |
molecules with respect to the center of the mass of the system, has |
| 645 |
been a method suggested to obtain the forces required for the force |
| 646 |
auto-correlation calculation.\cite{Marrink1994} However, simply |
| 647 |
resetting the coordinate will move the center of the mass of the |
| 648 |
whole system. To avoid this problem, we reset the forces of |
| 649 |
z-constrained molecules as well as subtract the total constraint |
| 650 |
forces from the rest of the system after the force calculation at |
| 651 |
each time step instead of resetting the coordinate. After the force |
| 652 |
calculation, we define $G_\alpha$ as |
| 653 |
\begin{equation} |
| 654 |
G_{\alpha} = \sum_i F_{\alpha i}, \label{oopseEq:zc1} |
| 655 |
\end{equation} |
| 656 |
where $F_{\alpha i}$ is the force in the z direction of atom $i$ in |
| 657 |
z-constrained molecule $\alpha$. The forces of the z constrained |
| 658 |
molecule are then set to: |
| 659 |
\begin{equation} |
| 660 |
F_{\alpha i} = F_{\alpha i} - |
| 661 |
\frac{m_{\alpha i} G_{\alpha}}{\sum_i m_{\alpha i}}. |
| 662 |
\end{equation} |
| 663 |
Here, $m_{\alpha i}$ is the mass of atom $i$ in the z-constrained |
| 664 |
molecule. Having rescaled the forces, the velocities must also be |
| 665 |
rescaled to subtract out any center of mass velocity in the z |
| 666 |
direction. |
| 667 |
\begin{equation} |
| 668 |
v_{\alpha i} = v_{\alpha i} - |
| 669 |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}, |
| 670 |
\end{equation} |
| 671 |
where $v_{\alpha i}$ is the velocity of atom $i$ in the z direction. |
| 672 |
Lastly, all of the accumulated z constrained forces must be |
| 673 |
subtracted from the system to keep the system center of mass from |
| 674 |
drifting. |
| 675 |
\begin{equation} |
| 676 |
F_{\beta i} = F_{\beta i} - \frac{m_{\beta i} \sum_{\alpha} |
| 677 |
G_{\alpha}} |
| 678 |
{\sum_{\beta}\sum_i m_{\beta i}}, |
| 679 |
\end{equation} |
| 680 |
where $\beta$ are all of the unconstrained molecules in the system. |
| 681 |
Similarly, the velocities of the unconstrained molecules must also |
| 682 |
be scaled. |
| 683 |
\begin{equation} |
| 684 |
v_{\beta i} = v_{\beta i} + \sum_{\alpha} |
| 685 |
\frac{\sum_i m_{\alpha i} v_{\alpha i}}{\sum_i m_{\alpha i}}. |
| 686 |
\end{equation} |
| 687 |
At the very beginning of the simulation, the molecules may not be at |
| 688 |
their constrained positions. To move a z-constrained molecule to its |
| 689 |
specified position, a simple harmonic potential is used |
| 690 |
\begin{equation} |
| 691 |
U(t)=\frac{1}{2}k_{\text{Harmonic}}(z(t)-z_{\text{cons}})^{2},% |
| 692 |
\end{equation} |
| 693 |
where $k_{\text{Harmonic}}$ is the harmonic force constant, $z(t)$ |
| 694 |
is the current $z$ coordinate of the center of mass of the |
| 695 |
constrained molecule, and $z_{\text{cons}}$ is the constrained |
| 696 |
position. The harmonic force operating on the z-constrained molecule |
| 697 |
at time $t$ can be calculated by |
| 698 |
\begin{equation} |
| 699 |
F_{z_{\text{Harmonic}}}(t)=-\frac{\partial U(t)}{\partial z(t)}= |
| 700 |
-k_{\text{Harmonic}}(z(t)-z_{\text{cons}}). |
| 701 |
\end{equation} |
