| 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 |
| 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 |
| 521 |
|
the total surface area of the cluter 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 |
| 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. } |
| 657 |
|
to molecules with oxygen directed toward the exterior of the cluster, |
| 658 |
|
forming a dangling hydrogen bond acceptor site. |
| 659 |
|
|
| 660 |
< |
In the absence of an electrostatic contribution from the exterior |
| 664 |
< |
bath, the orientational distribution of water molecules included in |
| 665 |
< |
the Langevin Hull will slightly resemble the distribution at a neat |
| 666 |
< |
water liquid/vapor interface. Previous molecular dynamics simulations |
| 667 |
< |
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 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. |
| 661 |
|
|
| 662 |
< |
The orientational preference exhibited by hull molecules in the |
| 663 |
< |
Langevin hull is significantly weaker than the preference caused by an |
| 680 |
< |
explicit hydrophobic bounding potential. Additionally, the Langevin |
| 681 |
< |
Hull does not require that the orientation of any molecules be fixed |
| 682 |
< |
in order to maintain bulk-like structure, even at the cluster surface. |
| 662 |
> |
Previous molecular dynamics simulations |
| 663 |
> |
of SPC/E water using 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 while we are unable to comment on the orientational preference of vapor phase molecules in a Langevin Hull simulation, we achieve good agreement for the orientation of liquid phase molecules at the interface. |
| 664 |
|
|
| 665 |
|
\subsection{Heterogeneous nanoparticle / water mixtures} |
| 666 |
|
|
| 667 |
|
To further test the method, we simulated gold nanopartices ($r = 18$ |
| 668 |
|
\AA) solvated by explicit SPC/E water clusters using the Langevin |
| 669 |
< |
hull. This was done at pressures of 1, 2, 5, 10, 20, 50 and 100 atm |
| 669 |
> |
Hull. This was done at pressures of 1, 2, 5, 10, 20, 50, 100 and 200 atm |
| 670 |
|
in order to observe the effects of pressure on the ordering of water |
| 671 |
|
ordering at the surface. In Fig. \ref{fig:RhoR} we show the density |
| 672 |
< |
of water adjacent to the surface as a function of pressure, as well as |
| 673 |
< |
the orientational ordering of water at the surface of the |
| 693 |
< |
nanoparticle. |
| 672 |
> |
of water adjacent to the surface and |
| 673 |
> |
the density of gold at the surface as a function of pressure. |
| 674 |
|
|
| 675 |
< |
\begin{figure} |
| 675 |
> |
Higher applied pressures de-structure the outermost layer of the gold nanoparticle and the water at the metal/water interface. Simulations at increased pressures have greater overlap of the gold and water densities, indicating a less well-defined interfacial surface. |
| 676 |
|
|
| 677 |
< |
\caption{interesting plot showing cluster behavior} |
| 677 |
> |
\begin{figure} |
| 678 |
> |
\includegraphics[width=\linewidth]{RhoR} |
| 679 |
> |
\caption{Densities of gold and water at the nanoparticle surface. Higher applied pressures de-structure both the gold nanoparticle surface and water at the metal/water interface.} |
| 680 |
|
\label{fig:RhoR} |
| 681 |
|
\end{figure} |
| 682 |
|
|
| 683 |
< |
At higher pressures, problems with the gold - water interaction |
| 683 |
> |
Indeed, at even higher pressures, problems with the gold - water interaction |
| 684 |
|
potential became apparent. The model we are using (due to Spohr) was |
| 685 |
|
intended for relatively low pressures; it utilizes both shifted Morse |
| 686 |
|
and repulsive Morse potentials to model the Au/O and Au/H |
| 697 |
|
The Langevin Hull samples the isobaric-isothermal ensemble for |
| 698 |
|
non-periodic systems by coupling the system to a bath characterized by |
| 699 |
|
pressure, temperature, and solvent viscosity. This enables the |
| 700 |
< |
simulation of heterogeneous systems composed of materials of |
| 700 |
> |
simulation of heterogeneous systems composed of materials with |
| 701 |
|
significantly different compressibilities. Because the boundary is |
| 702 |
|
dynamically determined during the simulation and the molecules |
| 703 |
< |
interacting with the boundary can change, the method and has minimal |
| 703 |
> |
interacting with the boundary can change, the method inflicts minimal |
| 704 |
|
perturbations on the behavior of molecules at the edges of the |
| 705 |
|
simulation. Further work on this method will involve implicit |
| 706 |
|
electrostatics at the boundary (which is missing in the current |
| 755 |
|
The individual hull operations scale with |
| 756 |
|
$\mathcal{O}(\frac{n}{p}\log\frac{n}{p})$ where $n$ is the total |
| 757 |
|
number of sites, and $p$ is the number of processors. These local |
| 758 |
< |
hull operations create a set of $p$ hulls each with approximately |
| 759 |
< |
$\frac{n}{3pr}$ sites (for a cluster of radius $r$). The worst-case |
| 758 |
> |
hull operations create a set of $p$ hulls, each with approximately |
| 759 |
> |
$\frac{n}{3pr}$ sites for a cluster of radius $r$. The worst-case |
| 760 |
|
communication cost for using a ``gather'' operation to distribute this |
| 761 |
|
information to all processors is $\mathcal{O}( \alpha (p-1) + \frac{n |
| 762 |
|
\beta (p-1)}{3 r p^2})$, while the final computation of the system |
| 764 |
|
|
| 765 |
|
For a large number of atoms on a moderately parallel machine, the |
| 766 |
|
total costs are dominated by the computations of the individual hulls, |
| 767 |
< |
and communication of these hulls to so the Langevin hull sees roughly |
| 767 |
> |
and communication of these hulls to create the Langevin Hull sees roughly |
| 768 |
|
linear speed-up with increasing processor counts. |
| 769 |
|
|
| 770 |
|
\section*{Acknowledgments} |
