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 and Nakahara developed a time-reversible |
20 |
< |
integrator for rigid bodies in quaternion representation. Although |
21 |
< |
it is not symplectic, this integrator still demonstrates a better |
22 |
< |
long-time energy conservation than traditional methods because of |
23 |
< |
the time-reversible nature. Extending Trotter-Suzuki to general |
24 |
< |
system with a flat phase space, Miller and his colleagues devised an |
25 |
< |
novel symplectic, time-reversible and volume-preserving integrator |
26 |
< |
in quaternion representation, which was shown to be superior to the |
27 |
< |
time-reversible integrator of Matubayasi and Nakahara. However, all |
28 |
< |
of the integrators in quaternion representation suffer from the |
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 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 |
< |
{\sc oopse} automatically updates ${\bf u}$ 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{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 |
158 |
|
increasing time step. For each time step, the dotted line is total |
159 |
|
energy using the DLM integrator, and the solid line comes from the |
160 |
|
quaternion integrator. The larger time step plots are shifted up |
161 |
< |
from the true energy baseline for clarity.} \label{timestep} |
161 |
> |
from the true energy baseline for clarity.} |
162 |
> |
\label{methodFig:timestep} |
163 |
|
\end{figure} |
164 |
|
|
156 |
– |
In Fig.~\ref{timestep}, the resulting energy drift at various time |
157 |
– |
steps for both the DLM and quaternion integration schemes is |
158 |
– |
compared. All of the 1000 molecule water simulations started with |
159 |
– |
the same configuration, and the only difference was the method for |
160 |
– |
handling rotational motion. At time steps of 0.1 and 0.5 fs, both |
161 |
– |
methods for propagating molecule rotation conserve energy fairly |
162 |
– |
well, with the quaternion method showing a slight energy drift over |
163 |
– |
time in the 0.5 fs time step simulation. At time steps of 1 and 2 |
164 |
– |
fs, the energy conservation benefits of the DLM method are clearly |
165 |
– |
demonstrated. Thus, while maintaining the same degree of energy |
166 |
– |
conservation, one can take considerably longer time steps, leading |
167 |
– |
to an overall reduction in computation time. |
168 |
– |
|
165 |
|
\subsection{\label{methodSection:NVT}Nos\'{e}-Hoover Thermostatting} |
166 |
|
|
167 |
< |
The Nos\'e-Hoover equations of motion are given by\cite{Hoover85} |
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} |
182 |
– |
|
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 |
192 |
< |
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 + |
203 |
< |
\sum_{i=1}^{N_{\mathrm{orient}}} \frac{1}{2} {\bf j}_i^T \cdot |
204 |
< |
\overleftrightarrow{\mathsf{I}}_i^{-1} \cdot {\bf j}_i. |
205 |
< |
\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 |
|
|
207 |
– |
In eq.(\ref{eq:nosehooverext}), $\tau_T$ is the time constant for |
208 |
– |
relaxation of the temperature to the target value. To set values |
209 |
– |
for $\tau_T$ or $T_{\mathrm{target}}$ in a simulation, one would use |
210 |
– |
the {\tt tauThermostat} and {\tt targetTemperature} keywords in the |
211 |
– |
{\tt .bass} file. The units for {\tt tauThermostat} are fs, and the |
212 |
– |
units for the {\tt targetTemperature} are degrees K. The |
213 |
– |
integration of the equations of motion is carried out in a |
214 |
– |
velocity-Verlet style 2 part algorithm: |
215 |
– |
|
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*} |
239 |
– |
|
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 |
< |
|
252 |
< |
Once the forces and torques have been obtained at the new time step, |
253 |
< |
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*} |
275 |
– |
|
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 relative tolerance for the self-consistency |
259 |
< |
check defaults to a value of $\mbox{10}^{-6}$, but {\sc oopse} will |
260 |
< |
terminate the iteration after 4 loops even if the consistency check |
283 |
< |
has not been satisfied. |
284 |
< |
|
285 |
< |
The Nos\'e-Hoover algorithm is known to conserve a Hamiltonian for |
286 |
< |
the extended system that is, to within a constant, identical to the |
287 |
< |
Helmholtz free energy,\cite{melchionna93} |
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 |
296 |
< |
of the integration. |
267 |
> |
$H_{\mathrm{NVT}}$, so the conserved quantity should be checked |
268 |
> |
periodically to verify the quality of the integration. |
269 |
|
|
298 |
– |
Bond constraints are applied at the end of both the {\tt moveA} and |
299 |
– |
{\tt moveB} portions of the algorithm. Details on the constraint |
300 |
– |
algorithms are given in section \ref{oopseSec:rattle}. |
301 |
– |
|
270 |
|
\subsection{\label{methodSection:NPTi}Constant-pressure integration with |
271 |
< |
isotropic box deformations (NPTi)} |
271 |
> |
isotropic box (NPTi)} |
272 |
|
|
273 |
< |
To carry out isobaric-isothermal ensemble calculations {\sc oopse} |
274 |
< |
implements the Melchionna modifications to the |
275 |
< |
Nos\'e-Hoover-Andersen equations of motion,\cite{melchionna93} |
308 |
< |
|
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} |
325 |
– |
|
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 |
< |
|
327 |
< |
In eq.(\ref{eq:melchionna1}), $\tau_B$ is the time constant for |
362 |
< |
relaxation of the pressure to the target value. To set values for |
363 |
< |
$\tau_B$ or $P_{\mathrm{target}}$ in a simulation, one would use the |
364 |
< |
{\tt tauBarostat} and {\tt targetPressure} keywords in the {\tt |
365 |
< |
.bass} file. The units for {\tt tauBarostat} are fs, and the units |
366 |
< |
for the {\tt targetPressure} are atmospheres. Like in the NVT |
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 |
|
|
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*} |
404 |
– |
|
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. {\sc oopse} carries out |
368 |
< |
this step iteratively (with a limit of 5 passes through the |
369 |
< |
iterative loop). Also, the simulation box $\mathsf{H}$ is scaled |
370 |
< |
uniformly for one full time step by an exponential factor that |
371 |
< |
depends on the value of $\eta$ at time $t + h / 2$. Reshaping the |
412 |
< |
box uniformly also scales the volume of the box by |
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} |
417 |
– |
|
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*} |
449 |
– |
|
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. The relative |
455 |
< |
tolerance for the self-consistency check defaults to a value of |
456 |
< |
$\mbox{10}^{-6}$, but {\sc oopse} will terminate the iteration after |
457 |
< |
4 loops even if the consistency check has not been satisfied. |
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, |
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 |
470 |
< |
checks on the quality of the integration. It is also known that |
471 |
< |
this algorithm samples the equilibrium distribution for the enthalpy |
472 |
< |
(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) + |
427 |
|
P_{\mathrm{target}} \mathcal{V}(t). |
428 |
|
\end{equation} |
429 |
|
|
479 |
– |
Bond constraints are applied at the end of both the {\tt moveA} and |
480 |
– |
{\tt moveB} portions of the algorithm. Details on the constraint |
481 |
– |
algorithms are given in section \ref{oopseSec:rattle}. |
482 |
– |
|
430 |
|
\subsection{\label{methodSection:NPTf}Constant-pressure integration with a |
431 |
|
flexible box (NPTf)} |
432 |
|
|
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 |
|
% |
500 |
|
\mathsf{H}(t + h) &\leftarrow \mathsf{H}(t) \cdot e^{-h |
501 |
|
\overleftrightarrow{\eta}(t + h / 2)} . |
502 |
|
\end{align*} |
503 |
< |
{\sc oopse} uses a power series expansion truncated at second order |
504 |
< |
for the exponential operation which scales the simulation box. |
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 |
|
|
559 |
– |
The {\tt moveB} portion of the algorithm is largely unchanged from |
560 |
– |
the NPTi integrator: |
561 |
– |
|
508 |
|
{\tt moveB:} |
509 |
|
\begin{align*} |
510 |
|
T(t + h) &\leftarrow \left\{{\bf v}(t + h)\right\}, |
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*} |
591 |
– |
|
537 |
|
The iterative schemes for both {\tt moveA} and {\tt moveB} are |
538 |
< |
identical to those described for the NPTi integrator. |
539 |
< |
|
540 |
< |
The NPTf integrator is known to conserve the following Hamiltonian: |
541 |
< |
\begin{equation} |
597 |
< |
H_{\mathrm{NPTf}} = V + K + f k_B T_{\mathrm{target}} \left( |
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) + P_{\mathrm{target}} \mathcal{V}(t) + \frac{f k_B |
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{equation} |
603 |
< |
|
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 |
553 |
|
finds most use in simulating crystals or liquid crystals which |
554 |
|
assume non-orthorhombic geometries. |
555 |
|
|
556 |
< |
\subsection{\label{methodSection:otherSpecialEnsembles}Other Special Ensembles} |
556 |
> |
\subsection{\label{methodSection:NPAT}NPAT Ensemble} |
557 |
|
|
558 |
< |
\subsubsection{\label{methodSection:NPAT}NPAT Ensemble} |
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 |
|
|
616 |
– |
A comprehensive understanding of structure¨Cfunction relations of |
617 |
– |
biological membrane system ultimately relies on structure and |
618 |
– |
dynamics of lipid bilayer, which are strongly affected by the |
619 |
– |
interfacial interaction between lipid molecules and surrounding |
620 |
– |
media. One quantity to describe the interfacial interaction is so |
621 |
– |
called the average surface area per lipid. Constat area and constant |
622 |
– |
lateral pressure simulation can be achieved by extending the |
623 |
– |
standard NPT ensemble with a different pressure control strategy |
624 |
– |
|
568 |
|
\begin{equation} |
569 |
< |
\.{\overleftrightarrow{{\eta _{\alpha \beta}}}}=\left\{\begin{array}{ll} |
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)$}\\ |
571 |
> |
& \mbox{if $ \alpha = \beta = z$}\\ |
572 |
|
0 & \mbox{otherwise}\\ |
573 |
|
\end{array} |
574 |
|
\right. |
575 |
|
\end{equation} |
633 |
– |
|
576 |
|
Note that the iterative schemes for NPAT are identical to those |
577 |
|
described for the NPTi integrator. |
578 |
|
|
579 |
< |
\subsubsection{\label{methodSection:NPrT}NP$\gamma$T Ensemble} |
579 |
> |
\subsection{\label{methodSection:NPrT}NP$\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 $A$ |
642 |
< |
\[ |
643 |
< |
\gamma = \frac{{\partial G}}{{\partial A}}. |
644 |
< |
\] |
645 |
< |
However, a surface tension of zero is not appropriate for relatively |
646 |
< |
small patches of membrane. In order to eliminate the edge effect of |
647 |
< |
the membrane simulation, a special ensemble, NP$\gamma$T, is |
648 |
< |
proposed to maintain the lateral surface tension and normal |
649 |
< |
pressure. The equation of motion for cell size control tensor, |
650 |
< |
$\eta$, in $NP\gamma T$ is |
584 |
> |
minimum with respect to surface area $A$, |
585 |
|
\begin{equation} |
586 |
< |
\.{\overleftrightarrow{{\eta _{\alpha \beta}}}}=\left\{\begin{array}{ll} |
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{$\alpha = \beta = z$} \\ |
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 |
606 |
< |
(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} |
663 |
< |
\over P} _{\alpha \alpha } - P_{{\rm{target}}} ) |
605 |
> |
\gamma _\alpha = - h_z( \overleftrightarrow{P} _{\alpha \alpha } |
606 |
> |
- P_{{\rm{target}}} ) |
607 |
|
\label{methodEquation:instantaneousSurfaceTensor} |
608 |
|
\end{equation} |
666 |
– |
|
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:constraintMethod}Constraint Method} |
617 |
> |
\section{\label{methodSection:zcons}The Z-Constraint Method} |
618 |
|
|
619 |
< |
%\subsection{\label{methodSection:bondConstraint}Bond Constraint for Rigid Body} |
620 |
< |
|
621 |
< |
%\subsection{\label{methodSection:zcons}Z-constraint Method} |
622 |
< |
|
623 |
< |
\section{\label{methodSection:langevin}Integrators for Langevin Dynamics of Rigid Bodies} |
624 |
< |
|
625 |
< |
\subsection{\label{methodSection:temperature}Temperature Control} |
626 |
< |
|
627 |
< |
\subsection{\label{methodSection:pressureControl}Pressure Control} |
628 |
< |
|
629 |
< |
\section{\label{methodSection:hydrodynamics}Hydrodynamics} |
630 |
< |
|
631 |
< |
%\section{\label{methodSection:coarseGrained}Coarse-Grained Modeling} |
632 |
< |
|
633 |
< |
%\section{\label{methodSection:moleculeScale}Molecular-Scale Modeling} |
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} |