| 121 |
|
protein like hen egg white lysozyme (PDB code: 1LYZ) yields an |
| 122 |
|
effective protein concentration of 100 mg/mL.\cite{Asthagiri20053300} |
| 123 |
|
|
| 124 |
< |
{\it Yotal} protein concentrations in the cell are typically on the |
| 124 |
> |
{\it Total} protein concentrations in the cell are typically on the |
| 125 |
|
order of 160-310 mg/ml,\cite{Brown1991195} and individual proteins |
| 126 |
|
have concentrations orders of magnitude lower than this in the |
| 127 |
|
cellular environment. The effective concentrations of single proteins |
| 251 |
|
\caption{The external temperature and pressure bath interacts only |
| 252 |
|
with those atoms on the convex hull (grey surface). The hull is |
| 253 |
|
computed dynamically at each time step, and molecules can move |
| 254 |
< |
between the interior (Newtonian) region and the Langevin hull.} |
| 254 |
> |
between the interior (Newtonian) region and the Langevin Hull.} |
| 255 |
|
\label{fig:hullSample} |
| 256 |
|
\end{figure} |
| 257 |
|
|
| 266 |
|
potential energy. For atoms on the exterior of the cluster |
| 267 |
|
(i.e. those that occupy one of the vertices of the convex hull), the |
| 268 |
|
equation of motion is modified with an external force, ${\mathbf |
| 269 |
< |
F}_i^{\mathrm ext}$, |
| 269 |
> |
F}_i^{\mathrm ext}$: |
| 270 |
|
\begin{equation} |
| 271 |
|
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U + {\mathbf F}_i^{\mathrm ext}. |
| 272 |
|
\end{equation} |
| 412 |
|
To test the new method, we have carried out simulations using the |
| 413 |
|
Langevin Hull on: 1) a crystalline system (gold nanoparticles), 2) a |
| 414 |
|
liquid droplet (SPC/E water),\cite{Berendsen1987} and 3) a |
| 415 |
< |
heterogeneous mixture (gold nanoparticles in a water droplet). In each |
| 416 |
< |
case, we have computed properties that depend on the external applied |
| 417 |
< |
pressure. Of particular interest for the single-phase systems is the |
| 418 |
< |
isothermal compressibility, |
| 415 |
> |
heterogeneous mixture (gold nanoparticles in an SPC/E water droplet). In each case, we have computed properties that depend on the external applied pressure. Of particular interest for the single-phase systems is the isothermal compressibility, |
| 416 |
|
\begin{equation} |
| 417 |
|
\kappa_{T} = -\frac{1}{V} \left ( \frac{\partial V}{\partial P} \right |
| 418 |
|
)_{T}. |
| 421 |
|
|
| 422 |
|
One problem with eliminating periodic boundary conditions and |
| 423 |
|
simulation boxes is that the volume of a three-dimensional point cloud |
| 424 |
< |
is not well-defined. In order to compute the compressibility of a |
| 424 |
> |
is not well-defined. In order to compute the compressibility of a |
| 425 |
|
bulk material, we make an assumption that the number density, $\rho = |
| 426 |
< |
\frac{N}{V}$, is uniform within some region of the point cloud. The |
| 426 |
> |
\frac{N}{V}$, is uniform within some region of the point cloud. The |
| 427 |
|
compressibility can then be expressed in terms of the average number |
| 428 |
|
of particles in that region, |
| 429 |
|
\begin{equation} |
| 430 |
|
\kappa_{T} = -\frac{1}{N} \left ( \frac{\partial N}{\partial P} \right |
| 431 |
< |
)_{T} |
| 431 |
> |
)_{T}. |
| 432 |
|
\label{eq:BMN} |
| 433 |
|
\end{equation} |
| 434 |
< |
The region we used is a spherical volume of 10 \AA\ radius centered in |
| 434 |
> |
The region we used is a spherical volume of 20 \AA\ radius centered in |
| 435 |
|
the middle of the cluster. $N$ is the average number of molecules |
| 436 |
|
found within this region throughout a given simulation. The geometry |
| 437 |
|
and size of the region is arbitrary, and any bulk-like portion of the |
| 518 |
|
temperature respond to the Langevin Hull for nanoparticles that were |
| 519 |
|
initialized far from the target pressure and temperature. As |
| 520 |
|
expected, the rate at which thermal equilibrium is achieved depends on |
| 521 |
< |
the total surface area of the cluter exposed to the bath as well as |
| 521 |
> |
the total surface area of the cluster exposed to the bath as well as |
| 522 |
|
the bath viscosity. Pressure that is applied suddenly to a cluster |
| 523 |
|
can excite breathing vibrations, but these rapidly damp out (on time |
| 524 |
< |
scales of 30-50 ps). |
| 524 |
> |
scales of 30 -- 50 ps). |
| 525 |
|
|
| 526 |
|
\subsection{Compressibility of SPC/E water clusters} |
| 527 |
|
|
| 545 |
|
and previous simulation work throughout the 1 -- 1000 atm pressure |
| 546 |
|
regime. Compressibilities computed using the Hull volume, however, |
| 547 |
|
deviate dramatically from the experimental values at low applied |
| 548 |
< |
pressures. The reason for this deviation is quite simple; at low |
| 548 |
> |
pressures. The reason for this deviation is quite simple: at low |
| 549 |
|
applied pressures, the liquid is in equilibrium with a vapor phase, |
| 550 |
|
and it is entirely possible for one (or a few) molecules to drift away |
| 551 |
|
from the liquid cluster (see Fig. \ref{fig:coneOfShame}). At low |
| 575 |
|
different pressures must be done to compute the first derivatives. It |
| 576 |
|
is also possible to compute the compressibility using the fluctuation |
| 577 |
|
dissipation theorem using either fluctuations in the |
| 578 |
< |
volume,\cite{Debenedetti1986}, |
| 578 |
> |
volume,\cite{Debenedetti1986} |
| 579 |
|
\begin{equation} |
| 580 |
|
\kappa_{T} = \frac{\left \langle V^{2} \right \rangle - \left \langle |
| 581 |
|
V \right \rangle ^{2}}{V \, k_{B} \, T}, |
| 585 |
|
fixed region, |
| 586 |
|
\begin{equation} |
| 587 |
|
\kappa_{T} = \frac{\left \langle N^{2} \right \rangle - \left \langle |
| 588 |
< |
N \right \rangle ^{2}}{N \, k_{B} \, T}, |
| 588 |
> |
N \right \rangle ^{2}}{N \, k_{B} \, T}. |
| 589 |
|
\label{eq:BMNfluct} |
| 590 |
|
\end{equation} |
| 591 |
|
Thus, the compressibility of each simulation can be calculated |
| 613 |
|
hydrophobic boundary, or orientational or radial constraints. |
| 614 |
|
Therefore, the orientational correlations of the molecules in water |
| 615 |
|
clusters are of particular interest in testing this method. Ideally, |
| 616 |
< |
the water molecules on the surfaces of the clusterss will have enough |
| 616 |
> |
the water molecules on the surfaces of the clusters will have enough |
| 617 |
|
mobility into and out of the center of the cluster to maintain |
| 618 |
|
bulk-like orientational distribution in the absence of orientational |
| 619 |
|
and radial constraints. However, since the number of hydrogen bonding |
| 643 |
|
\includegraphics[width=\linewidth]{pAngle} |
| 644 |
|
\caption{Distribution of $\cos{\theta}$ values for molecules on the |
| 645 |
|
interior of the cluster (squares) and for those participating in the |
| 646 |
< |
convex hull (circles) at a variety of pressures. The Langevin hull |
| 646 |
> |
convex hull (circles) at a variety of pressures. The Langevin Hull |
| 647 |
|
exhibits minor dewetting behavior with exposed oxygen sites on the |
| 648 |
|
hull water molecules. The orientational preference for exposed |
| 649 |
|
oxygen appears to be independent of applied pressure. } |
| 655 |
|
orientations. Molecules included in the convex hull show a slight |
| 656 |
|
preference for values of $\cos{\theta} < 0.$ These values correspond |
| 657 |
|
to molecules with oxygen directed toward the exterior of the cluster, |
| 658 |
< |
forming a dangling hydrogen bond acceptor site. |
| 658 |
> |
forming dangling hydrogen bond acceptor sites. |
| 659 |
|
|
| 660 |
< |
Previous molecular dynamics simulations |
| 661 |
< |
of SPC/E water that use periodic boundary conditions have shown that molecules on the liquid side of the liquid/vapor interface favor a similar orientation where oxygen is directed away from the bulk.\cite{Taylor1996} These simulations had both a liquid phase and a well-defined vapor phase in equilibrium and showed that vapor molecules generally had one hydrogen protruding from the surface, forming a dangling hydrogen bond donor. Our water cluster simulations do not have a true lasting vapor phase, but rather a few transient molecules that leave the liquid droplet. Thus we are unable to comment on the orientational preference of vapor phase molecules in a Langevin Hull simulation. |
| 660 |
> |
The orientational preference exhibited by water molecules on the hull |
| 661 |
> |
is significantly weaker than the preference caused by an explicit |
| 662 |
> |
hydrophobic bounding potential. Additionally, the Langevin Hull does |
| 663 |
> |
not require that the orientation of any molecules be fixed in order to |
| 664 |
> |
maintain bulk-like structure, even near the cluster surface. |
| 665 |
|
|
| 666 |
< |
However, the orientational preference exhibited by liquid phase hull molecules in the Langevin hull is significantly weaker than the preference caused by an explicit hydrophobic bounding potential. Additionally, the Langevin Hull does not require that the orientation of any molecules be fixed in order to maintain bulk-like structure, even at the cluster surface. |
| 666 |
> |
Previous molecular dynamics simulations of SPC/E liquid / vapor |
| 667 |
> |
interfaces using periodic boundary conditions have shown that |
| 668 |
> |
molecules on the liquid side of interface favor a similar orientation |
| 669 |
> |
where oxygen is directed away from the bulk.\cite{Taylor1996} These |
| 670 |
> |
simulations had well-defined liquid and vapor phase regions |
| 671 |
> |
equilibrium and it was observed that {\it vapor} molecules generally |
| 672 |
> |
had one hydrogen protruding from the surface, forming a dangling |
| 673 |
> |
hydrogen bond donor. Our water clusters do not have a true vapor |
| 674 |
> |
region, but rather a few transient molecules that leave the liquid |
| 675 |
> |
droplet (and which return to the droplet relatively quickly). |
| 676 |
> |
Although we cannot obtain an orientational preference of vapor phase |
| 677 |
> |
molecules in a Langevin Hull simulation, but we do agree with previous |
| 678 |
> |
estimates of the orientation of {\it liquid phase} molecules at the |
| 679 |
> |
interface. |
| 680 |
|
|
| 681 |
|
\subsection{Heterogeneous nanoparticle / water mixtures} |
| 682 |
|
|
| 683 |
|
To further test the method, we simulated gold nanopartices ($r = 18$ |
| 684 |
< |
\AA) solvated by explicit SPC/E water clusters using the Langevin |
| 685 |
< |
hull. This was done at pressures of 1, 2, 5, 10, 20, 50, 100 and 200 atm |
| 686 |
< |
in order to observe the effects of pressure on the ordering of water |
| 687 |
< |
ordering at the surface. In Fig. \ref{fig:RhoR} we show the density |
| 688 |
< |
of water adjacent to the surface as a function of pressure, as well as |
| 689 |
< |
the orientational ordering of water at the surface of the |
| 690 |
< |
nanoparticle. |
| 684 |
> |
\AA) solvated by explicit SPC/E water clusters using a model for the |
| 685 |
> |
gold / water interactions that has been used by Dou {\it et. al.} for |
| 686 |
> |
investigating the separation of water films near hot metal |
| 687 |
> |
surfaces.\cite{ISI:000167766600035} The Langevin Hull was used to |
| 688 |
> |
sample pressures of 1, 2, 5, 10, 20, 50, 100 and 200 atm, while all |
| 689 |
> |
simulations were done at a temperature of 300 K. At these |
| 690 |
> |
temperatures and pressures, there is no observed separation of the |
| 691 |
> |
water film from the surface. |
| 692 |
|
|
| 693 |
< |
\begin{figure} |
| 693 |
> |
In Fig. \ref{fig:RhoR} we show the density of water and gold as a |
| 694 |
> |
function of the distance from the center of the nanoparticle. Higher |
| 695 |
> |
applied pressures appear to destroy structural correlations in the |
| 696 |
> |
outermost monolayer of the gold nanoparticle as well as in the water |
| 697 |
> |
at the near the metal / water interface. Simulations at increased |
| 698 |
> |
pressures exhibit significant overlap of the gold and water densities, |
| 699 |
> |
indicating a less well-defined interfacial surface. |
| 700 |
|
|
| 701 |
< |
\caption{interesting plot showing cluster behavior} |
| 701 |
> |
\begin{figure} |
| 702 |
> |
\includegraphics[width=\linewidth]{RhoR} |
| 703 |
> |
\caption{Density profiles of gold and water at the nanoparticle |
| 704 |
> |
surface. Each curve has been normalized by the average density in |
| 705 |
> |
the bulk-like region available to the corresponding material. Higher applied pressures |
| 706 |
> |
de-structure both the gold nanoparticle surface and water at the |
| 707 |
> |
metal/water interface.} |
| 708 |
|
\label{fig:RhoR} |
| 709 |
|
\end{figure} |
| 710 |
|
|
| 711 |
< |
At higher pressures, problems with the gold - water interaction |
| 712 |
< |
potential became apparent. The model we are using (due to Spohr) was |
| 713 |
< |
intended for relatively low pressures; it utilizes both shifted Morse |
| 714 |
< |
and repulsive Morse potentials to model the Au/O and Au/H |
| 715 |
< |
interactions, respectively. The repulsive wall of the Morse potential |
| 716 |
< |
does not diverge quickly enough at short distances to prevent water |
| 717 |
< |
from diffusing into the center of the gold nanoparticles. This |
| 718 |
< |
behavior is likely not a realistic description of the real physics of |
| 719 |
< |
the situation. A better model of the gold-water adsorption behavior |
| 720 |
< |
appears to require harder repulsive walls to prevent this behavior. |
| 711 |
> |
At even higher pressures (500 atm and above), problems with the metal |
| 712 |
> |
- water interaction potential became quite clear. The model we are |
| 713 |
> |
using appears to have been parameterized for relatively low pressures; |
| 714 |
> |
it utilizes both shifted Morse and repulsive Morse potentials to model |
| 715 |
> |
the Au/O and Au/H interactions, respectively. The repulsive wall of |
| 716 |
> |
the Morse potential does not diverge quickly enough at short distances |
| 717 |
> |
to prevent water from diffusing into the center of the gold |
| 718 |
> |
nanoparticles. This behavior is likely not a realistic description of |
| 719 |
> |
the real physics of the situation. A better model of the gold-water |
| 720 |
> |
adsorption behavior appears to require harder repulsive walls to |
| 721 |
> |
prevent this behavior. |
| 722 |
|
|
| 723 |
|
\section{Discussion} |
| 724 |
|
\label{sec:discussion} |
| 793 |
|
|
| 794 |
|
For a large number of atoms on a moderately parallel machine, the |
| 795 |
|
total costs are dominated by the computations of the individual hulls, |
| 796 |
< |
and communication of these hulls to create the Langevin hull sees roughly |
| 796 |
> |
and communication of these hulls to create the Langevin Hull sees roughly |
| 797 |
|
linear speed-up with increasing processor counts. |
| 798 |
|
|
| 799 |
|
\section*{Acknowledgments} |
