| 121 |
|
protein like hen egg white lysozyme (PDB code: 1LYZ) yields an |
| 122 |
|
effective protein concentration of 100 mg/mL.\cite{Asthagiri20053300} |
| 123 |
|
|
| 124 |
< |
Typically {\it total} protein concentrations in the cell are on the |
| 124 |
> |
{\it Yotal} 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}$: |
| 270 |
|
\begin{equation} |
| 271 |
|
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U + {\mathbf F}_i^{\mathrm ext}. |
| 272 |
|
\end{equation} |
| 387 |
|
\item The convex hull is computed and facets are identified. |
| 388 |
|
\item For each facet: |
| 389 |
|
\begin{itemize} |
| 390 |
< |
\item[a.] The force from the pressure bath ($-PA_f\hat{n}_f$) is |
| 390 |
> |
\item[a.] The force from the pressure bath ($-\hat{n}_fPA_f$) is |
| 391 |
|
computed. |
| 392 |
|
\item[b.] The resistance tensor ($\Xi_f(t)$) is computed using the |
| 393 |
|
viscosity ($\eta$) of the bath. |
| 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 |
| 495 |
|
is 180.32 GPa, while previous calculations on the QSC potential in |
| 496 |
|
periodic-boundary simulations of the bulk crystal have yielded values |
| 497 |
|
of 175.53 GPa.\cite{QSC} Using the same force field, we have performed |
| 498 |
< |
a series of relatively short (200 ps) simulations on 40 \AA~ radius |
| 498 |
> |
a series of 1 ns simulations on 40 \AA~ radius |
| 499 |
|
nanoparticles under the Langevin Hull at a variety of applied |
| 500 |
|
pressures ranging from 0 -- 10 GPa. We obtain a value of 177.55 GPa |
| 501 |
|
for the bulk modulus of gold using this technique, in close agreement |
| 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 |
|
|
| 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 |
| 599 |
|
\subsection{Molecular orientation distribution at cluster boundary} |
| 600 |
|
|
| 601 |
|
In order for a non-periodic boundary method to be widely applicable, |
| 602 |
< |
they must be constructed in such a way that they allow a finite system |
| 602 |
> |
it must be constructed in such a way that they allow a finite system |
| 603 |
|
to replicate the properties of the bulk. Early non-periodic simulation |
| 604 |
|
methods (e.g. hydrophobic boundary potentials) induced spurious |
| 605 |
|
orientational correlations deep within the simulated |
| 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 |
| 621 |
|
likely that there will be an effective hydrophobicity of the hull. |
| 622 |
|
|
| 623 |
|
To determine the extent of these effects, we examined the |
| 624 |
< |
orientationations exhibited by SPC/E water in a cluster of 1372 |
| 624 |
> |
orientations exhibited by SPC/E water in a cluster of 1372 |
| 625 |
|
molecules at 300 K and at pressures ranging from 1 -- 1000 atm. The |
| 626 |
< |
orientational angle of a water molecule is described |
| 626 |
> |
orientational angle of a water molecule is described by |
| 627 |
|
\begin{equation} |
| 628 |
|
\cos{\theta}=\frac{\vec{r}_i\cdot\vec{\mu}_i}{|\vec{r}_i||\vec{\mu}_i|} |
| 629 |
|
\end{equation} |
| 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 |
< |
In the absence of an electrostatic contribution from the exterior |
| 661 |
< |
bath, the orientational distribution of water molecules included in |
| 662 |
< |
the Langevin Hull will slightly resemble the distribution at a neat |
| 663 |
< |
water liquid/vapor interface. Previous molecular dynamics simulations |
| 664 |
< |
of SPC/E water \cite{Taylor1996} have shown that molecules at the |
| 668 |
< |
liquid/vapor interface favor an orientation where one hydrogen |
| 669 |
< |
protrudes from the liquid phase. This behavior is demonstrated by |
| 670 |
< |
experiments \cite{Du1994} \cite{Scatena2001} showing that |
| 671 |
< |
approximately one-quarter of water molecules at the liquid/vapor |
| 672 |
< |
interface form dangling hydrogen bonds. The negligible preference |
| 673 |
< |
shown in these cluster simulations could be removed through the |
| 674 |
< |
introduction of an implicit solvent model, which would provide the |
| 675 |
< |
missing electrostatic interactions between the cluster molecules and |
| 676 |
< |
the surrounding temperature/pressure bath. |
| 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 |
< |
The orientational preference exhibited by hull molecules in the |
| 667 |
< |
Langevin hull is significantly weaker than the preference caused by an |
| 668 |
< |
explicit hydrophobic bounding potential. Additionally, the Langevin |
| 669 |
< |
Hull does not require that the orientation of any molecules be fixed |
| 670 |
< |
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 and 100 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} |
| 726 |
|
The Langevin Hull samples the isobaric-isothermal ensemble for |
| 727 |
|
non-periodic systems by coupling the system to a bath characterized by |
| 728 |
|
pressure, temperature, and solvent viscosity. This enables the |
| 729 |
< |
simulation of heterogeneous systems composed of materials of |
| 729 |
> |
simulation of heterogeneous systems composed of materials with |
| 730 |
|
significantly different compressibilities. Because the boundary is |
| 731 |
|
dynamically determined during the simulation and the molecules |
| 732 |
< |
interacting with the boundary can change, the method and has minimal |
| 732 |
> |
interacting with the boundary can change, the method inflicts minimal |
| 733 |
|
perturbations on the behavior of molecules at the edges of the |
| 734 |
|
simulation. Further work on this method will involve implicit |
| 735 |
|
electrostatics at the boundary (which is missing in the current |
| 784 |
|
The individual hull operations scale with |
| 785 |
|
$\mathcal{O}(\frac{n}{p}\log\frac{n}{p})$ where $n$ is the total |
| 786 |
|
number of sites, and $p$ is the number of processors. These local |
| 787 |
< |
hull operations create a set of $p$ hulls each with approximately |
| 788 |
< |
$\frac{n}{3pr}$ sites (for a cluster of radius $r$). The worst-case |
| 787 |
> |
hull operations create a set of $p$ hulls, each with approximately |
| 788 |
> |
$\frac{n}{3pr}$ sites for a cluster of radius $r$. The worst-case |
| 789 |
|
communication cost for using a ``gather'' operation to distribute this |
| 790 |
|
information to all processors is $\mathcal{O}( \alpha (p-1) + \frac{n |
| 791 |
|
\beta (p-1)}{3 r p^2})$, while the final computation of the system |
| 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 so 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} |
