12 |
|
are not currently suporrted by {\sc oopse}. |
13 |
|
|
14 |
|
The second most basic building block of a simulation is the |
15 |
< |
molecule. The molecule is a way for {\sc oopse} to keep track of the atoms |
16 |
< |
in a simulation in logical manner. This particular unit will store the |
17 |
< |
identities of all the atoms associated with itself and is responsible |
18 |
< |
for the evaluation of its own bonded interaction (i.e.~bonds, bends, |
19 |
< |
and torsions). |
15 |
> |
molecule. The molecule is a way for {\sc oopse} to keep track of the |
16 |
> |
atoms in a simulation in logical manner. This particular unit will |
17 |
> |
store the identities of all the atoms associated with itself and is |
18 |
> |
responsible for the evaluation of its own bonded interaction |
19 |
> |
(i.e.~bonds, bends, and torsions). |
20 |
|
|
21 |
< |
As stated in the previously, one of the features that sets OOPSE apart |
21 |
> |
As stated previously, one of the features that sets {\sc OOPSE} apart |
22 |
|
from most of the current molecular simulation packages is the ability |
23 |
|
to handle rigid body dynamics. Rigid bodies are non-spherical |
24 |
|
particles or collections of particles that have a constant internal |
25 |
|
potential and move collectively.\cite{Goldstein01} They are not |
26 |
< |
included in many standard simulation packages because of the need to |
26 |
> |
included in most simulation packages because of the need to |
27 |
|
consider orientational degrees of freedom and include an integrator |
28 |
|
that accurately propagates these motions in time. |
29 |
|
|
31 |
|
torque applied by the surroundings, which directly affect the |
32 |
|
translation and rotation in turn. In order to accumulate the total |
33 |
|
force on a rigid body, the external forces must first be calculated |
34 |
< |
for all the interal particles. The total force on the rigid body is |
34 |
> |
for all the internal particles. The total force on the rigid body is |
35 |
|
simply the sum of these external forces. Accumulation of the total |
36 |
< |
torque on the rigid body is similar to the force in that it is a sum |
37 |
< |
of the torque applied on each internal particle, mapped onto the |
38 |
< |
center of mass of the rigid body. |
36 |
> |
torque on the rigid body is more complex than the force in that it is |
37 |
> |
the torque applied on the center of mass that dictates rotational |
38 |
> |
motion. The summation of this torque is given by |
39 |
> |
\begin{equation} |
40 |
> |
\mathbf{\tau}_i= |
41 |
> |
\sum_{a}(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia}, |
42 |
> |
\label{eq:torqueAccumulate} |
43 |
> |
\end{equation} |
44 |
> |
where $\mathbf{\tau}_i$ and $\mathbf{r}_i$ are the torque about and |
45 |
> |
position of the center of mass respectively, while $\mathbf{f}_{ia}$ |
46 |
> |
and $\mathbf{r}_{ia}$ are the force on and position of the component |
47 |
> |
particles of the rigid body.\cite{allen87:csl} |
48 |
|
|
49 |
|
The application of the total torque is done in the body fixed axis of |
50 |
|
the rigid body. In order to move between the space fixed and body |
51 |
< |
fixed coordinate axes, parameters describing the orientation be |
52 |
< |
maintained for each rigid body. At a minimum, the rotation matrix can |
53 |
< |
be described and propagated by the three Euler |
54 |
< |
angles.\cite{Goldstein01} In order to avoid rotational limitations |
55 |
< |
when using Euler angles, the four parameter ``quaternion'' scheme can |
56 |
< |
be used instead.\cite{allen87:csl} Use of quaternions also leads to |
57 |
< |
performance enhancements, particularly for very small |
58 |
< |
systems.\cite{Evans77} OOPSE utilizes a relatively new scheme that |
59 |
< |
propagates the entire nine parameter rotation matrix. Further |
60 |
< |
discussion on this choice can be found in Sec.~\ref{sec:integrate}. |
51 |
> |
fixed coordinate axes, parameters describing the orientation must be |
52 |
> |
maintained for each rigid body. At a minimum, the rotation matrix |
53 |
> |
(\textbf{A}) can be described and propagated by the three Euler angles |
54 |
> |
($\phi, \theta, \text{and} \psi$), where \textbf{A} is composed of |
55 |
> |
trigonometric operations involving $\phi, \theta,$ and |
56 |
> |
$\psi$.\cite{Goldstein01} In order to avoid rotational limitations |
57 |
> |
inherent in using the Euler angles, the four parameter ``quaternion'' |
58 |
> |
scheme can be used instead, where \textbf{A} is composed of arithmetic |
59 |
> |
operations involving the four components of a quaternion ($q_0, q_1, |
60 |
> |
q_2, \text{and} q_3$).\cite{allen87:csl} Use of quaternions also leads |
61 |
> |
to performance enhancements, particularly for very small |
62 |
> |
systems.\cite{Evans77} |
63 |
|
|
64 |
+ |
{\sc OOPSE} utilizes a relatively new scheme that uses the entire nine |
65 |
+ |
parameter rotation matrix internally. Further discussion on this |
66 |
+ |
choice can be found in Sec.~\ref{sec:integrate}. |
67 |
+ |
|
68 |
|
\subsection{\label{sec:LJPot}The Lennard Jones Potential} |
69 |
|
|
70 |
|
The most basic force field implemented in OOPSE is the Lennard-Jones |
278 |
|
$i$ it takes its orientation from $\boldsymbol{\Omega}_i$. |
279 |
|
|
280 |
|
|
281 |
< |
\subsection{\label{sec:SSD}Water Model: SSD and Derivatives} |
281 |
> |
\subsection{\label{sec:SSD}The {\sc DUFF} Water Models: SSD/E and SSD/RF} |
282 |
|
|
283 |
< |
In the interest of computational efficiency, the native solvent used |
283 |
> |
In the interest of computational efficiency, the default solvent used |
284 |
|
in {\sc oopse} is the Soft Sticky Dipole (SSD) water model. SSD was |
285 |
|
developed by Ichiye \emph{et al.} as a modified form of the |
286 |
|
hard-sphere water model proposed by Bratko, Blum, and |
290 |
|
solvation shell. Thus, the interaction between two SSD water molecules |
291 |
|
\emph{i} and \emph{j} is given by the potential |
292 |
|
\begin{equation} |
293 |
< |
u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp} |
294 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
295 |
< |
u_{ij}^{sp} |
296 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
293 |
> |
V_{ij} = |
294 |
> |
V_{ij}^{LJ} (r_{ij})\ + V_{ij}^{dp} |
295 |
> |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
296 |
> |
V_{ij}^{sp} |
297 |
> |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
298 |
> |
\label{eq:ssdPot} |
299 |
|
\end{equation} |
300 |
|
where the $\mathbf{r}_{ij}$ is the position vector between molecules |
301 |
< |
\emph{i} and \emph{j} with magnitude equal to the distance $r_ij$, and |
301 |
> |
\emph{i} and \emph{j} with magnitude equal to the distance $r_{ij}$, and |
302 |
|
$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the |
303 |
< |
orientations of the respective molecules. The Lennard-Jones, dipole, |
304 |
< |
and sticky parts of the potential are giving by the following |
305 |
< |
equations, |
303 |
> |
orientations of the respective molecules. The Lennard-Jones and dipole |
304 |
> |
parts of the potential are given by equations \ref{eq:lennardJonesPot} |
305 |
> |
and \ref{eq:dipolePot} respectively. The sticky part is described by |
306 |
> |
the following, |
307 |
|
\begin{equation} |
308 |
< |
u_{ij}^{LJ}(r_{ij}) = 4\epsilon \left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right], |
308 |
> |
u_{ij}^{sp}(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)= |
309 |
> |
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij}, |
310 |
> |
\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) + |
311 |
> |
s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij}, |
312 |
> |
\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
313 |
> |
\label{eq:stickyPot} |
314 |
|
\end{equation} |
315 |
< |
\begin{equation} |
316 |
< |
u_{ij}^{dp} = \frac{\boldsymbol{\mu}_i\cdot\boldsymbol{\mu}_j}{r_{ij}^3}-\frac{3(\boldsymbol{\mu}_i\cdot\mathbf{r}_{ij})(\boldsymbol{\mu}_j\cdot\mathbf{r}_{ij})}{r_{ij}^5}\ , |
317 |
< |
\end{equation} |
315 |
> |
where $\nu_0$ is a strength parameter for the sticky potential, and |
316 |
> |
$s$ and $s^\prime$ are cubic switching functions which turn off the |
317 |
> |
sticky interaction beyond the first solvation shell. The $w$ function |
318 |
> |
can be thought of as an attractive potential with tetrahedral |
319 |
> |
geometry: |
320 |
|
\begin{equation} |
321 |
< |
\begin{split} |
322 |
< |
u_{ij}^{sp} |
323 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) |
299 |
< |
&= |
300 |
< |
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\\ |
301 |
< |
& \quad \ + |
302 |
< |
s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
303 |
< |
\end{split} |
321 |
> |
w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
322 |
> |
\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
323 |
> |
\label{eq:stickyW} |
324 |
|
\end{equation} |
325 |
< |
where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole |
326 |
< |
unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D, |
307 |
< |
$\nu_0$ scales the strength of the overall sticky potential, $s$ and |
308 |
< |
$s^\prime$ are cubic switching functions. The $w$ and $w^\prime$ |
309 |
< |
functions take the following forms, |
325 |
> |
while the $w^\prime$ function counters the normal aligned and |
326 |
> |
anti-aligned structures favored by point dipoles: |
327 |
|
\begin{equation} |
328 |
< |
w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
328 |
> |
w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
329 |
> |
(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0, |
330 |
> |
\label{eq:stickyWprime} |
331 |
|
\end{equation} |
332 |
< |
\begin{equation} |
333 |
< |
w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0, |
334 |
< |
\end{equation} |
335 |
< |
where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive |
336 |
< |
term that promotes hydrogen bonding orientations within the first |
337 |
< |
solvation shell, and $w^\prime$ is a dipolar repulsion term that |
338 |
< |
repels unrealistic dipolar arrangements within the first solvation |
320 |
< |
shell. A more detailed description of the functional parts and |
321 |
< |
variables in this potential can be found in other |
322 |
< |
articles.\cite{liu96:new_model,chandra99:ssd_md} |
332 |
> |
It should be noted that $w$ is proportional to the sum of the $Y_3^2$ |
333 |
> |
and $Y_3^{-2}$ spherical harmonics (a linear combination which |
334 |
> |
enhances the tetrahedral geometry for hydrogen bonded structures), |
335 |
> |
while $w^\prime$ is a purely empirical function. A more detailed |
336 |
> |
description of the functional parts and variables in this potential |
337 |
> |
can be found in the original SSD |
338 |
> |
articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} |
339 |
|
|
340 |
< |
Since SSD is a one-site point dipole model, the force calculations are |
341 |
< |
simplified significantly from a computational standpoint, in that the |
342 |
< |
number of long-range interactions is dramatically reduced. In the |
343 |
< |
original Monte Carlo simulations using this model, Ichiye \emph{et |
344 |
< |
al.} reported a calculation speed up of up to an order of magnitude |
345 |
< |
over other comparable models while maintaining the structural behavior |
346 |
< |
of water.\cite{liu96:new_model} In the original molecular dynamics studies of |
347 |
< |
SSD, it was shown that it actually improves upon the prediction of |
348 |
< |
water's dynamical properties over TIP3P and SPC/E.\cite{chandra99:ssd_md} |
340 |
> |
Since SSD is a single-point {\it dipolar} model, the force |
341 |
> |
calculations are simplified significantly relative to the standard |
342 |
> |
{\it charged} multi-point models. In the original Monte Carlo |
343 |
> |
simulations using this model, Ichiye {\it et al.} reported that using |
344 |
> |
SSD decreased computer time by a factor of 6-7 compared to other |
345 |
> |
models.\cite{Ichiye96} What is most impressive is that this savings |
346 |
> |
did not come at the expense of accurate depiction of the liquid state |
347 |
> |
properties. Indeed, SSD maintains reasonable agreement with the Soper |
348 |
> |
data for the structural features of liquid |
349 |
> |
water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties |
350 |
> |
exhibited by SSD agree with experiment better than those of more |
351 |
> |
computationally expensive models (like TIP3P and |
352 |
> |
SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction |
353 |
> |
of solvent properties makes SSD a very attractive model for the |
354 |
> |
simulation of large scale biochemical simulations. |
355 |
|
|
356 |
|
Recent constant pressure simulations revealed issues in the original |
357 |
|
SSD model that led to lower than expected densities at all target |
358 |
< |
pressures.\cite{Ichiye03,Gezelter04} Reparameterizations of the |
359 |
< |
original SSD have resulted in improved density behavior, as well as |
360 |
< |
alterations in the water structure and transport behavior. {\sc oopse} is |
361 |
< |
easily modified to impliment these new potential parameter sets for |
362 |
< |
the derivative water models: SSD1, SSD/E, and SSD/RF. All of the |
363 |
< |
variable parameters are listed in the accompanying BASS file, and |
364 |
< |
these parameters simply need to be changed to the updated values. |
358 |
> |
pressures.\cite{Ichiye03,Gezelter04} The default model in {\sc oopse} |
359 |
> |
is SSD/E, a density corrected derivative of SSD that exhibits improved |
360 |
> |
liquid structure and transport behavior. If the use of a reaction |
361 |
> |
field long-range interaction correction is desired, it is recommended |
362 |
> |
that the parameters be modified to those of the SSD/RF model. Solvent |
363 |
> |
parameters can be easily modified in an accompanying {\sc BASS} file |
364 |
> |
as illustrated in the scheme below. A table of the parameter values |
365 |
> |
and the drawbacks and benefits of the different density corrected SSD |
366 |
> |
models can be found in reference \ref{Gezelter04}. |
367 |
|
|
368 |
+ |
!!!Place a {\sc BASS} scheme showing SSD parameters around here!!! |
369 |
|
|
370 |
|
\subsection{\label{sec:eam}Embedded Atom Model} |
371 |
|
|