| 2 |
|
|
| 3 |
|
\section{\label{methodSection:rigidBodyIntegrators}Integrators for Rigid Body Motion in Molecular Dynamics} |
| 4 |
|
|
| 5 |
< |
In order to mimic the experiments, which are usually performed under |
| 5 |
> |
In order to mimic experiments which are usually performed under |
| 6 |
|
constant temperature and/or pressure, extended Hamiltonian system |
| 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 |
| 11 |
< |
normal directions of membranes. The $NPAT$ ensemble, in which the |
| 12 |
< |
normal pressure and the lateral surface area of the membrane are |
| 13 |
< |
kept constant, and the $NP\gamma T$ ensemble, in which the normal |
| 14 |
< |
pressure and the lateral surface tension are kept constant were |
| 15 |
< |
proposed to address this issue. |
| 8 |
> |
as the canonical and isobaric-isothermal ensembles. In addition to |
| 9 |
> |
the standard ensemble, specific ensembles have been developed to |
| 10 |
> |
account for the anisotropy between the lateral and normal directions |
| 11 |
> |
of membranes. The $NPAT$ ensemble, in which the normal pressure and |
| 12 |
> |
the lateral surface area of the membrane are kept constant, and the |
| 13 |
> |
$NP\gamma T$ ensemble, in which the normal pressure and the lateral |
| 14 |
> |
surface tension are kept constant were proposed to address the |
| 15 |
> |
issues. |
| 16 |
|
|
| 17 |
< |
Integration schemes for rotational motion of the rigid molecules in |
| 18 |
< |
microcanonical ensemble have been extensively studied in the last |
| 19 |
< |
two decades. Matubayasi developed a time-reversible integrator for |
| 20 |
< |
rigid bodies in quaternion representation. Although it is not |
| 21 |
< |
symplectic, this integrator still demonstrates a better long-time |
| 22 |
< |
energy conservation than traditional methods because of the |
| 23 |
< |
time-reversible nature. Extending Trotter-Suzuki to general system |
| 24 |
< |
with a flat phase space, Miller and his colleagues devised an novel |
| 25 |
< |
symplectic, time-reversible and volume-preserving integrator in |
| 17 |
> |
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 |
| 20 |
> |
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 |
| 23 |
> |
methods because of the time-reversible nature. Extending the |
| 24 |
> |
Trotter-Suzuki factorization to general system with a flat phase |
| 25 |
> |
space, Miller\cite{Miller2002} and his colleagues devised a novel |
| 26 |
> |
symplectic, time-reversible and volume-preserving integrator in the |
| 27 |
|
quaternion representation, which was shown to be superior to the |
| 28 |
|
Matubayasi's time-reversible integrator. However, all of the |
| 29 |
< |
integrators in quaternion representation suffer from the |
| 29 |
> |
integrators in the quaternion representation suffer from the |
| 30 |
|
computational penalty of constructing a rotation matrix from |
| 31 |
|
quaternions to evolve coordinates and velocities at every time step. |
| 32 |
< |
An alternative integration scheme utilizing rotation matrix directly |
| 33 |
< |
proposed by Dullweber, Leimkuhler and McLachlan (DLM) also preserved |
| 34 |
< |
the same structural properties of the Hamiltonian flow. In this |
| 35 |
< |
section, the integration scheme of DLM method will be reviewed and |
| 36 |
< |
extended to other ensembles. |
| 32 |
> |
An alternative integration scheme utilizing the rotation matrix |
| 33 |
> |
directly proposed by Dullweber, Leimkuhler and McLachlan (DLM) also |
| 34 |
> |
preserved the same structural properties of the Hamiltonian |
| 35 |
> |
propagator.\cite{Dullweber1997} In this section, the integration |
| 36 |
> |
scheme of DLM method will be reviewed and extended to other |
| 37 |
> |
ensembles. |
| 38 |
|
|
| 39 |
< |
\subsection{\label{methodSection:DLM}Integrating the Equations of Motion: the |
| 40 |
< |
DLM method} |
| 39 |
> |
\subsection{\label{methodSection:DLM}Integrating the Equations of Motion: The |
| 40 |
> |
DLM Method} |
| 41 |
|
|
| 42 |
|
The DLM method uses a Trotter factorization of the orientational |
| 43 |
|
propagator. This has three effects: |
| 46 |
|
{\it symplectic}), |
| 47 |
|
\item the integrator is time-{\it reversible}, making it suitable for Hybrid |
| 48 |
|
Monte Carlo applications, and |
| 49 |
< |
\item the error for a single time step is of order $\mathcal{O}\left(h^4\right)$ |
| 49 |
> |
\item the error for a single time step is of order $\mathcal{O}\left(h^3\right)$ |
| 50 |
|
for timesteps of length $h$. |
| 51 |
|
\end{enumerate} |
| 50 |
– |
|
| 52 |
|
The integration of the equations of motion is carried out in a |
| 53 |
|
velocity-Verlet style 2-part algorithm, where $h= \delta t$: |
| 54 |
|
|
| 63 |
|
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
| 64 |
|
+ \frac{h}{2} {\bf \tau}^b(t), \\ |
| 65 |
|
% |
| 66 |
< |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
| 66 |
> |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
| 67 |
|
(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
| 68 |
|
\end{align*} |
| 68 |
– |
|
| 69 |
|
In this context, the $\mathrm{rotate}$ function is the reversible |
| 70 |
|
product of the three body-fixed rotations, |
| 71 |
|
\begin{equation} |
| 74 |
|
/ 2) \cdot \mathsf{G}_x(a_x /2), |
| 75 |
|
\end{equation} |
| 76 |
|
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
| 77 |
< |
rotates both the rotation matrix ($\mathsf{A}$) and the body-fixed |
| 78 |
< |
angular momentum (${\bf j}$) by an angle $\theta$ around body-fixed |
| 77 |
> |
rotates both the rotation matrix $\mathsf{Q}$ and the body-fixed |
| 78 |
> |
angular momentum ${\bf j}$ by an angle $\theta$ around body-fixed |
| 79 |
|
axis $\alpha$, |
| 80 |
|
\begin{equation} |
| 81 |
|
\mathsf{G}_\alpha( \theta ) = \left\{ |
| 82 |
|
\begin{array}{lcl} |
| 83 |
< |
\mathsf{A}(t) & \leftarrow & \mathsf{A}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
| 83 |
> |
\mathsf{Q}(t) & \leftarrow & \mathsf{Q}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
| 84 |
|
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
| 85 |
|
j}(0). |
| 86 |
|
\end{array} |
| 100 |
|
\end{array} |
| 101 |
|
\right). |
| 102 |
|
\end{equation} |
| 103 |
< |
All other rotations follow in a straightforward manner. |
| 103 |
> |
All other rotations follow in a straightforward manner. After the |
| 104 |
> |
first part of the propagation, the forces and body-fixed torques are |
| 105 |
> |
calculated at the new positions and orientations |
| 106 |
|
|
| 105 |
– |
After the first part of the propagation, the forces and body-fixed |
| 106 |
– |
torques are calculated at the new positions and orientations |
| 107 |
– |
|
| 107 |
|
{\tt doForces:} |
| 108 |
|
\begin{align*} |
| 109 |
|
{\bf f}(t + h) &\leftarrow |
| 110 |
|
- \left(\frac{\partial V}{\partial {\bf r}}\right)_{{\bf r}(t + h)}, \\ |
| 111 |
|
% |
| 112 |
|
{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h) |
| 113 |
< |
\times \frac{\partial V}{\partial {\bf u}}, \\ |
| 113 |
> |
\times (\frac{\partial V}{\partial {\bf u}})_{u(t+h)}, \\ |
| 114 |
|
% |
| 115 |
< |
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) |
| 115 |
> |
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{Q}(t + h) |
| 116 |
|
\cdot {\bf \tau}^s(t + h). |
| 117 |
|
\end{align*} |
| 118 |
< |
|
| 119 |
< |
${\bf u}$ will be automatically updated when the rotation matrix |
| 121 |
< |
$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
| 118 |
> |
${\bf u}$ is automatically updated when the rotation matrix |
| 119 |
> |
$\mathsf{Q}$ is calculated in {\tt moveA}. Once the forces and |
| 120 |
|
torques have been obtained at the new time step, the velocities can |
| 121 |
|
be advanced to the same time value. |
| 122 |
|
|
| 130 |
|
\right) |
| 131 |
|
+ \frac{h}{2} {\bf \tau}^b(t + h) . |
| 132 |
|
\end{align*} |
| 135 |
– |
|
| 133 |
|
The matrix rotations used in the DLM method end up being more costly |
| 134 |
|
computationally than the simpler arithmetic quaternion propagation. |
| 135 |
|
With the same time step, a 1000-molecule water simulation shows an |
| 136 |
|
average 7\% increase in computation time using the DLM method in |
| 137 |
|
place of quaternions. This cost is more than justified when |
| 138 |
|
comparing the energy conservation of the two methods as illustrated |
| 139 |
< |
in Fig.~\ref{methodFig:timestep}. |
| 139 |
> |
in Fig.~\ref{methodFig:timestep} where the resulting energy drifts at |
| 140 |
> |
various time steps for both the DLM and quaternion integration |
| 141 |
> |
schemes are compared. All of the 1000 molecule water simulations |
| 142 |
> |
started with the same configuration, and the only difference was the |
| 143 |
> |
method for handling rotational motion. At time steps of 0.1 and 0.5 |
| 144 |
> |
fs, both methods for propagating molecule rotation conserve energy |
| 145 |
> |
fairly well, with the quaternion method showing a slight energy |
| 146 |
> |
drift over time in the 0.5 fs time step simulation. At time steps of |
| 147 |
> |
1 and 2 fs, the energy conservation benefits of the DLM method are |
| 148 |
> |
clearly demonstrated. Thus, while maintaining the same degree of |
| 149 |
> |
energy conservation, one can take considerably longer time steps, |
| 150 |
> |
leading to an overall reduction in computation time. |
| 151 |
|
|
| 152 |
|
\begin{figure} |
| 153 |
|
\centering |
| 162 |
|
\label{methodFig:timestep} |
| 163 |
|
\end{figure} |
| 164 |
|
|
| 157 |
– |
In Fig.~\ref{methodFig:timestep}, the resulting energy drift at |
| 158 |
– |
various time steps for both the DLM and quaternion integration |
| 159 |
– |
schemes is compared. All of the 1000 molecule water simulations |
| 160 |
– |
started with the same configuration, and the only difference was the |
| 161 |
– |
method for handling rotational motion. At time steps of 0.1 and 0.5 |
| 162 |
– |
fs, both methods for propagating molecule rotation conserve energy |
| 163 |
– |
fairly well, with the quaternion method showing a slight energy |
| 164 |
– |
drift over time in the 0.5 fs time step simulation. At time steps of |
| 165 |
– |
1 and 2 fs, the energy conservation benefits of the DLM method are |
| 166 |
– |
clearly demonstrated. Thus, while maintaining the same degree of |
| 167 |
– |
energy conservation, one can take considerably longer time steps, |
| 168 |
– |
leading to an overall reduction in computation time. |
| 169 |
– |
|
| 165 |
|
\subsection{\label{methodSection:NVT}Nos\'{e}-Hoover Thermostatting} |
| 166 |
|
|
| 167 |
|
The Nos\'e-Hoover equations of motion are given by\cite{Hoover1985} |
| 168 |
|
\begin{eqnarray} |
| 169 |
|
\dot{{\bf r}} & = & {\bf v}, \\ |
| 170 |
|
\dot{{\bf v}} & = & \frac{{\bf f}}{m} - \chi {\bf v} ,\\ |
| 171 |
< |
\dot{\mathsf{A}} & = & \mathsf{A} \cdot |
| 171 |
> |
\dot{\mathsf{Q}} & = & \mathsf{Q} \cdot |
| 172 |
|
\mbox{ skew}\left(\overleftrightarrow{\mathsf{I}}^{-1} \cdot {\bf j}\right), \\ |
| 173 |
|
\dot{{\bf j}} & = & {\bf j} \times \left( |
| 174 |
|
\overleftrightarrow{\mathsf{I}}^{-1} \cdot {\bf j} \right) - \mbox{ |
| 175 |
< |
rot}\left(\mathsf{A}^{T} \cdot \frac{\partial V}{\partial |
| 176 |
< |
\mathsf{A}} \right) - \chi {\bf j}. \label{eq:nosehoovereom} |
| 175 |
> |
rot}\left(\mathsf{Q}^{T} \cdot \frac{\partial V}{\partial |
| 176 |
> |
\mathsf{Q}} \right) - \chi {\bf j}. \label{eq:nosehoovereom} |
| 177 |
|
\end{eqnarray} |
| 183 |
– |
|
| 178 |
|
$\chi$ is an ``extra'' variable included in the extended system, and |
| 179 |
|
it is propagated using the first order equation of motion |
| 180 |
|
\begin{equation} |
| 181 |
|
\dot{\chi} = \frac{1}{\tau_{T}^2} \left( |
| 182 |
|
\frac{T}{T_{\mathrm{target}}} - 1 \right). \label{eq:nosehooverext} |
| 183 |
|
\end{equation} |
| 184 |
< |
|
| 185 |
< |
The instantaneous temperature $T$ is proportional to the total |
| 186 |
< |
kinetic energy (both translational and orientational) and is given |
| 193 |
< |
by |
| 184 |
> |
where $\tau_T$ is the time constant for relaxation of the |
| 185 |
> |
temperature to the target value, and the instantaneous temperature |
| 186 |
> |
$T$ is given by |
| 187 |
|
\begin{equation} |
| 188 |
< |
T = \frac{2 K}{f k_B} |
| 188 |
> |
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} |
| 192 |
|
f = 3 N + 3 N_{\mathrm{orient}} - N_{\mathrm{constraints}}, |
| 193 |
|
\end{equation} |
| 194 |
< |
and $K$ is the total kinetic energy, |
| 195 |
< |
\begin{equation} |
| 196 |
< |
K = \sum_{i=1}^{N} \frac{1}{2} m_i {\bf v}_i^T \cdot {\bf v}_i + |
| 204 |
< |
\sum_{i=1}^{N_{\mathrm{orient}}} \frac{1}{2} {\bf j}_i^T \cdot |
| 205 |
< |
\overleftrightarrow{\mathsf{I}}_i^{-1} \cdot {\bf j}_i. |
| 206 |
< |
\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 |
|
|
| 208 |
– |
In eq.(\ref{eq:nosehooverext}), $\tau_T$ is the time constant for |
| 209 |
– |
relaxation of the temperature to the target value. To set values |
| 210 |
– |
for $\tau_T$ or $T_{\mathrm{target}}$ in a simulation, one would use |
| 211 |
– |
the {\tt tauThermostat} and {\tt targetTemperature} keywords in the |
| 212 |
– |
{\tt .bass} file. The units for {\tt tauThermostat} are fs, and the |
| 213 |
– |
units for the {\tt targetTemperature} are degrees K. The |
| 214 |
– |
integration of the equations of motion is carried out in a |
| 215 |
– |
velocity-Verlet style 2 part algorithm: |
| 216 |
– |
|
| 198 |
|
{\tt moveA:} |
| 199 |
|
\begin{align*} |
| 200 |
|
T(t) &\leftarrow \left\{{\bf v}(t)\right\}, \left\{{\bf j}(t)\right\} ,\\ |
| 210 |
|
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 211 |
|
\chi(t) \right) ,\\ |
| 212 |
|
% |
| 213 |
< |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate} |
| 214 |
< |
\left(h * {\bf j}(t + h / 2) |
| 213 |
> |
\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*} |
| 240 |
– |
|
| 221 |
|
Here $\mathrm{rotate}(h * {\bf j} |
| 222 |
< |
\overleftrightarrow{\mathsf{I}}^{-1})$ is the same symplectic |
| 223 |
< |
Trotter factorization of the three rotation operations that was |
| 224 |
< |
discussed in the section on the DLM integrator. Note that this |
| 225 |
< |
operation modifies both the rotation matrix $\mathsf{A}$ and the |
| 226 |
< |
angular momentum ${\bf j}$. {\tt moveA} propagates velocities by a |
| 227 |
< |
half time step, and positional degrees of freedom by a full time |
| 228 |
< |
step. The new positions (and orientations) are then used to |
| 229 |
< |
calculate a new set of forces and torques in exactly the same way |
| 230 |
< |
they are calculated in the {\tt doForces} portion of the DLM |
| 231 |
< |
integrator. |
| 232 |
< |
|
| 253 |
< |
Once the forces and torques have been obtained at the new time step, |
| 254 |
< |
the temperature, velocities, and the extended system variable can be |
| 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:} |
| 251 |
|
\left( {\bf \tau}^b(t + h) - {\bf j}(t + h) |
| 252 |
|
\chi(t + h) \right) . |
| 253 |
|
\end{align*} |
| 276 |
– |
|
| 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. |
| 259 |
< |
|
| 260 |
< |
The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for |
| 284 |
< |
the extended system that is, to within a constant, identical to the |
| 285 |
< |
Helmholtz free energy,\cite{Melchionna1993} |
| 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 is maintained in the |
| 268 |
< |
last column of the {\tt .stat} file to allow checks on the quality |
| 294 |
< |
of the integration. |
| 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 |
| 271 |
< |
isotropic box deformations (NPTi)} |
| 271 |
> |
isotropic box (NPTi)} |
| 272 |
|
|
| 273 |
< |
Isobaric-isothermal ensemble integrator is implemented using the |
| 274 |
< |
Melchionna modifications to the Nos\'e-Hoover-Andersen equations of |
| 275 |
< |
motion,\cite{Melchionna1993} |
| 302 |
< |
|
| 273 |
> |
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} |
| 276 |
|
\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{A}} & = & \mathsf{A} \cdot |
| 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{A}^{T} \cdot \frac{\partial |
| 284 |
< |
V}{\partial \mathsf{A}} \right) - \chi {\bf j}, \\ |
| 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 |
| 289 |
|
P_{\mathrm{target}} \right), \\ |
| 290 |
|
\dot{\mathcal{V}} & = & 3 \mathcal{V} \eta . \label{eq:melchionna1} |
| 291 |
|
\end{eqnarray} |
| 319 |
– |
|
| 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 |
| 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, |
| 321 |
> |
the pressure tensor, |
| 322 |
|
\begin{equation} |
| 323 |
|
P(t) = \frac{1}{3} \mathrm{Tr} \left( |
| 324 |
< |
\overleftrightarrow{\mathsf{P}}(t). \right) |
| 324 |
> |
\overleftrightarrow{\mathsf{P}}(t) \right) . |
| 325 |
|
\end{equation} |
| 326 |
< |
|
| 355 |
< |
In eq.(\ref{eq:melchionna1}), $\tau_B$ is the time constant for |
| 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: |
| 342 |
|
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 343 |
|
\chi(t) \right), \\ |
| 344 |
|
% |
| 345 |
< |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
| 345 |
> |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
| 346 |
|
{\bf j}(t + h / 2) \overleftrightarrow{\mathsf{I}}^{-1} |
| 347 |
|
\right) ,\\ |
| 348 |
|
% |
| 362 |
|
\mathsf{H}(t + h) &\leftarrow e^{-h \eta(t + h / 2)} |
| 363 |
|
\mathsf{H}(t). |
| 364 |
|
\end{align*} |
| 394 |
– |
|
| 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 |
| 373 |
|
\mathcal{V}(t + h) \leftarrow e^{ - 3 h \eta(t + h /2)}. |
| 374 |
|
\mathcal{V}(t) |
| 375 |
|
\end{equation} |
| 406 |
– |
|
| 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 |
| 404 |
|
\tau}^b(t + h) - {\bf j}(t + h) |
| 405 |
|
\chi(t + h) \right) . |
| 406 |
|
\end{align*} |
| 438 |
– |
|
| 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 $t + h$. |
| 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 |
< |
Poor choices of $\delta t$, $\tau_T$, or $\tau_B$ can result in |
| 422 |
< |
non-conservation of $H_{\mathrm{NPTi}}$, so the conserved quantity |
| 423 |
< |
is maintained in the last column of the {\tt .stat} file to allow |
| 456 |
< |
checks on the quality of the integration. It is also known that |
| 457 |
< |
this algorithm samples the equilibrium distribution for the enthalpy |
| 458 |
< |
(including contributions for the thermostat and barostat), |
| 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) + |
| 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{A}} & = & \mathsf{A} \cdot |
| 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{A}^{T} \cdot \frac{\partial |
| 449 |
< |
V}{\partial \mathsf{A}} \right) - \chi {\bf j} ,\\ |
| 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 |
| 479 |
|
+ \frac{h}{2} \left( {\bf \tau}^b(t) - {\bf j}(t) |
| 480 |
|
\chi(t) \right), \\ |
| 481 |
|
% |
| 482 |
< |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
| 482 |
> |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left(h * |
| 483 |
|
{\bf j}(t + h / 2) \overleftrightarrow{\mathsf{I}}^{-1} |
| 484 |
|
\right), \\ |
| 485 |
|
% |
| 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. |
| 505 |
< |
|
| 506 |
< |
The {\tt moveB} portion of the algorithm is largely unchanged from |
| 542 |
< |
the NPTi integrator: |
| 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*} |
| 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*} |
| 573 |
– |
|
| 537 |
|
The iterative schemes for both {\tt moveA} and {\tt moveB} are |
| 538 |
< |
identical to those described for the NPTi integrator. |
| 539 |
< |
|
| 577 |
< |
The NPTf integrator is known to conserve the following Hamiltonian: |
| 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 |
| 545 |
< |
k_B T_{\mathrm{target}}}{2} |
| 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*} |
| 586 |
– |
|
| 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 |
| 553 |
|
finds most use in simulating crystals or liquid crystals which |
| 554 |
|
assume non-orthorhombic geometries. |
| 555 |
|
|
| 556 |
< |
\subsubsection{\label{methodSection:NPAT}NPAT Ensemble} |
| 556 |
> |
\subsection{\label{methodSection:NPAT}NPAT Ensemble} |
| 557 |
|
|
| 558 |
< |
A comprehensive understanding of structure¨Cfunction relations of |
| 559 |
< |
biological membrane system ultimately relies on structure and |
| 560 |
< |
dynamics of lipid bilayer, which are strongly affected by the |
| 561 |
< |
interfacial interaction between lipid molecules and surrounding |
| 562 |
< |
media. One quantity to describe the interfacial interaction is so |
| 563 |
< |
called the average surface area per lipid. Constat area and constant |
| 564 |
< |
lateral pressure simulation can be achieved by extending the |
| 565 |
< |
standard NPT ensemble with a different pressure control strategy |
| 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} |
| 573 |
|
\end{array} |
| 574 |
|
\right. |
| 575 |
|
\end{equation} |
| 614 |
– |
|
| 576 |
|
Note that the iterative schemes for NPAT are identical to those |
| 577 |
|
described for the NPTi integrator. |
| 578 |
|
|
| 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 $A$ |
| 624 |
< |
\[ |
| 625 |
< |
\gamma = \frac{{\partial G}}{{\partial A}}. |
| 626 |
< |
\] |
| 627 |
< |
However, a surface tension of zero is not appropriate for relatively |
| 628 |
< |
small patches of membrane. In order to eliminate the edge effect of |
| 629 |
< |
the membrane simulation, a special ensemble, NP$\gamma$T, is |
| 630 |
< |
proposed to maintain the lateral surface tension and normal |
| 631 |
< |
pressure. The equation of motion for cell size control tensor, |
| 632 |
< |
$\eta$, in $NP\gamma T$ is |
| 584 |
> |
minimum with respect to surface area $A$, |
| 585 |
|
\begin{equation} |
| 586 |
+ |
\gamma = \frac{{\partial G}}{{\partial A}}=0. |
| 587 |
+ |
\end{equation} |
| 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{$\alpha = \beta = z$} \\ |
| 606 |
|
- P_{{\rm{target}}} ) |
| 607 |
|
\label{methodEquation:instantaneousSurfaceTensor} |
| 608 |
|
\end{equation} |
| 648 |
– |
|
| 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. |
| 615 |
> |
$\gamma$ is set to zero, and if $x$ and $y$ can move independently. |
| 616 |
|
|
| 617 |
< |
\section{\label{methodSection:zcons}Z-Constraint Method} |
| 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}. |
| 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. |
| 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} |
| 628 |
|
\begin{equation} |
| 629 |
|
\delta F(z,t)=F(z,t)-\langle F(z,t)\rangle. |
| 630 |
|
\end{equation} |
| 671 |
– |
|
| 631 |
|
If the time-dependent friction decays rapidly, the static friction |
| 632 |
|
coefficient can be approximated by |
| 633 |
|
\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} |
| 684 |
– |
|
| 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 |
| 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. |
| 652 |
< |
|
| 695 |
< |
After the force calculation, define $G_\alpha$ as |
| 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} |
| 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} |
| 730 |
– |
|
| 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 |
