| 48 |
|
\section{Introduction} |
| 49 |
|
|
| 50 |
|
%applications of langevin dynamics |
| 51 |
< |
As alternative to Newtonian dynamics, Langevin dynamics, which |
| 52 |
< |
mimics a simple heat bath with stochastic and dissipative forces, |
| 53 |
< |
has been applied in a variety of studies. The stochastic treatment |
| 54 |
< |
of the solvent enables us to carry out substantially longer time |
| 55 |
< |
simulations. Implicit solvent Langevin dynamics simulations of |
| 56 |
< |
met-enkephalin not only outperform explicit solvent simulations for |
| 57 |
< |
computational efficiency, but also agrees very well with explicit |
| 58 |
< |
solvent simulations for dynamical properties.\cite{Shen2002} |
| 59 |
< |
Recently, applying Langevin dynamics with the UNRES model, Liow and |
| 60 |
< |
his coworkers suggest that protein folding pathways can be possibly |
| 61 |
< |
explored within a reasonable amount of time.\cite{Liwo2005} The |
| 62 |
< |
stochastic nature of the Langevin dynamics also enhances the |
| 63 |
< |
sampling of the system and increases the probability of crossing |
| 64 |
< |
energy barriers.\cite{Banerjee2004, Cui2003} Combining Langevin |
| 65 |
< |
dynamics with Kramers's theory, Klimov and Thirumalai identified |
| 66 |
< |
free-energy barriers by studying the viscosity dependence of the |
| 67 |
< |
protein folding rates.\cite{Klimov1997} In order to account for |
| 68 |
< |
solvent induced interactions missing from implicit solvent model, |
| 69 |
< |
Kaya incorporated desolvation free energy barrier into implicit |
| 70 |
< |
coarse-grained solvent model in protein folding/unfolding studies |
| 71 |
< |
and discovered a higher free energy barrier between the native and |
| 72 |
< |
denatured states. Because of its stability against noise, Langevin |
| 73 |
< |
dynamics is very suitable for studying remagnetization processes in |
| 74 |
< |
various systems.\cite{Palacios1998,Berkov2002,Denisov2003} For |
| 51 |
> |
Langevin dynamics, which mimics a simple heat bath with stochastic and |
| 52 |
> |
dissipative forces, has been applied in a variety of situations as an |
| 53 |
> |
alternative to molecular dynamics with explicit solvent molecules. |
| 54 |
> |
The stochastic treatment of the solvent allows the use of simulations |
| 55 |
> |
with substantially longer time and length scales. In general, the |
| 56 |
> |
dynamic and structural properties obtained from Langevin simulations |
| 57 |
> |
agree quite well with similar properties obtained from explicit |
| 58 |
> |
solvent simulations. |
| 59 |
> |
|
| 60 |
> |
Recent examples of the usefulness of Langevin simulations include a |
| 61 |
> |
study of met-enkephalin in which Langevin simulations predicted |
| 62 |
> |
dynamical properties that were largely in agreement with explicit |
| 63 |
> |
solvent simulations.\cite{Shen2002} By applying Langevin dynamics with |
| 64 |
> |
the UNRES model, Liow and his coworkers suggest that protein folding |
| 65 |
> |
pathways can be explored within a reasonable amount of |
| 66 |
> |
time.\cite{Liwo2005} |
| 67 |
> |
|
| 68 |
> |
The stochastic nature of Langevin dynamics also enhances the sampling |
| 69 |
> |
of the system and increases the probability of crossing energy |
| 70 |
> |
barriers.\cite{Cui2003,Banerjee2004} Combining Langevin dynamics with |
| 71 |
> |
Kramers's theory, Klimov and Thirumalai identified free-energy |
| 72 |
> |
barriers by studying the viscosity dependence of the protein folding |
| 73 |
> |
rates.\cite{Klimov1997} In order to account for solvent induced |
| 74 |
> |
interactions missing from the implicit solvent model, Kaya |
| 75 |
> |
incorporated a desolvation free energy barrier into protein |
| 76 |
> |
folding/unfolding studies and discovered a higher free energy barrier |
| 77 |
> |
between the native and denatured states.\cite{XXX} |
| 78 |
> |
|
| 79 |
> |
Because of its stability against noise, Langevin dynamics has also |
| 80 |
> |
proven useful for studying remagnetization processes in various |
| 81 |
> |
systems.\cite{Palacios1998,Berkov2002,Denisov2003} [Check: For |
| 82 |
|
instance, the oscillation power spectrum of nanoparticles from |
| 83 |
< |
Langevin dynamics simulation has the same peak frequencies for |
| 84 |
< |
different wave vectors, which recovers the property of magnetic |
| 85 |
< |
excitations in small finite structures.\cite{Berkov2005a} |
| 83 |
> |
Langevin dynamics has the same peak frequencies for different wave |
| 84 |
> |
vectors, which recovers the property of magnetic excitations in small |
| 85 |
> |
finite structures.\cite{Berkov2005a}] |
| 86 |
|
|
| 87 |
< |
%review rigid body dynamics |
| 88 |
< |
Rigid bodies are frequently involved in the modeling of different |
| 89 |
< |
areas, from engineering, physics, to chemistry. For example, |
| 90 |
< |
missiles and vehicle are usually modeled by rigid bodies. The |
| 91 |
< |
movement of the objects in 3D gaming engine or other physics |
| 92 |
< |
simulator is governed by the rigid body dynamics. In molecular |
| 93 |
< |
simulation, rigid body is used to simplify the model in |
| 94 |
< |
protein-protein docking study{\cite{Gray2003}}. |
| 87 |
> |
In typical LD simulations, the friction and random forces on |
| 88 |
> |
individual atoms are taken from the Stokes-Einstein hydrodynamic |
| 89 |
> |
approximation, |
| 90 |
> |
\begin{eqnarray} |
| 91 |
> |
m \dot{v}(t) & = & -\nabla U(x) - \xi m v(t) + R(t) \\ |
| 92 |
> |
\langle R(t) \rangle & = & 0 \\ |
| 93 |
> |
\langle R(t) R(t') \rangle & = & 2 k_B T \xi m \delta(t - t') |
| 94 |
> |
\end{eqnarray} |
| 95 |
> |
where $\xi \approx 6 \pi \eta a$. Here $\eta$ is the viscosity of the |
| 96 |
> |
implicit solvent, and $a$ is the hydrodynamic radius of the atom. |
| 97 |
|
|
| 98 |
< |
It is very important to develop stable and efficient methods to |
| 99 |
< |
integrate the equations of motion for orientational degrees of |
| 100 |
< |
freedom. Euler angles are the natural choice to describe the |
| 101 |
< |
rotational degrees of freedom. However, due to $\frac {1}{sin |
| 102 |
< |
\theta}$ singularities, the numerical integration of corresponding |
| 103 |
< |
equations of these motion is very inefficient and inaccurate. |
| 104 |
< |
Although an alternative integrator using multiple sets of Euler |
| 105 |
< |
angles can overcome this difficulty\cite{Barojas1973}, the |
| 106 |
< |
computational penalty and the loss of angular momentum conservation |
| 107 |
< |
still remain. A singularity-free representation utilizing |
| 108 |
< |
quaternions was developed by Evans in 1977.\cite{Evans1977} |
| 109 |
< |
Unfortunately, this approach used a nonseparable Hamiltonian |
| 110 |
< |
resulting from the quaternion representation, which prevented the |
| 111 |
< |
symplectic algorithm from being utilized. Another different approach |
| 112 |
< |
is to apply holonomic constraints to the atoms belonging to the |
| 104 |
< |
rigid body. Each atom moves independently under the normal forces |
| 105 |
< |
deriving from potential energy and constraint forces which are used |
| 106 |
< |
to guarantee the rigidness. However, due to their iterative nature, |
| 107 |
< |
the SHAKE and Rattle algorithms also converge very slowly when the |
| 108 |
< |
number of constraints increases.\cite{Ryckaert1977, Andersen1983} |
| 98 |
> |
The use of rigid substructures,\cite{???} |
| 99 |
> |
coarse-graining,\cite{Ayton,Sun,Zannoni} and ellipsoidal |
| 100 |
> |
representations of protein side chains~\cite{Schulten} has made the |
| 101 |
> |
use of the Stokes-Einstein approximation problematic. A rigid |
| 102 |
> |
substructure moves as a single unit with orientational as well as |
| 103 |
> |
translational degrees of freedom. This requires a more general |
| 104 |
> |
treatment of the hydrodynamics than the spherical approximation |
| 105 |
> |
provides. The atoms involved in a rigid or coarse-grained structure |
| 106 |
> |
should properly have solvent-mediated interactions with each |
| 107 |
> |
other. The theory of interactions {\it between} bodies moving through |
| 108 |
> |
a fluid has been developed over the past century and has been applied |
| 109 |
> |
to simulations of Brownian |
| 110 |
> |
motion.\cite{MarshallNewton,GarciaDeLaTorre} There a need to have a |
| 111 |
> |
more thorough treatment of hydrodynamics included in the library of |
| 112 |
> |
methods available for performing Langevin simulations. |
| 113 |
|
|
| 114 |
< |
A break-through in geometric literature suggests that, in order to |
| 114 |
> |
\subsection{Rigid Body Dynamics} |
| 115 |
> |
Rigid bodies are frequently involved in the modeling of large |
| 116 |
> |
collections of particles that move as a single unit. In molecular |
| 117 |
> |
simulations, rigid bodies have been used to simplify protein-protein |
| 118 |
> |
docking,\cite{Gray2003} and lipid bilayer simulations.\cite{Sun2008} |
| 119 |
> |
Many of the water models in common use are also rigid-body |
| 120 |
> |
models,\cite{TIPs,SPC/E} although they are typically evolved using |
| 121 |
> |
constraints rather than rigid body equations of motion. |
| 122 |
> |
|
| 123 |
> |
Euler angles are a natural choice to describe the rotational |
| 124 |
> |
degrees of freedom. However, due to $1 \over \sin \theta$ |
| 125 |
> |
singularities, the numerical integration of corresponding equations of |
| 126 |
> |
these motion can become inaccurate (and inefficient). Although an |
| 127 |
> |
alternative integrator using multiple sets of Euler angles can |
| 128 |
> |
overcome this problem,\cite{Barojas1973} the computational penalty and |
| 129 |
> |
the loss of angular momentum conservation remain. A singularity-free |
| 130 |
> |
representation utilizing quaternions was developed by Evans in |
| 131 |
> |
1977.\cite{Evans1977} Unfortunately, this approach uses a nonseparable |
| 132 |
> |
Hamiltonian resulting from the quaternion representation, which |
| 133 |
> |
prevented symplectic algorithms from being utilized until very |
| 134 |
> |
recently.\cite{Miller2002} Another approach is the application of |
| 135 |
> |
holonomic constraints to the atoms belonging to the rigid body. Each |
| 136 |
> |
atom moves independently under the normal forces deriving from |
| 137 |
> |
potential energy and constraint forces which are used to guarantee the |
| 138 |
> |
rigidness. However, due to their iterative nature, the SHAKE and |
| 139 |
> |
Rattle algorithms also converge very slowly when the number of |
| 140 |
> |
constraints increases.\cite{Ryckaert1977,Andersen1983} |
| 141 |
> |
|
| 142 |
> |
A breakthrough in geometric literature suggests that, in order to |
| 143 |
|
develop a long-term integration scheme, one should preserve the |
| 144 |
|
symplectic structure of the propagator. By introducing a conjugate |
| 145 |
|
momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's |
| 146 |
< |
equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was |
| 147 |
< |
proposed to evolve the Hamiltonian system in a constraint manifold |
| 148 |
< |
by iteratively satisfying the orthogonality constraint $Q^T Q = 1$. |
| 149 |
< |
An alternative method using the quaternion representation was |
| 150 |
< |
developed by Omelyan.\cite{Omelyan1998} However, both of these |
| 151 |
< |
methods are iterative and inefficient. In this section, we descibe a |
| 152 |
< |
symplectic Lie-Poisson integrator for rigid bodies developed by |
| 153 |
< |
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
| 146 |
> |
equation, a symplectic integrator, RSHAKE,\cite{Kol1997} was proposed |
| 147 |
> |
to evolve the Hamiltonian system in a constraint manifold by |
| 148 |
> |
iteratively satisfying the orthogonality constraint $Q^T Q = 1$. An |
| 149 |
> |
alternative method using the quaternion representation was developed |
| 150 |
> |
by Omelyan.\cite{Omelyan1998} However, both of these methods are |
| 151 |
> |
iterative and suffer from some related inefficiencies. A symplectic |
| 152 |
> |
Lie-Poisson integrator for rigid bodies developed by Dullweber {\it et |
| 153 |
> |
al.}\cite{Dullweber1997} gets around most of the limitations mentioned |
| 154 |
> |
above and has become the basis for our Langevin integrator. |
| 155 |
|
|
| 156 |
< |
%review langevin/browninan dynamics for arbitrarily shaped rigid body |
| 157 |
< |
Combining Langevin or Brownian dynamics with rigid body dynamics, |
| 158 |
< |
one can study slow processes in biomolecular systems. Modeling DNA |
| 159 |
< |
as a chain of rigid beads, which are subject to harmonic potentials |
| 160 |
< |
as well as excluded volume potentials, Mielke and his coworkers |
| 161 |
< |
discovered rapid superhelical stress generations from the stochastic |
| 162 |
< |
simulation of twin supercoiling DNA with response to induced |
| 163 |
< |
torques.\cite{Mielke2004} Membrane fusion is another key biological |
| 164 |
< |
process which controls a variety of physiological functions, such as |
| 165 |
< |
release of neurotransmitters \textit{etc}. A typical fusion event |
| 166 |
< |
happens on the time scale of a millisecond, which is impractical to |
| 167 |
< |
study using atomistic models with newtonian mechanics. With the help |
| 168 |
< |
of coarse-grained rigid body model and stochastic dynamics, the |
| 169 |
< |
fusion pathways were explored by many |
| 170 |
< |
researchers.\cite{Noguchi2001,Noguchi2002,Shillcock2005} Due to the |
| 171 |
< |
difficulty of numerical integration of anisotropic rotation, most of |
| 172 |
< |
the rigid body models are simply modeled using spheres, cylinders, |
| 173 |
< |
ellipsoids or other regular shapes in stochastic simulations. In an |
| 174 |
< |
effort to account for the diffusion anisotropy of arbitrary |
| 175 |
< |
particles, Fernandes and de la Torre improved the original Brownian |
| 176 |
< |
dynamics simulation algorithm\cite{Ermak1978,Allison1991} by |
| 177 |
< |
incorporating a generalized $6\times6$ diffusion tensor and |
| 178 |
< |
introducing a simple rotation evolution scheme consisting of three |
| 179 |
< |
consecutive rotations.\cite{Fernandes2002} Unfortunately, unexpected |
| 180 |
< |
errors and biases are introduced into the system due to the |
| 156 |
> |
|
| 157 |
> |
\subsection{The Hydrodynamic tensor and Brownian dynamics} |
| 158 |
> |
Combining Brownian dynamics with rigid substructures, one can study |
| 159 |
> |
slow processes in biomolecular systems. Modeling DNA as a chain of |
| 160 |
> |
beads which are subject to harmonic potentials as well as excluded |
| 161 |
> |
volume potentials, Mielke and his coworkers discovered rapid |
| 162 |
> |
superhelical stress generations from the stochastic simulation of twin |
| 163 |
> |
supercoiling DNA with response to induced torques.\cite{Mielke2004} |
| 164 |
> |
Membrane fusion is another key biological process which controls a |
| 165 |
> |
variety of physiological functions, such as release of |
| 166 |
> |
neurotransmitters \textit{etc}. A typical fusion event happens on the |
| 167 |
> |
time scale of a millisecond, which is impractical to study using |
| 168 |
> |
atomistic models with newtonian mechanics. With the help of |
| 169 |
> |
coarse-grained rigid body model and stochastic dynamics, the fusion |
| 170 |
> |
pathways were explored by Noguchi and others.\cite{Noguchi2001,Noguchi2002,Shillcock2005} |
| 171 |
> |
|
| 172 |
> |
Due to the difficulty of numerically integrating anisotropic |
| 173 |
> |
rotational motion, most of the coarse-grained rigid body models are |
| 174 |
> |
treated as spheres, cylinders, ellipsoids or other regular shapes in |
| 175 |
> |
stochastic simulations. In an effort to account for the diffusion |
| 176 |
> |
anisotropy of arbitrarily-shaped particles, Fernandes and Garc\'{i}a |
| 177 |
> |
de la Torre improved the original Brownian dynamics simulation |
| 178 |
> |
algorithm~\cite{Ermak1978,Allison1991} by incorporating a generalized |
| 179 |
> |
$6\times6$ diffusion tensor and introducing a rotational evolution |
| 180 |
> |
scheme consisting of three consecutive rotations.\cite{Fernandes2002} |
| 181 |
> |
Unfortunately, biases are introduced into the system due to the |
| 182 |
|
arbitrary order of applying the noncommuting rotation |
| 183 |
|
operators.\cite{Beard2003} Based on the observation the momentum |
| 184 |
|
relaxation time is much less than the time step, one may ignore the |
| 185 |
< |
inertia in Brownian dynamics. However, the assumption of zero |
| 186 |
< |
average acceleration is not always true for cooperative motion which |
| 187 |
< |
is common in protein motion. An inertial Brownian dynamics (IBD) was |
| 188 |
< |
proposed to address this issue by adding an inertial correction |
| 185 |
> |
inertia in Brownian dynamics. However, the assumption of zero average |
| 186 |
> |
acceleration is not always true for cooperative motion which is common |
| 187 |
> |
in proteins. An inertial Brownian dynamics (IBD) was proposed to |
| 188 |
> |
address this issue by adding an inertial correction |
| 189 |
|
term.\cite{Beard2000} As a complement to IBD which has a lower bound |
| 190 |
|
in time step because of the inertial relaxation time, long-time-step |
| 191 |
|
inertial dynamics (LTID) can be used to investigate the inertial |
| 192 |
|
behavior of the polymer segments in low friction |
| 193 |
|
regime.\cite{Beard2000} LTID can also deal with the rotational |
| 194 |
|
dynamics for nonskew bodies without translation-rotation coupling by |
| 195 |
< |
separating the translation and rotation motion and taking advantage |
| 196 |
< |
of the analytical solution of hydrodynamics properties. However, |
| 197 |
< |
typical nonskew bodies like cylinders and ellipsoids are inadequate |
| 198 |
< |
to represent most complex macromolecule assemblies. These intricate |
| 165 |
< |
molecules have been represented by a set of beads and their |
| 166 |
< |
hydrodynamic properties can be calculated using variants on the |
| 167 |
< |
standard hydrodynamic interaction tensors. |
| 195 |
> |
separating the translation and rotation motion and taking advantage of |
| 196 |
> |
the analytical solution of hydrodynamics properties. However, typical |
| 197 |
> |
nonskew bodies like cylinders and ellipsoids are inadequate to |
| 198 |
> |
represent most complex macromolecular assemblies. |
| 199 |
|
|
| 200 |
|
The goal of the present work is to develop a Langevin dynamics |
| 201 |
|
algorithm for arbitrary-shaped rigid particles by integrating the |
| 202 |
< |
accurate estimation of friction tensor from hydrodynamics theory |
| 203 |
< |
into the sophisticated rigid body dynamics algorithms. |
| 202 |
> |
accurate estimation of friction tensor from hydrodynamics theory into |
| 203 |
> |
a symplectic rigid body dynamics propagator. In the sections below, |
| 204 |
> |
we review some of the theory of hydrodynamic tensors developed for |
| 205 |
> |
Brownian simulations of rigid multi-particle systems, we then present |
| 206 |
> |
our integration method for a set of generalized Langevin equations of |
| 207 |
> |
motion, and we compare the behavior of the new Langevin integrator to |
| 208 |
> |
dynamical quantities obtained via explicit solvent molecular dynamics. |
| 209 |
|
|
| 210 |
< |
\subsection{\label{introSection:frictionTensor}Friction Tensor} |
| 211 |
< |
Theoretically, the friction kernel can be determined using the |
| 210 |
> |
\subsection{\label{introSection:frictionTensor}The Friction Tensor} |
| 211 |
> |
Theoretically, a complete friction kernel can be determined using the |
| 212 |
|
velocity autocorrelation function. However, this approach becomes |
| 213 |
< |
impractical when the system becomes more and more complicated. |
| 214 |
< |
Instead, various approaches based on hydrodynamics have been |
| 215 |
< |
developed to calculate the friction coefficients. In general, the |
| 216 |
< |
friction tensor $\Xi$ is a $6\times 6$ matrix given by |
| 217 |
< |
\[ |
| 213 |
> |
impractical when the solute becomes complex. Instead, various |
| 214 |
> |
approaches based on hydrodynamics have been developed to calculate the |
| 215 |
> |
friction coefficients. In general, the friction tensor $\Xi$ is a |
| 216 |
> |
$6\times 6$ matrix given by |
| 217 |
> |
\begin{equation} |
| 218 |
|
\Xi = \left( {\begin{array}{*{20}c} |
| 219 |
|
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
| 220 |
|
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
| 221 |
|
\end{array}} \right). |
| 222 |
< |
\] |
| 223 |
< |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are $3 \times 3$ |
| 224 |
< |
translational friction tensor and rotational resistance (friction) |
| 225 |
< |
tensor respectively, while ${\Xi^{tr} }$ is translation-rotation |
| 226 |
< |
coupling tensor and $ {\Xi^{rt} }$ is rotation-translation coupling |
| 227 |
< |
tensor. When a particle moves in a fluid, it may experience friction |
| 228 |
< |
force or torque along the opposite direction of the velocity or |
| 229 |
< |
angular velocity, |
| 230 |
< |
\[ |
| 222 |
> |
\end{equation} |
| 223 |
> |
Here, $\Xi^{tt}$ and $\Xi^{rr}$ are $3 \times 3$ translational and |
| 224 |
> |
rotational resistance (friction) tensors respectively, while |
| 225 |
> |
$\Xi^{tr}$ is translation-rotation coupling tensor and $\Xi^{rt}$ is |
| 226 |
> |
rotation-translation coupling tensor. When a particle moves in a |
| 227 |
> |
fluid, it may experience friction force ($\mathbf{F}_f$) and torque |
| 228 |
> |
($\mathbf{\tau}_f$) in opposition to the directions of the velocity |
| 229 |
> |
($\mathbf{v}$) and body-fixed angular velocity ($\mathbf{\omega}$), |
| 230 |
> |
\begin{equation} |
| 231 |
|
\left( \begin{array}{l} |
| 232 |
< |
F_R \\ |
| 233 |
< |
\tau _R \\ |
| 232 |
> |
\mathbf{F}_f \\ |
| 233 |
> |
\mathbf{\tau}_f \\ |
| 234 |
|
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 235 |
< |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
| 236 |
< |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
| 235 |
> |
\Xi ^{tt} & \Xi ^{rt} \\ |
| 236 |
> |
\Xi ^{tr} & \Xi ^{rr} \\ |
| 237 |
|
\end{array}} \right)\left( \begin{array}{l} |
| 238 |
< |
v \\ |
| 239 |
< |
w \\ |
| 240 |
< |
\end{array} \right) |
| 241 |
< |
\] |
| 206 |
< |
where $F_r$ is the friction force and $\tau _R$ is the friction |
| 207 |
< |
torque. |
| 238 |
> |
\mathbf{v} \\ |
| 239 |
> |
\mathbf{\omega} \\ |
| 240 |
> |
\end{array} \right). |
| 241 |
> |
\end{equation} |
| 242 |
|
|
| 243 |
|
\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shapes}} |
| 244 |
< |
|
| 245 |
< |
For a spherical particle with slip boundary conditions, the |
| 246 |
< |
translational and rotational friction constant can be calculated |
| 247 |
< |
from Stoke's law, |
| 248 |
< |
\[ |
| 215 |
< |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
| 244 |
> |
For a spherical particle under ``stick'' boundary conditions, the |
| 245 |
> |
translational and rotational friction tensors can be calculated from |
| 246 |
> |
Stoke's law, |
| 247 |
> |
\begin{equation} |
| 248 |
> |
\Xi^{tt} = \left( {\begin{array}{*{20}c} |
| 249 |
|
{6\pi \eta R} & 0 & 0 \\ |
| 250 |
|
0 & {6\pi \eta R} & 0 \\ |
| 251 |
|
0 & 0 & {6\pi \eta R} \\ |
| 252 |
|
\end{array}} \right) |
| 253 |
< |
\] |
| 253 |
> |
\end{equation} |
| 254 |
|
and |
| 255 |
< |
\[ |
| 255 |
> |
\begin{equation} |
| 256 |
|
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
| 257 |
|
{8\pi \eta R^3 } & 0 & 0 \\ |
| 258 |
|
0 & {8\pi \eta R^3 } & 0 \\ |
| 259 |
|
0 & 0 & {8\pi \eta R^3 } \\ |
| 260 |
|
\end{array}} \right) |
| 261 |
< |
\] |
| 261 |
> |
\end{equation} |
| 262 |
|
where $\eta$ is the viscosity of the solvent and $R$ is the |
| 263 |
|
hydrodynamic radius. |
| 264 |
|
|
| 265 |
|
Other non-spherical shapes, such as cylinders and ellipsoids, are |
| 266 |
< |
widely used as references for developing new hydrodynamics theory, |
| 266 |
> |
widely used as references for developing new hydrodynamics theories, |
| 267 |
|
because their properties can be calculated exactly. In 1936, Perrin |
| 268 |
|
extended Stokes's law to general ellipsoids, also called a triaxial |
| 269 |
|
ellipsoid, which is given in Cartesian coordinates |
| 270 |
< |
by\cite{Perrin1934, Perrin1936} |
| 271 |
< |
\[ |
| 272 |
< |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
| 273 |
< |
}} = 1 |
| 274 |
< |
\] |
| 275 |
< |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
| 276 |
< |
due to the complexity of the elliptic integral, only the ellipsoid |
| 277 |
< |
with the restriction of two axes being equal, \textit{i.e.} |
| 278 |
< |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
| 279 |
< |
exactly. Introducing an elliptic integral parameter $S$ for prolate |
| 280 |
< |
ellipsoids : |
| 281 |
< |
\[ |
| 249 |
< |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 250 |
< |
} }}{b}, |
| 251 |
< |
\] |
| 270 |
> |
by\cite{Perrin1934,Perrin1936} |
| 271 |
> |
\begin{equation} |
| 272 |
> |
\frac{x^2 }{a^2} + \frac{y^2}{b^2} + \frac{z^2 }{c^2} = 1 |
| 273 |
> |
\end{equation} |
| 274 |
> |
where the semi-axes are of lengths $a$, $b$, and $c$. Due to the |
| 275 |
> |
complexity of the elliptic integral, only uniaxial ellipsoids, |
| 276 |
> |
{\it i.e.} prolate ($ a \ge b = c$) and oblate ($ a < b = c $), can |
| 277 |
> |
be solved exactly. Introducing an elliptic integral parameter $S$ for |
| 278 |
> |
prolate ellipsoids : |
| 279 |
> |
\begin{equation} |
| 280 |
> |
S = \frac{2}{\sqrt{a^2 - b^2}} \ln \frac{a + \sqrt{a^2 - b^2}}{b}, |
| 281 |
> |
\end{equation} |
| 282 |
|
and oblate ellipsoids: |
| 283 |
< |
\[ |
| 284 |
< |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 285 |
< |
}}{a}, |
| 256 |
< |
\] |
| 283 |
> |
\begin{equation} |
| 284 |
> |
S = \frac{2}{\sqrt {b^2 - a^2 }} \arctan \frac{\sqrt {b^2 - a^2}}{a}, |
| 285 |
> |
\end{equation} |
| 286 |
|
one can write down the translational and rotational resistance |
| 287 |
< |
tensors |
| 287 |
> |
tensors for oblate, |
| 288 |
|
\begin{eqnarray*} |
| 289 |
< |
\Xi _a^{tt} & = & 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
| 290 |
< |
\Xi _b^{tt} & = & \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
| 262 |
< |
2a}}, |
| 289 |
> |
\Xi_a^{tt} & = & 16\pi \eta \frac{a^2 - b^2}{(2a^2 - b^2 )S - 2a}. \\ |
| 290 |
> |
\Xi_b^{tt} = \Xi_c^{tt} & = & 32\pi \eta \frac{a^2 - b^2 }{(2a^2 - 3b^2 )S + 2a}, |
| 291 |
|
\end{eqnarray*} |
| 292 |
< |
and |
| 292 |
> |
and prolate, |
| 293 |
|
\begin{eqnarray*} |
| 294 |
< |
\Xi _a^{rr} & = & \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
| 295 |
< |
\Xi _b^{rr} & = & \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}}. |
| 294 |
> |
\Xi_a^{rr} & = & \frac{32\pi}{3} \eta \frac{(a^2 - b^2 )b^2}{2a - b^2 S}, \\ |
| 295 |
> |
\Xi_b^{rr} = \Xi_c^{rr} & = & \frac{32\pi}{3} \eta \frac{(a^4 - b^4)}{(2a^2 - b^2 )S - 2a} |
| 296 |
|
\end{eqnarray*} |
| 297 |
+ |
ellipsoids. For both spherical and ellipsoidal particles, the |
| 298 |
+ |
translation-rotation and rotation-translation coupling tensors are |
| 299 |
+ |
zero. |
| 300 |
|
|
| 301 |
|
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}} |
| 302 |
|
|
| 304 |
|
analytical solution for the friction tensor for arbitrarily shaped |
| 305 |
|
rigid molecules. The ellipsoid of revolution model and general |
| 306 |
|
triaxial ellipsoid model have been used to approximate the |
| 307 |
< |
hydrodynamic properties of rigid bodies. However, since the mapping |
| 308 |
< |
from all possible ellipsoidal spaces, $r$-space, to all possible |
| 309 |
< |
combination of rotational diffusion coefficients, $D$-space, is not |
| 310 |
< |
unique\cite{Wegener1979} as well as the intrinsic coupling between |
| 311 |
< |
translational and rotational motion of rigid bodies, general |
| 312 |
< |
ellipsoids are not always suitable for modeling arbitrarily shaped |
| 313 |
< |
rigid molecules. A number of studies have been devoted to |
| 307 |
> |
hydrodynamic properties of rigid bodies. However, the mapping from all |
| 308 |
> |
possible ellipsoidal spaces, $r$-space, to all possible combination of |
| 309 |
> |
rotational diffusion coefficients, $D$-space, is not |
| 310 |
> |
unique.\cite{Wegener1979} Additionally, because there is intrinsic |
| 311 |
> |
coupling between translational and rotational motion of rigid bodies, |
| 312 |
> |
general ellipsoids are not always suitable for modeling arbitrarily |
| 313 |
> |
shaped rigid molecules. A number of studies have been devoted to |
| 314 |
|
determining the friction tensor for irregularly shaped rigid bodies |
| 315 |
< |
using more advanced methods where the molecule of interest was |
| 316 |
< |
modeled by a combinations of spheres\cite{Carrasco1999} and the |
| 317 |
< |
hydrodynamics properties of the molecule can be calculated using the |
| 318 |
< |
hydrodynamic interaction tensor. Let us consider a rigid assembly of |
| 319 |
< |
$N$ beads immersed in a continuous medium. Due to hydrodynamic |
| 320 |
< |
interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different |
| 321 |
< |
than its unperturbed velocity $v_i$, |
| 315 |
> |
using more advanced methods where the molecule of interest was modeled |
| 316 |
> |
by a combinations of spheres\cite{Carrasco1999} and the hydrodynamics |
| 317 |
> |
properties of the molecule can be calculated using the hydrodynamic |
| 318 |
> |
interaction tensor. Let us consider a rigid assembly of $N$ beads |
| 319 |
> |
immersed in a continuous medium. Due to hydrodynamic interaction, the |
| 320 |
> |
``net'' velocity of $i$th bead, $v'_i$ is different than its |
| 321 |
> |
unperturbed velocity $v_i$, |
| 322 |
|
\[ |
| 323 |
|
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
| 324 |
|
\] |
