28 |
|
between the native and denatured states. Because of its stability |
29 |
|
against noise, Langevin dynamics is very suitable for studying |
30 |
|
remagnetization processes in various |
31 |
< |
systems\cite{Garcia-Palacios1998,Berkov2002,Denisov2003}. For |
32 |
< |
instance, the oscillation power spectrum of nanoparticles from |
33 |
< |
Langevin dynamics simulation has the same peak frequencies for |
34 |
< |
different wave vectors,which recovers the property of magnetic |
35 |
< |
excitations in small finite structures\cite{Berkov2005a}. In an |
36 |
< |
attempt to reduce the computational cost of simulation, multiple |
37 |
< |
time stepping (MTS) methods have been introduced and have been of |
38 |
< |
great interest to macromolecule and protein |
39 |
< |
community\cite{Tuckerman1992}. Relying on the observation that |
40 |
< |
forces between distant atoms generally demonstrate slower |
41 |
< |
fluctuations than forces between close atoms, MTS method are |
42 |
< |
generally implemented by evaluating the slowly fluctuating forces |
43 |
< |
less frequently than the fast ones. Unfortunately, nonlinear |
44 |
< |
instability resulting from increasing timestep in MTS simulation |
45 |
< |
have became a critical obstruction preventing the long time |
46 |
< |
simulation. Due to the coupling to the heat bath, Langevin dynamics |
47 |
< |
has been shown to be able to damp out the resonance artifact more |
48 |
< |
efficiently\cite{Sandu1999}. |
31 |
> |
systems\cite{Palacios1998,Berkov2002,Denisov2003}. For instance, the |
32 |
> |
oscillation power spectrum of nanoparticles from Langevin dynamics |
33 |
> |
simulation has the same peak frequencies for different wave |
34 |
> |
vectors,which recovers the property of magnetic excitations in small |
35 |
> |
finite structures\cite{Berkov2005a}. In an attempt to reduce the |
36 |
> |
computational cost of simulation, multiple time stepping (MTS) |
37 |
> |
methods have been introduced and have been of great interest to |
38 |
> |
macromolecule and protein community\cite{Tuckerman1992}. Relying on |
39 |
> |
the observation that forces between distant atoms generally |
40 |
> |
demonstrate slower fluctuations than forces between close atoms, MTS |
41 |
> |
method are generally implemented by evaluating the slowly |
42 |
> |
fluctuating forces less frequently than the fast ones. |
43 |
> |
Unfortunately, nonlinear instability resulting from increasing |
44 |
> |
timestep in MTS simulation have became a critical obstruction |
45 |
> |
preventing the long time simulation. Due to the coupling to the heat |
46 |
> |
bath, Langevin dynamics has been shown to be able to damp out the |
47 |
> |
resonance artifact more efficiently\cite{Sandu1999}. |
48 |
|
|
49 |
|
%review langevin/browninan dynamics for arbitrarily shaped rigid body |
50 |
|
Combining Langevin or Brownian dynamics with rigid body dynamics, |
78 |
|
average acceleration is not always true for cooperative motion which |
79 |
|
is common in protein motion. An inertial Brownian dynamics (IBD) was |
80 |
|
proposed to address this issue by adding an inertial correction |
81 |
< |
term\cite{Beard2001}. As a complement to IBD which has a lower bound |
81 |
> |
term\cite{Beard2000}. As a complement to IBD which has a lower bound |
82 |
|
in time step because of the inertial relaxation time, long-time-step |
83 |
|
inertial dynamics (LTID) can be used to investigate the inertial |
84 |
|
behavior of the polymer segments in low friction |
85 |
< |
regime\cite{Beard2001}. LTID can also deal with the rotational |
85 |
> |
regime\cite{Beard2000}. LTID can also deal with the rotational |
86 |
|
dynamics for nonskew bodies without translation-rotation coupling by |
87 |
|
separating the translation and rotation motion and taking advantage |
88 |
|
of the analytical solution of hydrodynamics properties. However, |
171 |
|
due to the complexity of the elliptic integral, only the ellipsoid |
172 |
|
with the restriction of two axes having to be equal, \textit{i.e.} |
173 |
|
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
174 |
< |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
174 |
> |
exactly. Introducing an elliptic integral parameter $S$ for prolate |
175 |
> |
: |
176 |
|
\[ |
177 |
|
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
178 |
|
} }}{b}, |
179 |
|
\] |
180 |
< |
and oblate, |
180 |
> |
and oblate : |
181 |
|
\[ |
182 |
|
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
183 |
|
}}{a} |
186 |
|
tensors |
187 |
|
\[ |
188 |
|
\begin{array}{l} |
189 |
< |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
190 |
< |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
191 |
< |
\end{array}, |
189 |
> |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
190 |
> |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
191 |
> |
2a}}, |
192 |
> |
\end{array} |
193 |
|
\] |
194 |
|
and |
195 |
|
\[ |
196 |
|
\begin{array}{l} |
197 |
< |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
197 |
> |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
198 |
|
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
199 |
|
\end{array}. |
200 |
|
\] |
542 |
|
explained by the finite size effect and relative slow relaxation |
543 |
|
rate at lower viscosity regime. |
544 |
|
\begin{table} |
545 |
< |
\caption{Average temperatures from Langevin dynamics simulations of |
546 |
< |
SSD water molecules with reference temperature at 300~K.} |
545 |
> |
\caption{AVERAGE TEMPERATURES FROM LANGEVIN DYNAMICS SIMULATIONS OF |
546 |
> |
SSD WATER MOLECULES WITH REFERENCE TEMPERATURE AT 300~K.} |
547 |
|
\label{langevin:viscosity} |
548 |
|
\begin{center} |
549 |
< |
\begin{tabular}{|l|l|l|} |
549 |
> |
\begin{tabular}{lll} |
550 |
|
\hline |
551 |
|
$\eta$ & $\text{T}_{\text{avg}}$ & $\text{T}_{\text{rms}}$ \\ |
552 |
+ |
\hline |
553 |
|
1 & 300.47 & 10.99 \\ |
554 |
|
0.1 & 301.19 & 11.136 \\ |
555 |
|
0.01 & 303.04 & 11.796 \\ |
580 |
|
temperature fluctuation versus time.} \label{langevin:temperature} |
581 |
|
\end{figure} |
582 |
|
|
583 |
< |
\subsection{Langevin Dynamics of Banana Shaped Molecule} |
583 |
> |
\subsection{Langevin Dynamics of Banana Shaped Molecules} |
584 |
|
|
585 |
|
In order to verify that Langevin dynamics can mimic the dynamics of |
586 |
|
the systems absent of explicit solvents, we carried out two sets of |
587 |
|
simulations and compare their dynamic properties. |
588 |
< |
|
589 |
< |
\subsubsection{Simulations Contain One Banana Shaped Molecule} |
588 |
< |
|
589 |
< |
Fig.~\ref{langevin:oneBanana} shows a snapshot of the simulation |
590 |
< |
made of 256 pentane molecules and one banana shaped molecule at |
588 |
> |
Fig.~\ref{langevin:twoBanana} shows a snapshot of the simulation |
589 |
> |
made of 256 pentane molecules and two banana shaped molecules at |
590 |
|
273~K. It has an equivalent implicit solvent system containing only |
591 |
< |
one banana shaped molecule with viscosity of 0.289 center poise. To |
591 |
> |
two banana shaped molecules with viscosity of 0.289 center poise. To |
592 |
|
calculate the hydrodynamic properties of the banana shaped molecule, |
593 |
|
we create a rough shell model (see Fig.~\ref{langevin:roughShell}), |
594 |
|
in which the banana shaped molecule is represented as a ``shell'' |
595 |
< |
made of 2266 small identical beads with size of 0.3 $\AA$ on the |
595 |
> |
made of 2266 small identical beads with size of 0.3 \AA on the |
596 |
|
surface. Applying the procedure described in |
597 |
|
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
598 |
|
identified the center of resistance at $(0, 0.7482, -0.1988)$, as |
599 |
|
well as the resistance tensor, |
600 |
|
\[ |
601 |
|
\left( {\begin{array}{*{20}c} |
602 |
< |
0.9261 & 1.310e-14 & -7.292e-15&5.067e-14&0.08585&0.2057\\ |
603 |
< |
3.968e-14& 0.9270&-0.007063& 0.08585&6.764e-14&4.846e-14\\ |
604 |
< |
-6.561e-16&-0.007063&0.7494&0.2057&4.846e-14&1.5036e-14\\ |
605 |
< |
5.067e-14&0.0858&0.2057& 58.64& 8.563e-13&-8.5736\\ |
606 |
< |
0.08585&6.764e-14&4.846e-14&1.555e-12&48.30&3.219&\\ |
607 |
< |
0.2057&4.846e-14&1.5036e-14&-3.904e-13&3.219&10.7373\\ |
602 |
> |
0.9261 & 0 & 0&0&0.08585&0.2057\\ |
603 |
> |
0& 0.9270&-0.007063& 0.08585&0&0\\ |
604 |
> |
0&0.007063&0.7494&0.2057&0&0\\ |
605 |
> |
0&0.0858&0.2057& 58.64& 0&-8.5736\\ |
606 |
> |
0.08585&0&0&0&48.30&3.219&\\ |
607 |
> |
0.2057&0&0&0&3.219&10.7373\\ |
608 |
|
\end{array}} \right). |
609 |
|
\] |
610 |
+ |
%\[ |
611 |
+ |
%\left( {\begin{array}{*{20}c} |
612 |
+ |
%0.9261 & 1.310e-14 & -7.292e-15&5.067e-14&0.08585&0.2057\\ |
613 |
+ |
%3.968e-14& 0.9270&-0.007063& 0.08585&6.764e-14&4.846e-14\\ |
614 |
+ |
%-6.561e-16&-0.007063&0.7494&0.2057&4.846e-14&1.5036e-14\\ |
615 |
+ |
%5.067e-14&0.0858&0.2057& 58.64& 8.563e-13&-8.5736\\ |
616 |
+ |
%0.08585&6.764e-14&4.846e-14&1.555e-12&48.30&3.219&\\ |
617 |
+ |
%0.2057&4.846e-14&1.5036e-14&-3.904e-13&3.219&10.7373\\ |
618 |
+ |
%\end{array}} \right). |
619 |
+ |
%\] |
620 |
|
|
621 |
+ |
Curves of velocity auto-correlation functions in |
622 |
+ |
Fig.~\ref{langevin:vacf} were shown to match each other very well. |
623 |
+ |
However, because of the stochastic nature, simulation using Langevin |
624 |
+ |
dynamics was shown to decay slightly fast. In order to study the |
625 |
+ |
rotational motion of the molecules, we also calculated the auto- |
626 |
+ |
correlation function of the principle axis of the second GB |
627 |
+ |
particle, $u$. |
628 |
+ |
|
629 |
|
\begin{figure} |
630 |
|
\centering |
631 |
|
\includegraphics[width=\linewidth]{roughShell.eps} |
633 |
|
model for banana shaped molecule.} \label{langevin:roughShell} |
634 |
|
\end{figure} |
635 |
|
|
619 |
– |
%\begin{figure} |
620 |
– |
%\centering |
621 |
– |
%\includegraphics[width=\linewidth]{oneBanana.eps} |
622 |
– |
%\caption[Snapshot from Simulation of One Banana Shaped Molecules and |
623 |
– |
%256 Pentane Molecules]{Snapshot from simulation of one Banana shaped |
624 |
– |
%molecules and 256 pentane molecules.} \label{langevin:oneBanana} |
625 |
– |
%\end{figure} |
626 |
– |
|
627 |
– |
\subsubsection{Simulations Contain Two Banana Shaped Molecules} |
628 |
– |
|
636 |
|
\begin{figure} |
637 |
|
\centering |
638 |
|
\includegraphics[width=\linewidth]{twoBanana.eps} |
641 |
|
molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
642 |
|
\end{figure} |
643 |
|
|
644 |
+ |
\begin{figure} |
645 |
+ |
\centering |
646 |
+ |
\includegraphics[width=\linewidth]{vacf.eps} |
647 |
+ |
\caption[Plots of Velocity Auto-correlation Functions]{Velocity |
648 |
+ |
auto-correlation functions in NVE (blue) and Langevin dynamics |
649 |
+ |
(red).} \label{langevin:vacf} |
650 |
+ |
\end{figure} |
651 |
+ |
|
652 |
+ |
\begin{figure} |
653 |
+ |
\centering |
654 |
+ |
\includegraphics[width=\linewidth]{uacf.eps} |
655 |
+ |
\caption[Auto-correlation functions of the principle axis of the |
656 |
+ |
middle GB particle]{Auto-correlation functions of the principle axis |
657 |
+ |
of the middle GB particle in NVE (blue) and Langevin dynamics |
658 |
+ |
(red).} \label{langevin:twoBanana} |
659 |
+ |
\end{figure} |
660 |
+ |
|
661 |
|
\section{Conclusions} |
662 |
+ |
|
663 |
+ |
We have presented a new Langevin algorithm by incorporating the |
664 |
+ |
hydrodynamics properties of arbitrary shaped molecules into an |
665 |
+ |
advanced symplectic integration scheme. The temperature control |
666 |
+ |
ability of this algorithm was demonstrated by a set of simulations |
667 |
+ |
with different viscosities. It was also shown to have significant |
668 |
+ |
advantage of producing rapid thermal equilibration over |
669 |
+ |
Nos\'{e}-Hoover method. Further studies in systems involving banana |
670 |
+ |
shaped molecules illustrated that the dynamic properties could be |
671 |
+ |
preserved by using this new algorithm as an implicit solvent model. |