| 11 |
|
simulations. Implicit solvent Langevin dynamics simulations of |
| 12 |
|
met-enkephalin not only outperform explicit solvent simulations for |
| 13 |
|
computational efficiency, but also agrees very well with explicit |
| 14 |
< |
solvent simulations for dynamical properties\cite{Shen2002}. |
| 14 |
> |
solvent simulations for dynamical properties.\cite{Shen2002} |
| 15 |
|
Recently, applying Langevin dynamics with the UNRES model, Liow and |
| 16 |
|
his coworkers suggest that protein folding pathways can be possibly |
| 17 |
< |
explored within a reasonable amount of time\cite{Liwo2005}. The |
| 17 |
> |
explored within a reasonable amount of time.\cite{Liwo2005} The |
| 18 |
|
stochastic nature of the Langevin dynamics also enhances the |
| 19 |
|
sampling of the system and increases the probability of crossing |
| 20 |
< |
energy barriers\cite{Banerjee2004, Cui2003}. Combining Langevin |
| 20 |
> |
energy barriers.\cite{Banerjee2004, Cui2003} Combining Langevin |
| 21 |
|
dynamics with Kramers's theory, Klimov and Thirumalai identified |
| 22 |
|
free-energy barriers by studying the viscosity dependence of the |
| 23 |
< |
protein folding rates\cite{Klimov1997}. In order to account for |
| 23 |
> |
protein folding rates.\cite{Klimov1997} In order to account for |
| 24 |
|
solvent induced interactions missing from implicit solvent model, |
| 25 |
|
Kaya incorporated desolvation free energy barrier into implicit |
| 26 |
|
coarse-grained solvent model in protein folding/unfolding studies |
| 27 |
|
and discovered a higher free energy barrier between the native and |
| 28 |
|
denatured states. Because of its stability against noise, Langevin |
| 29 |
|
dynamics is very suitable for studying remagnetization processes in |
| 30 |
< |
various systems\cite{Palacios1998,Berkov2002,Denisov2003}. For |
| 30 |
> |
various systems.\cite{Palacios1998,Berkov2002,Denisov2003} For |
| 31 |
|
instance, the oscillation power spectrum of nanoparticles from |
| 32 |
|
Langevin dynamics simulation has the same peak frequencies for |
| 33 |
|
different wave vectors, which recovers the property of magnetic |
| 34 |
< |
excitations in small finite structures\cite{Berkov2005a}. |
| 34 |
> |
excitations in small finite structures.\cite{Berkov2005a} |
| 35 |
|
|
| 36 |
|
%review langevin/browninan dynamics for arbitrarily shaped rigid body |
| 37 |
|
Combining Langevin or Brownian dynamics with rigid body dynamics, |
| 40 |
|
as well as excluded volume potentials, Mielke and his coworkers |
| 41 |
|
discovered rapid superhelical stress generations from the stochastic |
| 42 |
|
simulation of twin supercoiling DNA with response to induced |
| 43 |
< |
torques\cite{Mielke2004}. Membrane fusion is another key biological |
| 43 |
> |
torques.\cite{Mielke2004} Membrane fusion is another key biological |
| 44 |
|
process which controls a variety of physiological functions, such as |
| 45 |
|
release of neurotransmitters \textit{etc}. A typical fusion event |
| 46 |
< |
happens on the time scale of millisecond, which is impractical to |
| 46 |
> |
happens on the time scale of a millisecond, which is impractical to |
| 47 |
|
study using atomistic models with newtonian mechanics. With the help |
| 48 |
|
of coarse-grained rigid body model and stochastic dynamics, the |
| 49 |
|
fusion pathways were explored by many |
| 50 |
< |
researchers\cite{Noguchi2001,Noguchi2002,Shillcock2005}. Due to the |
| 50 |
> |
researchers.\cite{Noguchi2001,Noguchi2002,Shillcock2005} Due to the |
| 51 |
|
difficulty of numerical integration of anisotropic rotation, most of |
| 52 |
|
the rigid body models are simply modeled using spheres, cylinders, |
| 53 |
|
ellipsoids or other regular shapes in stochastic simulations. In an |
| 56 |
|
dynamics simulation algorithm\cite{Ermak1978,Allison1991} by |
| 57 |
|
incorporating a generalized $6\times6$ diffusion tensor and |
| 58 |
|
introducing a simple rotation evolution scheme consisting of three |
| 59 |
< |
consecutive rotations\cite{Fernandes2002}. Unfortunately, unexpected |
| 59 |
> |
consecutive rotations.\cite{Fernandes2002} Unfortunately, unexpected |
| 60 |
|
errors and biases are introduced into the system due to the |
| 61 |
|
arbitrary order of applying the noncommuting rotation |
| 62 |
< |
operators\cite{Beard2003}. Based on the observation the momentum |
| 62 |
> |
operators.\cite{Beard2003} Based on the observation the momentum |
| 63 |
|
relaxation time is much less than the time step, one may ignore the |
| 64 |
|
inertia in Brownian dynamics. However, the assumption of zero |
| 65 |
|
average acceleration is not always true for cooperative motion which |
| 66 |
|
is common in protein motion. An inertial Brownian dynamics (IBD) was |
| 67 |
|
proposed to address this issue by adding an inertial correction |
| 68 |
< |
term\cite{Beard2000}. As a complement to IBD which has a lower bound |
| 68 |
> |
term.\cite{Beard2000} As a complement to IBD which has a lower bound |
| 69 |
|
in time step because of the inertial relaxation time, long-time-step |
| 70 |
|
inertial dynamics (LTID) can be used to investigate the inertial |
| 71 |
|
behavior of the polymer segments in low friction |
| 72 |
< |
regime\cite{Beard2000}. LTID can also deal with the rotational |
| 72 |
> |
regime.\cite{Beard2000} LTID can also deal with the rotational |
| 73 |
|
dynamics for nonskew bodies without translation-rotation coupling by |
| 74 |
|
separating the translation and rotation motion and taking advantage |
| 75 |
|
of the analytical solution of hydrodynamics properties. However, |
| 172 |
|
one can write down the translational and rotational resistance |
| 173 |
|
tensors |
| 174 |
|
\begin{eqnarray*} |
| 175 |
< |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
| 176 |
< |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
| 175 |
> |
\Xi _a^{tt} & = & 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
| 176 |
> |
\Xi _b^{tt} & = & \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
| 177 |
|
2a}}, |
| 178 |
|
\end{eqnarray*} |
| 179 |
|
and |
| 180 |
|
\begin{eqnarray*} |
| 181 |
< |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
| 182 |
< |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}}. |
| 183 |
< |
\end{eqnarray} |
| 181 |
> |
\Xi _a^{rr} & = & \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
| 182 |
> |
\Xi _b^{rr} & = & \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}}. |
| 183 |
> |
\end{eqnarray*} |
| 184 |
|
|
| 185 |
|
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}} |
| 186 |
|
|
| 223 |
|
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
| 224 |
|
A second order expression for element of different size was |
| 225 |
|
introduced by Rotne and Prager\cite{Rotne1969} and improved by |
| 226 |
< |
Garc\'{i}a de la Torre and Bloomfield\cite{Torre1977}, |
| 226 |
> |
Garc\'{i}a de la Torre and Bloomfield,\cite{Torre1977} |
| 227 |
|
\begin{equation} |
| 228 |
|
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
| 229 |
|
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
| 278 |
|
bead $i$ and origin $O$, the elements of resistance tensor at |
| 279 |
|
arbitrary origin $O$ can be written as |
| 280 |
|
\begin{eqnarray} |
| 281 |
< |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\ |
| 282 |
< |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 283 |
< |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\ |
| 281 |
> |
\Xi _{}^{tt} & = & \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\ |
| 282 |
> |
\Xi _{}^{tr} & = & \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 283 |
> |
\Xi _{}^{rr} & = & - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\ |
| 284 |
|
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 285 |
|
\end{eqnarray} |
| 286 |
|
The resistance tensor depends on the origin to which they refer. The |
| 528 |
|
\end{center} |
| 529 |
|
\end{table} |
| 530 |
|
|
| 531 |
< |
Another set of calculation were performed to study the efficiency of |
| 531 |
> |
Another set of calculations were performed to study the efficiency of |
| 532 |
|
temperature control using different temperature coupling schemes. |
| 533 |
|
The starting configuration is cooled to 173~K and evolved using NVE, |
| 534 |
|
NVT, and Langevin dynamic with time step of 2 fs. |
| 538 |
|
Nos\'e-Hoover temperature scaling scheme with thermostat of 5 ps |
| 539 |
|
which gives reasonable tight coupling, while the blue one from |
| 540 |
|
Langevin dynamics with viscosity of 0.1 poise demonstrates a faster |
| 541 |
< |
scaling to the desire temperature. In extremely lower friction |
| 542 |
< |
regime (when $ \eta \approx 0$), Langevin dynamics becomes normal |
| 541 |
> |
scaling to the desire temperature. When $ \eta = 0$, Langevin dynamics becomes normal |
| 542 |
|
NVE (see orange curve in Fig.~\ref{langevin:temperature}) which |
| 543 |
|
loses the temperature control ability. |
| 544 |
|
|
| 564 |
|
made of 2266 small identical beads with size of 0.3 \AA on the |
| 565 |
|
surface. Applying the procedure described in |
| 566 |
|
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
| 567 |
< |
identified the center of resistance at $(0\AA, 0.7482\AA, |
| 568 |
< |
-0.1988\AA)$, as well as the resistance tensor, |
| 567 |
> |
identified the center of resistance at (0 $\rm{\AA}$, 0.7482 $\rm{\AA}$, |
| 568 |
> |
-0.1988 $\rm{\AA}$), as well as the resistance tensor, |
| 569 |
|
\[ |
| 570 |
|
\left( {\begin{array}{*{20}c} |
| 571 |
|
0.9261 & 0 & 0&0&0.08585&0.2057\\ |
| 572 |
|
0& 0.9270&-0.007063& 0.08585&0&0\\ |
| 573 |
< |
0&0.007063&0.7494&0.2057&0&0\\ |
| 574 |
< |
0&0.0858&0.2057& 58.64& 0&-8.5736\\ |
| 573 |
> |
0&-0.007063&0.7494&0.2057&0&0\\ |
| 574 |
> |
0&0.0858&0.2057& 58.64& 0&0\\ |
| 575 |
|
0.08585&0&0&0&48.30&3.219&\\ |
| 576 |
|
0.2057&0&0&0&3.219&10.7373\\ |
| 577 |
|
\end{array}} \right). |
| 578 |
|
\] |
| 579 |
< |
%\[ |
| 581 |
< |
%\left( {\begin{array}{*{20}c} |
| 582 |
< |
%0.9261 & 1.310e-14 & -7.292e-15&5.067e-14&0.08585&0.2057\\ |
| 583 |
< |
%3.968e-14& 0.9270&-0.007063& 0.08585&6.764e-14&4.846e-14\\ |
| 584 |
< |
%-6.561e-16&-0.007063&0.7494&0.2057&4.846e-14&1.5036e-14\\ |
| 585 |
< |
%5.067e-14&0.0858&0.2057& 58.64& 8.563e-13&-8.5736\\ |
| 586 |
< |
%0.08585&6.764e-14&4.846e-14&1.555e-12&48.30&3.219&\\ |
| 587 |
< |
%0.2057&4.846e-14&1.5036e-14&-3.904e-13&3.219&10.7373\\ |
| 588 |
< |
%\end{array}} \right). |
| 589 |
< |
%\] |
| 579 |
> |
where the units for translational, translation-rotation coupling and rotational tensors are $\frac{kcal \cdot fs}{mol \cdot \rm{\AA}^2}$, $\frac{kcal \cdot fs}{mol \cdot \rm{\AA} \cdot rad}$ and $\frac{kcal \cdot fs}{mol \cdot rad^2}$ respectively. |
| 580 |
|
|
| 581 |
|
Curves of the velocity auto-correlation functions in |
| 582 |
|
Fig.~\ref{langevin:vacf} were shown to match each other very well. |
| 583 |
|
However, because of the stochastic nature, simulation using Langevin |
| 584 |
|
dynamics was shown to decay slightly faster than MD. In order to |
| 585 |
|
study the rotational motion of the molecules, we also calculated the |
| 586 |
< |
auto- correlation function of the principle axis of the second GB |
| 586 |
> |
auto-correlation function of the principle axis of the second GB |
| 587 |
|
particle, $u$. The discrepancy shown in Fig.~\ref{langevin:uacf} was |
| 588 |
< |
probably due to the reason that the viscosity using in the |
| 599 |
< |
simulations only partially preserved the dynamics of the system. |
| 588 |
> |
probably due to the reason that we used the experimental viscosity directly instead of calculating bulk viscosity from simulation. |
| 589 |
|
|
| 590 |
|
\begin{figure} |
| 591 |
|
\centering |
| 606 |
|
\centering |
| 607 |
|
\includegraphics[width=\linewidth]{vacf.eps} |
| 608 |
|
\caption[Plots of Velocity Auto-correlation Functions]{Velocity |
| 609 |
< |
auto-correlation functions of NVE (explicit solvent) in blue) and |
| 609 |
> |
auto-correlation functions of NVE (explicit solvent) in blue and |
| 610 |
|
Langevin dynamics (implicit solvent) in red.} \label{langevin:vacf} |
| 611 |
|
\end{figure} |
| 612 |
|
|
