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22		\begin{document}
23
24	<	\title{An algorithm for performing Langevin dynamics on rigid bodies of arbitrary shape }
24	>	\title{Langevin dynamics for rigid bodies of arbitrary shape}
25
26	<	\author{Xiuquan Sun, Teng Lin and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail:
27	<	gezelter@nd.edu} \\
28	<	Department of Chemistry and Biochemistry\\
26	>	\author{Xiuquan Sun, Teng Lin and J. Daniel
27	>	Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\
28	>	Department of Chemistry and Biochemistry,\\
29		University of Notre Dame\\
30		Notre Dame, Indiana 46556}
31
32		\date{\today}
33
34	–	\maketitle \doublespacing
34
35	<	\begin{abstract}
35	>	\maketitle
36
37	+
38	+
39	+	\begin{abstract}
40	+	We present an algorithm for carrying out Langevin dynamics simulations
41	+	on complex rigid bodies by incorporating the hydrodynamic resistance
42	+	tensors for arbitrary shapes into an advanced symplectic integration
43	+	scheme.  The integrator gives quantitative agreement with both
44	+	analytic and approximate hydrodynamic theories for a number of model
45	+	rigid bodies, and works well at reproducing the solute dynamical
46	+	properties (diffusion constants, and orientational relaxation times)
47	+	obtained from explicitly-solvated simulations.
48		\end{abstract}
49
50		\newpage
51
52	+
53	+
54		%\narrowtext
55
56		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
57		%                          BODY OF TEXT
58		%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
59
60	+	\begin{doublespace}
61	+
62		\section{Introduction}
63
64		%applications of langevin dynamics
65	<	As alternative to Newtonian dynamics, Langevin dynamics, which
66	<	mimics a simple heat bath with stochastic and dissipative forces,
67	<	has been applied in a variety of studies. The stochastic treatment
68	<	of the solvent enables us to carry out substantially longer time
69	<	simulations. Implicit solvent Langevin dynamics simulations of
70	<	met-enkephalin not only outperform explicit solvent simulations for
71	<	computational efficiency, but also agrees very well with explicit
72	<	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
75	<	instance, the oscillation power spectrum of nanoparticles from
76	<	Langevin dynamics simulation has the same peak frequencies for
77	<	different wave vectors, which recovers the property of magnetic
78	<	excitations in small finite structures.\cite{Berkov2005a}
65	>	Langevin dynamics, which mimics a heat bath using both stochastic and
66	>	dissipative forces, has been applied in a variety of situations as an
67	>	alternative to molecular dynamics with explicit solvent molecules.
68	>	The stochastic treatment of the solvent allows the use of simulations
69	>	with substantially longer time and length scales. In general, the
70	>	dynamic and structural properties obtained from Langevin simulations
71	>	agree quite well with similar properties obtained from explicit
72	>	solvent simulations.
73
74	<	%review rigid body dynamics
75	<	Rigid bodies are frequently involved in the modeling of different
76	<	areas, from engineering, physics, to chemistry. For example,
77	<	missiles and vehicle are usually modeled by rigid bodies.  The
78	<	movement of the objects in 3D gaming engine or other physics
79	<	simulator is governed by the rigid body dynamics. In molecular
80	<	simulation, rigid body is used to simplify the model in
87	<	protein-protein docking study{\cite{Gray2003}}.
74	>	Recent examples of the usefulness of Langevin simulations include a
75	>	study of met-enkephalin in which Langevin simulations predicted
76	>	dynamical properties that were largely in agreement with explicit
77	>	solvent simulations.\cite{Shen2002} By applying Langevin dynamics with
78	>	the UNRES model, Liwo and his coworkers suggest that protein folding
79	>	pathways can be explored within a reasonable amount of
80	>	time.\cite{Liwo2005}
81
82	<	It is very important to develop stable and efficient methods to
83	<	integrate the equations of motion for orientational degrees of
84	<	freedom. Euler angles are the natural choice to describe the
85	<	rotational degrees of freedom. However, due to $\frac {1}{sin
86	<	\theta}$ singularities, the numerical integration of corresponding
87	<	equations of these motion is very inefficient and inaccurate.
88	<	Although an alternative integrator using multiple sets of Euler
89	<	angles can overcome this difficulty\cite{Barojas1973}, the
90	<	computational penalty and the loss of angular momentum conservation
91	<	still remain. A singularity-free representation utilizing
99	<	quaternions was developed by Evans in 1977.\cite{Evans1977}
100	<	Unfortunately, this approach used a nonseparable Hamiltonian
101	<	resulting from the quaternion representation, which prevented the
102	<	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}
82	>	The stochastic nature of Langevin dynamics also enhances the sampling
83	>	of the system and increases the probability of crossing energy
84	>	barriers.\cite{Cui2003,Banerjee2004} Combining Langevin dynamics with
85	>	Kramers' theory, Klimov and Thirumalai identified free-energy
86	>	barriers by studying the viscosity dependence of the protein folding
87	>	rates.\cite{Klimov1997} In order to account for solvent induced
88	>	interactions missing from the implicit solvent model, Kaya
89	>	incorporated a desolvation free energy barrier into protein
90	>	folding/unfolding studies and discovered a higher free energy barrier
91	>	between the native and denatured states.\cite{HuseyinKaya07012005}
92
93	<	A break-through in geometric literature suggests that, in order to
94	<	develop a long-term integration scheme, one should preserve the
95	<	symplectic structure of the propagator. By introducing a conjugate
96	<	momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's
97	<	equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was
98	<	proposed to evolve the Hamiltonian system in a constraint manifold
99	<	by iteratively satisfying the orthogonality constraint $Q^T Q = 1$.
100	<	An alternative method using the quaternion representation was
101	<	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.
93	>	In typical LD simulations, the friction and random ($f_r$) forces on
94	>	individual atoms are taken from Stokes' law,
95	>	\begin{eqnarray}
96	>	m \dot{v}(t) & = & -\nabla U(x) - \xi m v(t) + f_r(t) \notag \\
97	>	\langle f_r(t) \rangle & = & 0 \\
98	>	\langle f_r(t) f_r(t') \rangle & = & 2 k_B T \xi m \delta(t - t') \notag
99	>	\end{eqnarray}
100	>	where $\xi \approx 6 \pi \eta \rho$.  Here $\eta$ is the viscosity of the
101	>	implicit solvent, and $\rho$ is the hydrodynamic radius of the atom.
102
103	<	%review langevin/browninan dynamics for arbitrarily shaped rigid body
104	<	Combining Langevin or Brownian dynamics with rigid body dynamics,
105	<	one can study slow processes in biomolecular systems. Modeling DNA
106	<	as a chain of rigid beads, which are subject to harmonic potentials
107	<	as well as excluded volume potentials, Mielke and his coworkers
108	<	discovered rapid superhelical stress generations from the stochastic
109	<	simulation of twin supercoiling DNA with response to induced
110	<	torques.\cite{Mielke2004} Membrane fusion is another key biological
111	<	process which controls a variety of physiological functions, such as
112	<	release of neurotransmitters \textit{etc}. A typical fusion event
113	<	happens on the time scale of a millisecond, which is impractical to
114	<	study using atomistic models with newtonian mechanics. With the help
115	<	of coarse-grained rigid body model and stochastic dynamics, the
116	<	fusion pathways were explored by many
117	<	researchers.\cite{Noguchi2001,Noguchi2002,Shillcock2005} Due to the
118	<	difficulty of numerical integration of anisotropic rotation, most of
119	<	the rigid body models are simply modeled using spheres, cylinders,
120	<	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
103	>	The use of rigid substructures,\cite{Chun:2000fj}
104	>	coarse-graining,\cite{Ayton01,Golubkov06,Orlandi:2006fk,SunX._jp0762020}
105	>	and ellipsoidal representations of protein side
106	>	chains~\cite{Fogolari:1996lr} has made the use of the Stokes-Einstein
107	>	approximation problematic.  A rigid substructure moves as a single
108	>	unit with orientational as well as translational degrees of freedom.
109	>	This requires a more general treatment of the hydrodynamics than the
110	>	spherical approximation provides.  Also, the atoms involved in a rigid
111	>	or coarse-grained structure have solvent-mediated interactions with
112	>	each other, and these interactions are ignored if all atoms are
113	>	treated as separate spherical particles.  The theory of interactions
114	>	{\it between} bodies moving through a fluid has been developed over
115	>	the past century and has been applied to simulations of Brownian
116	>	motion.\cite{FIXMAN:1986lr,Ramachandran1996}
117	>
118	>	In order to account for the diffusion anisotropy of complex shapes,
119	>	Fernandes and Garc\'{i}a de la Torre improved an earlier Brownian
120	>	dynamics simulation algorithm~\cite{Ermak1978,Allison1991} by
121		incorporating a generalized $6\times6$ diffusion tensor and
122	<	introducing a simple rotation evolution scheme consisting of three
123	<	consecutive rotations.\cite{Fernandes2002} Unfortunately, unexpected
124	<	errors and biases are introduced into the system due to the
125	<	arbitrary order of applying the noncommuting rotation
126	<	operators.\cite{Beard2003} Based on the observation the momentum
127	<	relaxation time is much less than the time step, one may ignore the
128	<	inertia in Brownian dynamics. However, the assumption of zero
129	<	average acceleration is not always true for cooperative motion which
130	<	is common in protein motion. An inertial Brownian dynamics (IBD) was
131	<	proposed to address this issue by adding an inertial correction
132	<	term.\cite{Beard2000} As a complement to IBD which has a lower bound
133	<	in time step because of the inertial relaxation time, long-time-step
134	<	inertial dynamics (LTID) can be used to investigate the inertial
135	<	behavior of the polymer segments in low friction
136	<	regime.\cite{Beard2000} LTID can also deal with the rotational
137	<	dynamics for nonskew bodies without translation-rotation coupling by
138	<	separating the translation and rotation motion and taking advantage
139	<	of the analytical solution of hydrodynamics properties. However,
140	<	typical nonskew bodies like cylinders and ellipsoids are inadequate
141	<	to represent most complex macromolecule assemblies. These intricate
142	<	molecules have been represented by a set of beads and their
143	<	hydrodynamic properties can be calculated using variants on the
144	<	standard hydrodynamic interaction tensors.
122	>	introducing a rotational evolution scheme consisting of three
123	>	consecutive rotations.\cite{Fernandes2002} Unfortunately, biases are
124	>	introduced into the system due to the arbitrary order of applying the
125	>	noncommuting rotation operators.\cite{Beard2003} Based on the
126	>	observation the momentum relaxation time is much less than the time
127	>	step, one may ignore the inertia in Brownian dynamics.  However, the
128	>	assumption of zero average acceleration is not always true for
129	>	cooperative motion which is common in proteins. An inertial Brownian
130	>	dynamics (IBD) was proposed to address this issue by adding an
131	>	inertial correction term.\cite{Beard2000} As a complement to IBD,
132	>	which has a lower bound in time step because of the inertial
133	>	relaxation time, long-time-step inertial dynamics (LTID) can be used
134	>	to investigate the inertial behavior of linked polymer segments in a
135	>	low friction regime.\cite{Beard2000} LTID can also deal with the
136	>	rotational dynamics for nonskew bodies without translation-rotation
137	>	coupling by separating the translation and rotation motion and taking
138	>	advantage of the analytical solution of hydrodynamic
139	>	properties. However, typical nonskew bodies like cylinders and
140	>	ellipsoids are inadequate to represent most complex macromolecular
141	>	assemblies. Therefore, the goal of this work is to adapt some of the
142	>	hydrodynamic methodologies developed to treat Brownian motion of
143	>	complex assemblies into a Langevin integrator for rigid bodies with
144	>	arbitrary shapes.
145
146	+	\subsection{Rigid Body Dynamics}
147	+	Rigid bodies are frequently involved in the modeling of large
148	+	collections of particles that move as a single unit.  In molecular
149	+	simulations, rigid bodies have been used to simplify protein-protein
150	+	docking,\cite{Gray2003} and lipid bilayer
151	+	simulations.\cite{SunX._jp0762020} Many of the water models in common
152	+	use are also rigid-body
153	+	models,\cite{Jorgensen83,Berendsen81,Berendsen87} although they are
154	+	typically evolved in molecular dynamics simulations using constraints
155	+	rather than rigid body equations of motion.
156	+
157	+	Euler angles are a natural choice to describe the rotational degrees
158	+	of freedom.  However, due to $\frac{1}{\sin \theta}$ singularities, the
159	+	numerical integration of corresponding equations of these motion can
160	+	become inaccurate (and inefficient).  Although the use of multiple
161	+	sets of Euler angles can overcome this problem,\cite{Barojas1973} the
162	+	computational penalty and the loss of angular momentum conservation
163	+	remain.  A singularity-free representation utilizing quaternions was
164	+	developed by Evans in 1977.\cite{Evans1977} The Evans quaternion
165	+	approach uses a nonseparable Hamiltonian, and this has prevented
166	+	symplectic algorithms from being utilized until very
167	+	recently.\cite{Miller2002}
168	+
169	+	Another approach is the application of holonomic constraints to the
170	+	atoms belonging to the rigid body.  Each atom moves independently
171	+	under the normal forces deriving from potential energy and constraints
172	+	are used to guarantee rigidity. However, due to their iterative
173	+	nature, the SHAKE and RATTLE algorithms converge very slowly when the
174	+	number of constraints (and the number of particles that belong to the
175	+	rigid body) increases.\cite{Ryckaert1977,Andersen1983}
176	+
177	+	In order to develop a stable and efficient integration scheme that
178	+	preserves most constants of the motion, symplectic propagators are
179	+	necessary.  By introducing a conjugate momentum to the rotation matrix
180	+	${\bf Q}$ and re-formulating Hamilton's equations, a symplectic
181	+	orientational integrator, RSHAKE,\cite{Kol1997} was proposed to evolve
182	+	rigid bodies on a constraint manifold by iteratively satisfying the
183	+	orthogonality constraint ${\bf Q}^T {\bf Q} = 1$.  An alternative
184	+	method using the quaternion representation was developed by
185	+	Omelyan.\cite{Omelyan1998} However, both of these methods are
186	+	iterative and suffer from some related inefficiencies. A symplectic
187	+	Lie-Poisson integrator for rigid bodies developed by Dullweber {\it et
188	+	al.}\cite{Dullweber1997} removes most of the limitations mentioned
189	+	above and is therefore the basis for our Langevin integrator.
190	+
191		The goal of the present work is to develop a Langevin dynamics
192	<	algorithm for arbitrary-shaped rigid particles by integrating the
193	<	accurate estimation of friction tensor from hydrodynamics theory
194	<	into the sophisticated rigid body dynamics algorithms.
192	>	algorithm for arbitrary-shaped rigid particles by integrating an
193	>	accurate estimate of the friction tensor from hydrodynamics theory
194	>	into a symplectic rigid body dynamics propagator.  In the sections
195	>	below, we review some of the theory of hydrodynamic tensors developed
196	>	primarily for Brownian simulations of multi-particle systems, we then
197	>	present our integration method for a set of generalized Langevin
198	>	equations of motion, and we compare the behavior of the new Langevin
199	>	integrator to dynamical quantities obtained via explicit solvent
200	>	molecular dynamics.
201
202	<	\subsection{\label{introSection:frictionTensor}Friction Tensor}
203	<	Theoretically, the friction kernel can be determined using the
204	<	velocity autocorrelation function. However, this approach becomes
205	<	impractical when the system becomes more and more complicated.
206	<	Instead, various approaches based on hydrodynamics have been
207	<	developed to calculate the friction coefficients. In general, the
208	<	friction tensor $\Xi$ is a $6\times 6$ matrix given by
209	<	\[
210	<	\Xi  = \left( {\begin{array}{*{20}c}
211	<	{\Xi _{}^{tt} } & {\Xi _{}^{rt} }  \\
212	<	{\Xi _{}^{tr} } & {\Xi _{}^{rr} }  \\
213	<	\end{array}} \right).
214	<	\]
215	<	Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are $3 \times 3$
216	<	translational friction tensor and rotational resistance (friction)
217	<	tensor respectively, while ${\Xi^{tr} }$ is translation-rotation
218	<	coupling tensor and $ {\Xi^{rt} }$ is rotation-translation coupling
219	<	tensor. When a particle moves in a fluid, it may experience friction
220	<	force or torque along the opposite direction of the velocity or
221	<	angular velocity,
222	<	\[
202	>	\subsection{\label{introSection:frictionTensor}The Friction Tensor}
203	>	Theoretically, a complete friction kernel for a solute particle can be
204	>	determined using the velocity autocorrelation function from a
205	>	simulation with explicit solvent molecules. However, this approach
206	>	becomes impractical when the solute becomes complex.  Instead, various
207	>	approaches based on hydrodynamics have been developed to calculate
208	>	static friction coefficients. In general, the friction tensor $\Xi$ is
209	>	a $6\times 6$ matrix given by
210	>	\begin{equation}
211	>	\Xi  = \left( \begin{array}{*{20}c}
212	>	\Xi^{tt} & \Xi^{rt}  \\
213	>	\Xi^{tr} & \Xi^{rr}  \\
214	>	\end{array} \right).
215	>	\end{equation}
216	>	Here, $\Xi^{tt}$ and $\Xi^{rr}$ are $3 \times 3$ translational and
217	>	rotational resistance (friction) tensors respectively, while
218	>	$\Xi^{tr}$ is translation-rotation coupling tensor and $\Xi^{rt}$ is
219	>	rotation-translation coupling tensor. When a particle moves in a
220	>	fluid, it may experience a friction force ($\mathbf{f}_f$) and torque
221	>	($\mathbf{\tau}_f$) in opposition to the velocity ($\mathbf{v}$) and
222	>	body-fixed angular velocity ($\mathbf{\omega}$),
223	>	\begin{equation}
224		\left( \begin{array}{l}
225	<	F_R  \\
226	<	\tau _R  \\
227	<	\end{array} \right) =  - \left( {\begin{array}{*{20}c}
228	<	{\Xi ^{tt} } & {\Xi ^{rt} }  \\
229	<	{\Xi ^{tr} } & {\Xi ^{rr} }  \\
230	<	\end{array}} \right)\left( \begin{array}{l}
231	<	v \\
232	<	w \\
233	<	\end{array} \right)
234	<	\]
206	<	where $F_r$ is the friction force and $\tau _R$ is the friction
207	<	torque.
225	>	\mathbf{f}_f  \\
226	>	\mathbf{\tau}_f  \\
227	>	\end{array} \right) =  - \left( \begin{array}{*{20}c}
228	>	\Xi^{tt} & \Xi^{rt}  \\
229	>	\Xi^{tr} & \Xi^{rr}  \\
230	>	\end{array} \right)\left( \begin{array}{l}
231	>	\mathbf{v} \\
232	>	\mathbf{\omega} \\
233	>	\end{array} \right).
234	>	\end{equation}
235
236		\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shapes}}
237	<
238	<	For a spherical particle with slip boundary conditions, the
239	<	translational and rotational friction constant can be calculated
240	<	from Stoke's law,
241	<	\[
242	<	\Xi ^{tt}  = \left( {\begin{array}{*{20}c}
243	<	{6\pi \eta R} & 0 & 0  \\
244	<	0 & {6\pi \eta R} & 0  \\
245	<	0 & 0 & {6\pi \eta R}  \\
246	<	\end{array}} \right)
247	<	\]
237	>	For a  spherical body under ``stick'' boundary conditions,
238	>	the translational and rotational friction tensors can be calculated
239	>	from Stokes' law,
240	>	\begin{equation}
241	>	\label{eq:StokesTranslation}
242	>	\Xi^{tt}  = \left( \begin{array}{*{20}c}
243	>	{6\pi \eta \rho} & 0 & 0  \\
244	>	0 & {6\pi \eta \rho} & 0  \\
245	>	0 & 0 & {6\pi \eta \rho}  \\
246	>	\end{array} \right)
247	>	\end{equation}
248		and
249	<	\[
250	<	\Xi ^{rr}  = \left( {\begin{array}{*{20}c}
251	<	{8\pi \eta R^3 } & 0 & 0  \\
252	<	0 & {8\pi \eta R^3 } & 0  \\
253	<	0 & 0 & {8\pi \eta R^3 }  \\
254	<	\end{array}} \right)
255	<	\]
256	<	where $\eta$ is the viscosity of the solvent and $R$ is the
257	<	hydrodynamic radius.
249	>	\begin{equation}
250	>	\label{eq:StokesRotation}
251	>	\Xi^{rr}  = \left( \begin{array}{*{20}c}
252	>	{8\pi \eta \rho^3 } & 0 & 0  \\
253	>	0 & {8\pi \eta \rho^3 } & 0  \\
254	>	0 & 0 & {8\pi \eta \rho^3 }  \\
255	>	\end{array} \right)
256	>	\end{equation}
257	>	where $\eta$ is the viscosity of the solvent and $\rho$ is the
258	>	hydrodynamic radius.  The presence of the rotational resistance tensor
259	>	implies that the spherical body has internal structure and
260	>	orientational degrees of freedom that must be propagated in time.  For
261	>	non-structured spherical bodies (i.e. the atoms in a traditional
262	>	molecular dynamics simulation) these degrees of freedom do not exist.
263
264		Other non-spherical shapes, such as cylinders and ellipsoids, are
265	<	widely used as references for developing new hydrodynamics theory,
265	>	widely used as references for developing new hydrodynamic theories,
266		because their properties can be calculated exactly. In 1936, Perrin
267	<	extended Stokes's law to general ellipsoids, also called a triaxial
268	<	ellipsoid, which is given in Cartesian coordinates
269	<	by\cite{Perrin1934, Perrin1936}
270	<	\[
271	<	\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2
272	<	}} = 1
273	<	\]
274	<	where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately,
275	<	due to the complexity of the elliptic integral, only the ellipsoid
276	<	with the restriction of two axes being equal, \textit{i.e.}
277	<	prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved
278	<	exactly. Introducing an elliptic integral parameter $S$ for prolate
279	<	ellipsoids :
280	<	\[
281	<	S = \frac{2}{{\sqrt {a^2  - b^2 } }}\ln \frac{{a + \sqrt {a^2  - b^2
282	<	} }}{b},
283	<	\]
284	<	and oblate ellipsoids:
285	<	\[
286	<	S = \frac{2}{{\sqrt {b^2  - a^2 } }}arctg\frac{{\sqrt {b^2  - a^2 }
287	<	}}{a},
288	<	\]
289	<	one can write down the translational and rotational resistance
290	<	tensors
291	<	\begin{eqnarray*}
292	<	\Xi _a^{tt}  & = & 16\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - b^2 )S - 2a}}. \\
293	<	\Xi _b^{tt}  & = & \Xi _c^{tt}  = 32\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - 3b^2 )S +
294	<	2a}},
295	<	\end{eqnarray*}
296	<	and
297	<	\begin{eqnarray*}
266	<	\Xi _a^{rr} & = & \frac{{32\pi }}{3}\eta \frac{{(a^2  - b^2 )b^2 }}{{2a - b^2 S}}, \\
267	<	\Xi _b^{rr} & = & \Xi _c^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^4  - b^4 )}}{{(2a^2  - b^2 )S - 2a}}.
268	<	\end{eqnarray*}
267	>	extended Stokes' law to general
268	>	ellipsoids,\cite{Perrin1934,Perrin1936} described in Cartesian
269	>	coordinates as
270	>	\begin{equation}
271	>	\frac{x^2 }{a^2} + \frac{y^2}{b^2} + \frac{z^2 }{c^2} = 1.
272	>	\end{equation}
273	>	Here, the semi-axes are of lengths $a$, $b$, and $c$. Due to the
274	>	complexity of the elliptic integral, only uniaxial ellipsoids, either
275	>	prolate ($a \ge b = c$) or oblate ($a < b = c$), were solved
276	>	exactly. Introducing an elliptic integral parameter $S$ for prolate,
277	>	\begin{equation}
278	>	S = \frac{2}{\sqrt{a^2  - b^2}} \ln \frac{a + \sqrt{a^2  - b^2}}{b},
279	>	\end{equation}
280	>	and oblate,
281	>	\begin{equation}
282	>	S = \frac{2}{\sqrt {b^2  - a^2 }} \arctan \frac{\sqrt {b^2  - a^2}}{a},
283	>	\end{equation}
284	>	ellipsoids, it is possible to write down exact solutions for the
285	>	resistance tensors.  As is the case for spherical bodies, the translational,
286	>	\begin{eqnarray}
287	>	\Xi_a^{tt}  & = & 16\pi \eta \frac{a^2  - b^2}{(2a^2  - b^2 )S - 2a}. \\
288	>	\Xi_b^{tt} =  \Xi_c^{tt} & = & 32\pi \eta \frac{a^2  - b^2 }{(2a^2 - 3b^2 )S + 2a},
289	>	\end{eqnarray}
290	>	and rotational,
291	>	\begin{eqnarray}
292	>	\Xi_a^{rr} & = & \frac{32\pi}{3} \eta \frac{(a^2  - b^2 )b^2}{2a - b^2 S}, \\
293	>	\Xi_b^{rr} = \Xi_c^{rr} & = & \frac{32\pi}{3} \eta \frac{(a^4  - b^4)}{(2a^2  - b^2 )S - 2a}
294	>	\end{eqnarray}
295	>	resistance tensors are diagonal $3 \times 3$ matrices. For both
296	>	spherical and ellipsoidal particles, the translation-rotation and
297	>	rotation-translation coupling tensors are zero.
298
299		\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}}
300	+	There is no analytical solution for the friction tensor for rigid
301	+	molecules of arbitrary shape. The ellipsoid of revolution and general
302	+	triaxial ellipsoid models have been widely used to approximate the
303	+	hydrodynamic properties of rigid bodies. However, the mapping from all
304	+	possible ellipsoidal spaces ($r$-space) to all possible combinations
305	+	of rotational diffusion coefficients ($D$-space) is not
306	+	unique.\cite{Wegener1979} Additionally, because there is intrinsic
307	+	coupling between translational and rotational motion of {\it skew}
308	+	rigid bodies, general ellipsoids are not always suitable for modeling
309	+	rigid molecules.  A number of studies have been devoted to determining
310	+	the friction tensor for irregular shapes using methods in which the
311	+	molecule of interest is modeled with a combination of
312	+	spheres\cite{Carrasco1999} and the hydrodynamic properties of the
313	+	molecule are then calculated using a set of two-point interaction
314	+	tensors.  We have found the {\it bead} and {\it rough shell} models of
315	+	Carrasco and Garc\'{i}a de la Torre to be the most useful of these
316	+	methods,\cite{Carrasco1999} and we review the basic outline of the
317	+	rough shell approach here.  A more thorough explanation can be found
318	+	in Ref. \citen{Carrasco1999}.
319
320	<	Unlike spherical and other simply shaped molecules, there is no
321	<	analytical solution for the friction tensor for arbitrarily shaped
322	<	rigid molecules. The ellipsoid of revolution model and general
323	<	triaxial ellipsoid model have been used to approximate the
324	<	hydrodynamic properties of rigid bodies. However, since the mapping
277	<	from all possible ellipsoidal spaces, $r$-space, to all possible
278	<	combination of rotational diffusion coefficients, $D$-space, is not
279	<	unique\cite{Wegener1979} as well as the intrinsic coupling between
280	<	translational and rotational motion of rigid bodies, general
281	<	ellipsoids are not always suitable for modeling arbitrarily shaped
282	<	rigid molecules. A number of studies have been devoted to
283	<	determining the friction tensor for irregularly shaped rigid bodies
284	<	using more advanced methods where the molecule of interest was
285	<	modeled by a combinations of spheres\cite{Carrasco1999} and the
286	<	hydrodynamics properties of the molecule can be calculated using the
287	<	hydrodynamic interaction tensor. Let us consider a rigid assembly of
288	<	$N$ beads immersed in a continuous medium. Due to hydrodynamic
289	<	interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different
290	<	than its unperturbed velocity $v_i$,
291	<	\[
292	<	v'_i  = v_i  - \sum\limits_{j \ne i} {T_{ij} F_j }
293	<	\]
294	<	where $F_i$ is the frictional force, and $T_{ij}$ is the
295	<	hydrodynamic interaction tensor. The friction force of $i$th bead is
296	<	proportional to its ``net'' velocity
320	>	Consider a rigid assembly of $N$ small beads moving through a
321	>	continuous medium.  Due to hydrodynamic interactions between the
322	>	beads, the net velocity of the $i^\mathrm{th}$ bead relative to the
323	>	medium, ${\bf v}'_i$, is different than its unperturbed velocity ${\bf
324	>	v}_i$,
325		\begin{equation}
326	<	F_i  = \zeta _i v_i  - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }.
299	<	\label{introEquation:tensorExpression}
326	>	{\bf v}'_i  = {\bf v}_i  - \sum\limits_{j \ne i} {{\bf T}_{ij} {\bf F}_j }
327		\end{equation}
328	<	This equation is the basis for deriving the hydrodynamic tensor. In
329	<	1930, Oseen and Burgers gave a simple solution to
330	<	Eq.~\ref{introEquation:tensorExpression}
328	>	where ${\bf F}_j$ is the frictional force on the medium due to bead $j$, and
329	>	${\bf T}_{ij}$ is the hydrodynamic interaction tensor between the two beads.
330	>	The frictional force felt by the $i^\mathrm{th}$ bead is proportional to
331	>	its net velocity
332		\begin{equation}
333	<	T_{ij}  = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij}
334	<	R_{ij}^T }}{{R_{ij}^2 }}} \right). \label{introEquation:oseenTensor}
333	>	{\bf F}_i  = \xi_i {\bf v}_i  - \xi_i \sum\limits_{j \ne i} {{\bf T}_{ij} {\bf F}_j }.
334	>	\label{introEquation:tensorExpression}
335		\end{equation}
336	<	Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$.
337	<	A second order expression for element of different size was
338	<	introduced by Rotne and Prager\cite{Rotne1969} and improved by
336	>	Eq. (\ref{introEquation:tensorExpression}) defines the two-point
337	>	hydrodynamic tensor, ${\bf T}_{ij}$.  There have been many proposed
338	>	solutions to this equation, including the simple solution given by
339	>	Oseen and Burgers in 1930 for two beads of identical radius.  A second
340	>	order expression for beads of different hydrodynamic radii was
341	>	introduced by Rotne and Prager,\cite{Rotne1969} and improved by
342		Garc\'{i}a de la Torre and Bloomfield,\cite{Torre1977}
343		\begin{equation}
344	<	T_{ij}  = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I +
345	<	\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma
346	<	_i^2  + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} -
347	<	\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right].
344	>	{\bf T}_{ij}  = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {{\bf I} +
345	>	\frac{{{\bf R}_{ij} {\bf R}_{ij}^T }}{{R_{ij}^2 }}} \right) + \frac{{\rho
346	>	_i^2  + \rho_j^2 }}{{R_{ij}^2 }}\left( {\frac{{\bf I}}{3} -
347	>	\frac{{{\bf R}_{ij} {\bf R}_{ij}^T }}{{R_{ij}^2 }}} \right)} \right].
348		\label{introEquation:RPTensorNonOverlapped}
349		\end{equation}
350	<	Both of the Eq.~\ref{introEquation:oseenTensor} and
351	<	Eq.~\ref{introEquation:RPTensorNonOverlapped} have an assumption
352	<	$R_{ij} \ge \sigma _i  + \sigma _j$. An alternative expression for
353	<	overlapping beads with the same radius, $\sigma$, is given by
350	>	Here ${\bf R}_{ij}$ is the distance vector between beads $i$ and $j$.  Both
351	>	the Oseen-Burgers tensor and
352	>	Eq.~\ref{introEquation:RPTensorNonOverlapped} have an assumption that
353	>	the beads do not overlap ($R_{ij} \ge \rho_i + \rho_j$).
354	>
355	>	To calculate the resistance tensor for a body represented as the union
356	>	of many non-overlapping beads, we first pick an arbitrary origin $O$
357	>	and then construct a $3N \times 3N$ supermatrix consisting of $N
358	>	\times N$ ${\bf B}_{ij}$ blocks
359		\begin{equation}
360	<	T_{ij}  = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 -
361	<	\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I +
362	<	\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right]
363	<	\label{introEquation:RPTensorOverlapped}
360	>	{\bf B} = \left( \begin{array}{*{20}c}
361	>	{\bf B}_{11} &  \ldots  & {\bf B}_{1N}   \\
362	>	\vdots  &  \ddots  &  \vdots   \\
363	>	{\bf B}_{N1} &  \cdots  & {\bf B}_{NN}
364	>	\end{array} \right)
365		\end{equation}
366	<	To calculate the resistance tensor at an arbitrary origin $O$, we
367	<	construct a $3N \times 3N$ matrix consisting of $N \times N$
331	<	$B_{ij}$ blocks
366	>	${\bf B}_{ij}$ is a version of the hydrodynamic tensor which includes the
367	>	self-contributions for spheres,
368		\begin{equation}
369	<	B = \left( {\begin{array}{*{20}c}
370	<	{B_{11} } &  \ldots  & {B_{1N} }  \\
335	<	\vdots  &  \ddots  &  \vdots   \\
336	<	{B_{N1} } &  \cdots  & {B_{NN} }  \\
337	<	\end{array}} \right),
369	>	{\bf B}_{ij}  = \delta _{ij} \frac{{\bf I}}{{6\pi \eta R_{ij}}} + (1 - \delta_{ij}
370	>	){\bf T}_{ij}
371		\end{equation}
372	<	where $B_{ij}$ is given by
373	<	\[
374	<	B_{ij}  = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij}
375	<	)T_{ij}
376	<	\]
377	<	where $\delta _{ij}$ is the Kronecker delta function. Inverting the
378	<	$B$ matrix, we obtain
379	<	\[
380	<	C = B^{ - 1}  = \left( {\begin{array}{*{20}c}
381	<	{C_{11} } &  \ldots  & {C_{1N} }  \\
382	<	\vdots  &  \ddots  &  \vdots   \\
383	<	{C_{N1} } &  \cdots  & {C_{NN} }  \\
384	<	\end{array}} \right),
385	<	\]
386	<	which can be partitioned into $N \times N$ $3 \times 3$ block
387	<	$C_{ij}$. With the help of $C_{ij}$ and the skew matrix $U_i$
388	<	\[
389	<	U_i  = \left( {\begin{array}{*{20}c}
390	<	0 & { - z_i } & {y_i }  \\
391	<	{z_i } & 0 & { - x_i }  \\
359	<	{ - y_i } & {x_i } & 0  \\
360	<	\end{array}} \right)
361	<	\]
372	>	where $\delta_{ij}$ is the Kronecker delta function. Inverting the
373	>	${\bf B}$ matrix, we obtain
374	>	\begin{equation}
375	>	{\bf C} = {\bf B}^{ - 1}  = \left(\begin{array}{*{20}c}
376	>	{\bf C}_{11} &  \ldots  & {\bf C}_{1N} \\
377	>	\vdots  &  \ddots  &  \vdots   \\
378	>	{\bf C}_{N1} &  \cdots  & {\bf C}_{NN}
379	>	\end{array} \right),
380	>	\end{equation}
381	>	which can be partitioned into $N \times N$ blocks labeled ${\bf C}_{ij}$.
382	>	(Each of the ${\bf C}_{ij}$ blocks is a $3 \times 3$ matrix.)  Using the
383	>	skew matrix,
384	>	\begin{equation}
385	>	{\bf U}_i  = \left(\begin{array}{*{20}c}
386	>	0 & -z_i & y_i \\
387	>	z_i &  0   & - x_i \\
388	>	-y_i & x_i & 0
389	>	\end{array}\right)
390	>	\label{eq:skewMatrix}
391	>	\end{equation}
392		where $x_i$, $y_i$, $z_i$ are the components of the vector joining
393	<	bead $i$ and origin $O$, the elements of resistance tensor at
394	<	arbitrary origin $O$ can be written as
393	>	bead $i$ and origin $O$, the elements of the resistance tensor (at the
394	>	arbitrary origin $O$) can be written as
395		\begin{eqnarray}
366	–	\Xi _{}^{tt}  & = & \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\
367	–	\Xi _{}^{tr}  & = & \Xi _{}^{rt}  = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\
368	–	\Xi _{}^{rr}  & = &  - \sum\limits_i {\sum\limits_j {U_i C_{ij} } }
369	–	U_j  + 6 \eta V {\bf I}. \notag
396		\label{introEquation:ResistanceTensorArbitraryOrigin}
397	+	\Xi^{tt}  & = & \sum\limits_i {\sum\limits_j {{\bf C}_{ij} } } \notag , \\
398	+	\Xi^{tr}  = \Xi _{}^{rt}  & = & \sum\limits_i {\sum\limits_j {{\bf U}_i {\bf C}_{ij} } } , \\
399	+	\Xi^{rr}  & = &  -\sum\limits_i \sum\limits_j {\bf U}_i {\bf C}_{ij} {\bf U}_j + 6 \eta V {\bf I}. \notag
400		\end{eqnarray}
401	<	The final term in the expression for $\Xi^{rr}$ is correction that
402	<	accounts for errors in the rotational motion of certain kinds of bead
403	<	models. The additive correction uses the solvent viscosity ($\eta$)
404	<	as well as the total volume of the beads that contribute to the
376	<	hydrodynamic model,
401	>	The final term in the expression for $\Xi^{rr}$ is a correction that
402	>	accounts for errors in the rotational motion of the bead models. The
403	>	additive correction uses the solvent viscosity ($\eta$) as well as the
404	>	total volume of the beads that contribute to the hydrodynamic model,
405		\begin{equation}
406	<	V = \frac{4 \pi}{3} \sum_{i=1}^{N} \sigma_i^3,
406	>	V = \frac{4 \pi}{3} \sum_{i=1}^{N} \rho_i^3,
407		\end{equation}
408	<	where $\sigma_i$ is the radius of bead $i$.  This correction term was
408	>	where $\rho_i$ is the radius of bead $i$.  This correction term was
409		rigorously tested and compared with the analytical results for
410	<	two-sphere and ellipsoidal systems by Garcia de la Torre and
410	>	two-sphere and ellipsoidal systems by Garc\'{i}a de la Torre and
411		Rodes.\cite{Torre:1983lr}
412
413	<
414	<	The resistance tensor depends on the origin to which they refer. The
415	<	proper location for applying the friction force is the center of
416	<	resistance (or center of reaction), at which the trace of rotational
417	<	resistance tensor, $ \Xi ^{rr}$ reaches a minimum value.
418	<	Mathematically, the center of resistance is defined as an unique
419	<	point of the rigid body at which the translation-rotation coupling
392	<	tensors are symmetric,
413	>	In general, resistance tensors depend on the origin at which they were
414	>	computed.  However, the proper location for applying the friction
415	>	force is the center of resistance, the special point at which the
416	>	trace of rotational resistance tensor, $\Xi^{rr}$ reaches a minimum
417	>	value.  Mathematically, the center of resistance can also be defined
418	>	as the unique point for a rigid body at which the translation-rotation
419	>	coupling tensors are symmetric,
420		\begin{equation}
421	<	\Xi^{tr}  = \left( {\Xi^{tr} } \right)^T
421	>	\Xi^{tr}  = \left(\Xi^{tr}\right)^T
422		\label{introEquation:definitionCR}
423		\end{equation}
424	<	From Equation \ref{introEquation:ResistanceTensorArbitraryOrigin},
425	<	we can easily derive that the translational resistance tensor is
426	<	origin independent, while the rotational resistance tensor and
424	>	From Eq. \ref{introEquation:ResistanceTensorArbitraryOrigin}, we can
425	>	easily derive that the {\it translational} resistance tensor is origin
426	>	independent, while the rotational resistance tensor and
427		translation-rotation coupling resistance tensor depend on the
428	<	origin. Given the resistance tensor at an arbitrary origin $O$, and
429	<	a vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can
430	<	obtain the resistance tensor at $P$ by
431	<	\begin{equation}
432	<	\begin{array}{l}
433	<	\Xi _P^{tt}  = \Xi _O^{tt}  \\
434	<	\Xi _P^{tr}  = \Xi _P^{rt}  = \Xi _O^{tr}  - U_{OP} \Xi _O^{tt}  \\
435	<	\Xi _P^{rr}  = \Xi _O^{rr}  - U_{OP} \Xi _O^{tt} U_{OP}  + \Xi _O^{tr} U_{OP}  - U_{OP} \Xi _O^{{tr} ^{^T }}  \\
436	<	\end{array}
437	<	\label{introEquation:resistanceTensorTransformation}
438	<	\end{equation}
439	<	where
440	<	\[
441	<	U_{OP}  = \left( {\begin{array}{*{20}c}
442	<	0 & { - z_{OP} } & {y_{OP} }  \\
443	<	{z_i } & 0 & { - x_{OP} }  \\
444	<	{ - y_{OP} } & {x_{OP} } & 0  \\
445	<	\end{array}} \right)
446	<	\]
447	<	Using Eq.~\ref{introEquation:definitionCR} and
448	<	Eq.~\ref{introEquation:resistanceTensorTransformation}, one can
449	<	locate the position of center of resistance,
450	<	\begin{eqnarray*}
451	<	\left( \begin{array}{l}
452	<	x_{OR}  \\
453	<	y_{OR}  \\
454	<	z_{OR}  \\
455	<	\end{array} \right) & = &\left( {\begin{array}{*{20}c}
456	<	{(\Xi _O^{rr} )_{yy}  + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} }  \\
457	<	{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz}  + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} }  \\
458	<	{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx}  + (\Xi _O^{rr} )_{yy} }  \\
459	<	\end{array}} \right)^{ - 1}  \\
460	<	& & \left( \begin{array}{l}
434	<	(\Xi _O^{tr} )_{yz}  - (\Xi _O^{tr} )_{zy}  \\
435	<	(\Xi _O^{tr} )_{zx}  - (\Xi _O^{tr} )_{xz}  \\
436	<	(\Xi _O^{tr} )_{xy}  - (\Xi _O^{tr} )_{yx}  \\
437	<	\end{array} \right) \\
438	<	\end{eqnarray*}
439	<	where $x_OR$, $y_OR$, $z_OR$ are the components of the vector
428	>	origin. Given the resistance tensor at an arbitrary origin $O$, and a
429	>	vector ,${\bf r}_{OP} = (x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we
430	>	can obtain the resistance tensor at $P$ by
431	>	\begin{eqnarray}
432	>	\label{introEquation:resistanceTensorTransformation}
433	>	\Xi_P^{tt}  & = & \Xi_O^{tt}  \notag \\
434	>	\Xi_P^{tr}  = \Xi_P^{rt}  & = & \Xi_O^{tr}  - {\bf U}_{OP} \Xi _O^{tt}  \\
435	>	\Xi_P^{rr}  & = &\Xi_O^{rr}  - {\bf U}_{OP} \Xi_O^{tt} {\bf U}_{OP}
436	>	+ \Xi_O^{tr} {\bf U}_{OP}  - {\bf U}_{OP} \left( \Xi_O^{tr}
437	>	\right)^{^T} \notag
438	>	\end{eqnarray}
439	>	where ${\bf U}_{OP}$ is the skew matrix (Eq. (\ref{eq:skewMatrix}))
440	>	for the vector between the origin $O$ and the point $P$. Using
441	>	Eqs.~\ref{introEquation:definitionCR}~and~\ref{introEquation:resistanceTensorTransformation},
442	>	one can locate the position of center of resistance,
443	>	\begin{equation*}
444	>	\left(\begin{array}{l}
445	>	x_{OR} \\
446	>	y_{OR} \\
447	>	z_{OR}
448	>	\end{array}\right) =
449	>	\left(\begin{array}{*{20}c}
450	>	(\Xi_O^{rr})_{yy} + (\Xi_O^{rr})_{zz} & -(\Xi_O^{rr})_{xy} & -(\Xi_O^{rr})_{xz} \\
451	>	-(\Xi_O^{rr})_{xy} & (\Xi_O^{rr})_{zz} + (\Xi_O^{rr})_{xx} & -(\Xi_O^{rr})_{yz} \\
452	>	-(\Xi_O^{rr})_{xz} & -(\Xi_O^{rr})_{yz} & (\Xi_O^{rr})_{xx} + (\Xi_O^{rr})_{yy} \\
453	>	\end{array}\right)^{-1}
454	>	\left(\begin{array}{l}
455	>	(\Xi_O^{tr})_{yz} - (\Xi_O^{tr})_{zy} \\
456	>	(\Xi_O^{tr})_{zx} - (\Xi_O^{tr})_{xz} \\
457	>	(\Xi_O^{tr})_{xy} - (\Xi_O^{tr})_{yx}
458	>	\end{array}\right)
459	>	\end{equation*}
460	>	where $x_{OR}$, $y_{OR}$, $z_{OR}$ are the components of the vector
461		joining center of resistance $R$ and origin $O$.
462
463	+	For a general rigid molecular substructure, finding the $6 \times 6$
464	+	resistance tensor can be a computationally demanding task.  First, a
465	+	lattice of small beads that extends well beyond the boundaries of the
466	+	rigid substructure is created.  The lattice is typically composed of
467	+	0.25 \AA\ beads on a dense FCC lattice.  The lattice constant is taken
468	+	to be the bead diameter, so that adjacent beads are touching, but do
469	+	not overlap. To make a shape corresponding to the rigid structure,
470	+	beads that sit on lattice sites that are outside the van der Waals
471	+	radii of all of the atoms comprising the rigid body are excluded from
472	+	the calculation.
473
474	+	For large structures, most of the beads will be deep within the rigid
475	+	body and will not contribute to the hydrodynamic tensor.  In the {\it
476	+	rough shell} approach, beads which have all of their lattice neighbors
477	+	inside the structure are considered interior beads, and are removed
478	+	from the calculation.  After following this procedure, only those
479	+	beads in direct contact with the van der Waals surface of the rigid
480	+	body are retained.  For reasonably large molecular structures, this
481	+	truncation can still produce bead assemblies with thousands of
482	+	members.
483	+
484	+	If all of the {\it atoms} comprising the rigid substructure are
485	+	spherical and non-overlapping, the tensor in
486	+	Eq.~(\ref{introEquation:RPTensorNonOverlapped}) may be used directly
487	+	using the atoms themselves as the hydrodynamic beads.  This is a
488	+	variant of the {\it bead model} approach of Carrasco and Garc\'{i}a de
489	+	la Torre.\cite{Carrasco1999} In this case, the size of the ${\bf B}$
490	+	matrix can be quite small, and the calculation of the hydrodynamic
491	+	tensor is straightforward.
492	+
493	+	In general, the inversion of the ${\bf B}$ matrix is the most
494	+	computationally demanding task.  This inversion is done only once for
495	+	each type of rigid structure.  We have used straightforward
496	+	LU-decomposition to solve the linear system and to obtain the elements
497	+	of ${\bf C}$. Once ${\bf C}$ has been obtained, the location of the
498	+	center of resistance ($R$) is found and the resistance tensor at this
499	+	point is calculated.  The $3 \times 1$ vector giving the location of
500	+	the rigid body's center of resistance and the $6 \times 6$ resistance
501	+	tensor are then stored for use in the Langevin dynamics calculation.
502	+	These quantities depend on solvent viscosity and temperature and must
503	+	be recomputed if different simulation conditions are required.
504	+
505		\section{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}}
506	+
507		Consider the Langevin equations of motion in generalized coordinates
508		\begin{equation}
509	<	M_i \dot V_i (t) = F_{s,i} (t) + F_{f,i(t)}  + F_{r,i} (t)
509	>	{\bf M} \dot{{\bf V}}(t) = {\bf F}_{s}(t) +
510	>	{\bf F}_{f}(t)  + {\bf F}_{r}(t)
511		\label{LDGeneralizedForm}
512		\end{equation}
513	<	where $M_i$ is a $6\times6$ generalized diagonal mass (include mass
514	<	and moment of inertial) matrix and $V_i$ is a generalized velocity,
515	<	$V_i = V_i(v_i,\omega _i)$. The right side of
516	<	Eq.~\ref{LDGeneralizedForm} consists of three generalized forces in
517	<	lab-fixed frame, systematic force $F_{s,i}$, dissipative force
518	<	$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the
519	<	system in Newtownian mechanics typically refers to lab-fixed frame,
520	<	it is also convenient to handle the rotation of rigid body in
521	<	body-fixed frame. Thus the friction and random forces are calculated
522	<	in body-fixed frame and converted back to lab-fixed frame by:
523	<	\[
524	<	\begin{array}{l}
525	<	F_{f,i}^l (t) = Q^T F_{f,i}^b (t), \\
526	<	F_{r,i}^l (t) = Q^T F_{r,i}^b (t). \\
463	<	\end{array}
464	<	\]
465	<	Here, the body-fixed friction force $F_{r,i}^b$ is proportional to
466	<	the body-fixed velocity at center of resistance $v_{R,i}^b$ and
467	<	angular velocity $\omega _i$
513	>	where ${\bf M}$ is a $6 \times 6$ diagonal mass matrix (which
514	>	includes the mass of the rigid body as well as the moments of inertia
515	>	in the body-fixed frame) and ${\bf V}$ is a generalized velocity,
516	>	${\bf V} =
517	>	\left\{{\bf v},{\bf \omega}\right\}$. The right side of
518	>	Eq.~\ref{LDGeneralizedForm} consists of three generalized forces: a
519	>	system force (${\bf F}_{s}$), a frictional or dissipative force (${\bf
520	>	F}_{f}$) and a stochastic force (${\bf F}_{r}$). While the evolution
521	>	of the system in Newtonian mechanics is typically done in the lab
522	>	frame, it is convenient to handle the dynamics of rigid bodies in
523	>	body-fixed frames. Thus the friction and random forces on each
524	>	substructure are calculated in a body-fixed frame and may converted
525	>	back to the lab frame using that substructure's rotation matrix (${\bf
526	>	Q}$):
527		\begin{equation}
528	<	F_{r,i}^b (t) = \left( \begin{array}{l}
529	<	f_{r,i}^b (t) \\
530	<	\tau _{r,i}^b (t) \\
531	<	\end{array} \right) =  - \left( {\begin{array}{*{20}c}
532	<	{\Xi _{R,t} } & {\Xi _{R,c}^T }  \\
533	<	{\Xi _{R,c} } & {\Xi _{R,r} }  \\
534	<	\end{array}} \right)\left( \begin{array}{l}
535	<	v_{R,i}^b (t) \\
536	<	\omega _i (t) \\
537	<	\end{array} \right),
528	>	{\bf F}_{f,r} =
529	>	\left( \begin{array}{c}
530	>	{\bf f}_{f,r} \\
531	>	{\bf \tau}_{f,r}
532	>	\end{array} \right)
533	>	=
534	>	\left( \begin{array}{c}
535	>	{\bf Q}^{T} {\bf f}^{~b}_{f,r} \\
536	>	{\bf Q}^{T} {\bf \tau}^{~b}_{f,r}
537	>	\end{array} \right)
538		\end{equation}
539	<	while the random force $F_{r,i}^l$ is a Gaussian stochastic variable
540	<	with zero mean and variance
539	>	The body-fixed friction force, ${\bf F}_{f}^{~b}$, is proportional to
540	>	the (body-fixed) velocity at the center of resistance
541	>	${\bf v}_{R}^{~b}$ and the angular velocity ${\bf \omega}$
542		\begin{equation}
543	<	\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle  =
544	<	\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle  =
545	<	2k_B T\Xi _R \delta (t - t'). \label{randomForce}
546	<	\end{equation}
547	<	The equation of motion for $v_i$ can be written as
543	>	{\bf F}_{f}^{~b}(t) = \left( \begin{array}{l}
544	>	{\bf f}_{f}^{~b}(t) \\
545	>	{\bf \tau}_{f}^{~b}(t) \\
546	>	\end{array} \right) =  - \left( \begin{array}{*{20}c}
547	>	\Xi_{R}^{tt} & \Xi_{R}^{rt} \\
548	>	\Xi_{R}^{tr} & \Xi_{R}^{rr}    \\
549	>	\end{array} \right)\left( \begin{array}{l}
550	>	{\bf v}_{R}^{~b}(t) \\
551	>	{\bf \omega}(t) \\
552	>	\end{array} \right),
553	>	\end{equation}
554	>	while the random force, ${\bf F}_{r}$, is a Gaussian stochastic
555	>	variable with zero mean and variance,
556		\begin{equation}
557	<	m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) +
558	<	f_{r,i}^l (t)
557	>	\left\langle {{\bf F}_{r}(t) ({\bf F}_{r}(t'))^T } \right\rangle  =
558	>	\left\langle {{\bf F}_{r}^{~b} (t) ({\bf F}_{r}^{~b} (t'))^T } \right\rangle  =
559	>	2 k_B T \Xi_R \delta(t - t'). \label{eq:randomForce}
560		\end{equation}
561	<	Since the frictional force is applied at the center of resistance
562	<	which generally does not coincide with the center of mass, an extra
563	<	torque is exerted at the center of mass. Thus, the net body-fixed
564	<	frictional torque at the center of mass, $\tau _{n,i}^b (t)$, is
565	<	given by
561	>	$\Xi_R$ is the $6\times6$ resistance tensor at the center of
562	>	resistance.  Once this tensor is known for a given rigid body (as
563	>	described in the previous section) obtaining a stochastic vector that
564	>	has the properties in Eq. (\ref{eq:randomForce}) can be done
565	>	efficiently by carrying out a one-time Cholesky decomposition to
566	>	obtain the square root matrix of the resistance tensor,
567	>	\begin{equation}
568	>	\Xi_R = {\bf S} {\bf S}^{T},
569	>	\label{eq:Cholesky}
570	>	\end{equation}
571	>	where ${\bf S}$ is a lower triangular matrix.\cite{Schlick2002} A
572	>	vector with the statistics required for the random force can then be
573	>	obtained by multiplying ${\bf S}$ onto a random 6-vector ${\bf Z}$ which
574	>	has elements chosen from a Gaussian distribution, such that:
575		\begin{equation}
576	<	\tau _{r,i}^b = \tau _{r,i}^b +r_{MR} \times f_{r,i}^b
576	>	\langle {\bf Z}_i \rangle = 0, \hspace{1in} \langle {\bf Z}_i \cdot
577	>	{\bf Z}_j \rangle = \frac{2 k_B T}{\delta t} \delta_{ij},
578		\end{equation}
579	<	where $r_{MR}$ is the vector from the center of mass to the center
580	<	of the resistance. Instead of integrating the angular velocity in
581	<	lab-fixed frame, we consider the equation of angular momentum in
582	<	body-fixed frame
579	>	where $\delta t$ is the timestep in use during the simulation. The
580	>	random force, ${\bf F}_{r}^{~b} = {\bf S} {\bf Z}$, can be shown to have the
581	>	correct properties required by Eq. (\ref{eq:randomForce}).
582	>
583	>	The equation of motion for the translational velocity of the center of
584	>	mass (${\bf v}$) can be written as
585		\begin{equation}
586	<	\dot j_i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b (t)
587	<	+ \tau _{r,i}^b(t)
586	>	m \dot{{\bf v}} (t) =  {\bf f}_{s}(t) + {\bf f}_{f}(t) +
587	>	{\bf f}_{r}(t)
588		\end{equation}
589	<	Embedding the friction terms into force and torque, one can
590	<	integrate the langevin equations of motion for rigid body of
591	<	arbitrary shape in a velocity-Verlet style 2-part algorithm, where
592	<	$h= \delta t$:
589	>	Since the frictional and random forces are applied at the center of
590	>	resistance, which generally does not coincide with the center of mass,
591	>	extra torques are exerted at the center of mass. Thus, the net
592	>	body-fixed torque at the center of mass, $\tau^{~b}(t)$,
593	>	is given by
594	>	\begin{equation}
595	>	\tau^{~b} \leftarrow \tau_{s}^{~b} + \tau_{f}^{~b} + \tau_{r}^{~b} + {\bf r}_{MR} \times \left( {\bf f}_{f}^{~b} + {\bf f}_{r}^{~b} \right)
596	>	\end{equation}
597	>	where ${\bf r}_{MR}$ is the vector from the center of mass to the center of
598	>	resistance. Instead of integrating the angular velocity in lab-fixed
599	>	frame, we consider the equation of motion for the angular momentum
600	>	(${\bf j}$) in the body-fixed frame
601	>	\begin{equation}
602	>	\frac{\partial}{\partial t}{\bf j}(t) = \tau^{~b}(t)
603	>	\end{equation}
604	>	Embedding the friction and random forces into the the total force and
605	>	torque, one can integrate the Langevin equations of motion for a rigid
606	>	body of arbitrary shape in a velocity-Verlet style 2-part algorithm,
607	>	where $h = \delta t$:
608
609	<	{\tt moveA:}
609	>	{\tt move A:}
610		\begin{align*}
611		{\bf v}\left(t + h / 2\right)  &\leftarrow  {\bf v}(t)
612		+ \frac{h}{2} \left( {\bf f}(t) / m \right), \\
#	Line 519 | Line 615 | $h= \delta t$:
615		+ h  {\bf v}\left(t + h / 2 \right), \\
616		%
617		{\bf j}\left(t + h / 2 \right)  &\leftarrow {\bf j}(t)
618	<	+ \frac{h}{2} {\bf \tau}^b(t), \\
618	>	+ \frac{h}{2} {\bf \tau}^{~b}(t), \\
619		%
620	<	\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j}
620	>	{\bf Q}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j}
621		(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right).
622		\end{align*}
623	<	In this context, the $\mathrm{rotate}$ function is the reversible
624	<	product of the three body-fixed rotations,
623	>	In this context, $\overleftrightarrow{\mathsf{I}}$ is the diagonal
624	>	moment of inertia tensor, and the $\mathrm{rotate}$ function is the
625	>	reversible product of the three body-fixed rotations,
626		\begin{equation}
627		\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot
628		\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y
629		/ 2) \cdot \mathsf{G}_x(a_x /2),
630		\end{equation}
631		where each rotational propagator, $\mathsf{G}_\alpha(\theta)$,
632	<	rotates both the rotation matrix ($\mathsf{Q}$) and the body-fixed
632	>	rotates both the rotation matrix ($\mathbf{Q}$) and the body-fixed
633		angular momentum (${\bf j}$) by an angle $\theta$ around body-fixed
634		axis $\alpha$,
635		\begin{equation}
636		\mathsf{G}_\alpha( \theta ) = \left\{
637		\begin{array}{lcl}
638	<	\mathsf{Q}(t) & \leftarrow & \mathsf{Q}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\
638	>	\mathbf{Q}(t) & \leftarrow & \mathbf{Q}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\
639		{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf
640		j}(0).
641		\end{array}
#	Line 560 | Line 657 | calculated at the new positions and orientations
657		\end{equation}
658		All other rotations follow in a straightforward manner. After the
659		first part of the propagation, the forces and body-fixed torques are
660	<	calculated at the new positions and orientations
660	>	calculated at the new positions and orientations.  The system forces
661	>	and torques are derivatives of the total potential energy function
662	>	($U$) with respect to the rigid body positions (${\bf r}$) and the
663	>	columns of the transposed rotation matrix ${\bf Q}^T = \left({\bf
664	>	u}_x, {\bf u}_y, {\bf u}_z \right)$:
665
666	<	{\tt doForces:}
666	>	{\tt Forces:}
667		\begin{align*}
668	<	{\bf f}(t + h) &\leftarrow
669	<	- \left(\frac{\partial V}{\partial {\bf r}}\right)_{{\bf r}(t + h)}, \\
668	>	{\bf f}_{s}(t + h) & \leftarrow
669	>	- \left(\frac{\partial U}{\partial {\bf r}}\right)_{{\bf r}(t + h)} \\
670		%
671	<	{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h)
672	<	\times \frac{\partial V}{\partial {\bf u}}, \\
671	>	{\bf \tau}_{s}(t + h) &\leftarrow {\bf u}(t + h)
672	>	\times \frac{\partial U}{\partial {\bf u}} \\
673		%
674	<	{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{Q}(t + h)
675	<	\cdot {\bf \tau}^s(t + h).
674	>	{\bf v}^{b}_{R}(t+h) & \leftarrow \mathbf{Q}(t+h) \cdot \left({\bf v}(t+h) + {\bf \omega}(t+h) \times {\bf r}_{MR} \right) \\
675	>	%
676	>	{\bf f}_{R,f}^{b}(t+h) & \leftarrow - \Xi_{R}^{tt} \cdot
677	>	{\bf v}^{b}_{R}(t+h) - \Xi_{R}^{rt} \cdot {\bf \omega}(t+h) \\
678	>	%
679	>	{\bf \tau}_{R,f}^{b}(t+h) & \leftarrow - \Xi_{R}^{tr} \cdot
680	>	{\bf v}^{b}_{R}(t+h) - \Xi_{R}^{rr} \cdot {\bf \omega}(t+h) \\
681	>	%
682	>	Z & \leftarrow  {\tt GaussianNormal}(2 k_B T / h, 6) \\
683	>	{\bf F}_{R,r}^{b}(t+h) & \leftarrow {\bf S} \cdot Z \\
684	>	%
685	>	{\bf f}(t+h) & \leftarrow {\bf f}_{s}(t+h) + \mathbf{Q}^{T}(t+h)
686	>	\cdot \left({\bf f}_{R,f}^{~b} + {\bf f}_{R,r}^{~b} \right) \\
687	>	%
688	>	\tau(t+h) & \leftarrow \tau_{s}(t+h) + \mathbf{Q}^{T}(t+h) \cdot \left(\tau_{R,f}^{~b} + \tau_{R,r}^{~b} \right) + {\bf r}_{MR} \times \left({\bf f}_{f}(t+h) + {\bf f}_{r}(t+h) \right) \\
689	>	\tau^{~b}(t+h) & \leftarrow \mathbf{Q}(t+h) \cdot \tau(t+h) \\
690		\end{align*}
691	+	Frictional (and random) forces and torques must be computed at the
692	+	center of resistance, so there are additional steps required to find
693	+	the body-fixed velocity (${\bf v}_{R}^{~b}$) at this location.  Mapping
694	+	the frictional and random forces at the center of resistance back to
695	+	the center of mass also introduces an additional term in the torque
696	+	one obtains at the center of mass.
697	+
698		Once the forces and torques have been obtained at the new time step,
699		the velocities can be advanced to the same time value.
700
701	<	{\tt moveB:}
701	>	{\tt move B:}
702		\begin{align*}
703		{\bf v}\left(t + h \right)  &\leftarrow  {\bf v}\left(t + h / 2
704		\right)
#	Line 584 | Line 706 | the velocities can be advanced to the same time value.
706		%
707		{\bf j}\left(t + h \right)  &\leftarrow {\bf j}\left(t + h / 2
708		\right)
709	<	+ \frac{h}{2} {\bf \tau}^b(t + h) .
709	>	+ \frac{h}{2} {\bf \tau}^{~b}(t + h) .
710		\end{align*}
711
712		\section{Validating the Method\label{sec:validating}}
#	Line 647 | Line 769 | simulation box.  All simulations were equilibrated usi
769		We performed reference microcanonical simulations with explicit
770		solvents for each of the different model system.  In each case there
771		was one solute model and 1929 solvent molecules present in the
772	<	simulation box.  All simulations were equilibrated using a
772	>	simulation box.  All simulations were equilibrated for 5 ns using a
773		constant-pressure and temperature integrator with target values of 300
774		K for the temperature and 1 atm for pressure.  Following this stage,
775	<	further equilibration and sampling was done in a microcanonical
776	<	ensemble.  Since the model bodies are typically quite massive, we were
777	<	able to use a time step of 25 fs.
775	>	further equilibration (5 ns) and sampling (10 ns) was done in a
776	>	microcanonical ensemble.  Since the model bodies are typically quite
777	>	massive, we were able to use a time step of 25 fs.
778
779		The model systems studied used both Lennard-Jones spheres as well as
780		uniaxial Gay-Berne ellipoids. In its original form, the Gay-Berne
781		potential was a single site model for the interactions of rigid
782	<	ellipsoidal molecules.\cite{Gay81} It can be thought of as a
782	>	ellipsoidal molecules.\cite{Gay1981} It can be thought of as a
783		modification of the Gaussian overlap model originally described by
784		Berne and Pechukas.\cite{Berne72} The potential is constructed in the
785		familiar form of the Lennard-Jones function using
786		orientation-dependent $\sigma$ and $\epsilon$ parameters,
787		\begin{equation*}
788	<	V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat
789	<	r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j},
790	<	{\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u
788	>	V_{ij}({{\bf \hat u}_i}, {{\bf \hat u}_j}, {{\bf \hat
789	>	r}_{ij}}) = 4\epsilon ({{\bf \hat u}_i}, {{\bf \hat u}_j},
790	>	{{\bf \hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({{\bf \hat u
791		}_i},
792	<	{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12}
793	<	-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j},
794	<	{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right]
792	>	{{\bf \hat u}_j}, {{\bf \hat r}_{ij}})+\sigma_0}\right)^{12}
793	>	-\left(\frac{\sigma_0}{r_{ij}-\sigma({{\bf \hat u}_i}, {{\bf \hat u}_j},
794	>	{{\bf \hat r}_{ij}})+\sigma_0}\right)^6\right]
795		\label{eq:gb}
796		\end{equation*}
797
#	Line 686 | Line 808 | are given elsewhere,\cite{Luckhurst90,Golubkov06,SunGe
808		Additionally, a well depth aspect ratio, $\epsilon^r = \epsilon^e /
809		\epsilon^s$, describes the ratio between the well depths in the {\it
810		end-to-end} and side-by-side configurations.  Details of the potential
811	<	are given elsewhere,\cite{Luckhurst90,Golubkov06,SunGezelter08} and an
811	>	are given elsewhere,\cite{Luckhurst90,Golubkov06,SunX._jp0762020} and an
812		excellent overview of the computational methods that can be used to
813		efficiently compute forces and torques for this potential can be found
814		in Ref. \citen{Golubkov06}
#	Line 721 | Line 843 | A similar form exists for the bulk viscosity
843		\int_{t_0}^{t_0 + t} P_{xz}(t') dt' \right)^2 \right\rangle_{t_0}.
844		\label{eq:shear}
845		\end{equation}
846	<	A similar form exists for the bulk viscosity
847	<	\begin{equation}
726	<	\kappa = \lim_{t->\infty} \frac{V}{2 k_B T} \frac{d}{dt} \left\langle \left(
727	<	\int_{t_0}^{t_0 + t}
728	<	\left(P\left(t'\right)-\left\langle P \right\rangle \right)dt'
729	<	\right)^2 \right\rangle_{t_0}.
730	<	\end{equation}
731	<	Alternatively, the shear viscosity can also be calculated using a
732	<	Green-Kubo formula with the off-diagonal pressure tensor correlation function,
733	<	\begin{equation}
734	<	\eta = \frac{V}{k_B T} \int_0^{\infty} \left\langle P_{xz}(t_0) P_{xz}(t_0
735	<	+ t) \right\rangle_{t_0} dt,
736	<	\end{equation}
737	<	although this method converges extremely slowly and is not practical
738	<	for obtaining viscosities from molecular dynamics simulations.
846	>	which converges much more rapidly in molecular dynamics simulations
847	>	than the traditional Green-Kubo formula.
848
849		The Langevin dynamics for the different model systems were performed
850		at the same temperature as the average temperature of the
#	Line 761 | Line 870 | have used in this study, there are differences between
870		compute the diffusive behavior for motion parallel to each body-fixed
871		axis by projecting the displacement of the particle onto the
872		body-fixed reference frame at $t=0$.  With an isotropic solvent, as we
873	<	have used in this study, there are differences between the three
874	<	diffusion constants, but these must converge to the same value at
875	<	longer times.  Translational diffusion constants for the different
876	<	shaped models are shown in table \ref{tab:translation}.
873	>	have used in this study, there may be differences between the three
874	>	diffusion constants at short times, but these must converge to the
875	>	same value at longer times.  Translational diffusion constants for the
876	>	different shaped models are shown in table \ref{tab:translation}.
877
878		In general, the three eigenvalues ($D_1, D_2, D_3$) of the rotational
879		diffusion tensor (${\bf D}_{rr}$) measure the diffusion of an object
#	Line 844 | Line 953 | hydrodynamic flows is well known, giving translational
953		an arbitrary value of 0.8 kcal/mol.
954
955		The Stokes-Einstein behavior of large spherical particles in
956	<	hydrodynamic flows is well known, giving translational friction
957	<	coefficients of $6 \pi \eta R$ (stick boundary conditions) and
958	<	rotational friction coefficients of $8 \pi \eta R^3$.  Recently,
959	<	Schmidt and Skinner have computed the behavior of spherical tag
960	<	particles in molecular dynamics simulations, and have shown that {\it
961	<	slip} boundary conditions ($\Xi_{tt} = 4 \pi \eta R$) may be more
962	<	appropriate for molecule-sized spheres embedded in a sea of spherical
963	<	solvent particles.\cite{Schmidt:2004fj,Schmidt:2003kx}
956	>	hydrodynamic flows with ``stick'' boundary conditions is well known,
957	>	and is given in Eqs. (\ref{eq:StokesTranslation}) and
958	>	(\ref{eq:StokesRotation}).  Recently, Schmidt and Skinner have
959	>	computed the behavior of spherical tag particles in molecular dynamics
960	>	simulations, and have shown that {\it slip} boundary conditions
961	>	($\Xi_{tt} = 4 \pi \eta \rho$) may be more appropriate for
962	>	molecule-sized spheres embedded in a sea of spherical solvent
963	>	particles.\cite{Schmidt:2004fj,Schmidt:2003kx}
964
965		Our simulation results show similar behavior to the behavior observed
966		by Schmidt and Skinner.  The diffusion constant obtained from our
#	Line 876 | Line 985 | D = \frac{k_B T}{6 \pi \eta a} G(\rho),
985		can be combined to give a single translational diffusion
986		constant,\cite{Berne90}
987		\begin{equation}
988	<	D = \frac{k_B T}{6 \pi \eta a} G(\rho),
988	>	D = \frac{k_B T}{6 \pi \eta a} G(s),
989		\label{Dperrin}
990		\end{equation}
991		as well as a single rotational diffusion coefficient,
992		\begin{equation}
993	<	\Theta = \frac{3 k_B T}{16 \pi \eta a^3} \left\{ \frac{(2 - \rho^2)
994	<	G(\rho) - 1}{1 - \rho^4} \right\}.
993	>	\Theta = \frac{3 k_B T}{16 \pi \eta a^3} \left\{ \frac{(2 - s^2)
994	>	G(s) - 1}{1 - s^4} \right\}.
995		\label{ThetaPerrin}
996		\end{equation}
997	<	In these expressions, $G(\rho)$ is a function of the axial ratio
998	<	($\rho = b / a$), which for prolate ellipsoids, is
997	>	In these expressions, $G(s)$ is a function of the axial ratio
998	>	($s = b / a$), which for prolate ellipsoids, is
999		\begin{equation}
1000	<	G(\rho) = (1- \rho^2)^{-1/2} \ln \left\{ \frac{1 + (1 -
892	<	\rho^2)^{1/2}}{\rho} \right\}
1000	>	G(s) = (1- s^2)^{-1/2} \ln \left\{ \frac{1 + (1 - s^2)^{1/2}}{s} \right\}
1001		\label{GPerrin}
1002		\end{equation}
1003		Again, there is some uncertainty about the correct boundary conditions
#	Line 914 | Line 1022 | The translational diffusion constants from the microca
1022		exact treatment of the diffusion tensor as well as the rough-shell
1023		model for the ellipsoid.
1024
1025	<	The translational diffusion constants from the microcanonical simulations
1026	<	agree well with the predictions of the Perrin model, although the rotational
1027	<	correlation times are a factor of 2 shorter than expected from hydrodynamic
1028	<	theory.  One explanation for the slower rotation
1029	<	of explicitly-solvated ellipsoids is the possibility that solute-solvent
1030	<	collisions happen at both ends of the solute whenever the principal
1031	<	axis of the ellipsoid is turning. In the upper portion of figure
1032	<	\ref{fig:explanation} we sketch a physical picture of this explanation.
1033	<	Since our Langevin integrator is providing nearly quantitative agreement with
1034	<	the Perrin model, it also predicts orientational diffusion for ellipsoids that
1035	<	exceed explicitly solvated correlation times by a factor of two.
1025	>	The translational diffusion constants from the microcanonical
1026	>	simulations agree well with the predictions of the Perrin model,
1027	>	although the {\it rotational} correlation times are a factor of 2
1028	>	shorter than expected from hydrodynamic theory.  One explanation for
1029	>	the slower rotation of explicitly-solvated ellipsoids is the
1030	>	possibility that solute-solvent collisions happen at both ends of the
1031	>	solute whenever the principal axis of the ellipsoid is turning. In the
1032	>	upper portion of figure \ref{fig:explanation} we sketch a physical
1033	>	picture of this explanation.  Since our Langevin integrator is
1034	>	providing nearly quantitative agreement with the Perrin model, it also
1035	>	predicts orientational diffusion for ellipsoids that exceed explicitly
1036	>	solvated correlation times by a factor of two.
1037
1038		\subsection{Rigid dumbbells}
1039		Perhaps the only {\it composite} rigid body for which analytic
#	Line 932 | Line 1041 | model. Equation (\ref{introEquation:oseenTensor}) abov
1041		two-sphere dumbbell model.  This model consists of two non-overlapping
1042		spheres held by a rigid bond connecting their centers. There are
1043		competing expressions for the 6x6 resistance tensor for this
1044	<	model. Equation (\ref{introEquation:oseenTensor}) above gives the
1045	<	original Oseen tensor, while the second order expression introduced by
1046	<	Rotne and Prager,\cite{Rotne1969} and improved by Garc\'{i}a de la
938	<	Torre and Bloomfield,\cite{Torre1977} is given above as
1044	>	model. The second order expression introduced by Rotne and
1045	>	Prager,\cite{Rotne1969} and improved by Garc\'{i}a de la Torre and
1046	>	Bloomfield,\cite{Torre1977} is given above as
1047		Eq. (\ref{introEquation:RPTensorNonOverlapped}).  In our case, we use
1048		a model dumbbell in which the two spheres are identical Lennard-Jones
1049		particles ($\sigma$ = 6.5 \AA\ , $\epsilon$ = 0.8 kcal / mol) held at
#	Line 948 | Line 1056 | those derived from the 6 x 6 tensors mentioned above).
1056		motion in a flow {\it perpendicular} to the inter-sphere
1057		axis.\cite{Davis:1969uq} We know of no analytic solutions for the {\it
1058		orientational} correlation times for this model system (other than
1059	<	those derived from the 6 x 6 tensors mentioned above).
1059	>	those derived from the 6 x 6 tensor mentioned above).
1060
1061		The bead model for this model system comprises the two large spheres
1062		by themselves, while the rough shell approximation used 3368 separate
#	Line 1016 | Line 1124 | rotational diffusion.
1124		} \label{fig:explanation}
1125		\end{figure}
1126
1019	–
1020	–
1127		\subsection{Composite banana-shaped molecules}
1128		Banana-shaped rigid bodies composed of three Gay-Berne ellipsoids have
1129		been used by Orlandi {\it et al.} to observe mesophases in
#	Line 1029 | Line 1135 | A reference system composed of a single banana rigid b
1135		behavior of this model, we have left out the dipolar interactions of
1136		the original Orlandi model.
1137
1138	<	A reference system composed of a single banana rigid body embedded in a
1139	<	sea of 1929 solvent particles was created and run under standard
1140	<	(microcanonical) molecular dynamics.  The resulting viscosity of this
1141	<	mixture was 0.298 centipoise (as estimated using Eq. (\ref{eq:shear})).
1142	<	To calculate the hydrodynamic properties of the banana rigid body model,
1143	<	we created a rough shell (see Fig.~\ref{fig:roughShell}), in which
1144	<	the banana is represented as a ``shell'' made of 3321 identical beads
1145	<	(0.25 \AA\  in diameter) distributed on the surface.  Applying the
1146	<	procedure described in Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we
1147	<	identified the center of resistance, ${\bf r} = $(0 \AA, 0.81 \AA, 0 \AA), as
1148	<	well as the resistance tensor,
1149	<	\begin{equation*}
1044	<	\Xi =
1045	<	\left( {\begin{array}{*{20}c}
1046	<	0.9261 & 0 & 0&0&0.08585&0.2057\\
1047	<	0& 0.9270&-0.007063& 0.08585&0&0\\
1048	<	0&-0.007063&0.7494&0.2057&0&0\\
1049	<	0&0.0858&0.2057& 58.64& 0&0\\0.08585&0&0&0&48.30&3.219&\\0.2057&0&0&0&3.219&10.7373\\\end{array}} \right),
1050	<	\end{equation*}
1051	<	where the units for translational, translation-rotation coupling and
1052	<	rotational tensors are (kcal fs / mol \AA$^2$), (kcal fs / mol \AA\ rad),
1053	<	and (kcal fs / mol rad$^2$), respectively.
1138	>	A reference system composed of a single banana rigid body embedded in
1139	>	a sea of 1929 solvent particles was created and run under standard
1140	>	(microcanonical) molecular dynamics.  The resulting viscosity of this
1141	>	mixture was 0.298 centipoise (as estimated using
1142	>	Eq. (\ref{eq:shear})).  To calculate the hydrodynamic properties of
1143	>	the banana rigid body model, we created a rough shell (see
1144	>	Fig.~\ref{fig:roughShell}), in which the banana is represented as a
1145	>	``shell'' made of 3321 identical beads (0.25 \AA\ in diameter)
1146	>	distributed on the surface.  Applying the procedure described in
1147	>	Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we
1148	>	identified the center of resistance, ${\bf r} = $(0 \AA, 0.81 \AA, 0
1149	>	\AA).
1150
1151	<	The Langevin rigid-body integrator (and the hydrodynamic diffusion tensor)
1152	<	are essentially quantitative for translational diffusion of this model.
1153	<	Orientational correlation times under the Langevin rigid-body integrator
1154	<	are within 11\% of the values obtained from explicit solvent, but these
1155	<	models also exhibit some solvent inaccessible surface area in the
1156	<	explicitly-solvated case.
1151	>	The Langevin rigid-body integrator (and the hydrodynamic diffusion
1152	>	tensor) are essentially quantitative for translational diffusion of
1153	>	this model.  Orientational correlation times under the Langevin
1154	>	rigid-body integrator are within 11\% of the values obtained from
1155	>	explicit solvent, but these models also exhibit some solvent
1156	>	inaccessible surface area in the explicitly-solvated case.
1157
1158		\subsection{Composite sphero-ellipsoids}
1063	–	Spherical heads perched on the ends of Gay-Berne ellipsoids have been
1064	–	used recently as models for lipid molecules.\cite{SunGezelter08,Ayton01}
1159
1160	<	MORE DETAILS
1160	>	Spherical heads perched on the ends of Gay-Berne ellipsoids have been
1161	>	used recently as models for lipid
1162	>	molecules.\cite{SunX._jp0762020,Ayton01} A reference system composed of
1163	>	a single lipid rigid body embedded in a sea of 1929 solvent particles
1164	>	was created and run under a microcanonical ensemble.  The resulting
1165	>	viscosity of this mixture was 0.349 centipoise (as estimated using
1166	>	Eq. (\ref{eq:shear})).  To calculate the hydrodynamic properties of
1167	>	the lipid rigid body model, we created a rough shell (see
1168	>	Fig.~\ref{fig:roughShell}), in which the lipid is represented as a
1169	>	``shell'' made of 3550 identical beads (0.25 \AA\ in diameter)
1170	>	distributed on the surface.  Applying the procedure described in
1171	>	Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we
1172	>	identified the center of resistance, ${\bf r} = $(0 \AA, 0 \AA, 1.46
1173	>	\AA).
1174
1175	+	The translational diffusion constants and rotational correlation times
1176	+	obtained using the Langevin rigid-body integrator (and the
1177	+	hydrodynamic tensor) are essentially quantitative when compared with
1178	+	the explicit solvent simulations for this model system.
1179
1180	<	\subsection{Summary}
1181	<	According to our simulations, the langevin dynamics is a reliable
1182	<	theory to apply to replace the explicit solvents, especially for the
1183	<	translation properties. For large molecules, the rotation properties
1184	<	are also mimiced reasonablly well.
1180	>	\subsection{Summary of comparisons with explicit solvent simulations}
1181	>	The Langevin rigid-body integrator we have developed is a reliable way
1182	>	to replace explicit solvent simulations in cases where the detailed
1183	>	solute-solvent interactions do not greatly impact the behavior of the
1184	>	solute.  As such, it has the potential to greatly increase the length
1185	>	and time scales of coarse grained simulations of large solvated
1186	>	molecules.  In cases where the dielectric screening of the solvent, or
1187	>	specific solute-solvent interactions become important for structural
1188	>	or dynamic features of the solute molecule, this integrator may be
1189	>	less useful.  However, for the kinds of coarse-grained modeling that
1190	>	have become popular in recent years (ellipsoidal side chains, rigid
1191	>	bodies, and molecular-scale models), this integrator may prove itself
1192	>	to be quite valuable.
1193
1194	+	\begin{figure}
1195	+	\centering
1196	+	\includegraphics[width=\linewidth]{graph}
1197	+	\caption[Mean squared displacements and orientational
1198	+	correlation functions for each of the model rigid bodies.]{The
1199	+	mean-squared displacements ($\langle r^2(t) \rangle$) and
1200	+	orientational correlation functions ($C_2(t)$) for each of the model
1201	+	rigid bodies studied.  The circles are the results for microcanonical
1202	+	simulations with explicit solvent molecules, while the other data sets
1203	+	are results for Langevin dynamics using the different hydrodynamic
1204	+	tensor approximations.  The Perrin model for the ellipsoids is
1205	+	considered the ``exact'' hydrodynamic behavior (this can also be said
1206	+	for the translational motion of the dumbbell operating under the bead
1207	+	model). In most cases, the various hydrodynamics models reproduce
1208	+	each other quantitatively.}
1209	+	\label{fig:results}
1210	+	\end{figure}
1211	+
1212		\begin{table*}
1213		\begin{minipage}{\linewidth}
1214		\begin{center}
1215		\caption{Translational diffusion constants (D) for the model systems
1216		calculated using microcanonical simulations (with explicit solvent),
1217		theoretical predictions, and Langevin simulations (with implicit solvent).
1218	<	Analytical solutions for the exactly-solved hydrodynamics models are
1219	<	from Refs. \citen{Einstein05} (sphere), \citen{Perrin1934} and \citen{Perrin1936}
1218	>	Analytical solutions for the exactly-solved hydrodynamics models are obtained
1219	>	from: Stokes' law (sphere), and Refs. \citen{Perrin1934} and \citen{Perrin1936}
1220		(ellipsoid), \citen{Stimson:1926qy} and \citen{Davis:1969uq}
1221		(dumbbell). The other model systems have no known analytic solution.
1222	<	All  diffusion constants are reported in units of $10^{-3}$ cm$^2$ / ps (=
1222	>	All diffusion constants are reported in units of $10^{-3}$ cm$^2$ / ps (=
1223		$10^{-4}$ \AA$^2$  / fs). }
1224		\begin{tabular}{lccccccc}
1225		\hline
#	Line 1090 | Line 1227 | sphere    & 0.261  & ?    & & 2.59 & exact       & 2.5
1227		\cline{2-3} \cline{5-7}
1228		model & $\eta$ (centipoise)  & D & & Analytical & method & Hydrodynamics & simulation \\
1229		\hline
1230	<	sphere    & 0.261  & ?    & & 2.59 & exact       & 2.59 & 2.56 \\
1230	>	sphere    & 0.279  & 3.06 & & 2.42 & exact       & 2.42 & 2.33 \\
1231		ellipsoid & 0.255  & 2.44 & & 2.34 & exact       & 2.34 & 2.37 \\
1232		& 0.255  & 2.44 & & 2.34 & rough shell & 2.36 & 2.28 \\
1233	<	dumbbell  & 0.322  & ?    & & 1.57 & bead model  & 1.57 & 1.57 \\
1234	<	& 0.322  & ?    & & 1.57 & rough shell & ?    & ?    \\
1233	>	dumbbell  & 0.308  & 2.06 & & 1.64 & bead model  & 1.65 & 1.62 \\
1234	>	& 0.308  & 2.06 & & 1.64 & rough shell & 1.59 & 1.62 \\
1235		banana    & 0.298  & 1.53 & &      & rough shell & 1.56 & 1.55 \\
1236	<	lipid     & 0.349  & 0.96 & &      & rough shell & 1.33 & 1.32 \\
1236	>	lipid     & 0.349  & 1.41 & &      & rough shell & 1.33 & 1.32 \\
1237		\end{tabular}
1238		\label{tab:translation}
1239		\end{center}
#	Line 1119 | Line 1256 | sphere    & 0.261  &      & & 9.06 & exact       & 9.0
1256		\cline{2-3} \cline{5-7}
1257		model & $\eta$ (centipoise) & $\tau$ & & Perrin & method & Hydrodynamic  & simulation \\
1258		\hline
1259	<	sphere    & 0.261  &      & & 9.06 & exact       & 9.06 & 9.11 \\
1259	>	sphere    & 0.279  &      & & 9.69 & exact       & 9.69 & 9.64 \\
1260		ellipsoid & 0.255  & 46.7 & & 22.0 & exact       & 22.0 & 22.2 \\
1261		& 0.255  & 46.7 & & 22.0 & rough shell & 22.6 & 22.2 \\
1262	<	dumbbell  & 0.322  & 14.0 & &      & bead model  & 52.3 & 52.8 \\
1263	<	& 0.322  & 14.0 & &      & rough shell & ?    & ?    \\
1262	>	dumbbell  & 0.308  & 14.1 & &      & bead model  & 50.0 & 50.1 \\
1263	>	& 0.308  & 14.1 & &      & rough shell & 41.5 & 41.3 \\
1264		banana    & 0.298  & 63.8 & &      & rough shell & 70.9 & 70.9 \\
1265		lipid     & 0.349  & 78.0 & &      & rough shell & 76.9 & 77.9 \\
1266		\hline
#	Line 1135 | Line 1272 | The Langevin dynamics integrator was applied to study
1272
1273		\section{Application: A rigid-body lipid bilayer}
1274
1275	<	The Langevin dynamics integrator was applied to study the formation of
1276	<	corrugated structures emerging from simulations of the coarse grained
1277	<	lipid molecular models presented above.  The initial configuration is
1278	<	taken from our molecular dynamics studies on lipid bilayers with
1279	<	lennard-Jones sphere solvents. The solvent molecules were excluded
1280	<	from the system, and the experimental value for the viscosity of water
1281	<	at 20C ($\eta = 1.00$ cp) was used to mimic the hydrodynamic effects
1282	<	of the solvent.  The absence of explicit solvent molecules and the
1283	<	stability of the integrator allowed us to take timesteps of 50 fs.  A
1284	<	total simulation run time of 100 ns was sampled.
1285	<	Fig. \ref{fig:bilayer} shows the configuration of the system after 100
1286	<	ns, and the ripple structure remains stable during the entire
1287	<	trajectory.  Compared with using explicit bead-model solvent
1288	<	molecules, the efficiency of the simulation has increased by an order
1289	<	of magnitude.
1275	>	To test the accuracy and efficiency of the new integrator, we applied
1276	>	it to study the formation of corrugated structures emerging from
1277	>	simulations of the coarse grained lipid molecular models presented
1278	>	above.  The initial configuration is taken from earlier molecular
1279	>	dynamics studies on lipid bilayers which had used spherical
1280	>	(Lennard-Jones) solvent particles and moderate (480 solvated lipid
1281	>	molecules) system sizes.\cite{SunX._jp0762020} the solvent molecules
1282	>	were excluded from the system and the box was replicated three times
1283	>	in the x- and y- axes (giving a single simulation cell containing 4320
1284	>	lipids).  The experimental value for the viscosity of water at 20C
1285	>	($\eta = 1.00$ cp) was used with the Langevin integrator to mimic the
1286	>	hydrodynamic effects of the solvent.  The absence of explicit solvent
1287	>	molecules and the stability of the integrator allowed us to take
1288	>	timesteps of 50 fs. A simulation run time of 30 ns was sampled to
1289	>	calculate structural properties.  Fig. \ref{fig:bilayer} shows the
1290	>	configuration of the system after 30 ns.  Structural properties of the
1291	>	bilayer (e.g. the head and body $P_2$ order parameters) are nearly
1292	>	identical to those obtained via solvated molecular dynamics. The
1293	>	ripple structure remained stable during the entire trajectory.
1294	>	Compared with using explicit bead-model solvent molecules, the 30 ns
1295	>	trajectory for 4320 lipids with the Langevin integrator is now {\it
1296	>	faster} on the same hardware than the same length trajectory was for
1297	>	the 480-lipid system previously studied.
1298
1299		\begin{figure}
1300		\centering
1301		\includegraphics[width=\linewidth]{bilayer}
1302		\caption[Snapshot of a bilayer of rigid-body models for lipids]{A
1303	<	snapshot of a bilayer composed of rigid-body models for lipid
1303	>	snapshot of a bilayer composed of 4320 rigid-body models for lipid
1304		molecules evolving using the Langevin integrator described in this
1305		work.} \label{fig:bilayer}
1306		\end{figure}
1307
1308		\section{Conclusions}
1309
1310	<	We have presented a new Langevin algorithm by incorporating the
1311	<	hydrodynamics properties of arbitrary shaped molecules into an
1312	<	advanced symplectic integration scheme. Further studies in systems
1313	<	involving banana shaped molecules illustrated that the dynamic
1314	<	properties could be preserved by using this new algorithm as an
1315	<	implicit solvent model.
1310	>	We have presented a new algorithm for carrying out Langevin dynamics
1311	>	simulations on complex rigid bodies by incorporating the hydrodynamic
1312	>	resistance tensors for arbitrary shapes into an advanced symplectic
1313	>	integration scheme.  The integrator gives quantitative agreement with
1314	>	both analytic and approximate hydrodynamic theories, and works
1315	>	reasonably well at reproducing the solute dynamical properties
1316	>	(diffusion constants, and orientational relaxation times) from
1317	>	explicitly-solvated simulations.  For the cases where there are
1318	>	discrepancies between our Langevin integrator and the explicit solvent
1319	>	simulations, two features of molecular simulations help explain the
1320	>	differences.
1321
1322	+	First, the use of ``stick'' boundary conditions for molecular-sized
1323	+	solutes in a sea of similarly-sized solvent particles may be
1324	+	problematic.  We are certainly not the first group to notice this
1325	+	difference between hydrodynamic theories and explicitly-solvated
1326	+	molecular
1327	+	simulations.\cite{Schmidt:2004fj,Schmidt:2003kx,Ravichandran:1999fk,TANG:1993lr}
1328	+	The problem becomes particularly noticable in both the translational
1329	+	diffusion of the spherical particles and the rotational diffusion of
1330	+	the ellipsoids.  In both of these cases it is clear that the
1331	+	approximations that go into hydrodynamics are the source of the error,
1332	+	and not the integrator itself.
1333
1334	+	Second, in the case of structures which have substantial surface area
1335	+	that is inaccessible to solvent particles, the hydrodynamic theories
1336	+	(and the Langevin integrator) may overestimate the effects of solvent
1337	+	friction because they overestimate the exposed surface area of the
1338	+	rigid body.  This is particularly noticable in the rotational
1339	+	diffusion of the dumbbell model.  We believe that given a solvent of
1340	+	known radius, it may be possible to modify the rough shell approach to
1341	+	place beads on solvent-accessible surface, instead of on the geometric
1342	+	surface defined by the van der Waals radii of the components of the
1343	+	rigid body.  Further work to confirm the behavior of this new
1344	+	approximation is ongoing.
1345	+
1346		\section{Acknowledgments}
1347		Support for this project was provided by the National Science
1348		Foundation under grant CHE-0134881. T.L. also acknowledges the
1349	<	financial support from center of applied mathematics at University
1350	<	of Notre Dame.
1349	>	financial support from Center of Applied Mathematics at University of
1350	>	Notre Dame.
1351	>
1352	>	\end{doublespace}
1353		\newpage
1354
1355	<	\bibliographystyle{jcp}
1355	>	\bibliographystyle{jcp2}
1356		\bibliography{langevin}
1182	–
1357		\end{document}

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