trunk/fennellDissertation/waterChapter.tex

\chapter{\label{chap:water}SIMPLE MODELS FOR WATER}

One of the most important tasks in the simulation of biochemical
systems is the proper depiction of the aqueous environment around the
molecules of interest.  In some cases (such as in the simulation of
phospholipid bilayers), the majority of the calculations that are
performed involve interactions with or between solvent molecules.
Thus, the motion and behavior of molecules in biochemical simulations
are highly dependent on the properties of the water model that is
chosen as the solvent.
\begin{figure}
\includegraphics[width=\linewidth]{./figures/waterModels.pdf}
\caption{Partial-charge geometries for the TIP3P, TIP4P, TIP5P, and 
SPC/E rigid body water models.\cite{Jorgensen83,Mahoney00,Berendsen87}
In the case of the TIP models, the depiction of water improves with 
increasing number of point charges; however, computational performance 
simultaneously degrades due to the increasing number of distances and 
interactions that need to be calculated.}
\label{fig:waterModels}
\end{figure}

As discussed in the previous chapter, water it typically modeled with
fixed geometries of point charges shielded by the repulsive part of a
Lennard-Jones interaction. Some of the common water models are shown
in figure \ref{fig:waterModels}. The various models all have their
benefits and drawbacks, and these primarily focus on the balance
between computational efficiency and the ability to accurately predict
the properties of bulk water. For example, the TIP5P model improves on
the structural and transport properties of water relative to the TIP3P
and TIP4P models, yet this comes at a greater than 50\% increase in
computational cost.\cite{Mahoney00,Mahoney01} This cost is entirely
due to the additional distance and electrostatic calculations that
come from the extra point charges in the model description. Thus, the
criteria for choosing a water model are capturing the liquid state
properties and having the fewest number of points to insure efficient
performance. As researchers have begun to study larger systems, such
as entire viruses, the choice readily shifts towards efficiency over
accuracy in order to make the calculations
feasible.\cite{Freddolino06}

\section{Soft Sticky Dipole Model for Water}

One recently developed model that largely succeeds in retaining the
accuracy of bulk properties while greatly reducing the computational
cost is the Soft Sticky Dipole (SSD) water
model.\cite{Liu96,Liu96b,Chandra99,Tan03} The SSD model was developed
as a modified form of the hard-sphere water model proposed by Bratko,
Blum, and Luzar.\cite{Bratko85,Bratko95} SSD is a {\it single point}
model which has an interaction site that is both a point dipole and a
Lennard-Jones core.  However, since the normal aligned and
anti-aligned geometries favored by point dipoles are poor mimics of
local structure in liquid water, a short ranged ``sticky'' potential
is also added.  The sticky potential directs the molecules to assume
the proper hydrogen bond orientation in the first solvation shell.

The interaction between two SSD water molecules \emph{i} and \emph{j}
is given by the potential
\begin{equation}
u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp}
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)\ +
u_{ij}^{sp}
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j),
\end{equation}
where the ${\bf r}_{ij}$ is the position vector between molecules
\emph{i} and \emph{j} with magnitude $r_{ij}$, and
${\bf \Omega}_i$ and ${\bf \Omega}_j$ represent the orientations of
the two molecules. The Lennard-Jones and dipole interactions are given
by the following familiar forms:
\begin{equation}
u_{ij}^{LJ}(r_{ij}) = 4\epsilon
\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right]
\ ,
\end{equation}
and
\begin{equation}
u_{ij}^{dp} = \frac{|\mu_i||\mu_j|}{4 \pi \epsilon_0 r_{ij}^3} \left(
\hat{\bf u}_i \cdot \hat{\bf u}_j - 3(\hat{\bf u}_i\cdot\hat{\bf
r}_{ij})(\hat{\bf u}_j\cdot\hat{\bf r}_{ij}) \right)\ ,
\end{equation}
where $\hat{\bf u}_i$ and $\hat{\bf u}_j$ are the unit vectors along
the dipoles of molecules $i$ and $j$ respectively. $|\mu_i|$ and
$|\mu_j|$ are the strengths of the dipole moments, and $\hat{\bf
r}_{ij}$ is the unit vector pointing from molecule $j$ to molecule
$i$.

The sticky potential is somewhat less familiar:
\begin{equation}
u_{ij}^{sp}
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) =
\frac{\nu_0}{2}[s(r_{ij})w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)
+ s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf
\Omega}_j)]\ .
\label{eq:stickyfunction}
\end{equation}
Here, $\nu_0$ is a strength parameter for the sticky potential, and
$s$ and $s^\prime$ are cubic switching functions which turn off the
sticky interaction beyond the first solvation shell. The $w$ function
can be thought of as an attractive potential with tetrahedral
geometry:
\begin{equation}
w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
\end{equation}
while the $w^\prime$ function counters the normal aligned and
anti-aligned structures favored by point dipoles:
\begin{equation}
w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^\circ,
\end{equation}
It should be noted that $w$ is proportional to the sum of the $Y_3^2$
and $Y_3^{-2}$ spherical harmonics (a linear combination which
enhances the tetrahedral geometry for hydrogen bonded structures),
while $w^\prime$ is a purely empirical function.  A more detailed
description of the functional parts and variables in this potential
can be found in the original SSD
articles.\cite{Liu96,Liu96b,Chandra99,Tan03}

Since SSD is a single-point {\it dipolar} model, the force
calculations are simplified significantly relative to the standard
{\it charged} multi-point models. In the original Monte Carlo
simulations using this model, Liu and Ichiye reported that using SSD
decreased computer time by a factor of 6-7 compared to other
models.\cite{Liu96b} What is most impressive is that this savings did
not come at the expense of accurate depiction of the liquid state
properties.  Indeed, SSD maintains reasonable agreement with the Soper
data for the structural features of liquid water.\cite{Soper86,Liu96b}
Additionally, the dynamical properties exhibited by SSD agree with
experiment better than those of more computationally expensive models
(like TIP3P and SPC/E).\cite{Chandra99} The combination of speed and
accurate depiction of solvent properties makes SSD a very attractive
model for the simulation of large scale biochemical simulations.

It is important to note that the SSD model was originally developed
for use with the Ewald summation for handling long-range
electrostatics.\cite{Ewald21} In applying this water model in a
variety of molecular systems, it would be useful to know its
properties and behavior under the more computationally efficient
reaction field (RF) technique, the correction techniques discussed in
the previous chapter, or even a simple
cutoff.\cite{Onsager36,Fennell06} This study addresses these issues by
looking at the structural and transport behavior of SSD over a variety
of temperatures with the purpose of utilizing the RF correction
technique.  We then suggest modifications to the parameters that
result in more realistic bulk phase behavior.  It should be noted that
in a recent publication, some of the original investigators of the SSD
water model have suggested adjustments to the SSD water model to
address abnormal density behavior (also observed here), calling the
corrected model SSD1.\cite{Tan03} In the later sections of this
chapter, we compare our modified variants of SSD with both the
original SSD and SSD1 models and discuss how our changes improve the
depiction of water.

\section{Simulation Methods}

Most of the calculations in this particular study were performed using
a internally developed simulation code that was one of the precursors
of the {\sc oopse} molecular dynamics (MD) package.\cite{Meineke05}
All of the capabilities of this code have been efficiently
incorporated into {\sc oopse}, and calculation results are consistent
between the two simulation packages. The later calculations involving
the damped shifted force ({\sc sf}) techniques were performed using
{\sc oopse}.

In the primary simulations of this study, long-range dipole-dipole
interaction corrections were accounted for by using either the
reaction field technique or a simple cubic switching function at the
cutoff radius. Interestingly, one of the early applications of a
reaction field was in Monte Carlo simulations of liquid
water.\cite{Barker73} In this method, the magnitude of the reaction
field acting on dipole $i$ is
\begin{equation}
\mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1}
\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} s(r_{ij}),
\label{eq:rfequation}
\end{equation}
where $\mathcal{R}$ is the cavity defined by the cutoff radius
($r_{c}$), $\varepsilon_{s}$ is the dielectric constant imposed on the
system, ${\bf\mu}_{j}$ is the dipole moment vector of particle $j$,
and $s(r_{ij})$ is a cubic switching function.\cite{Allen87} The
reaction field contribution to the total energy by particle $i$ is
given by $-\frac{1}{2}{\bf\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque
on dipole $i$ by ${\bf\mu}_{i}\times\mathcal{E}_{i}$.\cite{Allen87} An
applied reaction field will alter the bulk orientational properties of
simulated water, and there is particular sensitivity of these
properties on changes in the length of the cutoff
radius.\cite{vanderSpoel98} This variable behavior makes reaction
field a less attractive method than the Ewald sum; however, for very
large systems, the computational benefit of reaction field is is
significant.
 
In contrast to the simulations with a reaction field, we have also
performed a companion set of simulations {\it without} a surrounding
dielectric (i.e. using a simple cubic switching function at the cutoff
radius). As a result, we have developed two reparametrizations of SSD
which can be used either with or without an active reaction field.

To determine the preferred densities of the models, we performed
simulations in the isobaric-isothermal ({\it NPT}) ensemble. All
dynamical properties for these models were then obtained from
microcanonical ({\it NVE}) simulations done at densities matching the
{\it NPT} density for a particular target temperature.  The constant
pressure simulations were implemented using an integral thermostat and
barostat as outlined by Hoover.\cite{Hoover85,Hoover86} All molecules
were treated as non-linear rigid bodies. Vibrational constraints are
not necessary in simulations of SSD, because there are no explicit
hydrogen atoms, and thus no molecular vibrational modes need to be
considered.

The symplectic splitting method proposed by Dullweber, Leimkuhler, and
McLachlan ({\sc dlm}, see section \ref{sec:IntroIntegrate}) was used
to carry out the integration of the equations of motion in place of
the more prevalent quaternion
method.\cite{Dullweber97,Evans77,Evans77b,Allen87} The reason behind
this decision was that, in {\it NVE} simulations, the energy drift
when using quaternions was substantially greater than when using the
{\sc dlm} method (Fig. \ref{fig:timeStepIntegration}). This steady
drift in the total energy has also been observed in other
studies.\cite{Kol97}

\begin{figure}
\centering
\includegraphics[width=\linewidth]{./figures/timeStepIntegration.pdf}
\caption{Energy conservation using both quaternion-based integration 
and the {\sc dlm} method with increasing time step. The larger time
step plots are shifted from the true energy baseline (that of $\Delta
t$ = 0.1fs) for clarity.}
\label{fig:timeStepIntegration}
\end{figure}

The {\sc dlm} method allows for Verlet style integration orientational
motion of rigid bodies via a sequence of rotation matrix
operations. Because these matrix operations are more costly than the
simpler arithmetic operations for quaternion propagation, typical SSD
particle simulations using {\sc dlm} are 5-10\% slower than
simulations using the quaternion method and an identical time
step. This additional expense is justified because of the ability to
use time steps that are more that twice as long and still achieve the
same energy conservation.

Figure \ref{fig:timeStepIntegration} shows the resulting energy drift
at various time steps for both {\sc dlm} and quaternion
integration. All of the 1000 SSD particle simulations started with the
same configuration, and the only difference was the method used to
handle orientational motion. At time steps of 0.1 and 0.5fs, both
methods for propagating the orientational degrees of freedom conserve
energy fairly well, with the quaternion method showing a slight energy
drift over time in the 0.5fs time step simulation. Time steps of 1 and
2fs clearly demonstrate the benefits in energy conservation that come
with the {\sc dlm} method. Thus, while maintaining the same degree of
energy conservation, one can take considerably longer time steps,
leading to an overall reduction in computation time.

Energy drifts in water simulations using {\sc dlm} integration were
unnoticeable for time steps up to 3fs. We observed a slight energy
drift on the order of 0.012~kcal/mol per nanosecond with a time step
of 4fs. As expected, this drift increases dramatically with increasing
time step. To insure accuracy in our {\it NVE} simulations, time steps
were set at 2fs and were also kept at this value for {\it NPT}
simulations.

Proton-disordered ice crystals in both the I$_\textrm{h}$ and
I$_\textrm{c}$ lattices were generated as starting points for all
simulations. The I$_\textrm{h}$ crystals were formed by first
arranging the centers of mass of the SSD particles into a
``hexagonal'' ice lattice of 1024 particles. Because of the crystal
structure of I$_\textrm{h}$ ice, the simulation boxes were
orthorhombic in shape with an edge length ratio of approximately
1.00$\times$1.06$\times$1.23. We then allowed the particles to orient
freely about their fixed lattice positions with angular momenta values
randomly sampled at 400K. The rotational temperature was then scaled
down in stages to slowly cool the crystals to 25K. The particles were
then allowed to translate with fixed orientations at a constant
pressure of 1atm for 50ps at 25K. Finally, all constraints were
removed and the ice crystals were allowed to equilibrate for 50ps at
25K and a constant pressure of 1atm.  This procedure resulted in
structurally stable I$_\textrm{h}$ ice crystals that obey the
Bernal-Fowler rules.\cite{Bernal33,Rahman72} This method was also
utilized in the making of diamond lattice I$_\textrm{c}$ ice crystals,
with each cubic simulation box consisting of either 512 or 1000
particles. Only isotropic volume fluctuations were performed under
constant pressure, so the ratio of edge lengths remained constant
throughout the simulations.

\section{SSD Density Behavior}

Melting studies were performed on the randomized ice crystals using
the {\it NPT} ensemble. During melting simulations, the melting
transition and the density maximum can both be observed, provided that
the density maximum occurs in the liquid and not the supercooled
regime. It should be noted that the calculated melting temperature
($T_\textrm{m}$) will not be the true $T_\textrm{m}$ because of
super-heating due to the relatively short time scales in molecular
simulations. This behavior results in inflated $T_\textrm{m}$ values;
however, these values provide a reasonable initial estimate of
$T_\textrm{m}$. 

An ensemble average from five separate melting simulations was
acquired, each starting from different ice crystals generated as
described previously. All simulations were equilibrated for 100ps
prior to a 200ps data collection run at each temperature setting. The
temperature range of study spanned from 25 to 400K, with a maximum
degree increment of 25K. For regions of interest along this stepwise
progression, the temperature increment was decreased from 25K to 10
and 5K.  The above equilibration and production times were sufficient
in that the fluctuations in the volume autocorrelation function damped
out in all of the simulations in under 20ps.

Our initial simulations focused on the original SSD water model, and
an average density versus temperature plot is shown in figure
\ref{fig:ssdDense}. Note that the density maximum when using a
reaction field appears between 255 and 265K.  There were smaller
fluctuations in the density at 260K than at either 255 or 265K, so we
report this value as the location of the density maximum. Figure
\ref{fig:ssdDense} was constructed using ice I$_\textrm{h}$ crystals
for the initial configuration; though not pictured, the simulations
starting from ice I$_\textrm{c}$ crystal configurations showed similar
results, with a liquid-phase density maximum at the same temperature.

\begin{figure}
\centering
\includegraphics[width=\linewidth]{./figures/ssdDense.pdf}
\caption{ Density versus temperature for TIP3P, SPC/E, TIP4P, SSD,
SSD with a reaction field, and
experiment.\cite{Jorgensen98b,Clancy94,CRC80}. Note that using a
reaction field lowers the density more than the already lowered SSD
densities. The lower than expected densities for the SSD model
prompted the original reparametrization of SSD to SSD1.\cite{Tan03}}
\label{fig:ssdDense}
\end{figure}

The density maximum for SSD compares quite favorably to other simple
water models. Figure \ref{fig:ssdDense} also shows calculated
densities of several other models and experiment obtained from other
sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water
models, SSD has a temperature closest to the experimentally observed
density maximum. Of the {\it charge-based} models in figure
\ref{fig:ssdDense}, TIP4P has a density maximum behavior most like
that seen in SSD. Though not included in this plot, it is useful to
note that TIP5P has a density maximum nearly identical to the
experimentally measured temperature (see section
\ref{sec:t5peDensity}.

Liquid state densities in water have been observed to be dependent on
the cutoff radius ($R_\textrm{c}$), both with and without the use of a
reaction field.\cite{vanderSpoel98} In order to address the possible
effect of $R_\textrm{c}$, simulations were performed with a cutoff
radius of 12\AA\, complementing the 9\AA\ $R_\textrm{c}$ used in the
previous SSD simulations. All of the resulting densities overlapped
within error and showed no significant trend toward lower or higher
densities in simulations both with and without reaction field. 

The key feature to recognize in figure \ref{fig:ssdDense} is the
density scaling of SSD relative to other common models at any given
temperature. SSD assumes a lower density than any of the other listed
models at the same pressure, behavior which is especially apparent at
temperatures greater than 300K. Lower than expected densities have
been observed for other systems using a reaction field for long-range
electrostatic interactions, so the most likely reason for the reduced
densities is the presence of the reaction
field.\cite{vanderSpoel98,Nezbeda02} In order to test the effect of
the reaction field on the density of the systems, the simulations were
repeated without a reaction field present. The results of these
simulations are also displayed in figure \ref{fig:ssdDense}. Without
the reaction field, the densities increase to more experimentally
reasonable values, especially around the freezing point of liquid
water. The shape of the curve is similar to the curve produced from
SSD simulations using reaction field, specifically the rapidly
decreasing densities at higher temperatures; however, a shift in the
density maximum location, down to 245K, is observed. This is a more
accurate comparison to the other listed water models, in that no long
range corrections were applied in those
simulations.\cite{Clancy94,Jorgensen98b} However, even without the
reaction field, the density around 300K is still significantly lower
than experiment and comparable water models. This anomalous behavior
was what lead Tan {\it et al.} to recently reparametrize
SSD.\cite{Tan03} Throughout the remainder of the paper our
reparametrizations of SSD will be compared with their newer SSD1
model.

\section{SSD Transport Behavior}

Accurate dynamical properties of a water model are particularly
important when using the model to study permeation or transport across
biological membranes.  In order to probe transport in bulk water, {\it
NVE} simulations were performed at the average densities obtained from
the {\it NPT} simulations at an identical target
temperature. Simulations started with randomized velocities and
underwent 50ps of temperature scaling and 50ps of constant energy
equilibration before a 200ps data collection run. Diffusion constants
were calculated via linear fits to the long-time behavior of the
mean-square displacement as a function of time.\cite{Allen87} The
averaged results from five sets of {\it NVE} simulations are displayed
in figure \ref{fig:ssdDiffuse}, alongside experimental, SPC/E, and TIP5P
results.\cite{Gillen72,Holz00,Clancy94,Mahoney01} 

\begin{figure}
\centering
\includegraphics[width=\linewidth]{./figures/ssdDiffuse.pdf} 
\caption{ Average self-diffusion constant as a function of temperature for 
SSD, SPC/E, and TIP5P compared with experimental
data.\cite{Clancy94,Mahoney01,Gillen72,Holz00} Of the three water
models shown, SSD has the least deviation from the experimental
values. The rapidly increasing diffusion constants for TIP5P and SSD
correspond to significant decreases in density at the higher
temperatures.}
\label{fig:ssdDiffuse}
\end{figure} 

The observed values for the diffusion constant point out one of the
strengths of the SSD model. Of the three models shown, the SSD model
has the most accurate depiction of self-diffusion in both the
supercooled and liquid regimes.  SPC/E does a respectable job by
reproducing values similar to experiment around 290K; however, it
deviates at both higher and lower temperatures, failing to predict the
correct thermal trend. TIP5P and SSD both start off low at colder
temperatures and tend to diffuse too rapidly at higher temperatures.
This behavior at higher temperatures is not particularly surprising
since the densities of both TIP5P and SSD are lower than experimental
water densities at higher temperatures.  When calculating the
diffusion coefficients for SSD at experimental densities (instead of
the densities from the {\it NPT} simulations), the resulting values
fall more in line with experiment at these temperatures.

\section{Structural Changes and Characterization}

By starting the simulations from the crystalline state, we can get an
estimation of the $T_\textrm{m}$ of the ice structure, and beyond the
melting point, we study the phase behavior of the liquid. The constant
pressure heat capacity ($C_\textrm{p}$) was monitored to locate
$T_\textrm{m}$ in each of the simulations. In the melting simulations
of the 1024 particle ice I$_\textrm{h}$ simulations, a large spike in
$C_\textrm{p}$ occurs at 245K, indicating a first order phase
transition for the melting of these ice crystals (see figure
\ref{fig:ssdCp}. When the reaction field is turned off, the melting
transition occurs at 235K.  These melting transitions are considerably
lower than the experimental value of 273K, indicating that the solid
ice I$_\textrm{h}$ is not thermodynamically preferred relative to the
liquid state at these lower temperatures.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{./figures/ssdCp.pdf}
\caption{Heat capacity versus temperature for the SSD model with an 
active reaction field. Note the large spike in $C_p$ around 245K,
indicating a phase transition from the ordered crystal to disordered
liquid.}
\label{fig:ssdCp}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=\linewidth]{./figures/fullContour.pdf} 
\caption{ Contour plots of 2D angular pair correlation functions for
512 SSD molecules at 100K (A \& B) and 300K (C \& D). Dark areas
signify regions of enhanced density while light areas signify
depletion relative to the bulk density. White areas have pair
correlation values below 0.5 and black areas have values above 1.5.}
\label{fig:contour} 
\end{figure} 

\begin{figure}
\centering
\includegraphics[width=2.5in]{./figures/corrDiag.pdf} 
\caption{ An illustration of angles involved in the correlations observed in figure \ref{fig:contour}.}
\label{fig:corrAngle}
\end{figure}

Additional analysis of the melting process was performed using
two-dimensional structure and dipole angle correlations. Expressions
for these correlations are as follows:

\begin{equation}
g_{\text{AB}}(r,\cos\theta) = \frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\theta-\cos\theta_{ij})\delta(r-\left|{\bf r}_{ij}\right|)\rangle\ ,
\end{equation}
\begin{equation}
g_{\text{AB}}(r,\cos\omega) = 
\frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\omega-\cos\omega_{ij})\delta(r-\left|{\bf r}_{ij}\right|)\rangle\ ,
\end{equation}
where $\theta$ and $\omega$ refer to the angles shown in figure
\ref{fig:corrAngle}. By binning over both distance and the cosine of the
desired angle between the two dipoles, the $g(r)$ can be analyzed to
determine the common dipole arrangements that constitute the peaks and
troughs in the standard one-dimensional $g(r)$ plots. Frames A and B
of figure \ref{fig:contour} show results from an ice I$_\textrm{c}$
simulation. The first peak in the $g(r)$ consists primarily of the
preferred hydrogen bonding arrangements as dictated by the tetrahedral
sticky potential - one peak for the hydrogen bond donor and the other
for the hydrogen bond acceptor.  Due to the high degree of
crystallinity of the sample, the second and third solvation shells
show a repeated peak arrangement which decays at distances around the
fourth solvation shell, near the imposed cutoff for the Lennard-Jones
and dipole-dipole interactions.  In the higher temperature simulation
shown in frames C and D, these long-range features deteriorate
rapidly. The first solvation shell still shows the strong effect of
the sticky-potential, although it covers a larger area, extending to
include a fraction of aligned dipole peaks within the first solvation
shell. The latter peaks lose due to thermal motion and as the
competing dipole force overcomes the sticky potential's tight
tetrahedral structuring of the crystal.

This complex interplay between dipole and sticky interactions was
remarked upon as a possible reason for the split second peak in the
oxygen-oxygen pair correlation function,
$g_\textrm{OO}(r)$.\cite{Liu96b} At low temperatures, the second
solvation shell peak appears to have two distinct components that
blend together to form one observable peak. At higher temperatures,
this split character alters to show the leading 4\AA\ peak dominated
by equatorial anti-parallel dipole orientations. There is also a
tightly bunched group of axially arranged dipoles that most likely
consist of the smaller fraction of aligned dipole pairs. The trailing
component of the split peak at 5\AA\ is dominated by aligned dipoles
that assume hydrogen bond arrangements similar to those seen in the
first solvation shell. This evidence indicates that the dipole pair
interaction begins to dominate outside of the range of the dipolar
repulsion term.  The energetically favorable dipole arrangements
populate the region immediately outside this repulsion region (around
4\AA), while arrangements that seek to satisfy both the sticky and
dipole forces locate themselves just beyond this initial buildup
(around 5\AA).

This analysis indicates that the split second peak is primarily the
product of the dipolar repulsion term of the sticky potential. In
fact, the inner peak can be pushed out and merged with the outer split
peak just by extending the switching function ($s^\prime(r_{ij})$)
from its normal 4\AA\ cutoff to values of 4.5 or even 5\AA. This
type of correction is not recommended for improving the liquid
structure, since the second solvation shell would still be shifted too
far out. In addition, this would have an even more detrimental effect
on the system densities, leading to a liquid with a more open
structure and a density considerably lower than the already low SSD
density.  A better correction would be to include the
quadrupole-quadrupole interactions for the water particles outside of
the first solvation shell, but this would remove the simplicity and
speed advantage of SSD.

\section{Adjusted Potentials: SSD/RF and SSD/E}

The propensity of SSD to adopt lower than expected densities under
varying conditions is troubling, especially at higher temperatures. In
order to correct this model for use with a reaction field, it is
necessary to adjust the force field parameters for the primary
intermolecular interactions. In undergoing a reparametrization, it is
important not to focus on just one property and neglect the others. In
this case, it would be ideal to correct the densities while
maintaining the accurate transport behavior.

The parameters available for tuning include the $\sigma$ and
$\epsilon$ Lennard-Jones parameters, the dipole strength ($\mu$), the
strength of the sticky potential ($\nu_0$), and the cutoff distances
for the sticky attractive and dipole repulsive cubic switching
function cutoffs ($r_l$, $r_u$ and $r_l^\prime$, $r_u^\prime$
respectively). The results of the reparametrizations are shown in
table \ref{tab:ssdParams}. We are calling these reparametrizations the
Soft Sticky Dipole Reaction Field (SSD/RF - for use with a reaction
field) and Soft Sticky Dipole Extended (SSD/E - an attempt to improve
the liquid structure in simulations without a long-range correction).

\begin{table}
\caption{PARAMETERS FOR THE ORIGINAL AND ADJUSTED SSD MODELS}

\centering
\begin{tabular}{ lcccc } 
\toprule
\toprule
Parameters & SSD & SSD1 & SSD/E & SSD/RF \\
\midrule
$\sigma$ (\AA)  & 3.051 & 3.016 & 3.035 & 3.019\\
$\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\
$\mu$ (D) & 2.35 & 2.35 & 2.42 & 2.48\\
$\nu_0$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\
$\omega^\circ$ & 0.07715 & 0.07715 & 0.07715 & 0.07715\\
$r_l$ (\AA) & 2.75 & 2.75 & 2.40 & 2.40\\
$r_u$ (\AA) & 3.35 & 3.35 & 3.80 & 3.80\\
$r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\
$r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\
\bottomrule
\end{tabular}
\label{tab:ssdParams}
\end{table}

\begin{figure}
\centering 
\includegraphics[width=4.5in]{./figures/newGofRCompare.pdf} 
\caption{ Plots showing the experimental $g(r)$ (Ref. \cite{Hura00}) 
with SSD/E and SSD1 without reaction field (top), as well as SSD/RF
and SSD1 with reaction field turned on (bottom). The changes in
parameters have lowered and broadened the first peak of SSD/E and
SSD/RF, resulting in a better fit to the first solvation shell.}
\label{fig:gofrCompare} 
\end{figure} 

\begin{figure}
\centering
\includegraphics[width=\linewidth]{./figures/dualPotentials.pdf} 
\caption{ Positive and negative isosurfaces of the sticky potential for
SSD and SSD1 (A) and SSD/E \& SSD/RF (B). Gold areas correspond to the
tetrahedral attractive component, and blue areas correspond to the
dipolar repulsive component.}
\label{fig:isosurface} 
\end{figure} 

In the original paper detailing the development of SSD, Liu and Ichiye
placed particular emphasis on an accurate description of the first
solvation shell. This resulted in a somewhat tall and narrow first
peak in $g(r)$ that integrated to give similar coordination numbers to
the experimental data obtained by Soper and
Phillips.\cite{Liu96b,Soper86} New experimental x-ray scattering data
from Hura {\it et al.} indicates a slightly lower and shifted first
peak in the $g_\textrm{OO}(r)$, so our adjustments to SSD were made
after taking into consideration the new experimental
findings.\cite{Hura00} Figure \ref{fig:gofrCompare} shows the
relocation of the first peak of the oxygen-oxygen $g(r)$ by comparing
the revised SSD model (SSD1), SSD/E, and SSD/RF to the new
experimental results. Both modified water models have shorter peaks
that match more closely to the experimental peak (as seen in the
insets of figure \ref{fig:gofrCompare}).  This structural alteration
was accomplished by the combined reduction in the Lennard-Jones
$\sigma$ variable and adjustment of the sticky potential strength and
cutoffs.  As can be seen in table \ref{tab:ssdParams}, the cutoffs for
the tetrahedral attractive and dipolar repulsive terms were nearly
swapped with each other.  Isosurfaces of the original and modified
sticky potentials are shown in figure \ref{fig:isosurface}. In these
isosurfaces, it is easy to see how altering the cutoffs changes the
repulsive and attractive character of the particles. With a reduced
repulsive surface, the particles can move closer to one another,
increasing the density for the overall system.  This change in
interaction cutoff also results in a more gradual orientational motion
by allowing the particles to maintain preferred dipolar arrangements
before they begin to feel the pull of the tetrahedral
restructuring. As the particles move closer together, the dipolar
repulsion term becomes active and excludes unphysical nearest-neighbor
arrangements. This compares with how SSD and SSD1 exclude preferred
dipole alignments before the particles feel the pull of the ``hydrogen
bonds''. Aside from improving the shape of the first peak in the
$g(r)$, this modification improves the densities considerably by
allowing the persistence of full dipolar character below the previous
4\AA\ cutoff.

While adjusting the location and shape of the first peak of $g(r)$
improves the densities, these changes alone are insufficient to bring
the system densities up to the values observed experimentally.  To
further increase the densities, the dipole moments were increased in
both of our adjusted models. Since SSD is a dipole based model, the
structure and transport are very sensitive to changes in the dipole
moment. The original SSD simply used the dipole moment calculated from
the TIP3P water model, which at 2.35~D is significantly greater than
the experimental gas phase value of 1.84~D. The larger dipole moment
is a more realistic value and improves the dielectric properties of
the fluid. Both theoretical and experimental measurements indicate a
liquid phase dipole moment ranging from 2.4~D to values as high as
3.11~D, providing a substantial range of reasonable values for a
dipole moment.\cite{Sprik91,Gubskaya02,Badyal00,Barriol64} Moderately
increasing the dipole moments to 2.42 and 2.48~D for SSD/E and SSD/RF,
respectively, leads to significant changes in the density and
transport of the water models.

\subsection{Density Behavior}

In order to demonstrate the benefits of these reparametrizations, we
performed a series of {\it NPT} and {\it NVE} simulations to probe the
density and transport properties of the adapted models and compare the
results to the original SSD model. This comparison involved full {\it
NPT} melting sequences for both SSD/E and SSD/RF, as well as {\it NVE}
transport calculations at the calculated self-consistent
densities. Again, the results were obtained from five separate
simulations of 1024 particle systems, and the melting sequences were
started from different ice I$_\textrm{h}$ crystals constructed as
described previously. Each {\it NPT} simulation was equilibrated for
100ps before a 200ps data collection run at each temperature step,
and the final configuration from the previous temperature simulation
was used as a starting point. All {\it NVE} simulations had the same
thermalization, equilibration, and data collection times as stated
previously.

\begin{figure}
\centering
\includegraphics[width=\linewidth]{./figures/ssdeDense.pdf} 
\caption{ Comparison of densities calculated with SSD/E to 
SSD1 without a reaction field, TIP3P, SPC/E, TIP5P, and
experiment.\cite{Jorgensen98b,Clancy94,Mahoney00,CRC80} Both SSD1 and
SSD/E show good agreement with experiment when the long-range
correction is neglected.}
\label{fig:ssdeDense} 
\end{figure} 

Figure \ref{fig:ssdeDense} shows the density profiles for SSD/E, SSD1,
TIP3P, TIP4P, and SPC/E alongside the experimental results. The
calculated densities for both SSD/E and SSD1 have increased
significantly over the original SSD model (see figure
\ref{fig:ssdDense}) and are in better agreement with the experimental
values. At 298 K, the densities of SSD/E and SSD1 without a long-range
correction are 0.996 g/cm$^3$ and 0.999 g/cm$^3$ respectively.  These
both compare well with the experimental value of 0.997 g/cm$^3$, and
they are considerably better than the SSD value of 0.967 g/cm$^3$. The
changes to the dipole moment and sticky switching functions have
improved the structuring of the liquid (as seen in figure
\ref{fig:gofrCompare}), but they have shifted the density maximum to
much lower temperatures. This comes about via an increase in the
liquid disorder through the weakening of the sticky potential and
strengthening of the dipolar character. However, this increasing
disorder in the SSD/E model has little effect on the melting
transition. By monitoring $C_p$ throughout these simulations, we
observed a melting transition for SSD/E at 235K, the same as SSD and
SSD1.

\begin{figure}
\centering
\includegraphics[width=\linewidth]{./figures/ssdrfDense.pdf} 
\caption{ Comparison of densities calculated with SSD/RF to 
SSD1 with a reaction field, TIP3P, SPC/E, TIP5P, and
experiment.\cite{Jorgensen98b,Clancy94,Mahoney00,CRC80} This plot
shows the benefit afforded by the reparametrization for use with a
reaction field correction - SSD/RF provides significantly more
accurate densities than SSD1 when performing room temperature
simulations.}
\label{fig:ssdrfDense} 
\end{figure} 

Including the reaction field long-range correction results in a more
interesting comparison.  A density profile including SSD/RF and SSD1
with an active reaction field is shown in figure \ref{fig:ssdrfDense}.
As observed in the simulations without a reaction field, the densities
of SSD/RF and SSD1 show a dramatic increase over normal SSD (see
figure \ref{fig:ssdDense}). At 298 K, SSD/RF has a density of 0.997
g/cm$^3$, directly in line with experiment and considerably better
than the original SSD value of 0.941 g/cm$^3$ and the SSD1 value of
0.972 g/cm$^3$. These results further emphasize the importance of
reparametrization in order to model the density properly under
different simulation conditions.  Again, these changes have only a
minor effect on the melting point, which observed at 245K for SSD/RF,
is identical to SSD and only 5K lower than SSD1 with a reaction
field. Additionally, the difference in density maxima is not as
extreme, with SSD/RF showing a density maximum at 255K, fairly close
to the density maxima of 260K and 265K, shown by SSD and SSD1
respectively.

\subsection{Transport Behavior}

\begin{figure}
\centering
\includegraphics[width=\linewidth]{./figures/ssdeDiffuse.pdf} 
\caption{ The diffusion constants calculated from SSD/E and 
SSD1 (both without a reaction field) along with experimental
results.\cite{Gillen72,Holz00} The {\it NVE} calculations were
performed at the average densities from the {\it NPT} simulations for
the respective models. SSD/E is slightly more mobile than experiment
at all of the temperatures, but it is closer to experiment at
biologically relevant temperatures than SSD1 without a long-range
correction.}
\label{fig:ssdeDiffuse} 
\end{figure} 

The reparametrization of the SSD water model, both for use with and
without an applied long-range correction, brought the densities up to
what is expected for proper simulation of liquid water. In addition to
improving the densities, it is important that the diffusive behavior
of SSD be maintained or improved. Figure \ref{fig:ssdeDiffuse}
compares the temperature dependence of the diffusion constant of SSD/E
to SSD1 without an active reaction field at the densities calculated
from their respective {\it NPT} simulations at 1 atm. The diffusion
constant for SSD/E is consistently higher than experiment, while SSD1
remains lower than experiment until relatively high temperatures
(around 360K). Both models follow the shape of the experimental curve
below 300K but tend to diffuse too rapidly at higher temperatures, as
seen in SSD1 crossing above 360K.  This increasing diffusion relative
to the experimental values is caused by the rapidly decreasing system
density with increasing temperature.  Both SSD1 and SSD/E show this
deviation in particle mobility, but this trend has different
implications on the diffusive behavior of the models.  While SSD1
shows more experimentally accurate diffusive behavior in the high
temperature regimes, SSD/E shows more accurate behavior in the
supercooled and biologically relevant temperature ranges.  Thus, the
changes made to improve the liquid structure may have had an adverse
affect on the density maximum, but they improve the transport behavior
of SSD/E relative to SSD1 under the most commonly simulated
conditions.

\begin{figure}
\centering
\includegraphics[width=\linewidth]{./figures/ssdrfDiffuse.pdf} 
\caption{ The diffusion constants calculated from SSD/RF and 
SSD1 (both with an active reaction field) along with experimental
results.\cite{Gillen72,Holz00} The {\it NVE} calculations were
performed at the average densities from the {\it NPT} simulations for
both of the models. SSD/RF captures the self-diffusion of water
throughout most of this temperature range. The increasing diffusion
constants at high temperatures for both models can be attributed to
lower calculated densities than those observed in experiment.}
\label{fig:ssdrfDiffuse} 
\end{figure} 

In figure \ref{fig:ssdrfDiffuse}, the diffusion constants for SSD/RF are
compared to SSD1 with an active reaction field. Note that SSD/RF
tracks the experimental results quantitatively, identical within error
throughout most of the temperature range shown and exhibiting only a
slight increasing trend at higher temperatures. SSD1 tends to diffuse
more slowly at low temperatures and deviates to diffuse too rapidly at
temperatures greater than 330K.  As stated above, this deviation away
from the ideal trend is due to a rapid decrease in density at higher
temperatures. SSD/RF does not suffer from this problem as much as SSD1
because the calculated densities are closer to the experimental
values. These results again emphasize the importance of careful
reparametrization when using an altered long-range correction.

\subsection{Summary of Liquid State Properties}

\begin{table}
\caption{PROPERTIES OF THE SINGLE-POINT WATER MODELS COMPARED WITH
EXPERIMENTAL DATA AT AMBIENT CONDITIONS}
\footnotesize
\centering
\begin{tabular}{ lccccc } 
\toprule
\toprule
& SSD1 & SSD/E & SSD1 (RF) & SSD/RF & Expt. \\
\midrule
$\rho$ (g/cm$^3$) & 0.999(1) & 0.996(1) & 0.972(2) & 0.997(1) & 0.997 \\
$C_p$ (cal/mol K) & 28.80(11) & 25.45(9) & 28.28(6) & 23.83(16) & 17.98 \\
$D$ ($10^{-5}$ cm$^2$/s) & 1.78(7) & 2.51(18) & 2.00(17) & 2.32(6) & 2.299\\
Coordination Number ($n_C$) & 3.9 & 4.3 & 3.8 & 4.4 & 4.7 \\ 
H-bonds per particle ($n_H$) & 3.7 & 3.6 & 3.7 & 3.7 & 3.5 \\
$\tau_1$ (ps) & 10.9(6) & 7.3(4) & 7.5(7) & 7.2(4) & 5.7 \\
$\tau_2$ (ps) & 4.7(4) & 3.1(2) & 3.5(3) & 3.2(2) & 2.3 \\
\bottomrule
\end{tabular}
\label{tab:liquidProperties}
\end{table}

Table \ref{tab:liquidProperties} gives a synopsis of the liquid state
properties of the water models compared in this study along with the
experimental values for liquid water at ambient conditions. The
coordination number ($n_C$) and number of hydrogen bonds per particle
($n_H$) were calculated by integrating the following relations:
\begin{equation}
n_C = 4\pi\rho_{\text{OO}}\int_{0}^{a}r^2g_{\textrm{OO}}(r)dr,
\end{equation}
\begin{equation}
n_H = 4\pi\rho_{\text{OH}}\int_{0}^{b}r^2g_{\textrm{OH}}(r)dr,
\end{equation}
where $\rho$ is the number density of the specified pair interactions,
$a$ and $b$ are the radial locations of the minima following the first
peak in $g_\textrm{OO}(r)$ or $g_\textrm{OH}(r)$ respectively. The
number of hydrogen bonds stays relatively constant across all of the
models, but the coordination numbers of SSD/E and SSD/RF show an
improvement over SSD1.  This improvement is primarily due to extension
of the first solvation shell in the new parameter sets.  Because $n_H$
and $n_C$ are nearly identical in SSD1, it appears that all molecules
in the first solvation shell are involved in hydrogen bonds.  Since
$n_H$ and $n_C$ differ in the newly parameterized models, the
orientations in the first solvation shell are a bit more ``fluid''.
Therefore SSD1 over-structures the first solvation shell and our
reparametrizations have returned this shell to more realistic
liquid-like behavior.

The time constants for the orientational autocorrelation functions
are also displayed in Table \ref{tab:liquidProperties}. The dipolar
orientational time correlation functions ($C_{l}$) are described
by:
\begin{equation}
C_{l}(t) = \langle P_l[\hat{\mathbf{u}}_j(0)\cdot\hat{\mathbf{u}}_j(t)]\rangle,
\end{equation}
where $P_l$ are Legendre polynomials of order $l$ and
$\hat{\mathbf{u}}_j$ is the unit vector pointing along the molecular
dipole.\cite{Rahman71} Note that this is identical to equation
(\ref{eq:OrientCorr}) were $\alpha$ is equal to $z$. From these
correlation functions, the orientational relaxation time of the dipole
vector can be calculated from an exponential fit in the long-time
regime ($t >
\tau_l$).\cite{Rothschild84} Calculation of these time constants were
averaged over five detailed {\it NVE} simulations performed at the ambient
conditions for each of the respective models. It should be noted that
the commonly cited value of 1.9 ps for $\tau_2$ was determined from
the NMR data in Ref. \cite{Krynicki66} at a temperature near
34$^\circ$C.\cite{Rahman71} Because of the strong temperature
dependence of $\tau_2$, it is necessary to recalculate it at 298K to
make proper comparisons. The value shown in Table
\ref{tab:liquidProperties} was calculated from the same NMR data in the 
fashion described in Ref. \cite{Krynicki66}. Similarly, $\tau_1$ was
recomputed for 298K from the data in Ref. \cite{Eisenberg69}.
Again, SSD/E and SSD/RF show improved behavior over SSD1, both with
and without an active reaction field. Turning on the reaction field
leads to much improved time constants for SSD1; however, these results
also include a corresponding decrease in system density.
Orientational relaxation times published in the original SSD dynamics
paper are smaller than the values observed here, and this difference
can be attributed to the use of the Ewald sum.\cite{Chandra99}

\subsection{SSD/RF and Damped Electrostatics}

\begin{table}
\caption{PROPERTIES OF SSD/RF WHEN USING VARIOUS ELECTROSTATIC CORRECTION METHODS}
\footnotesize
\centering
\begin{tabular}{ lccc } 
\toprule
\toprule
& Reaction Field & Damped Electrostatics & Expt. \\
& $\epsilon = 80$ & $\alpha = 0.2125\AA$ & \\
\midrule
$\rho$ (g/cm$^3$) & 0.997(1) & 1.004(4) & 0.997 \\
$C_p$ (cal/mol K) & 23.8(2) & 27(1) & 17.98 \\
$D$ ($10^{-5}$ cm$^2$/s) & 2.32(6) & 2.33(2) & 2.299\\
Coordination Number ($n_C$) & 4.4 & 4.3 & 4.7 \\ 
H-bonds per particle ($n_H$) & 3.7 & 3.7 & 3.5 \\
$\tau_1$ (ps) & 7.2(4) & 5.82(1) & 5.7 \\
$\tau_2$ (ps) & 3.2(2) & 2.42(1) & 2.3 \\
\bottomrule
\end{tabular}
\label{tab:dampedSSDRF}
\end{table}

In addition to the properties tabulated in table
\ref{tab:dampedSSDRF}, we calculated the static dielectric constant 
from a 5ns simulation of SSD/RF using the damped electrostatics. The
resulting value of 82.6(6) compares very favorably with the
experimental value of 78.3.\cite{Malmberg56} This value is closer to
the experimental value than what was expected according to figure
\ref{fig:dielectricMap}, raising some questions as to the accuracy of
the visual contours in the figure. This simply enforces the
qualitative nature of contour plotting.

\section{Tetrahedrally Restructured Elongated Dipole (TRED) Water Model}

\section{Conclusions}

In the above sections, the density maximum and temperature dependence
of the self-diffusion constant were studied for the SSD water model,
both with and without the use of reaction field, via a series of {\it
NPT} and {\it NVE} simulations. The constant pressure simulations
showed a density maximum near 260K. In most cases, the calculated
densities were significantly lower than the densities obtained from
other water models (and experiment). Analysis of self-diffusion showed
SSD to capture the transport properties of water well in both the
liquid and supercooled liquid regimes.

In order to correct the density behavior, we reparametrized the
original SSD model for use both with and without a reaction field
(SSD/RF and SSD/E), and made comparisons with SSD1, an alternate
density corrected version of SSD. Both models improve the liquid
structure, densities, and diffusive properties under their respective
simulation conditions, indicating the necessity of reparametrization
when changing the method of calculating long-range electrostatic
interactions.  

These simple water models are excellent choices for representing
explicit water in large scale simulations of biochemical systems. They
are more computationally efficient than the common charge based water
models, and, in many cases, exhibit more realistic bulk phase fluid
properties. These models are one of the few cases in which maximizing
efficiency does not result in a loss in realistic representation.
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