| 4 |
|
\section{Introduction} |
| 5 |
|
|
| 6 |
|
%applications of langevin dynamics |
| 7 |
< |
As an excellent alternative to newtonian dynamics, Langevin |
| 8 |
< |
dynamics, which mimics a simple heat bath with stochastic and |
| 9 |
< |
dissipative forces, has been applied in a variety of studies. The |
| 10 |
< |
stochastic treatment of the solvent enables us to carry out |
| 11 |
< |
substantially longer time simulation. Implicit solvent Langevin |
| 12 |
< |
dynamics simulation of met-enkephalin not only outperforms explicit |
| 13 |
< |
solvent simulation on computation efficiency, but also agrees very |
| 14 |
< |
well with explicit solvent simulation on dynamics |
| 15 |
< |
properties\cite{Shen2002}. Recently, applying Langevin dynamics with |
| 16 |
< |
UNRES model, Liow and his coworkers suggest that protein folding |
| 17 |
< |
pathways can be possibly exploited within a reasonable amount of |
| 18 |
< |
time\cite{Liwo2005}. The stochastic nature of the Langevin dynamics |
| 19 |
< |
also enhances the sampling of the system and increases the |
| 20 |
< |
probability of crossing energy barrier\cite{Banerjee2004, Cui2003}. |
| 21 |
< |
Combining Langevin dynamics with Kramers's theory, Klimov and |
| 22 |
< |
Thirumalai identified the free-energy barrier by studying the |
| 23 |
< |
viscosity dependence of the protein folding rates\cite{Klimov1997}. |
| 24 |
< |
In order to account for solvent induced interactions missing from |
| 25 |
< |
implicit solvent model, Kaya incorporated desolvation free energy |
| 26 |
< |
barrier into implicit coarse-grained solvent model in protein |
| 27 |
< |
folding/unfolding study and discovered a higher free energy barrier |
| 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{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}. |
| 7 |
> |
As alternative to Newtonian dynamics, Langevin dynamics, which |
| 8 |
> |
mimics a simple heat bath with stochastic and dissipative forces, |
| 9 |
> |
has been applied in a variety of studies. The stochastic treatment |
| 10 |
> |
of the solvent enables us to carry out substantially longer time |
| 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}. |
| 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 |
| 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 |
| 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 |
| 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 |
| 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}. |
| 35 |
|
|
| 36 |
|
%review langevin/browninan dynamics for arbitrarily shaped rigid body |
| 37 |
|
Combining Langevin or Brownian dynamics with rigid body dynamics, |
| 38 |
< |
one can study the slow processes in biomolecular systems. Modeling |
| 39 |
< |
the DNA as a chain of rigid spheres beads, which subject to harmonic |
| 40 |
< |
potentials as well as excluded volume potentials, Mielke and his |
| 41 |
< |
coworkers discover rapid superhelical stress generations from the |
| 42 |
< |
stochastic simulation of twin supercoiling DNA with response to |
| 43 |
< |
induced torques\cite{Mielke2004}. Membrane fusion is another key |
| 44 |
< |
biological process which controls a variety of physiological |
| 45 |
< |
functions, such as release of neurotransmitters \textit{etc}. A |
| 46 |
< |
typical fusion event happens on the time scale of millisecond, which |
| 47 |
< |
is impracticable to study using all atomistic model with newtonian |
| 48 |
< |
mechanics. With the help of coarse-grained rigid body model and |
| 49 |
< |
stochastic dynamics, the fusion pathways were exploited by many |
| 38 |
> |
one can study slow processes in biomolecular systems. Modeling DNA |
| 39 |
> |
as a chain of rigid beads, which are subject to harmonic potentials |
| 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 |
| 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 |
| 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 |
| 51 |
< |
difficulty of numerical integration of anisotropy rotation, most of |
| 52 |
< |
the rigid body models are simply modeled by sphere, cylinder, |
| 53 |
< |
ellipsoid or other regular shapes in stochastic simulations. In an |
| 54 |
< |
effort to account for the diffusion anisotropy of the arbitrary |
| 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 |
| 54 |
> |
effort to account for the diffusion anisotropy of arbitrary |
| 55 |
|
particles, Fernandes and de la Torre improved the original Brownian |
| 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 |
| 60 |
< |
error and bias are introduced into the system due to the arbitrary |
| 61 |
< |
order of applying the noncommuting rotation |
| 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 |
| 63 |
|
relaxation time is much less than the time step, one may ignore the |
| 64 |
< |
inertia in Brownian dynamics. However, assumption of the zero |
| 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 |
| 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, |
| 76 |
< |
typical nonskew bodies like cylinder and ellipsoid are inadequate to |
| 77 |
< |
represent most complex macromolecule assemblies. These intricate |
| 76 |
> |
typical nonskew bodies like cylinders and ellipsoids are inadequate |
| 77 |
> |
to represent most complex macromolecule assemblies. These intricate |
| 78 |
|
molecules have been represented by a set of beads and their |
| 79 |
< |
hydrodynamics properties can be calculated using variant |
| 80 |
< |
hydrodynamic interaction tensors. |
| 79 |
> |
hydrodynamic properties can be calculated using variants on the |
| 80 |
> |
standard hydrodynamic interaction tensors. |
| 81 |
|
|
| 82 |
|
The goal of the present work is to develop a Langevin dynamics |
| 83 |
< |
algorithm for arbitrary rigid particles by integrating the accurate |
| 84 |
< |
estimation of friction tensor from hydrodynamics theory into the |
| 85 |
< |
sophisticated rigid body dynamics. |
| 83 |
> |
algorithm for arbitrary-shaped rigid particles by integrating the |
| 84 |
> |
accurate estimation of friction tensor from hydrodynamics theory |
| 85 |
> |
into the sophisticated rigid body dynamics algorithms. |
| 86 |
|
|
| 87 |
|
\section{Computational Methods{\label{methodSec}}} |
| 88 |
|
|
| 89 |
|
\subsection{\label{introSection:frictionTensor}Friction Tensor} |
| 90 |
|
Theoretically, the friction kernel can be determined using the |
| 91 |
< |
velocity autocorrelation function. However, this approach become |
| 92 |
< |
impractical when the system become more and more complicate. |
| 91 |
> |
velocity autocorrelation function. However, this approach becomes |
| 92 |
> |
impractical when the system becomes more and more complicated. |
| 93 |
|
Instead, various approaches based on hydrodynamics have been |
| 94 |
< |
developed to calculate the friction coefficients. In general, |
| 94 |
> |
developed to calculate the friction coefficients. In general, the |
| 95 |
|
friction tensor $\Xi$ is a $6\times 6$ matrix given by |
| 96 |
|
\[ |
| 97 |
|
\Xi = \left( {\begin{array}{*{20}c} |
| 99 |
|
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
| 100 |
|
\end{array}} \right). |
| 101 |
|
\] |
| 102 |
< |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
| 103 |
< |
tensor and rotational resistance (friction) tensor respectively, |
| 104 |
< |
while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ |
| 105 |
< |
{\Xi^{rt} }$ is rotation-translation coupling tensor. When a |
| 106 |
< |
particle moves in a fluid, it may experience friction force or |
| 107 |
< |
torque along the opposite direction of the velocity or angular |
| 108 |
< |
velocity, |
| 102 |
> |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are $3 \times 3$ |
| 103 |
> |
translational friction tensor and rotational resistance (friction) |
| 104 |
> |
tensor respectively, while ${\Xi^{tr} }$ is translation-rotation |
| 105 |
> |
coupling tensor and $ {\Xi^{rt} }$ is rotation-translation coupling |
| 106 |
> |
tensor. When a particle moves in a fluid, it may experience friction |
| 107 |
> |
force or torque along the opposite direction of the velocity or |
| 108 |
> |
angular velocity, |
| 109 |
|
\[ |
| 110 |
|
\left( \begin{array}{l} |
| 111 |
|
F_R \\ |
| 119 |
|
\end{array} \right) |
| 120 |
|
\] |
| 121 |
|
where $F_r$ is the friction force and $\tau _R$ is the friction |
| 122 |
< |
toque. |
| 122 |
> |
torque. |
| 123 |
|
|
| 124 |
< |
\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shape}} |
| 124 |
> |
\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shapes}} |
| 125 |
|
|
| 126 |
|
For a spherical particle with slip boundary conditions, the |
| 127 |
|
translational and rotational friction constant can be calculated |
| 142 |
|
\end{array}} \right) |
| 143 |
|
\] |
| 144 |
|
where $\eta$ is the viscosity of the solvent and $R$ is the |
| 145 |
< |
hydrodynamics radius. |
| 145 |
> |
hydrodynamic radius. |
| 146 |
|
|
| 147 |
< |
Other non-spherical shape, such as cylinder and ellipsoid |
| 148 |
< |
\textit{etc}, are widely used as reference for developing new |
| 149 |
< |
hydrodynamics theory, because their properties can be calculated |
| 150 |
< |
exactly. In 1936, Perrin extended Stokes's law to general |
| 151 |
< |
ellipsoids, also called a triaxial ellipsoid, which is given in |
| 152 |
< |
Cartesian coordinates by\cite{Perrin1934, Perrin1936} |
| 147 |
> |
Other non-spherical shapes, such as cylinders and ellipsoids, are |
| 148 |
> |
widely used as references for developing new hydrodynamics theory, |
| 149 |
> |
because their properties can be calculated exactly. In 1936, Perrin |
| 150 |
> |
extended Stokes's law to general ellipsoids, also called a triaxial |
| 151 |
> |
ellipsoid, which is given in Cartesian coordinates |
| 152 |
> |
by\cite{Perrin1934, Perrin1936} |
| 153 |
|
\[ |
| 154 |
|
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
| 155 |
|
}} = 1 |
| 156 |
|
\] |
| 157 |
|
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
| 158 |
|
due to the complexity of the elliptic integral, only the ellipsoid |
| 159 |
< |
with the restriction of two axes having to be equal, \textit{i.e.} |
| 159 |
> |
with the restriction of two axes being equal, \textit{i.e.} |
| 160 |
|
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
| 161 |
|
exactly. Introducing an elliptic integral parameter $S$ for prolate |
| 162 |
< |
: |
| 162 |
> |
ellipsoids : |
| 163 |
|
\[ |
| 164 |
|
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 165 |
|
} }}{b}, |
| 166 |
|
\] |
| 167 |
< |
and oblate : |
| 167 |
> |
and oblate ellipsoids: |
| 168 |
|
\[ |
| 169 |
|
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 170 |
< |
}}{a}. |
| 170 |
> |
}}{a}, |
| 171 |
|
\] |
| 172 |
|
one can write down the translational and rotational resistance |
| 173 |
|
tensors |
| 174 |
< |
\[ |
| 188 |
< |
\begin{array}{l} |
| 174 |
> |
\begin{eqnarray*} |
| 175 |
|
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
| 176 |
|
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
| 177 |
|
2a}}, |
| 178 |
< |
\end{array} |
| 193 |
< |
\] |
| 178 |
> |
\end{eqnarray*} |
| 179 |
|
and |
| 180 |
< |
\[ |
| 196 |
< |
\begin{array}{l} |
| 180 |
> |
\begin{eqnarray*} |
| 181 |
|
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
| 182 |
< |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
| 183 |
< |
\end{array}. |
| 200 |
< |
\] |
| 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 |
|
|
| 192 |
|
from all possible ellipsoidal spaces, $r$-space, to all possible |
| 193 |
|
combination of rotational diffusion coefficients, $D$-space, is not |
| 194 |
|
unique\cite{Wegener1979} as well as the intrinsic coupling between |
| 195 |
< |
translational and rotational motion of rigid body, general |
| 195 |
> |
translational and rotational motion of rigid bodies, general |
| 196 |
|
ellipsoids are not always suitable for modeling arbitrarily shaped |
| 197 |
< |
rigid molecule. A number of studies have been devoted to determine |
| 198 |
< |
the friction tensor for irregularly shaped rigid bodies using more |
| 199 |
< |
advanced method where the molecule of interest was modeled by |
| 200 |
< |
combinations of spheres(beads)\cite{Carrasco1999} and the |
| 197 |
> |
rigid molecules. A number of studies have been devoted to |
| 198 |
> |
determining the friction tensor for irregularly shaped rigid bodies |
| 199 |
> |
using more advanced methods where the molecule of interest was |
| 200 |
> |
modeled by a combinations of spheres\cite{Carrasco1999} and the |
| 201 |
|
hydrodynamics properties of the molecule can be calculated using the |
| 202 |
|
hydrodynamic interaction tensor. Let us consider a rigid assembly of |
| 203 |
< |
$N$ beads immersed in a continuous medium. Due to hydrodynamics |
| 203 |
> |
$N$ beads immersed in a continuous medium. Due to hydrodynamic |
| 204 |
|
interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different |
| 205 |
|
than its unperturbed velocity $v_i$, |
| 206 |
|
\[ |
| 256 |
|
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
| 257 |
|
)T_{ij} |
| 258 |
|
\] |
| 259 |
< |
where $\delta _{ij}$ is Kronecker delta function. Inverting the $B$ |
| 260 |
< |
matrix, we obtain |
| 259 |
> |
where $\delta _{ij}$ is the Kronecker delta function. Inverting the |
| 260 |
> |
$B$ matrix, we obtain |
| 261 |
|
\[ |
| 262 |
|
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 263 |
|
{C_{11} } & \ldots & {C_{1N} } \\ |
| 266 |
|
\end{array}} \right), |
| 267 |
|
\] |
| 268 |
|
which can be partitioned into $N \times N$ $3 \times 3$ block |
| 269 |
< |
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
| 269 |
> |
$C_{ij}$. With the help of $C_{ij}$ and the skew matrix $U_i$ |
| 270 |
|
\[ |
| 271 |
|
U_i = \left( {\begin{array}{*{20}c} |
| 272 |
|
0 & { - z_i } & {y_i } \\ |
| 275 |
|
\end{array}} \right) |
| 276 |
|
\] |
| 277 |
|
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
| 278 |
< |
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
| 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 , \\ |
| 283 |
|
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\ |
| 284 |
|
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 285 |
|
\end{eqnarray} |
| 303 |
– |
|
| 286 |
|
The resistance tensor depends on the origin to which they refer. The |
| 287 |
< |
proper location for applying friction force is the center of |
| 287 |
> |
proper location for applying the friction force is the center of |
| 288 |
|
resistance (or center of reaction), at which the trace of rotational |
| 289 |
|
resistance tensor, $ \Xi ^{rr}$ reaches a minimum value. |
| 290 |
|
Mathematically, the center of resistance is defined as an unique |
| 291 |
|
point of the rigid body at which the translation-rotation coupling |
| 292 |
< |
tensor are symmetric, |
| 292 |
> |
tensors are symmetric, |
| 293 |
|
\begin{equation} |
| 294 |
|
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
| 295 |
|
\label{introEquation:definitionCR} |
| 296 |
|
\end{equation} |
| 297 |
|
From Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 298 |
< |
we can easily find out that the translational resistance tensor is |
| 298 |
> |
we can easily derive that the translational resistance tensor is |
| 299 |
|
origin independent, while the rotational resistance tensor and |
| 300 |
|
translation-rotation coupling resistance tensor depend on the |
| 301 |
|
origin. Given the resistance tensor at an arbitrary origin $O$, and |
| 341 |
|
|
| 342 |
|
\subsection{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}} |
| 343 |
|
|
| 344 |
< |
Consider a Langevin equation of motions in generalized coordinates |
| 344 |
> |
Consider the Langevin equations of motion in generalized coordinates |
| 345 |
|
\begin{equation} |
| 346 |
|
M_i \dot V_i (t) = F_{s,i} (t) + F_{f,i(t)} + F_{r,i} (t) |
| 347 |
|
\label{LDGeneralizedForm} |
| 358 |
|
in body-fixed frame and converted back to lab-fixed frame by: |
| 359 |
|
\[ |
| 360 |
|
\begin{array}{l} |
| 361 |
< |
F_{f,i}^l (t) = A^T F_{f,i}^b (t), \\ |
| 362 |
< |
F_{r,i}^l (t) = A^T F_{r,i}^b (t). \\ |
| 361 |
> |
F_{f,i}^l (t) = Q^T F_{f,i}^b (t), \\ |
| 362 |
> |
F_{r,i}^l (t) = Q^T F_{r,i}^b (t). \\ |
| 363 |
|
\end{array} |
| 364 |
|
\] |
| 365 |
|
Here, the body-fixed friction force $F_{r,i}^b$ is proportional to |
| 398 |
|
\tau _{r,i}^b = \tau _{r,i}^b +r_{MR} \times f_{r,i}^b |
| 399 |
|
\end{equation} |
| 400 |
|
where $r_{MR}$ is the vector from the center of mass to the center |
| 401 |
< |
of the resistance. Instead of integrating angular velocity in |
| 402 |
< |
lab-fixed frame, we consider the equation of motion of angular |
| 403 |
< |
momentum in body-fixed frame |
| 401 |
> |
of the resistance. Instead of integrating the angular velocity in |
| 402 |
> |
lab-fixed frame, we consider the equation of angular momentum in |
| 403 |
> |
body-fixed frame |
| 404 |
|
\begin{equation} |
| 405 |
|
\dot j_i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b (t) |
| 406 |
|
+ \tau _{r,i}^b(t) |
| 421 |
|
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
| 422 |
|
+ \frac{h}{2} {\bf \tau}^b(t), \\ |
| 423 |
|
% |
| 424 |
< |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
| 424 |
> |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
| 425 |
|
(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
| 426 |
|
\end{align*} |
| 427 |
|
In this context, the $\mathrm{rotate}$ function is the reversible |
| 432 |
|
/ 2) \cdot \mathsf{G}_x(a_x /2), |
| 433 |
|
\end{equation} |
| 434 |
|
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
| 435 |
< |
rotates both the rotation matrix ($\mathsf{A}$) and the body-fixed |
| 435 |
> |
rotates both the rotation matrix ($\mathsf{Q}$) and the body-fixed |
| 436 |
|
angular momentum (${\bf j}$) by an angle $\theta$ around body-fixed |
| 437 |
|
axis $\alpha$, |
| 438 |
|
\begin{equation} |
| 439 |
|
\mathsf{G}_\alpha( \theta ) = \left\{ |
| 440 |
|
\begin{array}{lcl} |
| 441 |
< |
\mathsf{A}(t) & \leftarrow & \mathsf{A}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
| 441 |
> |
\mathsf{Q}(t) & \leftarrow & \mathsf{Q}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
| 442 |
|
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
| 443 |
|
j}(0). |
| 444 |
|
\end{array} |
| 470 |
|
{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h) |
| 471 |
|
\times \frac{\partial V}{\partial {\bf u}}, \\ |
| 472 |
|
% |
| 473 |
< |
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) |
| 473 |
> |
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{Q}(t + h) |
| 474 |
|
\cdot {\bf \tau}^s(t + h). |
| 475 |
|
\end{align*} |
| 476 |
|
Once the forces and torques have been obtained at the new time step, |
| 490 |
|
\section{Results and Discussion} |
| 491 |
|
|
| 492 |
|
The Langevin algorithm described in previous section has been |
| 493 |
< |
implemented in {\sc oopse}\cite{Meineke2005} and applied to the |
| 494 |
< |
studies of kinetics and thermodynamic properties in several systems. |
| 493 |
> |
implemented in {\sc oopse}\cite{Meineke2005} and applied to studies |
| 494 |
> |
of the static and dynamic properties in several systems. |
| 495 |
|
|
| 496 |
|
\subsection{Temperature Control} |
| 497 |
|
|
| 502 |
|
the algorithm. In order to verify the stability of this new |
| 503 |
|
algorithm, a series of simulations are performed on system |
| 504 |
|
consisiting of 256 SSD water molecules with different viscosities. |
| 505 |
< |
Initial configuration for the simulations is taken from a 1ns NVT |
| 506 |
< |
simulation with a cubic box of 19.7166~\AA. All simulation are |
| 505 |
> |
The initial configuration for the simulations is taken from a 1ns |
| 506 |
> |
NVT simulation with a cubic box of 19.7166~\AA. All simulation are |
| 507 |
|
carried out with cutoff radius of 9~\AA and 2 fs time step for 1 ns |
| 508 |
< |
with reference temperature at 300~K. Average temperature as a |
| 508 |
> |
with reference temperature at 300~K. The average temperature as a |
| 509 |
|
function of $\eta$ is shown in Table \ref{langevin:viscosity} where |
| 510 |
|
the temperatures range from 303.04~K to 300.47~K for $\eta = 0.01 - |
| 511 |
|
1$ poise. The better temperature control at higher viscosity can be |
| 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 |
| 543 |
< |
NVE (see green curve in Fig.~\ref{langevin:temperature}) which loses |
| 544 |
< |
the temperature control ability. |
| 543 |
> |
NVE (see orange curve in Fig.~\ref{langevin:temperature}) which |
| 544 |
> |
loses the temperature control ability. |
| 545 |
|
|
| 546 |
|
\begin{figure} |
| 547 |
|
\centering |
| 560 |
|
273~K. It has an equivalent implicit solvent system containing only |
| 561 |
|
two banana shaped molecules with viscosity of 0.289 center poise. To |
| 562 |
|
calculate the hydrodynamic properties of the banana shaped molecule, |
| 563 |
< |
we create a rough shell model (see Fig.~\ref{langevin:roughShell}), |
| 563 |
> |
we created a rough shell model (see Fig.~\ref{langevin:roughShell}), |
| 564 |
|
in which the banana shaped molecule is represented as a ``shell'' |
| 565 |
|
made of 2266 small identical beads with size of 0.3 \AA on the |
| 566 |
|
surface. Applying the procedure described in |
| 567 |
|
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
| 568 |
< |
identified the center of resistance at $(0, 0.7482, -0.1988)$, as |
| 569 |
< |
well as the resistance tensor, |
| 568 |
> |
identified the center of resistance at $(0\AA, 0.7482\AA, |
| 569 |
> |
-0.1988\AA)$, as well as the resistance tensor, |
| 570 |
|
\[ |
| 571 |
|
\left( {\begin{array}{*{20}c} |
| 572 |
|
0.9261 & 0 & 0&0&0.08585&0.2057\\ |
| 588 |
|
%\end{array}} \right). |
| 589 |
|
%\] |
| 590 |
|
|
| 591 |
< |
Curves of velocity auto-correlation functions in |
| 591 |
> |
Curves of the velocity auto-correlation functions in |
| 592 |
|
Fig.~\ref{langevin:vacf} were shown to match each other very well. |
| 593 |
|
However, because of the stochastic nature, simulation using Langevin |
| 594 |
< |
dynamics was shown to decay slightly fast. In order to study the |
| 595 |
< |
rotational motion of the molecules, we also calculated the auto- |
| 596 |
< |
correlation function of the principle axis of the second GB |
| 597 |
< |
particle, $u$. |
| 594 |
> |
dynamics was shown to decay slightly faster than MD. In order to |
| 595 |
> |
study the rotational motion of the molecules, we also calculated the |
| 596 |
> |
auto- correlation function of the principle axis of the second GB |
| 597 |
> |
particle, $u$. The discrepancy shown in Fig.~\ref{langevin:uacf} was |
| 598 |
> |
probably due to the reason that the viscosity using in the |
| 599 |
> |
simulations only partially preserved the dynamics of the system. |
| 600 |
|
|
| 601 |
|
\begin{figure} |
| 602 |
|
\centering |
| 617 |
|
\centering |
| 618 |
|
\includegraphics[width=\linewidth]{vacf.eps} |
| 619 |
|
\caption[Plots of Velocity Auto-correlation Functions]{Velocity |
| 620 |
< |
auto-correlation functions in NVE (blue) and Langevin dynamics |
| 621 |
< |
(red).} \label{langevin:vacf} |
| 620 |
> |
auto-correlation functions of NVE (explicit solvent) in blue) and |
| 621 |
> |
Langevin dynamics (implicit solvent) in red.} \label{langevin:vacf} |
| 622 |
|
\end{figure} |
| 623 |
|
|
| 624 |
|
\begin{figure} |
| 626 |
|
\includegraphics[width=\linewidth]{uacf.eps} |
| 627 |
|
\caption[Auto-correlation functions of the principle axis of the |
| 628 |
|
middle GB particle]{Auto-correlation functions of the principle axis |
| 629 |
< |
of the middle GB particle in NVE (blue) and Langevin dynamics |
| 630 |
< |
(red).} \label{langevin:twoBanana} |
| 629 |
> |
of the middle GB particle of NVE (blue) and Langevin dynamics |
| 630 |
> |
(red).} \label{langevin:uacf} |
| 631 |
|
\end{figure} |
| 632 |
|
|
| 633 |
|
\section{Conclusions} |
