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\section{Introduction} |
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|
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%applications of langevin dynamics |
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As alternative to Newtonian dynamics, Langevin dynamics, which |
52 |
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mimics a simple heat bath with stochastic and dissipative forces, |
53 |
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has been applied in a variety of studies. The stochastic treatment |
54 |
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of the solvent enables us to carry out substantially longer time |
55 |
< |
simulations. Implicit solvent Langevin dynamics simulations of |
56 |
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met-enkephalin not only outperform explicit solvent simulations for |
57 |
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computational efficiency, but also agrees very well with explicit |
58 |
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solvent simulations for dynamical properties.\cite{Shen2002} |
59 |
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Recently, applying Langevin dynamics with the UNRES model, Liow and |
60 |
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his coworkers suggest that protein folding pathways can be possibly |
61 |
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explored within a reasonable amount of time.\cite{Liwo2005} The |
62 |
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stochastic nature of the Langevin dynamics also enhances the |
63 |
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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 |
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free-energy barriers by studying the viscosity dependence of the |
67 |
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protein folding rates.\cite{Klimov1997} In order to account for |
68 |
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solvent induced interactions missing from implicit solvent model, |
69 |
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Kaya incorporated desolvation free energy barrier into implicit |
70 |
< |
coarse-grained solvent model in protein folding/unfolding studies |
71 |
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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 |
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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 |
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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' 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{HuseyinKaya07012005} |
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 |
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Langevin dynamics has the same peak frequencies for different wave |
84 |
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vectors, which recovers the property of magnetic excitations in small |
85 |
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finite structures.\cite{Berkov2005a}] |
86 |
|
|
87 |
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%review rigid body dynamics |
88 |
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Rigid bodies are frequently involved in the modeling of different |
89 |
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areas, from engineering, physics, to chemistry. For example, |
90 |
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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 |
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individual atoms are taken from Stokes' law, |
89 |
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\begin{eqnarray} |
90 |
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m \dot{v}(t) & = & -\nabla U(x) - \xi m v(t) + R(t) \\ |
91 |
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\langle R(t) \rangle & = & 0 \\ |
92 |
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\langle R(t) R(t') \rangle & = & 2 k_B T \xi m \delta(t - t') |
93 |
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\end{eqnarray} |
94 |
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where $\xi \approx 6 \pi \eta a$. Here $\eta$ is the viscosity of the |
95 |
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implicit solvent, and $a$ is the hydrodynamic radius of the atom. |
96 |
|
|
97 |
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It is very important to develop stable and efficient methods to |
98 |
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integrate the equations of motion for orientational degrees of |
99 |
< |
freedom. Euler angles are the natural choice to describe the |
100 |
< |
rotational degrees of freedom. However, due to $\frac {1}{sin |
101 |
< |
\theta}$ singularities, the numerical integration of corresponding |
102 |
< |
equations of these motion is very inefficient and inaccurate. |
103 |
< |
Although an alternative integrator using multiple sets of Euler |
104 |
< |
angles can overcome this difficulty\cite{Barojas1973}, the |
97 |
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The use of rigid substructures,\cite{Chun:2000fj} |
98 |
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coarse-graining,\cite{Ayton01,Golubkov06,Orlandi:2006fk,SunGezelter08} |
99 |
> |
and ellipsoidal representations of protein side chains~\cite{Fogolari:1996lr} |
100 |
> |
has made the use of the Stokes-Einstein approximation problematic. A |
101 |
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rigid substructure moves as a single unit with orientational as well |
102 |
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as translational degrees of freedom. This requires a more general |
103 |
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treatment of the hydrodynamics than the spherical approximation |
104 |
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provides. The atoms involved in a rigid or coarse-grained structure |
105 |
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should properly have solvent-mediated interactions with each |
106 |
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other. The theory of interactions {\it between} bodies moving through |
107 |
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a fluid has been developed over the past century and has been applied |
108 |
> |
to simulations of Brownian |
109 |
> |
motion.\cite{FIXMAN:1986lr,Ramachandran1996} |
110 |
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|
111 |
> |
In order to account for the diffusion anisotropy of arbitrarily-shaped |
112 |
> |
particles, Fernandes and Garc\'{i}a de la Torre improved the original |
113 |
> |
Brownian dynamics simulation algorithm~\cite{Ermak1978,Allison1991} by |
114 |
> |
incorporating a generalized $6\times6$ diffusion tensor and |
115 |
> |
introducing a rotational evolution scheme consisting of three |
116 |
> |
consecutive rotations.\cite{Fernandes2002} Unfortunately, biases are |
117 |
> |
introduced into the system due to the arbitrary order of applying the |
118 |
> |
noncommuting rotation operators.\cite{Beard2003} Based on the |
119 |
> |
observation the momentum relaxation time is much less than the time |
120 |
> |
step, one may ignore the inertia in Brownian dynamics. However, the |
121 |
> |
assumption of zero average acceleration is not always true for |
122 |
> |
cooperative motion which is common in proteins. An inertial Brownian |
123 |
> |
dynamics (IBD) was proposed to address this issue by adding an |
124 |
> |
inertial correction term.\cite{Beard2000} As a complement to IBD which |
125 |
> |
has a lower bound in time step because of the inertial relaxation |
126 |
> |
time, long-time-step inertial dynamics (LTID) can be used to |
127 |
> |
investigate the inertial behavior of linked polymer segments in a low |
128 |
> |
friction regime.\cite{Beard2000} LTID can also deal with the |
129 |
> |
rotational dynamics for nonskew bodies without translation-rotation |
130 |
> |
coupling by separating the translation and rotation motion and taking |
131 |
> |
advantage of the analytical solution of hydrodynamics |
132 |
> |
properties. However, typical nonskew bodies like cylinders and |
133 |
> |
ellipsoids are inadequate to represent most complex macromolecular |
134 |
> |
assemblies. There is therefore a need for incorporating the |
135 |
> |
hydrodynamics of complex (and potentially skew) rigid bodies in the |
136 |
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library of methods available for performing Langevin simulations. |
137 |
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|
138 |
> |
\subsection{Rigid Body Dynamics} |
139 |
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Rigid bodies are frequently involved in the modeling of large |
140 |
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collections of particles that move as a single unit. In molecular |
141 |
> |
simulations, rigid bodies have been used to simplify protein-protein |
142 |
> |
docking,\cite{Gray2003} and lipid bilayer |
143 |
> |
simulations.\cite{SunGezelter08} Many of the water models in common |
144 |
> |
use are also rigid-body |
145 |
> |
models,\cite{Jorgensen83,Berendsen81,Berendsen87} although they are |
146 |
> |
typically evolved using constraints rather than rigid body equations |
147 |
> |
of motion. |
148 |
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|
149 |
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Euler angles are a natural choice to describe the rotational degrees |
150 |
> |
of freedom. However, due to $\frac{1}{\sin \theta}$ singularities, the |
151 |
> |
numerical integration of corresponding equations of these motion can |
152 |
> |
become inaccurate (and inefficient). Although the use of multiple |
153 |
> |
sets of Euler angles can overcome this problem,\cite{Barojas1973} the |
154 |
|
computational penalty and the loss of angular momentum conservation |
155 |
< |
still remain. A singularity-free representation utilizing |
156 |
< |
quaternions was developed by Evans in 1977.\cite{Evans1977} |
157 |
< |
Unfortunately, this approach used a nonseparable Hamiltonian |
158 |
< |
resulting from the quaternion representation, which prevented the |
159 |
< |
symplectic algorithm from being utilized. Another different approach |
103 |
< |
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} |
155 |
> |
remain. A singularity-free representation utilizing quaternions was |
156 |
> |
developed by Evans in 1977.\cite{Evans1977} The Evans quaternion |
157 |
> |
approach uses a nonseparable Hamiltonian, and this has prevented |
158 |
> |
symplectic algorithms from being utilized until very |
159 |
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recently.\cite{Miller2002} |
160 |
|
|
161 |
< |
A break-through in geometric literature suggests that, in order to |
162 |
< |
develop a long-term integration scheme, one should preserve the |
163 |
< |
symplectic structure of the propagator. By introducing a conjugate |
164 |
< |
momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's |
165 |
< |
equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was |
166 |
< |
proposed to evolve the Hamiltonian system in a constraint manifold |
167 |
< |
by iteratively satisfying the orthogonality constraint $Q^T Q = 1$. |
117 |
< |
An alternative method using the quaternion representation was |
118 |
< |
developed by Omelyan.\cite{Omelyan1998} However, both of these |
119 |
< |
methods are iterative and inefficient. In this section, we descibe a |
120 |
< |
symplectic Lie-Poisson integrator for rigid bodies developed by |
121 |
< |
Dullweber and his coworkers\cite{Dullweber1997} in depth. |
161 |
> |
Another approach is the application of holonomic constraints to the |
162 |
> |
atoms belonging to the rigid body. Each atom moves independently |
163 |
> |
under the normal forces deriving from potential energy and constraints |
164 |
> |
are used to guarantee rigidity. However, due to their iterative |
165 |
> |
nature, the SHAKE and RATTLE algorithms converge very slowly when the |
166 |
> |
number of constraints (and the number of particles that belong to the |
167 |
> |
rigid body) increases.\cite{Ryckaert1977,Andersen1983} |
168 |
|
|
169 |
< |
%review langevin/browninan dynamics for arbitrarily shaped rigid body |
170 |
< |
Combining Langevin or Brownian dynamics with rigid body dynamics, |
171 |
< |
one can study slow processes in biomolecular systems. Modeling DNA |
172 |
< |
as a chain of rigid beads, which are subject to harmonic potentials |
173 |
< |
as well as excluded volume potentials, Mielke and his coworkers |
174 |
< |
discovered rapid superhelical stress generations from the stochastic |
175 |
< |
simulation of twin supercoiling DNA with response to induced |
176 |
< |
torques.\cite{Mielke2004} Membrane fusion is another key biological |
177 |
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process which controls a variety of physiological functions, such as |
178 |
< |
release of neurotransmitters \textit{etc}. A typical fusion event |
179 |
< |
happens on the time scale of a millisecond, which is impractical to |
180 |
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study using atomistic models with newtonian mechanics. With the help |
181 |
< |
of coarse-grained rigid body model and stochastic dynamics, the |
136 |
< |
fusion pathways were explored by many |
137 |
< |
researchers.\cite{Noguchi2001,Noguchi2002,Shillcock2005} Due to the |
138 |
< |
difficulty of numerical integration of anisotropic rotation, most of |
139 |
< |
the rigid body models are simply modeled using spheres, cylinders, |
140 |
< |
ellipsoids or other regular shapes in stochastic simulations. In an |
141 |
< |
effort to account for the diffusion anisotropy of arbitrary |
142 |
< |
particles, Fernandes and de la Torre improved the original Brownian |
143 |
< |
dynamics simulation algorithm\cite{Ermak1978,Allison1991} by |
144 |
< |
incorporating a generalized $6\times6$ diffusion tensor and |
145 |
< |
introducing a simple rotation evolution scheme consisting of three |
146 |
< |
consecutive rotations.\cite{Fernandes2002} Unfortunately, unexpected |
147 |
< |
errors and biases are introduced into the system due to the |
148 |
< |
arbitrary order of applying the noncommuting rotation |
149 |
< |
operators.\cite{Beard2003} Based on the observation the momentum |
150 |
< |
relaxation time is much less than the time step, one may ignore the |
151 |
< |
inertia in Brownian dynamics. However, the assumption of zero |
152 |
< |
average acceleration is not always true for cooperative motion which |
153 |
< |
is common in protein motion. An inertial Brownian dynamics (IBD) was |
154 |
< |
proposed to address this issue by adding an inertial correction |
155 |
< |
term.\cite{Beard2000} As a complement to IBD which has a lower bound |
156 |
< |
in time step because of the inertial relaxation time, long-time-step |
157 |
< |
inertial dynamics (LTID) can be used to investigate the inertial |
158 |
< |
behavior of the polymer segments in low friction |
159 |
< |
regime.\cite{Beard2000} LTID can also deal with the rotational |
160 |
< |
dynamics for nonskew bodies without translation-rotation coupling by |
161 |
< |
separating the translation and rotation motion and taking advantage |
162 |
< |
of the analytical solution of hydrodynamics properties. However, |
163 |
< |
typical nonskew bodies like cylinders and ellipsoids are inadequate |
164 |
< |
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. |
169 |
> |
In order to develop a stable and efficient integration scheme that |
170 |
> |
preserves most constants of the motion, symplectic propagators are |
171 |
> |
necessary. By introducing a conjugate momentum to the rotation matrix |
172 |
> |
$Q$ and re-formulating Hamilton's equations, a symplectic |
173 |
> |
orientational integrator, RSHAKE,\cite{Kol1997} was proposed to evolve |
174 |
> |
rigid bodies on a constraint manifold by iteratively satisfying the |
175 |
> |
orthogonality constraint $Q^T Q = 1$. An alternative method using the |
176 |
> |
quaternion representation was developed by Omelyan.\cite{Omelyan1998} |
177 |
> |
However, both of these methods are iterative and suffer from some |
178 |
> |
related inefficiencies. A symplectic Lie-Poisson integrator for rigid |
179 |
> |
bodies developed by Dullweber {\it et al.}\cite{Dullweber1997} removes |
180 |
> |
most of the limitations mentioned above and is therefore the basis for |
181 |
> |
our Langevin integrator. |
182 |
|
|
183 |
|
The goal of the present work is to develop a Langevin dynamics |
184 |
|
algorithm for arbitrary-shaped rigid particles by integrating the |
185 |
< |
accurate estimation of friction tensor from hydrodynamics theory |
186 |
< |
into the sophisticated rigid body dynamics algorithms. |
185 |
> |
accurate estimation of friction tensor from hydrodynamics theory into |
186 |
> |
a symplectic rigid body dynamics propagator. In the sections below, |
187 |
> |
we review some of the theory of hydrodynamic tensors developed |
188 |
> |
primarily for Brownian simulations of multi-particle systems, we then |
189 |
> |
present our integration method for a set of generalized Langevin |
190 |
> |
equations of motion, and we compare the behavior of the new Langevin |
191 |
> |
integrator to dynamical quantities obtained via explicit solvent |
192 |
> |
molecular dynamics. |
193 |
|
|
194 |
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\subsection{\label{introSection:frictionTensor}Friction Tensor} |
195 |
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Theoretically, the friction kernel can be determined using the |
194 |
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\subsection{\label{introSection:frictionTensor}The Friction Tensor} |
195 |
> |
Theoretically, a complete friction kernel can be determined using the |
196 |
|
velocity autocorrelation function. However, this approach becomes |
197 |
< |
impractical when the system becomes more and more complicated. |
198 |
< |
Instead, various approaches based on hydrodynamics have been |
199 |
< |
developed to calculate the friction coefficients. In general, the |
200 |
< |
friction tensor $\Xi$ is a $6\times 6$ matrix given by |
201 |
< |
\[ |
202 |
< |
\Xi = \left( {\begin{array}{*{20}c} |
203 |
< |
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
204 |
< |
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
205 |
< |
\end{array}} \right). |
206 |
< |
\] |
207 |
< |
Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are $3 \times 3$ |
208 |
< |
translational friction tensor and rotational resistance (friction) |
209 |
< |
tensor respectively, while ${\Xi^{tr} }$ is translation-rotation |
210 |
< |
coupling tensor and $ {\Xi^{rt} }$ is rotation-translation coupling |
211 |
< |
tensor. When a particle moves in a fluid, it may experience friction |
212 |
< |
force or torque along the opposite direction of the velocity or |
213 |
< |
angular velocity, |
214 |
< |
\[ |
197 |
> |
impractical when the solute becomes complex. Instead, various |
198 |
> |
approaches based on hydrodynamics have been developed to calculate the |
199 |
> |
friction coefficients. In general, the friction tensor $\Xi$ is a |
200 |
> |
$6\times 6$ matrix given by |
201 |
> |
\begin{equation} |
202 |
> |
\Xi = \left( \begin{array}{*{20}c} |
203 |
> |
\Xi^{tt} & \Xi^{rt} \\ |
204 |
> |
\Xi^{tr} & \Xi^{rr} \\ |
205 |
> |
\end{array} \right). |
206 |
> |
\end{equation} |
207 |
> |
Here, $\Xi^{tt}$ and $\Xi^{rr}$ are $3 \times 3$ translational and |
208 |
> |
rotational resistance (friction) tensors respectively, while |
209 |
> |
$\Xi^{tr}$ is translation-rotation coupling tensor and $\Xi^{rt}$ is |
210 |
> |
rotation-translation coupling tensor. When a particle moves in a |
211 |
> |
fluid, it may experience friction force ($\mathbf{F}_f$) and torque |
212 |
> |
($\mathbf{\tau}_f$) in opposition to the directions of the velocity |
213 |
> |
($\mathbf{v}$) and body-fixed angular velocity ($\mathbf{\omega}$), |
214 |
> |
\begin{equation} |
215 |
|
\left( \begin{array}{l} |
216 |
< |
F_R \\ |
217 |
< |
\tau _R \\ |
218 |
< |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
219 |
< |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
220 |
< |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
221 |
< |
\end{array}} \right)\left( \begin{array}{l} |
222 |
< |
v \\ |
223 |
< |
w \\ |
224 |
< |
\end{array} \right) |
225 |
< |
\] |
206 |
< |
where $F_r$ is the friction force and $\tau _R$ is the friction |
207 |
< |
torque. |
216 |
> |
\mathbf{F}_f \\ |
217 |
> |
\mathbf{\tau}_f \\ |
218 |
> |
\end{array} \right) = - \left( \begin{array}{*{20}c} |
219 |
> |
\Xi^{tt} & \Xi^{rt} \\ |
220 |
> |
\Xi^{tr} & \Xi^{rr} \\ |
221 |
> |
\end{array} \right)\left( \begin{array}{l} |
222 |
> |
\mathbf{v} \\ |
223 |
> |
\mathbf{\omega} \\ |
224 |
> |
\end{array} \right). |
225 |
> |
\end{equation} |
226 |
|
|
227 |
|
\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shapes}} |
228 |
< |
|
229 |
< |
For a spherical particle with slip boundary conditions, the |
230 |
< |
translational and rotational friction constant can be calculated |
231 |
< |
from Stoke's law, |
232 |
< |
\[ |
215 |
< |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
228 |
> |
For a spherical particle under ``stick'' boundary conditions, the |
229 |
> |
translational and rotational friction tensors can be calculated from |
230 |
> |
Stokes' law, |
231 |
> |
\begin{equation} |
232 |
> |
\Xi^{tt} = \left( \begin{array}{*{20}c} |
233 |
|
{6\pi \eta R} & 0 & 0 \\ |
234 |
|
0 & {6\pi \eta R} & 0 \\ |
235 |
|
0 & 0 & {6\pi \eta R} \\ |
236 |
< |
\end{array}} \right) |
237 |
< |
\] |
236 |
> |
\end{array} \right) |
237 |
> |
\end{equation} |
238 |
|
and |
239 |
< |
\[ |
240 |
< |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
239 |
> |
\begin{equation} |
240 |
> |
\Xi^{rr} = \left( \begin{array}{*{20}c} |
241 |
|
{8\pi \eta R^3 } & 0 & 0 \\ |
242 |
|
0 & {8\pi \eta R^3 } & 0 \\ |
243 |
|
0 & 0 & {8\pi \eta R^3 } \\ |
244 |
< |
\end{array}} \right) |
245 |
< |
\] |
244 |
> |
\end{array} \right) |
245 |
> |
\end{equation} |
246 |
|
where $\eta$ is the viscosity of the solvent and $R$ is the |
247 |
|
hydrodynamic radius. |
248 |
|
|
249 |
|
Other non-spherical shapes, such as cylinders and ellipsoids, are |
250 |
< |
widely used as references for developing new hydrodynamics theory, |
250 |
> |
widely used as references for developing new hydrodynamics theories, |
251 |
|
because their properties can be calculated exactly. In 1936, Perrin |
252 |
< |
extended Stokes's law to general ellipsoids, also called a triaxial |
253 |
< |
ellipsoid, which is given in Cartesian coordinates |
254 |
< |
by\cite{Perrin1934, Perrin1936} |
255 |
< |
\[ |
256 |
< |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
257 |
< |
}} = 1 |
258 |
< |
\] |
259 |
< |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
260 |
< |
due to the complexity of the elliptic integral, only the ellipsoid |
261 |
< |
with the restriction of two axes being equal, \textit{i.e.} |
262 |
< |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
263 |
< |
exactly. Introducing an elliptic integral parameter $S$ for prolate |
264 |
< |
ellipsoids : |
265 |
< |
\[ |
266 |
< |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
267 |
< |
} }}{b}, |
268 |
< |
\] |
269 |
< |
and oblate ellipsoids: |
253 |
< |
\[ |
254 |
< |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
255 |
< |
}}{a}, |
256 |
< |
\] |
257 |
< |
one can write down the translational and rotational resistance |
258 |
< |
tensors |
252 |
> |
extended Stokes' law to general ellipsoids which are given in |
253 |
> |
Cartesian coordinates by~\cite{Perrin1934,Perrin1936} |
254 |
> |
\begin{equation} |
255 |
> |
\frac{x^2 }{a^2} + \frac{y^2}{b^2} + \frac{z^2 }{c^2} = 1. |
256 |
> |
\end{equation} |
257 |
> |
Here, the semi-axes are of lengths $a$, $b$, and $c$. Due to the |
258 |
> |
complexity of the elliptic integral, only uniaxial ellipsoids, either |
259 |
> |
prolate ($a \ge b = c$) or oblate ($a < b = c$), can be solved |
260 |
> |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
261 |
> |
\begin{equation} |
262 |
> |
S = \frac{2}{\sqrt{a^2 - b^2}} \ln \frac{a + \sqrt{a^2 - b^2}}{b}, |
263 |
> |
\end{equation} |
264 |
> |
and oblate, |
265 |
> |
\begin{equation} |
266 |
> |
S = \frac{2}{\sqrt {b^2 - a^2 }} \arctan \frac{\sqrt {b^2 - a^2}}{a}, |
267 |
> |
\end{equation} |
268 |
> |
ellipsoids, one can write down the translational and rotational |
269 |
> |
resistance tensors: |
270 |
|
\begin{eqnarray*} |
271 |
< |
\Xi _a^{tt} & = & 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
272 |
< |
\Xi _b^{tt} & = & \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
262 |
< |
2a}}, |
271 |
> |
\Xi_a^{tt} & = & 16\pi \eta \frac{a^2 - b^2}{(2a^2 - b^2 )S - 2a}. \\ |
272 |
> |
\Xi_b^{tt} = \Xi_c^{tt} & = & 32\pi \eta \frac{a^2 - b^2 }{(2a^2 - 3b^2 )S + 2a}, |
273 |
|
\end{eqnarray*} |
274 |
< |
and |
274 |
> |
for oblate, and |
275 |
|
\begin{eqnarray*} |
276 |
< |
\Xi _a^{rr} & = & \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
277 |
< |
\Xi _b^{rr} & = & \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}}. |
276 |
> |
\Xi_a^{rr} & = & \frac{32\pi}{3} \eta \frac{(a^2 - b^2 )b^2}{2a - b^2 S}, \\ |
277 |
> |
\Xi_b^{rr} = \Xi_c^{rr} & = & \frac{32\pi}{3} \eta \frac{(a^4 - b^4)}{(2a^2 - b^2 )S - 2a} |
278 |
|
\end{eqnarray*} |
279 |
+ |
for prolate ellipsoids. For both spherical and ellipsoidal particles, |
280 |
+ |
the translation-rotation and rotation-translation coupling tensors are |
281 |
+ |
zero. |
282 |
|
|
283 |
|
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}} |
271 |
– |
|
284 |
|
Unlike spherical and other simply shaped molecules, there is no |
285 |
|
analytical solution for the friction tensor for arbitrarily shaped |
286 |
|
rigid molecules. The ellipsoid of revolution model and general |
287 |
|
triaxial ellipsoid model have been used to approximate the |
288 |
< |
hydrodynamic properties of rigid bodies. However, since the mapping |
289 |
< |
from all possible ellipsoidal spaces, $r$-space, to all possible |
290 |
< |
combination of rotational diffusion coefficients, $D$-space, is not |
291 |
< |
unique\cite{Wegener1979} as well as the intrinsic coupling between |
292 |
< |
translational and rotational motion of rigid bodies, general |
293 |
< |
ellipsoids are not always suitable for modeling arbitrarily shaped |
294 |
< |
rigid molecules. A number of studies have been devoted to |
288 |
> |
hydrodynamic properties of rigid bodies. However, the mapping from all |
289 |
> |
possible ellipsoidal spaces, $r$-space, to all possible combination of |
290 |
> |
rotational diffusion coefficients, $D$-space, is not |
291 |
> |
unique.\cite{Wegener1979} Additionally, because there is intrinsic |
292 |
> |
coupling between translational and rotational motion of rigid bodies, |
293 |
> |
general ellipsoids are not always suitable for modeling arbitrarily |
294 |
> |
shaped rigid molecules. A number of studies have been devoted to |
295 |
|
determining the friction tensor for irregularly shaped rigid bodies |
296 |
< |
using more advanced methods where the molecule of interest was |
297 |
< |
modeled by a combinations of spheres\cite{Carrasco1999} and the |
298 |
< |
hydrodynamics properties of the molecule can be calculated using the |
299 |
< |
hydrodynamic interaction tensor. Let us consider a rigid assembly of |
300 |
< |
$N$ beads immersed in a continuous medium. Due to hydrodynamic |
301 |
< |
interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different |
302 |
< |
than its unperturbed velocity $v_i$, |
303 |
< |
\[ |
296 |
> |
using more advanced methods where the molecule of interest was modeled |
297 |
> |
by a combinations of spheres\cite{Carrasco1999} and the hydrodynamics |
298 |
> |
properties of the molecule can be calculated using the hydrodynamic |
299 |
> |
interaction tensor. |
300 |
> |
|
301 |
> |
Consider a rigid assembly of $N$ beads immersed in a continuous |
302 |
> |
medium. Due to hydrodynamic interaction, the ``net'' velocity of $i$th |
303 |
> |
bead, $v'_i$ is different than its unperturbed velocity $v_i$, |
304 |
> |
\begin{equation} |
305 |
|
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
306 |
< |
\] |
307 |
< |
where $F_i$ is the frictional force, and $T_{ij}$ is the |
308 |
< |
hydrodynamic interaction tensor. The friction force of $i$th bead is |
309 |
< |
proportional to its ``net'' velocity |
306 |
> |
\end{equation} |
307 |
> |
where $F_i$ is the frictional force, and $T_{ij}$ is the hydrodynamic |
308 |
> |
interaction tensor. The frictional force on the $i^\mathrm{th}$ bead |
309 |
> |
is proportional to its ``net'' velocity |
310 |
|
\begin{equation} |
311 |
|
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
312 |
|
\label{introEquation:tensorExpression} |
343 |
|
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
344 |
|
$B_{ij}$ blocks |
345 |
|
\begin{equation} |
346 |
< |
B = \left( {\begin{array}{*{20}c} |
347 |
< |
{B_{11} } & \ldots & {B_{1N} } \\ |
346 |
> |
B = \left( \begin{array}{*{20}c} |
347 |
> |
B_{11} & \ldots & B_{1N} \\ |
348 |
|
\vdots & \ddots & \vdots \\ |
349 |
< |
{B_{N1} } & \cdots & {B_{NN} } \\ |
350 |
< |
\end{array}} \right), |
349 |
> |
B_{N1} & \cdots & B_{NN} \\ |
350 |
> |
\end{array} \right), |
351 |
|
\end{equation} |
352 |
|
where $B_{ij}$ is given by |
353 |
< |
\[ |
353 |
> |
\begin{equation} |
354 |
|
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
355 |
|
)T_{ij} |
356 |
< |
\] |
356 |
> |
\end{equation} |
357 |
|
where $\delta _{ij}$ is the Kronecker delta function. Inverting the |
358 |
|
$B$ matrix, we obtain |
359 |
|
\[ |
438 |
|
x_{OR} \\ |
439 |
|
y_{OR} \\ |
440 |
|
z_{OR} \\ |
441 |
< |
\end{array} \right) & = &\left( {\begin{array}{*{20}c} |
441 |
> |
\end{array} \right) & = &\left( \begin{array}{*{20}c} |
442 |
|
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
443 |
|
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
444 |
|
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
445 |
< |
\end{array}} \right)^{ - 1} \\ |
445 |
> |
\end{array} \right)^{ - 1} \\ |
446 |
|
& & \left( \begin{array}{l} |
447 |
|
(\Xi _O^{tr} )_{yz} - (\Xi _O^{tr} )_{zy} \\ |
448 |
|
(\Xi _O^{tr} )_{zx} - (\Xi _O^{tr} )_{xz} \\ |
454 |
|
|
455 |
|
|
456 |
|
\section{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}} |
457 |
+ |
|
458 |
|
Consider the Langevin equations of motion in generalized coordinates |
459 |
|
\begin{equation} |
460 |
< |
M_i \dot V_i (t) = F_{s,i} (t) + F_{f,i(t)} + F_{r,i} (t) |
460 |
> |
\mathbf{M} \dot{\mathbf{V}}(t) = \mathbf{F}_{s}(t) + |
461 |
> |
\mathbf{F}_{f}(t) + \mathbf{R}(t) |
462 |
|
\label{LDGeneralizedForm} |
463 |
|
\end{equation} |
464 |
< |
where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
465 |
< |
and moment of inertial) matrix and $V_i$ is a generalized velocity, |
466 |
< |
$V_i = V_i(v_i,\omega _i)$. The right side of |
467 |
< |
Eq.~\ref{LDGeneralizedForm} consists of three generalized forces in |
468 |
< |
lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
469 |
< |
$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
470 |
< |
system in Newtownian mechanics typically refers to lab-fixed frame, |
471 |
< |
it is also convenient to handle the rotation of rigid body in |
472 |
< |
body-fixed frame. Thus the friction and random forces are calculated |
473 |
< |
in body-fixed frame and converted back to lab-fixed frame by: |
474 |
< |
\[ |
464 |
> |
where $\mathbf{M}$ is a $6 \times 6$ diagonal mass matrix (which |
465 |
> |
includes the mass of the rigid body as well as the moments of inertia |
466 |
> |
in the body-fixed frame) and $\mathbf{V}$ is a generalized velocity, |
467 |
> |
$\mathbf{V} = |
468 |
> |
\left\{\mathbf{v},\mathbf{\omega}\right\}$. The right side of |
469 |
> |
Eq.~\ref{LDGeneralizedForm} consists of three generalized forces: a |
470 |
> |
system force $\mathbf{F}_{s}$, a frictional or dissipative force |
471 |
> |
$\mathbf{F}_{f}$ and stochastic force $\mathbf{R}$. While the |
472 |
> |
evolution of the system in Newtownian mechanics is typically done in the |
473 |
> |
lab-fixed frame, it is convenient to handle the rotation of rigid |
474 |
> |
bodies in the body-fixed frame. Thus the friction and random forces are |
475 |
> |
calculated in body-fixed frame and converted back to lab-fixed frame |
476 |
> |
using the rigid body's rotation matrix ($Q$): |
477 |
> |
\begin{equation} |
478 |
|
\begin{array}{l} |
479 |
< |
F_{f,i}^l (t) = Q^T F_{f,i}^b (t), \\ |
480 |
< |
F_{r,i}^l (t) = Q^T F_{r,i}^b (t). \\ |
479 |
> |
\mathbf{F}_{f}(t) = Q^{T} \mathbf{F}_{f}^b (t), \\ |
480 |
> |
\mathbf{R}(t) = Q^{T} \mathbf{R}^b (t). \\ |
481 |
|
\end{array} |
482 |
< |
\] |
483 |
< |
Here, the body-fixed friction force $F_{r,i}^b$ is proportional to |
484 |
< |
the body-fixed velocity at center of resistance $v_{R,i}^b$ and |
485 |
< |
angular velocity $\omega _i$ |
482 |
> |
\end{equation} |
483 |
> |
Here, the body-fixed friction force $\mathbf{F}_{f,i}^b$ is proportional to |
484 |
> |
the body-fixed velocity at the center of resistance $\mathbf{v}_{R,i}^b$ and |
485 |
> |
angular velocity $\mathbf{\omega}_i$ |
486 |
|
\begin{equation} |
487 |
< |
F_{r,i}^b (t) = \left( \begin{array}{l} |
488 |
< |
f_{r,i}^b (t) \\ |
489 |
< |
\tau _{r,i}^b (t) \\ |
490 |
< |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
491 |
< |
{\Xi _{R,t} } & {\Xi _{R,c}^T } \\ |
492 |
< |
{\Xi _{R,c} } & {\Xi _{R,r} } \\ |
493 |
< |
\end{array}} \right)\left( \begin{array}{l} |
494 |
< |
v_{R,i}^b (t) \\ |
495 |
< |
\omega _i (t) \\ |
487 |
> |
\mathbf{F}_{f}^b (t) = \left( \begin{array}{l} |
488 |
> |
\mathbf{f}_{f}^b (t) \\ |
489 |
> |
\mathbf{\tau}_{f}^b (t) \\ |
490 |
> |
\end{array} \right) = - \left( \begin{array}{*{20}c} |
491 |
> |
\Xi_{R,t} & \Xi_{R,c}^T \\ |
492 |
> |
\Xi_{R,c} & \Xi_{R,r} \\ |
493 |
> |
\end{array} \right)\left( \begin{array}{l} |
494 |
> |
\mathbf{v}_{R}^b (t) \\ |
495 |
> |
\mathbf{\omega} (t) \\ |
496 |
|
\end{array} \right), |
497 |
|
\end{equation} |
498 |
< |
while the random force $F_{r,i}^l$ is a Gaussian stochastic variable |
498 |
> |
while the random force $\mathbf{R}^l$ is a Gaussian stochastic variable |
499 |
|
with zero mean and variance |
500 |
|
\begin{equation} |
501 |
< |
\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
502 |
< |
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
503 |
< |
2k_B T\Xi _R \delta (t - t'). \label{randomForce} |
501 |
> |
\left\langle {\mathbf{R}^l (t) (\mathbf{R}^l (t'))^T } \right\rangle = |
502 |
> |
\left\langle {\mathbf{R}^b (t) (\mathbf{R}^b (t'))^T } \right\rangle = |
503 |
> |
2 k_B T \Xi_R \delta(t - t'). \label{randomForce} |
504 |
|
\end{equation} |
505 |
< |
The equation of motion for $v_i$ can be written as |
505 |
> |
Once the $6\times6$ resistance tensor at the center of resistance |
506 |
> |
($\Xi_R$) is known, obtaining a stochastic vector that has the |
507 |
> |
properties in Eq. (\ref{eq:randomForce}) can be done efficiently by |
508 |
> |
carrying out a one-time Cholesky decomposition to obtain the square |
509 |
> |
root matrix of $\Xi_R$.\cite{SchlickBook} Each time a random force |
510 |
> |
vector is needed, a gaussian random vector is generated and then the |
511 |
> |
square root matrix is multiplied onto this vector. |
512 |
> |
|
513 |
> |
The equation of motion for $\mathbf{v}$ can be written as |
514 |
|
\begin{equation} |
515 |
< |
m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) + |
516 |
< |
f_{r,i}^l (t) |
515 |
> |
m \dot{\mathbf{v}} (t) = \mathbf{f}_{s} (t) + \mathbf{f}_{f}^l (t) + |
516 |
> |
\mathbf{R}^l (t) |
517 |
|
\end{equation} |
518 |
|
Since the frictional force is applied at the center of resistance |
519 |
|
which generally does not coincide with the center of mass, an extra |
520 |
|
torque is exerted at the center of mass. Thus, the net body-fixed |
521 |
< |
frictional torque at the center of mass, $\tau _{n,i}^b (t)$, is |
521 |
> |
frictional torque at the center of mass, $\tau_{f}^b (t)$, is |
522 |
|
given by |
523 |
|
\begin{equation} |
524 |
< |
\tau _{r,i}^b = \tau _{r,i}^b +r_{MR} \times f_{r,i}^b |
524 |
> |
\tau_{f}^b \leftarrow \tau_{f}^b + \mathbf{r}_{MR} \times \mathbf{f}_{r}^b |
525 |
|
\end{equation} |
526 |
|
where $r_{MR}$ is the vector from the center of mass to the center |
527 |
|
of the resistance. Instead of integrating the angular velocity in |
528 |
|
lab-fixed frame, we consider the equation of angular momentum in |
529 |
|
body-fixed frame |
530 |
|
\begin{equation} |
531 |
< |
\dot j_i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b (t) |
506 |
< |
+ \tau _{r,i}^b(t) |
531 |
> |
\dot j(t) = \tau_{s} (t) + \tau_{f}^b (t) + \mathbf{R}^b(t) |
532 |
|
\end{equation} |
533 |
< |
Embedding the friction terms into force and torque, one can |
534 |
< |
integrate the langevin equations of motion for rigid body of |
535 |
< |
arbitrary shape in a velocity-Verlet style 2-part algorithm, where |
511 |
< |
$h= \delta t$: |
533 |
> |
Embedding the friction terms into force and torque, one can integrate |
534 |
> |
the Langevin equations of motion for rigid body of arbitrary shape in |
535 |
> |
a velocity-Verlet style 2-part algorithm, where $h= \delta t$: |
536 |
|
|
537 |
|
{\tt moveA:} |
538 |
|
\begin{align*} |
1085 |
|
|
1086 |
|
\subsection{Composite sphero-ellipsoids} |
1087 |
|
Spherical heads perched on the ends of Gay-Berne ellipsoids have been |
1088 |
< |
used recently as models for lipid molecules.\cite{SunGezelter08,Ayton01} |
1089 |
< |
|
1088 |
> |
used recently as models for lipid |
1089 |
> |
molecules.\cite{SunGezelter08,Ayton01} |
1090 |
|
MORE DETAILS |
1091 |
|
|
1092 |
+ |
A reference system composed of a single lipid rigid body embedded in a |
1093 |
+ |
sea of 1929 solvent particles was created and run under standard |
1094 |
+ |
(microcanonical) molecular dynamics. The resulting viscosity of this |
1095 |
+ |
mixture was 0.349 centipoise (as estimated using |
1096 |
+ |
Eq. (\ref{eq:shear})). To calculate the hydrodynamic properties of |
1097 |
+ |
the lipid rigid body model, we created a rough shell (see |
1098 |
+ |
Fig.~\ref{fig:roughShell}), in which the lipid is represented as a |
1099 |
+ |
``shell'' made of 3550 identical beads (0.25 \AA\ in diameter) |
1100 |
+ |
distributed on the surface. Applying the procedure described in |
1101 |
+ |
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
1102 |
+ |
identified the center of resistance, ${\bf r} = $(0 \AA, 0 \AA, 1.46 |
1103 |
+ |
\AA). |
1104 |
|
|
1105 |
+ |
|
1106 |
|
\subsection{Summary} |
1107 |
|
According to our simulations, the langevin dynamics is a reliable |
1108 |
|
theory to apply to replace the explicit solvents, especially for the |
1109 |
|
translation properties. For large molecules, the rotation properties |
1110 |
|
are also mimiced reasonablly well. |
1111 |
|
|
1112 |
+ |
\begin{figure} |
1113 |
+ |
\centering |
1114 |
+ |
\includegraphics[width=\linewidth]{graph} |
1115 |
+ |
\caption[Mean squared displacements and orientational |
1116 |
+ |
correlation functions for each of the model rigid bodies.]{The |
1117 |
+ |
mean-squared displacements ($\langle r^2(t) \rangle$) and |
1118 |
+ |
orientational correlation functions ($C_2(t)$) for each of the model |
1119 |
+ |
rigid bodies studied. The circles are the results for microcanonical |
1120 |
+ |
simulations with explicit solvent molecules, while the other data sets |
1121 |
+ |
are results for Langevin dynamics using the different hydrodynamic |
1122 |
+ |
tensor approximations. The Perrin model for the ellipsoids is |
1123 |
+ |
considered the ``exact'' hydrodynamic behavior (this can also be said |
1124 |
+ |
for the translational motion of the dumbbell operating under the bead |
1125 |
+ |
model). In most cases, the various hydrodynamics models reproduce |
1126 |
+ |
each other quantitatively.} |
1127 |
+ |
\label{fig:results} |
1128 |
+ |
\end{figure} |
1129 |
+ |
|
1130 |
|
\begin{table*} |
1131 |
|
\begin{minipage}{\linewidth} |
1132 |
|
\begin{center} |
1145 |
|
\cline{2-3} \cline{5-7} |
1146 |
|
model & $\eta$ (centipoise) & D & & Analytical & method & Hydrodynamics & simulation \\ |
1147 |
|
\hline |
1148 |
< |
sphere & 0.261 & ? & & 2.59 & exact & 2.59 & 2.56 \\ |
1148 |
> |
sphere & 0.279 & 3.06 & & 2.42 & exact & 2.42 & 2.33 \\ |
1149 |
|
ellipsoid & 0.255 & 2.44 & & 2.34 & exact & 2.34 & 2.37 \\ |
1150 |
|
& 0.255 & 2.44 & & 2.34 & rough shell & 2.36 & 2.28 \\ |
1151 |
< |
dumbbell & 0.322 & ? & & 1.57 & bead model & 1.57 & 1.57 \\ |
1152 |
< |
& 0.322 & ? & & 1.57 & rough shell & ? & ? \\ |
1151 |
> |
dumbbell & 0.308 & 2.06 & & 1.64 & bead model & 1.65 & 1.62 \\ |
1152 |
> |
& 0.308 & 2.06 & & 1.64 & rough shell & 1.59 & 1.62 \\ |
1153 |
|
banana & 0.298 & 1.53 & & & rough shell & 1.56 & 1.55 \\ |
1154 |
|
lipid & 0.349 & 0.96 & & & rough shell & 1.33 & 1.32 \\ |
1155 |
|
\end{tabular} |
1174 |
|
\cline{2-3} \cline{5-7} |
1175 |
|
model & $\eta$ (centipoise) & $\tau$ & & Perrin & method & Hydrodynamic & simulation \\ |
1176 |
|
\hline |
1177 |
< |
sphere & 0.261 & & & 9.06 & exact & 9.06 & 9.11 \\ |
1177 |
> |
sphere & 0.279 & & & 9.69 & exact & 9.69 & 9.64 \\ |
1178 |
|
ellipsoid & 0.255 & 46.7 & & 22.0 & exact & 22.0 & 22.2 \\ |
1179 |
|
& 0.255 & 46.7 & & 22.0 & rough shell & 22.6 & 22.2 \\ |
1180 |
< |
dumbbell & 0.322 & 14.0 & & & bead model & 52.3 & 52.8 \\ |
1181 |
< |
& 0.322 & 14.0 & & & rough shell & ? & ? \\ |
1180 |
> |
dumbbell & 0.308 & 14.1 & & & bead model & 50.0 & 50.1 \\ |
1181 |
> |
& 0.308 & 14.1 & & & rough shell & 41.5 & 41.3 \\ |
1182 |
|
banana & 0.298 & 63.8 & & & rough shell & 70.9 & 70.9 \\ |
1183 |
|
lipid & 0.349 & 78.0 & & & rough shell & 76.9 & 77.9 \\ |
1184 |
|
\hline |