| 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 |
|
|
