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\begin{document} |
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\title{An algorithm for performing Langevin dynamics on rigid bodies of arbitrary shape } |
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\title{Langevin dynamics for rigid bodies of arbitrary shape} |
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\author{Xiuquan Sun, Teng Lin and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: |
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gezelter@nd.edu} \\ |
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Department of Chemistry and Biochemistry\\ |
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\author{Xiuquan Sun, Teng Lin and J. Daniel |
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Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
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Department of Chemistry and Biochemistry,\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\maketitle \doublespacing |
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\begin{abstract} |
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\maketitle |
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\begin{abstract} |
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We present an algorithm for carrying out Langevin dynamics simulations |
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on complex rigid bodies by incorporating the hydrodynamic resistance |
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tensors for arbitrary shapes into an advanced rotational integration |
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scheme. The integrator gives quantitative agreement with both |
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analytic and approximate hydrodynamic theories for a number of model |
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rigid bodies, and works well at reproducing the solute dynamical |
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properties (diffusion constants, and orientational relaxation times) |
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obtained from explicitly-solvated simulations. |
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\end{abstract} |
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\newpage |
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% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\begin{doublespace} |
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\section{Introduction} |
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%applications of langevin dynamics |
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As alternative to Newtonian dynamics, Langevin dynamics, which |
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mimics a simple heat bath with stochastic and dissipative forces, |
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has been applied in a variety of studies. The stochastic treatment |
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of the solvent enables us to carry out substantially longer time |
| 67 |
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simulations. Implicit solvent Langevin dynamics simulations of |
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met-enkephalin not only outperform explicit solvent simulations for |
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computational efficiency, but also agrees very well with explicit |
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solvent simulations for dynamical properties.\cite{Shen2002} |
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Recently, applying Langevin dynamics with the UNRES model, Liow and |
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his coworkers suggest that protein folding pathways can be possibly |
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explored within a reasonable amount of time.\cite{Liwo2005} The |
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stochastic nature of the Langevin dynamics also enhances the |
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sampling of the system and increases the probability of crossing |
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energy barriers.\cite{Banerjee2004, Cui2003} Combining Langevin |
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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 |
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solvent induced interactions missing from implicit solvent model, |
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Kaya incorporated desolvation free energy barrier into implicit |
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coarse-grained solvent model in protein folding/unfolding studies |
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and discovered a higher free energy barrier between the native and |
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denatured states. Because of its stability against noise, Langevin |
| 73 |
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dynamics is very suitable for studying remagnetization processes in |
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various systems.\cite{Palacios1998,Berkov2002,Denisov2003} For |
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instance, the oscillation power spectrum of nanoparticles from |
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Langevin dynamics simulation has the same peak frequencies for |
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different wave vectors, which recovers the property of magnetic |
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excitations in small finite structures.\cite{Berkov2005a} |
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Langevin dynamics, which mimics a heat bath using both stochastic and |
| 64 |
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dissipative forces, has been applied in a variety of situations as an |
| 65 |
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alternative to molecular dynamics with explicit solvent molecules. |
| 66 |
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The stochastic treatment of the solvent allows the use of simulations |
| 67 |
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with substantially longer time and length scales. In general, the |
| 68 |
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dynamic and structural properties obtained from Langevin simulations |
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agree quite well with similar properties obtained from explicit |
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solvent simulations. |
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|
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%review rigid body dynamics |
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Rigid bodies are frequently involved in the modeling of different |
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areas, from engineering, physics, to chemistry. For example, |
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missiles and vehicle are usually modeled by rigid bodies. The |
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movement of the objects in 3D gaming engine or other physics |
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simulator is governed by the rigid body dynamics. In molecular |
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simulation, rigid body is used to simplify the model in |
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protein-protein docking study{\cite{Gray2003}}. |
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Recent examples of the usefulness of Langevin simulations include a |
| 73 |
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study of met-enkephalin in which Langevin simulations predicted |
| 74 |
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dynamical properties that were largely in agreement with explicit |
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solvent simulations.\cite{Shen2002} By applying Langevin dynamics with |
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the UNRES model, Liwo and his coworkers suggest that protein folding |
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pathways can be explored within a reasonable amount of |
| 78 |
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time.\cite{Liwo2005} |
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|
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It is very important to develop stable and efficient methods to |
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integrate the equations of motion for orientational degrees of |
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freedom. Euler angles are the natural choice to describe the |
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rotational degrees of freedom. However, due to $\frac {1}{sin |
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\theta}$ singularities, the numerical integration of corresponding |
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equations of these motion is very inefficient and inaccurate. |
| 86 |
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Although an alternative integrator using multiple sets of Euler |
| 87 |
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angles can overcome this difficulty\cite{Barojas1973}, the |
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computational penalty and the loss of angular momentum conservation |
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still remain. A singularity-free representation utilizing |
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quaternions was developed by Evans in 1977.\cite{Evans1977} |
| 100 |
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Unfortunately, this approach used a nonseparable Hamiltonian |
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resulting from the quaternion representation, which prevented the |
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symplectic algorithm from being utilized. Another different approach |
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is to apply holonomic constraints to the atoms belonging to the |
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rigid body. Each atom moves independently under the normal forces |
| 105 |
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deriving from potential energy and constraint forces which are used |
| 106 |
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to guarantee the rigidness. However, due to their iterative nature, |
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the SHAKE and Rattle algorithms also converge very slowly when the |
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number of constraints increases.\cite{Ryckaert1977, Andersen1983} |
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The stochastic nature of Langevin dynamics also enhances the sampling |
| 81 |
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of the system and increases the probability of crossing energy |
| 82 |
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barriers.\cite{Cui2003,Banerjee2004} Combining Langevin dynamics with |
| 83 |
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Kramers' theory, Klimov and Thirumalai identified free-energy |
| 84 |
> |
barriers by studying the viscosity dependence of the protein folding |
| 85 |
> |
rates.\cite{Klimov1997} In order to account for solvent induced |
| 86 |
> |
interactions missing from the implicit solvent model, Kaya |
| 87 |
> |
incorporated a desolvation free energy barrier into protein |
| 88 |
> |
folding/unfolding studies and discovered a higher free energy barrier |
| 89 |
> |
between the native and denatured states.\cite{HuseyinKaya07012005} |
| 90 |
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|
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A break-through in geometric literature suggests that, in order to |
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develop a long-term integration scheme, one should preserve the |
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symplectic structure of the propagator. By introducing a conjugate |
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momentum to the rotation matrix $Q$ and re-formulating Hamiltonian's |
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equation, a symplectic integrator, RSHAKE\cite{Kol1997}, was |
| 96 |
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proposed to evolve the Hamiltonian system in a constraint manifold |
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by iteratively satisfying the orthogonality constraint $Q^T Q = 1$. |
| 98 |
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An alternative method using the quaternion representation was |
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developed by Omelyan.\cite{Omelyan1998} However, both of these |
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methods are iterative and inefficient. In this section, we descibe a |
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symplectic Lie-Poisson integrator for rigid bodies developed by |
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Dullweber and his coworkers\cite{Dullweber1997} in depth. |
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In typical LD simulations, the friction and random ($f_r$) forces on |
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individual atoms are taken from Stokes' law, |
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\begin{eqnarray} |
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m \dot{v}(t) & = & -\nabla U(x) - \xi m v(t) + f_r(t) \notag \\ |
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\langle f_r(t) \rangle & = & 0 \\ |
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\langle f_r(t) f_r(t') \rangle & = & 2 k_B T \xi m \delta(t - t') \notag |
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\end{eqnarray} |
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where $\xi \approx 6 \pi \eta \rho$. Here $\eta$ is the viscosity of the |
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implicit solvent, and $\rho$ is the hydrodynamic radius of the atom. |
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|
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%review langevin/browninan dynamics for arbitrarily shaped rigid body |
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Combining Langevin or Brownian dynamics with rigid body dynamics, |
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one can study slow processes in biomolecular systems. Modeling DNA |
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as a chain of rigid beads, which are subject to harmonic potentials |
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as well as excluded volume potentials, Mielke and his coworkers |
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discovered rapid superhelical stress generations from the stochastic |
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simulation of twin supercoiling DNA with response to induced |
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torques.\cite{Mielke2004} Membrane fusion is another key biological |
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process which controls a variety of physiological functions, such as |
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release of neurotransmitters \textit{etc}. A typical fusion event |
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happens on the time scale of a millisecond, which is impractical to |
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study using atomistic models with newtonian mechanics. With the help |
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of coarse-grained rigid body model and stochastic dynamics, the |
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fusion pathways were explored by many |
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researchers.\cite{Noguchi2001,Noguchi2002,Shillcock2005} Due to the |
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difficulty of numerical integration of anisotropic rotation, most of |
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the rigid body models are simply modeled using spheres, cylinders, |
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ellipsoids or other regular shapes in stochastic simulations. In an |
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effort to account for the diffusion anisotropy of arbitrary |
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particles, Fernandes and de la Torre improved the original Brownian |
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dynamics simulation algorithm\cite{Ermak1978,Allison1991} by |
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The use of rigid substructures,\cite{Chun:2000fj} |
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coarse-graining,\cite{Ayton01,Golubkov06,Orlandi:2006fk,SunX._jp0762020} |
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and ellipsoidal representations of protein side |
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chains~\cite{Fogolari:1996lr} has made the use of the Stokes-Einstein |
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approximation problematic. A rigid substructure moves as a single |
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unit with orientational as well as translational degrees of freedom. |
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This requires a more general treatment of the hydrodynamics than the |
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spherical approximation provides. Also, the atoms involved in a rigid |
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or coarse-grained structure have solvent-mediated interactions with |
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each other, and these interactions are ignored if all atoms are |
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treated as separate spherical particles. The theory of interactions |
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{\it between} bodies moving through a fluid has been developed over |
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the past century and has been applied to simulations of Brownian |
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motion.\cite{FIXMAN:1986lr,Ramachandran1996} |
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|
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In order to account for the diffusion anisotropy of complex shapes, |
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Fernandes and Garc\'{i}a de la Torre improved an earlier Brownian |
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dynamics simulation algorithm~\cite{Ermak1978,Allison1991} by |
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incorporating a generalized $6\times6$ diffusion tensor and |
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introducing a simple rotation evolution scheme consisting of three |
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consecutive rotations.\cite{Fernandes2002} Unfortunately, unexpected |
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errors and biases are introduced into the system due to the |
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arbitrary order of applying the noncommuting rotation |
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operators.\cite{Beard2003} Based on the observation the momentum |
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relaxation time is much less than the time step, one may ignore the |
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inertia in Brownian dynamics. However, the assumption of zero |
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average acceleration is not always true for cooperative motion which |
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is common in protein motion. An inertial Brownian dynamics (IBD) was |
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proposed to address this issue by adding an inertial correction |
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term.\cite{Beard2000} As a complement to IBD which has a lower bound |
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in time step because of the inertial relaxation time, long-time-step |
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inertial dynamics (LTID) can be used to investigate the inertial |
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behavior of the polymer segments in low friction |
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regime.\cite{Beard2000} LTID can also deal with the rotational |
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dynamics for nonskew bodies without translation-rotation coupling by |
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separating the translation and rotation motion and taking advantage |
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of the analytical solution of hydrodynamics properties. However, |
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typical nonskew bodies like cylinders and ellipsoids are inadequate |
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to represent most complex macromolecule assemblies. These intricate |
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molecules have been represented by a set of beads and their |
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hydrodynamic properties can be calculated using variants on the |
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standard hydrodynamic interaction tensors. |
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introducing a rotational evolution scheme consisting of three |
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consecutive rotations.\cite{Fernandes2002} Unfortunately, biases are |
| 122 |
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introduced into the system due to the arbitrary order of applying the |
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noncommuting rotation operators.\cite{Beard2003} Based on the |
| 124 |
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observation the momentum relaxation time is much less than the time |
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step, one may ignore the inertia in Brownian dynamics. However, the |
| 126 |
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assumption of zero average acceleration is not always true for |
| 127 |
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cooperative motion which is common in proteins. An inertial Brownian |
| 128 |
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dynamics (IBD) was proposed to address this issue by adding an |
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inertial correction term.\cite{Beard2000} As a complement to IBD, |
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which has a lower bound in time step because of the inertial |
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relaxation time, long-time-step inertial dynamics (LTID) can be used |
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to investigate the inertial behavior of linked polymer segments in a |
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low friction regime.\cite{Beard2000} LTID can also deal with the |
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rotational dynamics for nonskew bodies without translation-rotation |
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coupling by separating the translation and rotation motion and taking |
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advantage of the analytical solution of hydrodynamic |
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properties. However, typical nonskew bodies like cylinders and |
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ellipsoids are inadequate to represent most complex macromolecular |
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assemblies. Therefore, the goal of this work is to adapt some of the |
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hydrodynamic methodologies developed to treat Brownian motion of |
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complex assemblies into a Langevin integrator for rigid bodies with |
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arbitrary shapes. |
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|
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\subsection{Rigid Body Dynamics} |
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Rigid bodies are frequently involved in the modeling of large |
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collections of particles that move as a single unit. In molecular |
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simulations, rigid bodies have been used to simplify protein-protein |
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docking,\cite{Gray2003} and lipid bilayer |
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simulations.\cite{SunX._jp0762020} Many of the water models in common |
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use are also rigid-body |
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models,\cite{Jorgensen83,Berendsen81,Berendsen87} although they are |
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typically evolved in molecular dynamics simulations using constraints |
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rather than rigid body equations of motion. |
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|
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Euler angles are a natural choice to describe the rotational degrees |
| 156 |
+ |
of freedom. However, due to $\frac{1}{\sin \theta}$ singularities, the |
| 157 |
+ |
numerical integration of corresponding equations of these motion can |
| 158 |
+ |
become inaccurate (and inefficient). Although the use of multiple |
| 159 |
+ |
sets of Euler angles can overcome this problem,\cite{Barojas1973} the |
| 160 |
+ |
computational penalty and the loss of angular momentum conservation |
| 161 |
+ |
remain. A singularity-free representation utilizing quaternions was |
| 162 |
+ |
developed by Evans in 1977.\cite{Evans1977} The Evans quaternion |
| 163 |
+ |
approach uses a nonseparable Hamiltonian, and this has prevented |
| 164 |
+ |
symplectic algorithms from being utilized until very |
| 165 |
+ |
recently.\cite{Miller2002} |
| 166 |
+ |
|
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+ |
Another approach is the application of holonomic constraints to the |
| 168 |
+ |
atoms belonging to the rigid body. Each atom moves independently |
| 169 |
+ |
under the normal forces deriving from potential energy and constraints |
| 170 |
+ |
are used to guarantee rigidity. However, due to their iterative |
| 171 |
+ |
nature, the SHAKE and RATTLE algorithms converge very slowly when the |
| 172 |
+ |
number of constraints (and the number of particles that belong to the |
| 173 |
+ |
rigid body) increases.\cite{Ryckaert1977,Andersen1983} |
| 174 |
+ |
|
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In order to develop a stable and efficient integration scheme that |
| 176 |
+ |
preserves most constants of the motion in microcanonical simulations, |
| 177 |
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symplectic propagators are necessary. By introducing a conjugate |
| 178 |
+ |
momentum to the rotation matrix ${\bf Q}$ and re-formulating |
| 179 |
+ |
Hamilton's equations, a symplectic orientational integrator, |
| 180 |
+ |
RSHAKE,\cite{Kol1997} was proposed to evolve rigid bodies on a |
| 181 |
+ |
constraint manifold by iteratively satisfying the orthogonality |
| 182 |
+ |
constraint ${\bf Q}^T {\bf Q} = 1$. An alternative method using the |
| 183 |
+ |
quaternion representation was developed by Omelyan.\cite{Omelyan1998} |
| 184 |
+ |
However, both of these methods are iterative and suffer from some |
| 185 |
+ |
related inefficiencies. A symplectic Lie-Poisson integrator for rigid |
| 186 |
+ |
bodies developed by Dullweber {\it et al.}\cite{Dullweber1997} removes |
| 187 |
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most of the limitations mentioned above and is therefore the basis for |
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our Langevin integrator. |
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+ |
|
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The goal of the present work is to develop a Langevin dynamics |
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algorithm for arbitrary-shaped rigid particles by integrating the |
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accurate estimation of friction tensor from hydrodynamics theory |
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< |
into the sophisticated rigid body dynamics algorithms. |
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algorithm for arbitrary-shaped rigid particles by integrating an |
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accurate estimate of the friction tensor from hydrodynamics theory |
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into a stable and efficient rigid body dynamics propagator. In the |
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sections below, we review some of the theory of hydrodynamic tensors |
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developed primarily for Brownian simulations of multi-particle |
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systems, we then present our integration method for a set of |
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generalized Langevin equations of motion, and we compare the behavior |
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of the new Langevin integrator to dynamical quantities obtained via |
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explicit solvent molecular dynamics. |
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|
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\subsection{\label{introSection:frictionTensor}Friction Tensor} |
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Theoretically, the friction kernel can be determined using the |
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velocity autocorrelation function. However, this approach becomes |
| 204 |
< |
impractical when the system becomes more and more complicated. |
| 205 |
< |
Instead, various approaches based on hydrodynamics have been |
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< |
developed to calculate the friction coefficients. In general, the |
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friction tensor $\Xi$ is a $6\times 6$ matrix given by |
| 208 |
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\[ |
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\Xi = \left( {\begin{array}{*{20}c} |
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{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
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{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
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\end{array}} \right). |
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\] |
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Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are $3 \times 3$ |
| 215 |
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translational friction tensor and rotational resistance (friction) |
| 216 |
< |
tensor respectively, while ${\Xi^{tr} }$ is translation-rotation |
| 217 |
< |
coupling tensor and $ {\Xi^{rt} }$ is rotation-translation coupling |
| 218 |
< |
tensor. When a particle moves in a fluid, it may experience friction |
| 219 |
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force or torque along the opposite direction of the velocity or |
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angular velocity, |
| 221 |
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\[ |
| 201 |
> |
\subsection{\label{introSection:frictionTensor}The Friction Tensor} |
| 202 |
> |
Theoretically, a complete friction kernel for a solute particle can be |
| 203 |
> |
determined using the velocity autocorrelation function from a |
| 204 |
> |
simulation with explicit solvent molecules. However, this approach |
| 205 |
> |
becomes impractical when the solute becomes complex. Instead, various |
| 206 |
> |
approaches based on hydrodynamics have been developed to calculate |
| 207 |
> |
static friction coefficients. In general, the friction tensor $\Xi$ is |
| 208 |
> |
a $6\times 6$ matrix given by |
| 209 |
> |
\begin{equation} |
| 210 |
> |
\Xi = \left( \begin{array}{*{20}c} |
| 211 |
> |
\Xi^{tt} & \Xi^{rt} \\ |
| 212 |
> |
\Xi^{tr} & \Xi^{rr} \\ |
| 213 |
> |
\end{array} \right). |
| 214 |
> |
\end{equation} |
| 215 |
> |
Here, $\Xi^{tt}$ and $\Xi^{rr}$ are $3 \times 3$ translational and |
| 216 |
> |
rotational resistance (friction) tensors respectively, while |
| 217 |
> |
$\Xi^{tr}$ is translation-rotation coupling tensor and $\Xi^{rt}$ is |
| 218 |
> |
rotation-translation coupling tensor. When a particle moves in a |
| 219 |
> |
fluid, it may experience a friction force ($\mathbf{f}_f$) and torque |
| 220 |
> |
($\mathbf{\tau}_f$) in opposition to the velocity ($\mathbf{v}$) and |
| 221 |
> |
body-fixed angular velocity ($\mathbf{\omega}$), |
| 222 |
> |
\begin{equation} |
| 223 |
|
\left( \begin{array}{l} |
| 224 |
< |
F_R \\ |
| 225 |
< |
\tau _R \\ |
| 226 |
< |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 227 |
< |
{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
| 228 |
< |
{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
| 229 |
< |
\end{array}} \right)\left( \begin{array}{l} |
| 230 |
< |
v \\ |
| 231 |
< |
w \\ |
| 232 |
< |
\end{array} \right) |
| 233 |
< |
\] |
| 234 |
< |
where $F_r$ is the friction force and $\tau _R$ is the friction |
| 235 |
< |
torque. |
| 224 |
> |
\mathbf{f}_f \\ |
| 225 |
> |
\mathbf{\tau}_f \\ |
| 226 |
> |
\end{array} \right) = - \left( \begin{array}{*{20}c} |
| 227 |
> |
\Xi^{tt} & \Xi^{rt} \\ |
| 228 |
> |
\Xi^{tr} & \Xi^{rr} \\ |
| 229 |
> |
\end{array} \right)\left( \begin{array}{l} |
| 230 |
> |
\mathbf{v} \\ |
| 231 |
> |
\mathbf{\omega} \\ |
| 232 |
> |
\end{array} \right). |
| 233 |
> |
\end{equation} |
| 234 |
> |
For an arbitrary body moving in a fluid, Peters has derived a set of |
| 235 |
> |
fluctuation-dissipation relations for the friction |
| 236 |
> |
tensors,\cite{Peters:1999qy,Peters:1999uq,Peters:2000fk} |
| 237 |
> |
\begin{eqnarray} |
| 238 |
> |
\Xi^{tt} & = & \frac{1}{k_B T} \int_0^\infty \left[ \langle {\bf |
| 239 |
> |
F}(0) {\bf F}(-s) \rangle_{eq} - \langle {\bf F} \rangle_{eq}^2 |
| 240 |
> |
\right] ds \\ |
| 241 |
> |
\notag \\ |
| 242 |
> |
\Xi^{tr} & = & \frac{1}{k_B T} \int_0^\infty \left[ \langle {\bf |
| 243 |
> |
F}(0) {\bf \tau}(-s) \rangle_{eq} - \langle {\bf F} \rangle_{eq} |
| 244 |
> |
\langle {\bf \tau} \rangle_{eq} \right] ds \\ |
| 245 |
> |
\notag \\ |
| 246 |
> |
\Xi^{rt} & = & \frac{1}{k_B T} \int_0^\infty \left[ \langle {\bf |
| 247 |
> |
\tau}(0) {\bf F}(-s) \rangle_{eq} - \langle {\bf \tau} \rangle_{eq} |
| 248 |
> |
\langle {\bf F} \rangle_{eq} \right] ds \\ |
| 249 |
> |
\notag \\ |
| 250 |
> |
\Xi^{rr} & = & \frac{1}{k_B T} \int_0^\infty \left[ \langle {\bf |
| 251 |
> |
\tau}(0) {\bf \tau}(-s) \rangle_{eq} - \langle {\bf \tau} \rangle_{eq}^2 |
| 252 |
> |
\right] ds |
| 253 |
> |
\end{eqnarray} |
| 254 |
> |
In these expressions, the forces (${\bf F}$) and torques (${\bf |
| 255 |
> |
\tau}$) are those that arise solely from the interactions of the body with |
| 256 |
> |
the surrounding fluid. For a single solute body in an isotropic fluid, |
| 257 |
> |
the average forces and torques in these expressions ($\langle {\bf F} |
| 258 |
> |
\rangle_{eq}$ and $\langle {\bf \tau} \rangle_{eq}$) |
| 259 |
> |
vanish, and one obtains the simpler force-torque correlation formulae |
| 260 |
> |
of Nienhuis.\cite{Nienhuis:1970lr} Molecular dynamics simulations with |
| 261 |
> |
explicit solvent molecules can be used to obtain estimates of the |
| 262 |
> |
friction tensors with these formulae. In practice, however, one needs |
| 263 |
> |
relatively long simulations with frequently-stored force and torque |
| 264 |
> |
information to compute friction tensors, and this becomes |
| 265 |
> |
prohibitively expensive when there are large numbers of large solute |
| 266 |
> |
particles. For bodies with simple shapes, there are a number of |
| 267 |
> |
approximate expressions that allow computation of these tensors |
| 268 |
> |
without the need for expensive simulations that utilize explicit |
| 269 |
> |
solvent particles. |
| 270 |
|
|
| 271 |
|
\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shapes}} |
| 272 |
< |
|
| 273 |
< |
For a spherical particle with slip boundary conditions, the |
| 274 |
< |
translational and rotational friction constant can be calculated |
| 275 |
< |
from Stoke's law, |
| 276 |
< |
\[ |
| 277 |
< |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
| 278 |
< |
{6\pi \eta R} & 0 & 0 \\ |
| 279 |
< |
0 & {6\pi \eta R} & 0 \\ |
| 280 |
< |
0 & 0 & {6\pi \eta R} \\ |
| 281 |
< |
\end{array}} \right) |
| 282 |
< |
\] |
| 272 |
> |
For a spherical body under ``stick'' boundary conditions, the |
| 273 |
> |
translational and rotational friction tensors can be estimated from |
| 274 |
> |
Stokes' law, |
| 275 |
> |
\begin{equation} |
| 276 |
> |
\label{eq:StokesTranslation} |
| 277 |
> |
\Xi^{tt} = \left( \begin{array}{*{20}c} |
| 278 |
> |
{6\pi \eta \rho} & 0 & 0 \\ |
| 279 |
> |
0 & {6\pi \eta \rho} & 0 \\ |
| 280 |
> |
0 & 0 & {6\pi \eta \rho} \\ |
| 281 |
> |
\end{array} \right) |
| 282 |
> |
\end{equation} |
| 283 |
|
and |
| 284 |
< |
\[ |
| 285 |
< |
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
| 286 |
< |
{8\pi \eta R^3 } & 0 & 0 \\ |
| 287 |
< |
0 & {8\pi \eta R^3 } & 0 \\ |
| 288 |
< |
0 & 0 & {8\pi \eta R^3 } \\ |
| 289 |
< |
\end{array}} \right) |
| 290 |
< |
\] |
| 291 |
< |
where $\eta$ is the viscosity of the solvent and $R$ is the |
| 292 |
< |
hydrodynamic radius. |
| 284 |
> |
\begin{equation} |
| 285 |
> |
\label{eq:StokesRotation} |
| 286 |
> |
\Xi^{rr} = \left( \begin{array}{*{20}c} |
| 287 |
> |
{8\pi \eta \rho^3 } & 0 & 0 \\ |
| 288 |
> |
0 & {8\pi \eta \rho^3 } & 0 \\ |
| 289 |
> |
0 & 0 & {8\pi \eta \rho^3 } \\ |
| 290 |
> |
\end{array} \right) |
| 291 |
> |
\end{equation} |
| 292 |
> |
where $\eta$ is the viscosity of the solvent and $\rho$ is the |
| 293 |
> |
hydrodynamic radius. The presence of the rotational resistance tensor |
| 294 |
> |
implies that the spherical body has internal structure and |
| 295 |
> |
orientational degrees of freedom that must be propagated in time. For |
| 296 |
> |
non-structured spherical bodies (i.e. the atoms in a traditional |
| 297 |
> |
molecular dynamics simulation) these degrees of freedom do not exist. |
| 298 |
|
|
| 299 |
|
Other non-spherical shapes, such as cylinders and ellipsoids, are |
| 300 |
< |
widely used as references for developing new hydrodynamics theory, |
| 300 |
> |
widely used as references for developing new hydrodynamic theories, |
| 301 |
|
because their properties can be calculated exactly. In 1936, Perrin |
| 302 |
< |
extended Stokes's law to general ellipsoids, also called a triaxial |
| 303 |
< |
ellipsoid, which is given in Cartesian coordinates |
| 304 |
< |
by\cite{Perrin1934, Perrin1936} |
| 305 |
< |
\[ |
| 306 |
< |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
| 307 |
< |
}} = 1 |
| 308 |
< |
\] |
| 309 |
< |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
| 310 |
< |
due to the complexity of the elliptic integral, only the ellipsoid |
| 311 |
< |
with the restriction of two axes being equal, \textit{i.e.} |
| 312 |
< |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
| 313 |
< |
exactly. Introducing an elliptic integral parameter $S$ for prolate |
| 314 |
< |
ellipsoids : |
| 315 |
< |
\[ |
| 316 |
< |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 317 |
< |
} }}{b}, |
| 318 |
< |
\] |
| 319 |
< |
and oblate ellipsoids: |
| 320 |
< |
\[ |
| 321 |
< |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 322 |
< |
}}{a}, |
| 323 |
< |
\] |
| 324 |
< |
one can write down the translational and rotational resistance |
| 325 |
< |
tensors |
| 326 |
< |
\begin{eqnarray*} |
| 327 |
< |
\Xi _a^{tt} & = & 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
| 328 |
< |
\Xi _b^{tt} & = & \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
| 329 |
< |
2a}}, |
| 330 |
< |
\end{eqnarray*} |
| 331 |
< |
and |
| 332 |
< |
\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*} |
| 302 |
> |
extended Stokes' law to general |
| 303 |
> |
ellipsoids,\cite{Perrin1934,Perrin1936} described in Cartesian |
| 304 |
> |
coordinates as |
| 305 |
> |
\begin{equation} |
| 306 |
> |
\frac{x^2 }{a^2} + \frac{y^2}{b^2} + \frac{z^2 }{c^2} = 1. |
| 307 |
> |
\end{equation} |
| 308 |
> |
Here, the semi-axes are of lengths $a$, $b$, and $c$. Due to the |
| 309 |
> |
complexity of the elliptic integral, only uniaxial ellipsoids, either |
| 310 |
> |
prolate ($a \ge b = c$) or oblate ($a < b = c$), were solved |
| 311 |
> |
exactly. Introducing an elliptic integral parameter $S$ for prolate, |
| 312 |
> |
\begin{equation} |
| 313 |
> |
S = \frac{2}{\sqrt{a^2 - b^2}} \ln \frac{a + \sqrt{a^2 - b^2}}{b}, |
| 314 |
> |
\end{equation} |
| 315 |
> |
and oblate, |
| 316 |
> |
\begin{equation} |
| 317 |
> |
S = \frac{2}{\sqrt {b^2 - a^2 }} \arctan \frac{\sqrt {b^2 - a^2}}{a}, |
| 318 |
> |
\end{equation} |
| 319 |
> |
ellipsoids, it is possible to write down exact solutions for the |
| 320 |
> |
resistance tensors. As is the case for spherical bodies, the translational, |
| 321 |
> |
\begin{eqnarray} |
| 322 |
> |
\Xi_a^{tt} & = & 16\pi \eta \frac{a^2 - b^2}{(2a^2 - b^2 )S - 2a}. \\ |
| 323 |
> |
\Xi_b^{tt} = \Xi_c^{tt} & = & 32\pi \eta \frac{a^2 - b^2 }{(2a^2 - 3b^2 )S + 2a}, |
| 324 |
> |
\end{eqnarray} |
| 325 |
> |
and rotational, |
| 326 |
> |
\begin{eqnarray} |
| 327 |
> |
\Xi_a^{rr} & = & \frac{32\pi}{3} \eta \frac{(a^2 - b^2 )b^2}{2a - b^2 S}, \\ |
| 328 |
> |
\Xi_b^{rr} = \Xi_c^{rr} & = & \frac{32\pi}{3} \eta \frac{(a^4 - b^4)}{(2a^2 - b^2 )S - 2a} |
| 329 |
> |
\end{eqnarray} |
| 330 |
> |
resistance tensors are diagonal $3 \times 3$ matrices. For both |
| 331 |
> |
spherical and ellipsoidal particles, the translation-rotation and |
| 332 |
> |
rotation-translation coupling tensors are zero. |
| 333 |
|
|
| 334 |
|
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}} |
| 335 |
+ |
Other than the fluctuation dissipation formulae given by |
| 336 |
+ |
Peters,\cite{Peters:1999qy,Peters:1999uq,Peters:2000fk} there are no |
| 337 |
+ |
analytic solutions for the friction tensor for rigid molecules of |
| 338 |
+ |
arbitrary shape. The ellipsoid of revolution and general triaxial |
| 339 |
+ |
ellipsoid models have been widely used to approximate the hydrodynamic |
| 340 |
+ |
properties of rigid bodies. However, the mapping from all possible |
| 341 |
+ |
ellipsoidal spaces ($r$-space) to all possible combinations of |
| 342 |
+ |
rotational diffusion coefficients ($D$-space) is not |
| 343 |
+ |
unique.\cite{Wegener1979} Additionally, because there is intrinsic |
| 344 |
+ |
coupling between translational and rotational motion of {\it skew} |
| 345 |
+ |
rigid bodies, general ellipsoids are not always suitable for modeling |
| 346 |
+ |
rigid molecules. A number of studies have been devoted to determining |
| 347 |
+ |
the friction tensor for irregular shapes using methods in which the |
| 348 |
+ |
molecule of interest is modeled with a combination of |
| 349 |
+ |
spheres\cite{Carrasco1999} and the hydrodynamic properties of the |
| 350 |
+ |
molecule are then calculated using a set of two-point interaction |
| 351 |
+ |
tensors. We have found the {\it bead} and {\it rough shell} models of |
| 352 |
+ |
Carrasco and Garc\'{i}a de la Torre to be the most useful of these |
| 353 |
+ |
methods,\cite{Carrasco1999} and we review the basic outline of the |
| 354 |
+ |
rough shell approach here. A more thorough explanation can be found |
| 355 |
+ |
in Ref. \citen{Carrasco1999}. |
| 356 |
|
|
| 357 |
< |
Unlike spherical and other simply shaped molecules, there is no |
| 358 |
< |
analytical solution for the friction tensor for arbitrarily shaped |
| 359 |
< |
rigid molecules. The ellipsoid of revolution model and general |
| 360 |
< |
triaxial ellipsoid model have been used to approximate the |
| 361 |
< |
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 |
| 357 |
> |
Consider a rigid assembly of $N$ small beads moving through a |
| 358 |
> |
continuous medium. Due to hydrodynamic interactions between the |
| 359 |
> |
beads, the net velocity of the $i^\mathrm{th}$ bead relative to the |
| 360 |
> |
medium, ${\bf v}'_i$, is different than its unperturbed velocity ${\bf |
| 361 |
> |
v}_i$, |
| 362 |
|
\begin{equation} |
| 363 |
< |
F_i = \zeta _i v_i - \zeta _i \sum\limits_{j \ne i} {T_{ij} F_j }. |
| 299 |
< |
\label{introEquation:tensorExpression} |
| 363 |
> |
{\bf v}'_i = {\bf v}_i - \sum\limits_{j \ne i} {{\bf T}_{ij} {\bf F}_j } |
| 364 |
|
\end{equation} |
| 365 |
< |
This equation is the basis for deriving the hydrodynamic tensor. In |
| 366 |
< |
1930, Oseen and Burgers gave a simple solution to |
| 367 |
< |
Eq.~\ref{introEquation:tensorExpression} |
| 365 |
> |
where ${\bf F}_j$ is the frictional force on the medium due to bead $j$, and |
| 366 |
> |
${\bf T}_{ij}$ is the hydrodynamic interaction tensor between the two beads. |
| 367 |
> |
The frictional force felt by the $i^\mathrm{th}$ bead is proportional to |
| 368 |
> |
its net velocity |
| 369 |
|
\begin{equation} |
| 370 |
< |
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
| 371 |
< |
R_{ij}^T }}{{R_{ij}^2 }}} \right). \label{introEquation:oseenTensor} |
| 370 |
> |
{\bf F}_i = \xi_i {\bf v}_i - \xi_i \sum\limits_{j \ne i} {{\bf T}_{ij} {\bf F}_j }. |
| 371 |
> |
\label{introEquation:tensorExpression} |
| 372 |
|
\end{equation} |
| 373 |
< |
Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$. |
| 374 |
< |
A second order expression for element of different size was |
| 375 |
< |
introduced by Rotne and Prager\cite{Rotne1969} and improved by |
| 373 |
> |
Eq. (\ref{introEquation:tensorExpression}) defines the two-point |
| 374 |
> |
hydrodynamic tensor, ${\bf T}_{ij}$. There have been many proposed |
| 375 |
> |
solutions to this equation, including the simple solution given by |
| 376 |
> |
Oseen and Burgers in 1930 for two beads of identical radius. A second |
| 377 |
> |
order expression for beads of different hydrodynamic radii was |
| 378 |
> |
introduced by Rotne and Prager,\cite{Rotne1969} and improved by |
| 379 |
|
Garc\'{i}a de la Torre and Bloomfield,\cite{Torre1977} |
| 380 |
|
\begin{equation} |
| 381 |
< |
T_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I + |
| 382 |
< |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma |
| 383 |
< |
_i^2 + \sigma _j^2 }}{{r_{ij}^2 }}\left( {\frac{I}{3} - |
| 384 |
< |
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
| 381 |
> |
{\bf T}_{ij} = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {{\bf I} + |
| 382 |
> |
\frac{{{\bf R}_{ij} {\bf R}_{ij}^T }}{{R_{ij}^2 }}} \right) + \frac{{\rho |
| 383 |
> |
_i^2 + \rho_j^2 }}{{R_{ij}^2 }}\left( {\frac{{\bf I}}{3} - |
| 384 |
> |
\frac{{{\bf R}_{ij} {\bf R}_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
| 385 |
|
\label{introEquation:RPTensorNonOverlapped} |
| 386 |
|
\end{equation} |
| 387 |
< |
Both of the Eq.~\ref{introEquation:oseenTensor} and |
| 388 |
< |
Eq.~\ref{introEquation:RPTensorNonOverlapped} have an assumption |
| 389 |
< |
$R_{ij} \ge \sigma _i + \sigma _j$. An alternative expression for |
| 390 |
< |
overlapping beads with the same radius, $\sigma$, is given by |
| 387 |
> |
Here ${\bf R}_{ij}$ is the distance vector between beads $i$ and $j$. Both |
| 388 |
> |
the Oseen-Burgers tensor and |
| 389 |
> |
Eq.~\ref{introEquation:RPTensorNonOverlapped} have an assumption that |
| 390 |
> |
the beads do not overlap ($R_{ij} \ge \rho_i + \rho_j$). |
| 391 |
> |
|
| 392 |
> |
To calculate the resistance tensor for a body represented as the union |
| 393 |
> |
of many non-overlapping beads, we first pick an arbitrary origin $O$ |
| 394 |
> |
and then construct a $3N \times 3N$ supermatrix consisting of $N |
| 395 |
> |
\times N$ ${\bf B}_{ij}$ blocks |
| 396 |
|
\begin{equation} |
| 397 |
< |
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
| 398 |
< |
\frac{2}{{32}}\frac{{R_{ij} }}{\sigma }} \right)I + |
| 399 |
< |
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
| 400 |
< |
\label{introEquation:RPTensorOverlapped} |
| 397 |
> |
{\bf B} = \left( \begin{array}{*{20}c} |
| 398 |
> |
{\bf B}_{11} & \ldots & {\bf B}_{1N} \\ |
| 399 |
> |
\vdots & \ddots & \vdots \\ |
| 400 |
> |
{\bf B}_{N1} & \cdots & {\bf B}_{NN} |
| 401 |
> |
\end{array} \right) |
| 402 |
|
\end{equation} |
| 403 |
< |
To calculate the resistance tensor at an arbitrary origin $O$, we |
| 404 |
< |
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
| 331 |
< |
$B_{ij}$ blocks |
| 403 |
> |
${\bf B}_{ij}$ is a version of the hydrodynamic tensor which includes the |
| 404 |
> |
self-contributions for spheres, |
| 405 |
|
\begin{equation} |
| 406 |
< |
B = \left( {\begin{array}{*{20}c} |
| 407 |
< |
{B_{11} } & \ldots & {B_{1N} } \\ |
| 335 |
< |
\vdots & \ddots & \vdots \\ |
| 336 |
< |
{B_{N1} } & \cdots & {B_{NN} } \\ |
| 337 |
< |
\end{array}} \right), |
| 406 |
> |
{\bf B}_{ij} = \delta _{ij} \frac{{\bf I}}{{6\pi \eta R_{ij}}} + (1 - \delta_{ij} |
| 407 |
> |
){\bf T}_{ij} |
| 408 |
|
\end{equation} |
| 409 |
< |
where $B_{ij}$ is given by |
| 410 |
< |
\[ |
| 411 |
< |
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
| 412 |
< |
)T_{ij} |
| 413 |
< |
\] |
| 414 |
< |
where $\delta _{ij}$ is the Kronecker delta function. Inverting the |
| 415 |
< |
$B$ matrix, we obtain |
| 416 |
< |
\[ |
| 417 |
< |
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
| 418 |
< |
{C_{11} } & \ldots & {C_{1N} } \\ |
| 419 |
< |
\vdots & \ddots & \vdots \\ |
| 420 |
< |
{C_{N1} } & \cdots & {C_{NN} } \\ |
| 421 |
< |
\end{array}} \right), |
| 422 |
< |
\] |
| 423 |
< |
which can be partitioned into $N \times N$ $3 \times 3$ block |
| 424 |
< |
$C_{ij}$. With the help of $C_{ij}$ and the skew matrix $U_i$ |
| 425 |
< |
\[ |
| 426 |
< |
U_i = \left( {\begin{array}{*{20}c} |
| 427 |
< |
0 & { - z_i } & {y_i } \\ |
| 428 |
< |
{z_i } & 0 & { - x_i } \\ |
| 359 |
< |
{ - y_i } & {x_i } & 0 \\ |
| 360 |
< |
\end{array}} \right) |
| 361 |
< |
\] |
| 409 |
> |
where $\delta_{ij}$ is the Kronecker delta function. Inverting the |
| 410 |
> |
${\bf B}$ matrix, we obtain |
| 411 |
> |
\begin{equation} |
| 412 |
> |
{\bf C} = {\bf B}^{ - 1} = \left(\begin{array}{*{20}c} |
| 413 |
> |
{\bf C}_{11} & \ldots & {\bf C}_{1N} \\ |
| 414 |
> |
\vdots & \ddots & \vdots \\ |
| 415 |
> |
{\bf C}_{N1} & \cdots & {\bf C}_{NN} |
| 416 |
> |
\end{array} \right), |
| 417 |
> |
\end{equation} |
| 418 |
> |
which can be partitioned into $N \times N$ blocks labeled ${\bf C}_{ij}$. |
| 419 |
> |
(Each of the ${\bf C}_{ij}$ blocks is a $3 \times 3$ matrix.) Using the |
| 420 |
> |
skew matrix, |
| 421 |
> |
\begin{equation} |
| 422 |
> |
{\bf U}_i = \left(\begin{array}{*{20}c} |
| 423 |
> |
0 & -z_i & y_i \\ |
| 424 |
> |
z_i & 0 & - x_i \\ |
| 425 |
> |
-y_i & x_i & 0 |
| 426 |
> |
\end{array}\right) |
| 427 |
> |
\label{eq:skewMatrix} |
| 428 |
> |
\end{equation} |
| 429 |
|
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
| 430 |
< |
bead $i$ and origin $O$, the elements of resistance tensor at |
| 431 |
< |
arbitrary origin $O$ can be written as |
| 430 |
> |
bead $i$ and origin $O$, the elements of the resistance tensor (at the |
| 431 |
> |
arbitrary origin $O$) can be written as |
| 432 |
|
\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 |
| 433 |
|
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 434 |
+ |
\Xi^{tt} & = & \sum\limits_i {\sum\limits_j {{\bf C}_{ij} } } \notag , \\ |
| 435 |
+ |
\Xi^{tr} = \Xi _{}^{rt} & = & \sum\limits_i {\sum\limits_j {{\bf U}_i {\bf C}_{ij} } } , \\ |
| 436 |
+ |
\Xi^{rr} & = & -\sum\limits_i \sum\limits_j {\bf U}_i {\bf C}_{ij} {\bf U}_j + 6 \eta V {\bf I}. \notag |
| 437 |
|
\end{eqnarray} |
| 438 |
< |
The final term in the expression for $\Xi^{rr}$ is correction that |
| 439 |
< |
accounts for errors in the rotational motion of certain kinds of bead |
| 440 |
< |
models. The additive correction uses the solvent viscosity ($\eta$) |
| 441 |
< |
as well as the total volume of the beads that contribute to the |
| 376 |
< |
hydrodynamic model, |
| 438 |
> |
The final term in the expression for $\Xi^{rr}$ is a correction that |
| 439 |
> |
accounts for errors in the rotational motion of the bead models. The |
| 440 |
> |
additive correction uses the solvent viscosity ($\eta$) as well as the |
| 441 |
> |
total volume of the beads that contribute to the hydrodynamic model, |
| 442 |
|
\begin{equation} |
| 443 |
< |
V = \frac{4 \pi}{3} \sum_{i=1}^{N} \sigma_i^3, |
| 443 |
> |
V = \frac{4 \pi}{3} \sum_{i=1}^{N} \rho_i^3, |
| 444 |
|
\end{equation} |
| 445 |
< |
where $\sigma_i$ is the radius of bead $i$. This correction term was |
| 445 |
> |
where $\rho_i$ is the radius of bead $i$. This correction term was |
| 446 |
|
rigorously tested and compared with the analytical results for |
| 447 |
< |
two-sphere and ellipsoidal systems by Garcia de la Torre and |
| 447 |
> |
two-sphere and ellipsoidal systems by Garc\'{i}a de la Torre and |
| 448 |
|
Rodes.\cite{Torre:1983lr} |
| 449 |
|
|
| 450 |
< |
|
| 451 |
< |
The resistance tensor depends on the origin to which they refer. The |
| 452 |
< |
proper location for applying the friction force is the center of |
| 453 |
< |
resistance (or center of reaction), at which the trace of rotational |
| 454 |
< |
resistance tensor, $ \Xi ^{rr}$ reaches a minimum value. |
| 455 |
< |
Mathematically, the center of resistance is defined as an unique |
| 456 |
< |
point of the rigid body at which the translation-rotation coupling |
| 392 |
< |
tensors are symmetric, |
| 450 |
> |
In general, resistance tensors depend on the origin at which they were |
| 451 |
> |
computed. However, the proper location for applying the friction |
| 452 |
> |
force is the center of resistance, the special point at which the |
| 453 |
> |
trace of rotational resistance tensor, $\Xi^{rr}$ reaches a minimum |
| 454 |
> |
value. Mathematically, the center of resistance can also be defined |
| 455 |
> |
as the unique point for a rigid body at which the translation-rotation |
| 456 |
> |
coupling tensors are symmetric, |
| 457 |
|
\begin{equation} |
| 458 |
< |
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
| 458 |
> |
\Xi^{tr} = \left(\Xi^{tr}\right)^T |
| 459 |
|
\label{introEquation:definitionCR} |
| 460 |
|
\end{equation} |
| 461 |
< |
From Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
| 462 |
< |
we can easily derive that the translational resistance tensor is |
| 463 |
< |
origin independent, while the rotational resistance tensor and |
| 461 |
> |
From Eq. \ref{introEquation:ResistanceTensorArbitraryOrigin}, we can |
| 462 |
> |
easily derive that the {\it translational} resistance tensor is origin |
| 463 |
> |
independent, while the rotational resistance tensor and |
| 464 |
|
translation-rotation coupling resistance tensor depend on the |
| 465 |
< |
origin. Given the resistance tensor at an arbitrary origin $O$, and |
| 466 |
< |
a vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
| 467 |
< |
obtain the resistance tensor at $P$ by |
| 468 |
< |
\begin{equation} |
| 469 |
< |
\begin{array}{l} |
| 470 |
< |
\Xi _P^{tt} = \Xi _O^{tt} \\ |
| 471 |
< |
\Xi _P^{tr} = \Xi _P^{rt} = \Xi _O^{tr} - U_{OP} \Xi _O^{tt} \\ |
| 472 |
< |
\Xi _P^{rr} = \Xi _O^{rr} - U_{OP} \Xi _O^{tt} U_{OP} + \Xi _O^{tr} U_{OP} - U_{OP} \Xi _O^{{tr} ^{^T }} \\ |
| 473 |
< |
\end{array} |
| 474 |
< |
\label{introEquation:resistanceTensorTransformation} |
| 475 |
< |
\end{equation} |
| 476 |
< |
where |
| 477 |
< |
\[ |
| 478 |
< |
U_{OP} = \left( {\begin{array}{*{20}c} |
| 479 |
< |
0 & { - z_{OP} } & {y_{OP} } \\ |
| 480 |
< |
{z_i } & 0 & { - x_{OP} } \\ |
| 481 |
< |
{ - y_{OP} } & {x_{OP} } & 0 \\ |
| 482 |
< |
\end{array}} \right) |
| 483 |
< |
\] |
| 484 |
< |
Using Eq.~\ref{introEquation:definitionCR} and |
| 485 |
< |
Eq.~\ref{introEquation:resistanceTensorTransformation}, one can |
| 486 |
< |
locate the position of center of resistance, |
| 487 |
< |
\begin{eqnarray*} |
| 488 |
< |
\left( \begin{array}{l} |
| 489 |
< |
x_{OR} \\ |
| 490 |
< |
y_{OR} \\ |
| 491 |
< |
z_{OR} \\ |
| 492 |
< |
\end{array} \right) & = &\left( {\begin{array}{*{20}c} |
| 493 |
< |
{(\Xi _O^{rr} )_{yy} + (\Xi _O^{rr} )_{zz} } & { - (\Xi _O^{rr} )_{xy} } & { - (\Xi _O^{rr} )_{xz} } \\ |
| 494 |
< |
{ - (\Xi _O^{rr} )_{xy} } & {(\Xi _O^{rr} )_{zz} + (\Xi _O^{rr} )_{xx} } & { - (\Xi _O^{rr} )_{yz} } \\ |
| 495 |
< |
{ - (\Xi _O^{rr} )_{xz} } & { - (\Xi _O^{rr} )_{yz} } & {(\Xi _O^{rr} )_{xx} + (\Xi _O^{rr} )_{yy} } \\ |
| 496 |
< |
\end{array}} \right)^{ - 1} \\ |
| 497 |
< |
& & \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 |
| 465 |
> |
origin. Given the resistance tensor at an arbitrary origin $O$, and a |
| 466 |
> |
vector ,${\bf r}_{OP} = (x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we |
| 467 |
> |
can obtain the resistance tensor at $P$ by |
| 468 |
> |
\begin{eqnarray} |
| 469 |
> |
\label{introEquation:resistanceTensorTransformation} |
| 470 |
> |
\Xi_P^{tt} & = & \Xi_O^{tt} \notag \\ |
| 471 |
> |
\Xi_P^{tr} = \Xi_P^{rt} & = & \Xi_O^{tr} - {\bf U}_{OP} \Xi _O^{tt} \\ |
| 472 |
> |
\Xi_P^{rr} & = &\Xi_O^{rr} - {\bf U}_{OP} \Xi_O^{tt} {\bf U}_{OP} |
| 473 |
> |
+ \Xi_O^{tr} {\bf U}_{OP} - {\bf U}_{OP} \left( \Xi_O^{tr} |
| 474 |
> |
\right)^{^T} \notag |
| 475 |
> |
\end{eqnarray} |
| 476 |
> |
where ${\bf U}_{OP}$ is the skew matrix (Eq. (\ref{eq:skewMatrix})) |
| 477 |
> |
for the vector between the origin $O$ and the point $P$. Using |
| 478 |
> |
Eqs.~\ref{introEquation:definitionCR}~and~\ref{introEquation:resistanceTensorTransformation}, |
| 479 |
> |
one can locate the position of center of resistance, |
| 480 |
> |
\begin{equation*} |
| 481 |
> |
\left(\begin{array}{l} |
| 482 |
> |
x_{OR} \\ |
| 483 |
> |
y_{OR} \\ |
| 484 |
> |
z_{OR} |
| 485 |
> |
\end{array}\right) = |
| 486 |
> |
\left(\begin{array}{*{20}c} |
| 487 |
> |
(\Xi_O^{rr})_{yy} + (\Xi_O^{rr})_{zz} & -(\Xi_O^{rr})_{xy} & -(\Xi_O^{rr})_{xz} \\ |
| 488 |
> |
-(\Xi_O^{rr})_{xy} & (\Xi_O^{rr})_{zz} + (\Xi_O^{rr})_{xx} & -(\Xi_O^{rr})_{yz} \\ |
| 489 |
> |
-(\Xi_O^{rr})_{xz} & -(\Xi_O^{rr})_{yz} & (\Xi_O^{rr})_{xx} + (\Xi_O^{rr})_{yy} \\ |
| 490 |
> |
\end{array}\right)^{-1} |
| 491 |
> |
\left(\begin{array}{l} |
| 492 |
> |
(\Xi_O^{tr})_{yz} - (\Xi_O^{tr})_{zy} \\ |
| 493 |
> |
(\Xi_O^{tr})_{zx} - (\Xi_O^{tr})_{xz} \\ |
| 494 |
> |
(\Xi_O^{tr})_{xy} - (\Xi_O^{tr})_{yx} |
| 495 |
> |
\end{array}\right) |
| 496 |
> |
\end{equation*} |
| 497 |
> |
where $x_{OR}$, $y_{OR}$, $z_{OR}$ are the components of the vector |
| 498 |
|
joining center of resistance $R$ and origin $O$. |
| 499 |
|
|
| 500 |
+ |
For a general rigid molecular substructure, finding the $6 \times 6$ |
| 501 |
+ |
resistance tensor can be a computationally demanding task. First, a |
| 502 |
+ |
lattice of small beads that extends well beyond the boundaries of the |
| 503 |
+ |
rigid substructure is created. The lattice is typically composed of |
| 504 |
+ |
0.25 \AA\ beads on a dense FCC lattice. The lattice constant is taken |
| 505 |
+ |
to be the bead diameter, so that adjacent beads are touching, but do |
| 506 |
+ |
not overlap. To make a shape corresponding to the rigid structure, |
| 507 |
+ |
beads that sit on lattice sites that are outside the van der Waals |
| 508 |
+ |
radii of all of the atoms comprising the rigid body are excluded from |
| 509 |
+ |
the calculation. |
| 510 |
|
|
| 511 |
+ |
For large structures, most of the beads will be deep within the rigid |
| 512 |
+ |
body and will not contribute to the hydrodynamic tensor. In the {\it |
| 513 |
+ |
rough shell} approach, beads which have all of their lattice neighbors |
| 514 |
+ |
inside the structure are considered interior beads, and are removed |
| 515 |
+ |
from the calculation. After following this procedure, only those |
| 516 |
+ |
beads in direct contact with the van der Waals surface of the rigid |
| 517 |
+ |
body are retained. For reasonably large molecular structures, this |
| 518 |
+ |
truncation can still produce bead assemblies with thousands of |
| 519 |
+ |
members. |
| 520 |
+ |
|
| 521 |
+ |
If all of the {\it atoms} comprising the rigid substructure are |
| 522 |
+ |
spherical and non-overlapping, the tensor in |
| 523 |
+ |
Eq.~(\ref{introEquation:RPTensorNonOverlapped}) may be used directly |
| 524 |
+ |
using the atoms themselves as the hydrodynamic beads. This is a |
| 525 |
+ |
variant of the {\it bead model} approach of Carrasco and Garc\'{i}a de |
| 526 |
+ |
la Torre.\cite{Carrasco1999} In this case, the size of the ${\bf B}$ |
| 527 |
+ |
matrix can be quite small, and the calculation of the hydrodynamic |
| 528 |
+ |
tensor is straightforward. |
| 529 |
+ |
|
| 530 |
+ |
In general, the inversion of the ${\bf B}$ matrix is the most |
| 531 |
+ |
computationally demanding task. This inversion is done only once for |
| 532 |
+ |
each type of rigid structure. We have used straightforward |
| 533 |
+ |
LU-decomposition to solve the linear system and to obtain the elements |
| 534 |
+ |
of ${\bf C}$. Once ${\bf C}$ has been obtained, the location of the |
| 535 |
+ |
center of resistance ($R$) is found and the resistance tensor at this |
| 536 |
+ |
point is calculated. The $3 \times 1$ vector giving the location of |
| 537 |
+ |
the rigid body's center of resistance and the $6 \times 6$ resistance |
| 538 |
+ |
tensor are then stored for use in the Langevin dynamics calculation. |
| 539 |
+ |
These quantities depend on solvent viscosity and temperature and must |
| 540 |
+ |
be recomputed if different simulation conditions are required. |
| 541 |
+ |
|
| 542 |
|
\section{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}} |
| 543 |
+ |
|
| 544 |
|
Consider the Langevin equations of motion in generalized coordinates |
| 545 |
|
\begin{equation} |
| 546 |
< |
M_i \dot V_i (t) = F_{s,i} (t) + F_{f,i(t)} + F_{r,i} (t) |
| 546 |
> |
{\bf M} \dot{{\bf V}}(t) = {\bf F}_{s}(t) + |
| 547 |
> |
{\bf F}_{f}(t) + {\bf F}_{r}(t) |
| 548 |
|
\label{LDGeneralizedForm} |
| 549 |
|
\end{equation} |
| 550 |
< |
where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
| 551 |
< |
and moment of inertial) matrix and $V_i$ is a generalized velocity, |
| 552 |
< |
$V_i = V_i(v_i,\omega _i)$. The right side of |
| 553 |
< |
Eq.~\ref{LDGeneralizedForm} consists of three generalized forces in |
| 554 |
< |
lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
| 555 |
< |
$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
| 556 |
< |
system in Newtownian mechanics typically refers to lab-fixed frame, |
| 557 |
< |
it is also convenient to handle the rotation of rigid body in |
| 558 |
< |
body-fixed frame. Thus the friction and random forces are calculated |
| 559 |
< |
in body-fixed frame and converted back to lab-fixed frame by: |
| 560 |
< |
\[ |
| 561 |
< |
\begin{array}{l} |
| 562 |
< |
F_{f,i}^l (t) = Q^T F_{f,i}^b (t), \\ |
| 563 |
< |
F_{r,i}^l (t) = Q^T F_{r,i}^b (t). \\ |
| 463 |
< |
\end{array} |
| 464 |
< |
\] |
| 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 $\omega _i$ |
| 550 |
> |
where ${\bf M}$ is a $6 \times 6$ diagonal mass matrix (which |
| 551 |
> |
includes the mass of the rigid body as well as the moments of inertia |
| 552 |
> |
in the body-fixed frame) and ${\bf V}$ is a generalized velocity, |
| 553 |
> |
${\bf V} = |
| 554 |
> |
\left\{{\bf v},{\bf \omega}\right\}$. The right side of |
| 555 |
> |
Eq.~\ref{LDGeneralizedForm} consists of three generalized forces: a |
| 556 |
> |
system force (${\bf F}_{s}$), a frictional or dissipative force (${\bf |
| 557 |
> |
F}_{f}$) and a stochastic force (${\bf F}_{r}$). While the evolution |
| 558 |
> |
of the system in Newtonian mechanics is typically done in the lab |
| 559 |
> |
frame, it is convenient to handle the dynamics of rigid bodies in |
| 560 |
> |
body-fixed frames. Thus the friction and random forces on each |
| 561 |
> |
substructure are calculated in a body-fixed frame and may converted |
| 562 |
> |
back to the lab frame using that substructure's rotation matrix (${\bf |
| 563 |
> |
Q}$): |
| 564 |
|
\begin{equation} |
| 565 |
< |
F_{r,i}^b (t) = \left( \begin{array}{l} |
| 566 |
< |
f_{r,i}^b (t) \\ |
| 567 |
< |
\tau _{r,i}^b (t) \\ |
| 568 |
< |
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 569 |
< |
{\Xi _{R,t} } & {\Xi _{R,c}^T } \\ |
| 570 |
< |
{\Xi _{R,c} } & {\Xi _{R,r} } \\ |
| 571 |
< |
\end{array}} \right)\left( \begin{array}{l} |
| 572 |
< |
v_{R,i}^b (t) \\ |
| 573 |
< |
\omega _i (t) \\ |
| 574 |
< |
\end{array} \right), |
| 565 |
> |
{\bf F}_{f,r} = |
| 566 |
> |
\left( \begin{array}{c} |
| 567 |
> |
{\bf f}_{f,r} \\ |
| 568 |
> |
{\bf \tau}_{f,r} |
| 569 |
> |
\end{array} \right) |
| 570 |
> |
= |
| 571 |
> |
\left( \begin{array}{c} |
| 572 |
> |
{\bf Q}^{T} {\bf f}^{~b}_{f,r} \\ |
| 573 |
> |
{\bf Q}^{T} {\bf \tau}^{~b}_{f,r} |
| 574 |
> |
\end{array} \right) |
| 575 |
|
\end{equation} |
| 576 |
< |
while the random force $F_{r,i}^l$ is a Gaussian stochastic variable |
| 577 |
< |
with zero mean and variance |
| 576 |
> |
The body-fixed friction force, ${\bf F}_{f}^{~b}$, is proportional to |
| 577 |
> |
the (body-fixed) velocity at the center of resistance |
| 578 |
> |
${\bf v}_{R}^{~b}$ and the angular velocity ${\bf \omega}$ |
| 579 |
|
\begin{equation} |
| 580 |
< |
\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
| 581 |
< |
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
| 582 |
< |
2k_B T\Xi _R \delta (t - t'). \label{randomForce} |
| 583 |
< |
\end{equation} |
| 584 |
< |
The equation of motion for $v_i$ can be written as |
| 585 |
< |
\begin{equation} |
| 586 |
< |
m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) + |
| 587 |
< |
f_{r,i}^l (t) |
| 588 |
< |
\end{equation} |
| 589 |
< |
Since the frictional force is applied at the center of resistance |
| 590 |
< |
which generally does not coincide with the center of mass, an extra |
| 591 |
< |
torque is exerted at the center of mass. Thus, the net body-fixed |
| 592 |
< |
frictional torque at the center of mass, $\tau _{n,i}^b (t)$, is |
| 496 |
< |
given by |
| 580 |
> |
{\bf F}_{f}^{~b}(t) = \left( \begin{array}{l} |
| 581 |
> |
{\bf f}_{f}^{~b}(t) \\ |
| 582 |
> |
{\bf \tau}_{f}^{~b}(t) \\ |
| 583 |
> |
\end{array} \right) = - \left( \begin{array}{*{20}c} |
| 584 |
> |
\Xi_{R}^{tt} & \Xi_{R}^{rt} \\ |
| 585 |
> |
\Xi_{R}^{tr} & \Xi_{R}^{rr} \\ |
| 586 |
> |
\end{array} \right)\left( \begin{array}{l} |
| 587 |
> |
{\bf v}_{R}^{~b}(t) \\ |
| 588 |
> |
{\bf \omega}(t) \\ |
| 589 |
> |
\end{array} \right), |
| 590 |
> |
\end{equation} |
| 591 |
> |
while the random force, ${\bf F}_{r}$, is a Gaussian stochastic |
| 592 |
> |
variable with zero mean and variance, |
| 593 |
|
\begin{equation} |
| 594 |
< |
\tau _{r,i}^b = \tau _{r,i}^b +r_{MR} \times f_{r,i}^b |
| 594 |
> |
\left\langle {{\bf F}_{r}(t) ({\bf F}_{r}(t'))^T } \right\rangle = |
| 595 |
> |
\left\langle {{\bf F}_{r}^{~b} (t) ({\bf F}_{r}^{~b} (t'))^T } \right\rangle = |
| 596 |
> |
2 k_B T \Xi_R \delta(t - t'). \label{eq:randomForce} |
| 597 |
|
\end{equation} |
| 598 |
< |
where $r_{MR}$ is the vector from the center of mass to the center |
| 599 |
< |
of the resistance. Instead of integrating the angular velocity in |
| 600 |
< |
lab-fixed frame, we consider the equation of angular momentum in |
| 601 |
< |
body-fixed frame |
| 598 |
> |
$\Xi_R$ is the $6\times6$ resistance tensor at the center of |
| 599 |
> |
resistance. Once this tensor is known for a given rigid body (as |
| 600 |
> |
described in the previous section) obtaining a stochastic vector that |
| 601 |
> |
has the properties in Eq. (\ref{eq:randomForce}) can be done |
| 602 |
> |
efficiently by carrying out a one-time Cholesky decomposition to |
| 603 |
> |
obtain the square root matrix of the resistance tensor, |
| 604 |
> |
\begin{equation} |
| 605 |
> |
\Xi_R = {\bf S} {\bf S}^{T}, |
| 606 |
> |
\label{eq:Cholesky} |
| 607 |
> |
\end{equation} |
| 608 |
> |
where ${\bf S}$ is a lower triangular matrix.\cite{Schlick2002} A |
| 609 |
> |
vector with the statistics required for the random force can then be |
| 610 |
> |
obtained by multiplying ${\bf S}$ onto a random 6-vector ${\bf Z}$ which |
| 611 |
> |
has elements chosen from a Gaussian distribution, such that: |
| 612 |
|
\begin{equation} |
| 613 |
< |
\dot j_i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b (t) |
| 614 |
< |
+ \tau _{r,i}^b(t) |
| 613 |
> |
\langle {\bf Z}_i \rangle = 0, \hspace{1in} \langle {\bf Z}_i \cdot |
| 614 |
> |
{\bf Z}_j \rangle = \frac{2 k_B T}{\delta t} \delta_{ij}, |
| 615 |
|
\end{equation} |
| 616 |
< |
Embedding the friction terms into force and torque, one can |
| 617 |
< |
integrate the langevin equations of motion for rigid body of |
| 618 |
< |
arbitrary shape in a velocity-Verlet style 2-part algorithm, where |
| 511 |
< |
$h= \delta t$: |
| 616 |
> |
where $\delta t$ is the timestep in use during the simulation. The |
| 617 |
> |
random force, ${\bf F}_{r}^{~b} = {\bf S} {\bf Z}$, can be shown to have the |
| 618 |
> |
correct properties required by Eq. (\ref{eq:randomForce}). |
| 619 |
|
|
| 620 |
< |
{\tt moveA:} |
| 620 |
> |
The equation of motion for the translational velocity of the center of |
| 621 |
> |
mass (${\bf v}$) can be written as |
| 622 |
> |
\begin{equation} |
| 623 |
> |
m \dot{{\bf v}} (t) = {\bf f}_{s}(t) + {\bf f}_{f}(t) + |
| 624 |
> |
{\bf f}_{r}(t) |
| 625 |
> |
\end{equation} |
| 626 |
> |
Since the frictional and random forces are applied at the center of |
| 627 |
> |
resistance, which generally does not coincide with the center of mass, |
| 628 |
> |
extra torques are exerted at the center of mass. Thus, the net |
| 629 |
> |
body-fixed torque at the center of mass, $\tau^{~b}(t)$, |
| 630 |
> |
is given by |
| 631 |
> |
\begin{equation} |
| 632 |
> |
\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) |
| 633 |
> |
\end{equation} |
| 634 |
> |
where ${\bf r}_{MR}$ is the vector from the center of mass to the center of |
| 635 |
> |
resistance. Instead of integrating the angular velocity in lab-fixed |
| 636 |
> |
frame, we consider the equation of motion for the angular momentum |
| 637 |
> |
(${\bf j}$) in the body-fixed frame |
| 638 |
> |
\begin{equation} |
| 639 |
> |
\frac{\partial}{\partial t}{\bf j}(t) = \tau^{~b}(t) |
| 640 |
> |
\end{equation} |
| 641 |
> |
Embedding the friction and random forces into the the total force and |
| 642 |
> |
torque, one can integrate the Langevin equations of motion for a rigid |
| 643 |
> |
body of arbitrary shape in a velocity-Verlet style 2-part algorithm, |
| 644 |
> |
where $h = \delta t$: |
| 645 |
> |
|
| 646 |
> |
{\tt move A:} |
| 647 |
|
\begin{align*} |
| 648 |
|
{\bf v}\left(t + h / 2\right) &\leftarrow {\bf v}(t) |
| 649 |
|
+ \frac{h}{2} \left( {\bf f}(t) / m \right), \\ |
| 652 |
|
+ h {\bf v}\left(t + h / 2 \right), \\ |
| 653 |
|
% |
| 654 |
|
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
| 655 |
< |
+ \frac{h}{2} {\bf \tau}^b(t), \\ |
| 655 |
> |
+ \frac{h}{2} {\bf \tau}^{~b}(t), \\ |
| 656 |
|
% |
| 657 |
< |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
| 657 |
> |
{\bf Q}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
| 658 |
|
(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
| 659 |
|
\end{align*} |
| 660 |
< |
In this context, the $\mathrm{rotate}$ function is the reversible |
| 661 |
< |
product of the three body-fixed rotations, |
| 660 |
> |
In this context, $\overleftrightarrow{\mathsf{I}}$ is the diagonal |
| 661 |
> |
moment of inertia tensor, and the $\mathrm{rotate}$ function is the |
| 662 |
> |
reversible product of the three body-fixed rotations, |
| 663 |
|
\begin{equation} |
| 664 |
|
\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot |
| 665 |
|
\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y |
| 666 |
|
/ 2) \cdot \mathsf{G}_x(a_x /2), |
| 667 |
|
\end{equation} |
| 668 |
|
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
| 669 |
< |
rotates both the rotation matrix ($\mathsf{Q}$) and the body-fixed |
| 669 |
> |
rotates both the rotation matrix ($\mathbf{Q}$) and the body-fixed |
| 670 |
|
angular momentum (${\bf j}$) by an angle $\theta$ around body-fixed |
| 671 |
|
axis $\alpha$, |
| 672 |
|
\begin{equation} |
| 673 |
|
\mathsf{G}_\alpha( \theta ) = \left\{ |
| 674 |
|
\begin{array}{lcl} |
| 675 |
< |
\mathsf{Q}(t) & \leftarrow & \mathsf{Q}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
| 675 |
> |
\mathbf{Q}(t) & \leftarrow & \mathbf{Q}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
| 676 |
|
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
| 677 |
|
j}(0). |
| 678 |
|
\end{array} |
| 694 |
|
\end{equation} |
| 695 |
|
All other rotations follow in a straightforward manner. After the |
| 696 |
|
first part of the propagation, the forces and body-fixed torques are |
| 697 |
< |
calculated at the new positions and orientations |
| 697 |
> |
calculated at the new positions and orientations. The system forces |
| 698 |
> |
and torques are derivatives of the total potential energy function |
| 699 |
> |
($U$) with respect to the rigid body positions (${\bf r}$) and the |
| 700 |
> |
columns of the transposed rotation matrix ${\bf Q}^T = \left({\bf |
| 701 |
> |
u}_x, {\bf u}_y, {\bf u}_z \right)$: |
| 702 |
|
|
| 703 |
< |
{\tt doForces:} |
| 703 |
> |
{\tt Forces:} |
| 704 |
|
\begin{align*} |
| 705 |
< |
{\bf f}(t + h) &\leftarrow |
| 706 |
< |
- \left(\frac{\partial V}{\partial {\bf r}}\right)_{{\bf r}(t + h)}, \\ |
| 705 |
> |
{\bf f}_{s}(t + h) & \leftarrow |
| 706 |
> |
- \left(\frac{\partial U}{\partial {\bf r}}\right)_{{\bf r}(t + h)} \\ |
| 707 |
|
% |
| 708 |
< |
{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h) |
| 709 |
< |
\times \frac{\partial V}{\partial {\bf u}}, \\ |
| 708 |
> |
{\bf \tau}_{s}(t + h) &\leftarrow {\bf u}(t + h) |
| 709 |
> |
\times \frac{\partial U}{\partial {\bf u}} \\ |
| 710 |
|
% |
| 711 |
< |
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{Q}(t + h) |
| 712 |
< |
\cdot {\bf \tau}^s(t + h). |
| 711 |
> |
{\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) \\ |
| 712 |
> |
% |
| 713 |
> |
{\bf f}_{R,f}^{b}(t+h) & \leftarrow - \Xi_{R}^{tt} \cdot |
| 714 |
> |
{\bf v}^{b}_{R}(t+h) - \Xi_{R}^{rt} \cdot {\bf \omega}(t+h) \\ |
| 715 |
> |
% |
| 716 |
> |
{\bf \tau}_{R,f}^{b}(t+h) & \leftarrow - \Xi_{R}^{tr} \cdot |
| 717 |
> |
{\bf v}^{b}_{R}(t+h) - \Xi_{R}^{rr} \cdot {\bf \omega}(t+h) \\ |
| 718 |
> |
% |
| 719 |
> |
Z & \leftarrow {\tt GaussianNormal}(2 k_B T / h, 6) \\ |
| 720 |
> |
{\bf F}_{R,r}^{b}(t+h) & \leftarrow {\bf S} \cdot Z \\ |
| 721 |
> |
% |
| 722 |
> |
{\bf f}(t+h) & \leftarrow {\bf f}_{s}(t+h) + \mathbf{Q}^{T}(t+h) |
| 723 |
> |
\cdot \left({\bf f}_{R,f}^{~b} + {\bf f}_{R,r}^{~b} \right) \\ |
| 724 |
> |
% |
| 725 |
> |
\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) \\ |
| 726 |
> |
\tau^{~b}(t+h) & \leftarrow \mathbf{Q}(t+h) \cdot \tau(t+h) \\ |
| 727 |
|
\end{align*} |
| 728 |
+ |
Frictional (and random) forces and torques must be computed at the |
| 729 |
+ |
center of resistance, so there are additional steps required to find |
| 730 |
+ |
the body-fixed velocity (${\bf v}_{R}^{~b}$) at this location. Mapping |
| 731 |
+ |
the frictional and random forces at the center of resistance back to |
| 732 |
+ |
the center of mass also introduces an additional term in the torque |
| 733 |
+ |
one obtains at the center of mass. |
| 734 |
+ |
|
| 735 |
|
Once the forces and torques have been obtained at the new time step, |
| 736 |
|
the velocities can be advanced to the same time value. |
| 737 |
|
|
| 738 |
< |
{\tt moveB:} |
| 738 |
> |
{\tt move B:} |
| 739 |
|
\begin{align*} |
| 740 |
|
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t + h / 2 |
| 741 |
|
\right) |
| 743 |
|
% |
| 744 |
|
{\bf j}\left(t + h \right) &\leftarrow {\bf j}\left(t + h / 2 |
| 745 |
|
\right) |
| 746 |
< |
+ \frac{h}{2} {\bf \tau}^b(t + h) . |
| 746 |
> |
+ \frac{h}{2} {\bf \tau}^{~b}(t + h) . |
| 747 |
|
\end{align*} |
| 748 |
|
|
| 749 |
|
\section{Validating the Method\label{sec:validating}} |
| 797 |
|
\begin{figure} |
| 798 |
|
\centering |
| 799 |
|
\includegraphics[width=3in]{sketch} |
| 800 |
< |
\caption[Sketch of the model systems]{A sketch of the model systems |
| 801 |
< |
used in evaluating the behavior of the rigid body Langevin |
| 643 |
< |
integrator.} \label{fig:models} |
| 800 |
> |
\caption[A sketch of the model systems used in evaluating the behavior |
| 801 |
> |
of the rigid body Langevin integrator]{} \label{fig:models} |
| 802 |
|
\end{figure} |
| 803 |
|
|
| 804 |
|
\subsection{Simulation Methodology} |
| 805 |
|
We performed reference microcanonical simulations with explicit |
| 806 |
|
solvents for each of the different model system. In each case there |
| 807 |
|
was one solute model and 1929 solvent molecules present in the |
| 808 |
< |
simulation box. All simulations were equilibrated using a |
| 808 |
> |
simulation box. All simulations were equilibrated for 5 ns using a |
| 809 |
|
constant-pressure and temperature integrator with target values of 300 |
| 810 |
|
K for the temperature and 1 atm for pressure. Following this stage, |
| 811 |
< |
further equilibration and sampling was done in a microcanonical |
| 812 |
< |
ensemble. Since the model bodies are typically quite massive, we were |
| 813 |
< |
able to use a time step of 25 fs. |
| 811 |
> |
further equilibration (5 ns) and sampling (10 ns) was done in a |
| 812 |
> |
microcanonical ensemble. Since the model bodies are typically quite |
| 813 |
> |
massive, we were able to use a time step of 25 fs. |
| 814 |
|
|
| 815 |
|
The model systems studied used both Lennard-Jones spheres as well as |
| 816 |
|
uniaxial Gay-Berne ellipoids. In its original form, the Gay-Berne |
| 817 |
|
potential was a single site model for the interactions of rigid |
| 818 |
< |
ellipsoidal molecules.\cite{Gay81} It can be thought of as a |
| 818 |
> |
ellipsoidal molecules.\cite{Gay1981} It can be thought of as a |
| 819 |
|
modification of the Gaussian overlap model originally described by |
| 820 |
|
Berne and Pechukas.\cite{Berne72} The potential is constructed in the |
| 821 |
|
familiar form of the Lennard-Jones function using |
| 822 |
|
orientation-dependent $\sigma$ and $\epsilon$ parameters, |
| 823 |
|
\begin{equation*} |
| 824 |
< |
V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
| 825 |
< |
r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 826 |
< |
{\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u |
| 824 |
> |
V_{ij}({{\bf \hat u}_i}, {{\bf \hat u}_j}, {{\bf \hat |
| 825 |
> |
r}_{ij}}) = 4\epsilon ({{\bf \hat u}_i}, {{\bf \hat u}_j}, |
| 826 |
> |
{{\bf \hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({{\bf \hat u |
| 827 |
|
}_i}, |
| 828 |
< |
{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} |
| 829 |
< |
-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 830 |
< |
{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right] |
| 828 |
> |
{{\bf \hat u}_j}, {{\bf \hat r}_{ij}})+\sigma_0}\right)^{12} |
| 829 |
> |
-\left(\frac{\sigma_0}{r_{ij}-\sigma({{\bf \hat u}_i}, {{\bf \hat u}_j}, |
| 830 |
> |
{{\bf \hat r}_{ij}})+\sigma_0}\right)^6\right] |
| 831 |
|
\label{eq:gb} |
| 832 |
|
\end{equation*} |
| 833 |
|
|
| 844 |
|
Additionally, a well depth aspect ratio, $\epsilon^r = \epsilon^e / |
| 845 |
|
\epsilon^s$, describes the ratio between the well depths in the {\it |
| 846 |
|
end-to-end} and side-by-side configurations. Details of the potential |
| 847 |
< |
are given elsewhere,\cite{Luckhurst90,Golubkov06,SunGezelter08} and an |
| 847 |
> |
are given elsewhere,\cite{Luckhurst90,Golubkov06,SunX._jp0762020} and an |
| 848 |
|
excellent overview of the computational methods that can be used to |
| 849 |
|
efficiently compute forces and torques for this potential can be found |
| 850 |
|
in Ref. \citen{Golubkov06} |
| 879 |
|
\int_{t_0}^{t_0 + t} P_{xz}(t') dt' \right)^2 \right\rangle_{t_0}. |
| 880 |
|
\label{eq:shear} |
| 881 |
|
\end{equation} |
| 882 |
< |
A similar form exists for the bulk viscosity |
| 883 |
< |
\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. |
| 882 |
> |
which converges much more rapidly in molecular dynamics simulations |
| 883 |
> |
than the traditional Green-Kubo formula. |
| 884 |
|
|
| 885 |
|
The Langevin dynamics for the different model systems were performed |
| 886 |
|
at the same temperature as the average temperature of the |
| 906 |
|
compute the diffusive behavior for motion parallel to each body-fixed |
| 907 |
|
axis by projecting the displacement of the particle onto the |
| 908 |
|
body-fixed reference frame at $t=0$. With an isotropic solvent, as we |
| 909 |
< |
have used in this study, there are differences between the three |
| 910 |
< |
diffusion constants, but these must converge to the same value at |
| 911 |
< |
longer times. Translational diffusion constants for the different |
| 912 |
< |
shaped models are shown in table \ref{tab:translation}. |
| 909 |
> |
have used in this study, there may be differences between the three |
| 910 |
> |
diffusion constants at short times, but these must converge to the |
| 911 |
> |
same value at longer times. Translational diffusion constants for the |
| 912 |
> |
different shaped models are shown in table \ref{tab:translation}. |
| 913 |
|
|
| 914 |
|
In general, the three eigenvalues ($D_1, D_2, D_3$) of the rotational |
| 915 |
|
diffusion tensor (${\bf D}_{rr}$) measure the diffusion of an object |
| 989 |
|
an arbitrary value of 0.8 kcal/mol. |
| 990 |
|
|
| 991 |
|
The Stokes-Einstein behavior of large spherical particles in |
| 992 |
< |
hydrodynamic flows is well known, giving translational friction |
| 993 |
< |
coefficients of $6 \pi \eta R$ (stick boundary conditions) and |
| 994 |
< |
rotational friction coefficients of $8 \pi \eta R^3$. Recently, |
| 995 |
< |
Schmidt and Skinner have computed the behavior of spherical tag |
| 996 |
< |
particles in molecular dynamics simulations, and have shown that {\it |
| 997 |
< |
slip} boundary conditions ($\Xi_{tt} = 4 \pi \eta R$) may be more |
| 998 |
< |
appropriate for molecule-sized spheres embedded in a sea of spherical |
| 999 |
< |
solvent particles.\cite{Schmidt:2004fj,Schmidt:2003kx} |
| 992 |
> |
hydrodynamic flows with ``stick'' boundary conditions is well known, |
| 993 |
> |
and is given in Eqs. (\ref{eq:StokesTranslation}) and |
| 994 |
> |
(\ref{eq:StokesRotation}). Recently, Schmidt and Skinner have |
| 995 |
> |
computed the behavior of spherical tag particles in molecular dynamics |
| 996 |
> |
simulations, and have shown that {\it slip} boundary conditions |
| 997 |
> |
($\Xi_{tt} = 4 \pi \eta \rho$) may be more appropriate for |
| 998 |
> |
molecule-sized spheres embedded in a sea of spherical solvent |
| 999 |
> |
particles.\cite{Schmidt:2004fj,Schmidt:2003kx} |
| 1000 |
|
|
| 1001 |
|
Our simulation results show similar behavior to the behavior observed |
| 1002 |
|
by Schmidt and Skinner. The diffusion constant obtained from our |
| 1021 |
|
can be combined to give a single translational diffusion |
| 1022 |
|
constant,\cite{Berne90} |
| 1023 |
|
\begin{equation} |
| 1024 |
< |
D = \frac{k_B T}{6 \pi \eta a} G(\rho), |
| 1024 |
> |
D = \frac{k_B T}{6 \pi \eta a} G(s), |
| 1025 |
|
\label{Dperrin} |
| 1026 |
|
\end{equation} |
| 1027 |
|
as well as a single rotational diffusion coefficient, |
| 1028 |
|
\begin{equation} |
| 1029 |
< |
\Theta = \frac{3 k_B T}{16 \pi \eta a^3} \left\{ \frac{(2 - \rho^2) |
| 1030 |
< |
G(\rho) - 1}{1 - \rho^4} \right\}. |
| 1029 |
> |
\Theta = \frac{3 k_B T}{16 \pi \eta a^3} \left\{ \frac{(2 - s^2) |
| 1030 |
> |
G(s) - 1}{1 - s^4} \right\}. |
| 1031 |
|
\label{ThetaPerrin} |
| 1032 |
|
\end{equation} |
| 1033 |
< |
In these expressions, $G(\rho)$ is a function of the axial ratio |
| 1034 |
< |
($\rho = b / a$), which for prolate ellipsoids, is |
| 1033 |
> |
In these expressions, $G(s)$ is a function of the axial ratio |
| 1034 |
> |
($s = b / a$), which for prolate ellipsoids, is |
| 1035 |
|
\begin{equation} |
| 1036 |
< |
G(\rho) = (1- \rho^2)^{-1/2} \ln \left\{ \frac{1 + (1 - |
| 892 |
< |
\rho^2)^{1/2}}{\rho} \right\} |
| 1036 |
> |
G(s) = (1- s^2)^{-1/2} \ln \left\{ \frac{1 + (1 - s^2)^{1/2}}{s} \right\} |
| 1037 |
|
\label{GPerrin} |
| 1038 |
|
\end{equation} |
| 1039 |
|
Again, there is some uncertainty about the correct boundary conditions |
| 1058 |
|
exact treatment of the diffusion tensor as well as the rough-shell |
| 1059 |
|
model for the ellipsoid. |
| 1060 |
|
|
| 1061 |
< |
The translational diffusion constants from the microcanonical simulations |
| 1062 |
< |
agree well with the predictions of the Perrin model, although the rotational |
| 1063 |
< |
correlation times are a factor of 2 shorter than expected from hydrodynamic |
| 1064 |
< |
theory. One explanation for the slower rotation |
| 1065 |
< |
of explicitly-solvated ellipsoids is the possibility that solute-solvent |
| 1066 |
< |
collisions happen at both ends of the solute whenever the principal |
| 1067 |
< |
axis of the ellipsoid is turning. In the upper portion of figure |
| 1068 |
< |
\ref{fig:explanation} we sketch a physical picture of this explanation. |
| 1069 |
< |
Since our Langevin integrator is providing nearly quantitative agreement with |
| 1070 |
< |
the Perrin model, it also predicts orientational diffusion for ellipsoids that |
| 1071 |
< |
exceed explicitly solvated correlation times by a factor of two. |
| 1061 |
> |
The translational diffusion constants from the microcanonical |
| 1062 |
> |
simulations agree well with the predictions of the Perrin model, |
| 1063 |
> |
although the {\it rotational} correlation times are a factor of 2 |
| 1064 |
> |
shorter than expected from hydrodynamic theory. One explanation for |
| 1065 |
> |
the slower rotation of explicitly-solvated ellipsoids is the |
| 1066 |
> |
possibility that solute-solvent collisions happen at both ends of the |
| 1067 |
> |
solute whenever the principal axis of the ellipsoid is turning. In the |
| 1068 |
> |
upper portion of figure \ref{fig:explanation} we sketch a physical |
| 1069 |
> |
picture of this explanation. Since our Langevin integrator is |
| 1070 |
> |
providing nearly quantitative agreement with the Perrin model, it also |
| 1071 |
> |
predicts orientational diffusion for ellipsoids that exceed explicitly |
| 1072 |
> |
solvated correlation times by a factor of two. |
| 1073 |
|
|
| 1074 |
|
\subsection{Rigid dumbbells} |
| 1075 |
|
Perhaps the only {\it composite} rigid body for which analytic |
| 1077 |
|
two-sphere dumbbell model. This model consists of two non-overlapping |
| 1078 |
|
spheres held by a rigid bond connecting their centers. There are |
| 1079 |
|
competing expressions for the 6x6 resistance tensor for this |
| 1080 |
< |
model. Equation (\ref{introEquation:oseenTensor}) above gives the |
| 1081 |
< |
original Oseen tensor, while the second order expression introduced by |
| 1082 |
< |
Rotne and Prager,\cite{Rotne1969} and improved by Garc\'{i}a de la |
| 938 |
< |
Torre and Bloomfield,\cite{Torre1977} is given above as |
| 1080 |
> |
model. The second order expression introduced by Rotne and |
| 1081 |
> |
Prager,\cite{Rotne1969} and improved by Garc\'{i}a de la Torre and |
| 1082 |
> |
Bloomfield,\cite{Torre1977} is given above as |
| 1083 |
|
Eq. (\ref{introEquation:RPTensorNonOverlapped}). In our case, we use |
| 1084 |
|
a model dumbbell in which the two spheres are identical Lennard-Jones |
| 1085 |
|
particles ($\sigma$ = 6.5 \AA\ , $\epsilon$ = 0.8 kcal / mol) held at |
| 1092 |
|
motion in a flow {\it perpendicular} to the inter-sphere |
| 1093 |
|
axis.\cite{Davis:1969uq} We know of no analytic solutions for the {\it |
| 1094 |
|
orientational} correlation times for this model system (other than |
| 1095 |
< |
those derived from the 6 x 6 tensors mentioned above). |
| 1095 |
> |
those derived from the 6 x 6 tensor mentioned above). |
| 1096 |
|
|
| 1097 |
|
The bead model for this model system comprises the two large spheres |
| 1098 |
|
by themselves, while the rough shell approximation used 3368 separate |
| 1107 |
|
\begin{figure} |
| 1108 |
|
\centering |
| 1109 |
|
\includegraphics[width=2in]{RoughShell} |
| 1110 |
< |
\caption[Model rigid bodies and their rough shell approximations]{The |
| 1111 |
< |
model rigid bodies (left column) used to test this algorithm and their |
| 1112 |
< |
rough-shell approximations (right-column) that were used to compute |
| 1113 |
< |
the hydrodynamic tensors. The top two models (ellipsoid and dumbbell) |
| 1114 |
< |
have analytic solutions and were used to test the rough shell |
| 1115 |
< |
approximation. The lower two models (banana and lipid) were compared |
| 1116 |
< |
with explicitly-solvated molecular dynamics simulations. } |
| 1110 |
> |
\caption[The model rigid bodies (left column) used to test this |
| 1111 |
> |
algorithm and their rough-shell approximations (right-column) that |
| 1112 |
> |
were used to compute the hydrodynamic tensors. The top two models |
| 1113 |
> |
(ellipsoid and dumbbell) have analytic solutions and were used to test |
| 1114 |
> |
the rough shell approximation. The lower two models (banana and |
| 1115 |
> |
lipid) were compared with explicitly-solvated molecular dynamics |
| 1116 |
> |
simulations]{} |
| 1117 |
|
\label{fig:roughShell} |
| 1118 |
|
\end{figure} |
| 1119 |
|
|
| 1145 |
|
\centering |
| 1146 |
|
\includegraphics[width=6in]{explanation} |
| 1147 |
|
\caption[Explanations of the differences between orientational |
| 1004 |
– |
correlation times for explicitly-solvated models and hydrodynamics |
| 1005 |
– |
predictions]{Explanations of the differences between orientational |
| 1148 |
|
correlation times for explicitly-solvated models and hydrodynamic |
| 1149 |
< |
predictions. For the ellipsoids (upper figures), rotation of the |
| 1149 |
> |
predictions. For the ellipsoids (upper figures), rotation of the |
| 1150 |
|
principal axis can involve correlated collisions at both sides of the |
| 1151 |
|
solute. In the rigid dumbbell model (lower figures), the large size |
| 1152 |
|
of the explicit solvent spheres prevents them from coming in contact |
| 1154 |
|
Therefore, the explicit solvent only provides drag over a |
| 1155 |
|
substantially reduced surface area of this model, where the |
| 1156 |
|
hydrodynamic theories utilize the entire surface area for estimating |
| 1157 |
< |
rotational diffusion. |
| 1016 |
< |
} \label{fig:explanation} |
| 1157 |
> |
rotational diffusion]{} \label{fig:explanation} |
| 1158 |
|
\end{figure} |
| 1159 |
|
|
| 1019 |
– |
|
| 1020 |
– |
|
| 1160 |
|
\subsection{Composite banana-shaped molecules} |
| 1161 |
|
Banana-shaped rigid bodies composed of three Gay-Berne ellipsoids have |
| 1162 |
|
been used by Orlandi {\it et al.} to observe mesophases in |
| 1168 |
|
behavior of this model, we have left out the dipolar interactions of |
| 1169 |
|
the original Orlandi model. |
| 1170 |
|
|
| 1171 |
< |
A reference system composed of a single banana rigid body embedded in a |
| 1172 |
< |
sea of 1929 solvent particles was created and run under standard |
| 1173 |
< |
(microcanonical) molecular dynamics. The resulting viscosity of this |
| 1174 |
< |
mixture was 0.298 centipoise (as estimated using Eq. (\ref{eq:shear})). |
| 1175 |
< |
To calculate the hydrodynamic properties of the banana rigid body model, |
| 1176 |
< |
we created a rough shell (see Fig.~\ref{fig:roughShell}), in which |
| 1177 |
< |
the banana is represented as a ``shell'' made of 3321 identical beads |
| 1178 |
< |
(0.25 \AA\ in diameter) distributed on the surface. Applying the |
| 1179 |
< |
procedure described in Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
| 1180 |
< |
identified the center of resistance, ${\bf r} = $(0 \AA, 0.81 \AA, 0 \AA), as |
| 1181 |
< |
well as the resistance tensor, |
| 1182 |
< |
\begin{equation*} |
| 1044 |
< |
\Xi = |
| 1045 |
< |
\left( {\begin{array}{*{20}c} |
| 1046 |
< |
0.9261 & 0 & 0&0&0.08585&0.2057\\ |
| 1047 |
< |
0& 0.9270&-0.007063& 0.08585&0&0\\ |
| 1048 |
< |
0&-0.007063&0.7494&0.2057&0&0\\ |
| 1049 |
< |
0&0.0858&0.2057& 58.64& 0&0\\0.08585&0&0&0&48.30&3.219&\\0.2057&0&0&0&3.219&10.7373\\\end{array}} \right), |
| 1050 |
< |
\end{equation*} |
| 1051 |
< |
where the units for translational, translation-rotation coupling and |
| 1052 |
< |
rotational tensors are (kcal fs / mol \AA$^2$), (kcal fs / mol \AA\ rad), |
| 1053 |
< |
and (kcal fs / mol rad$^2$), respectively. |
| 1171 |
> |
A reference system composed of a single banana rigid body embedded in |
| 1172 |
> |
a sea of 1929 solvent particles was created and run under standard |
| 1173 |
> |
(microcanonical) molecular dynamics. The resulting viscosity of this |
| 1174 |
> |
mixture was 0.298 centipoise (as estimated using |
| 1175 |
> |
Eq. (\ref{eq:shear})). To calculate the hydrodynamic properties of |
| 1176 |
> |
the banana rigid body model, we created a rough shell (see |
| 1177 |
> |
Fig.~\ref{fig:roughShell}), in which the banana is represented as a |
| 1178 |
> |
``shell'' made of 3321 identical beads (0.25 \AA\ in diameter) |
| 1179 |
> |
distributed on the surface. Applying the procedure described in |
| 1180 |
> |
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
| 1181 |
> |
identified the center of resistance, ${\bf r} = $(0 \AA, 0.81 \AA, 0 |
| 1182 |
> |
\AA). |
| 1183 |
|
|
| 1184 |
< |
The Langevin rigid-body integrator (and the hydrodynamic diffusion tensor) |
| 1185 |
< |
are essentially quantitative for translational diffusion of this model. |
| 1186 |
< |
Orientational correlation times under the Langevin rigid-body integrator |
| 1187 |
< |
are within 11\% of the values obtained from explicit solvent, but these |
| 1188 |
< |
models also exhibit some solvent inaccessible surface area in the |
| 1189 |
< |
explicitly-solvated case. |
| 1184 |
> |
The Langevin rigid-body integrator (and the hydrodynamic diffusion |
| 1185 |
> |
tensor) are essentially quantitative for translational diffusion of |
| 1186 |
> |
this model. Orientational correlation times under the Langevin |
| 1187 |
> |
rigid-body integrator are within 11\% of the values obtained from |
| 1188 |
> |
explicit solvent, but these models also exhibit some solvent |
| 1189 |
> |
inaccessible surface area in the explicitly-solvated case. |
| 1190 |
|
|
| 1191 |
|
\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} |
| 1192 |
|
|
| 1193 |
< |
MORE DETAILS |
| 1193 |
> |
Spherical heads perched on the ends of Gay-Berne ellipsoids have been |
| 1194 |
> |
used recently as models for lipid |
| 1195 |
> |
molecules.\cite{SunX._jp0762020,Ayton01} A reference system composed |
| 1196 |
> |
of a single lipid rigid body embedded in a sea of 1929 solvent |
| 1197 |
> |
particles was created and run under a microcanonical ensemble. The |
| 1198 |
> |
resulting viscosity of this mixture was 0.349 centipoise (as estimated |
| 1199 |
> |
using Eq. (\ref{eq:shear})). To calculate the hydrodynamic properties |
| 1200 |
> |
of the lipid rigid body model, we created a rough shell (see |
| 1201 |
> |
Fig.~\ref{fig:roughShell}), in which the lipid is represented as a |
| 1202 |
> |
``shell'' made of 3550 identical beads (0.25 \AA\ in diameter) |
| 1203 |
> |
distributed on the surface. Applying the procedure described by |
| 1204 |
> |
Eq. (\ref{introEquation:ResistanceTensorArbitraryOrigin}), we |
| 1205 |
> |
identified the center of resistance, ${\bf r} = $(0 \AA, 0 \AA, 1.46 |
| 1206 |
> |
\AA). |
| 1207 |
|
|
| 1208 |
+ |
The translational diffusion constants and rotational correlation times |
| 1209 |
+ |
obtained using the Langevin rigid-body integrator (and the |
| 1210 |
+ |
hydrodynamic tensor) are essentially quantitative when compared with |
| 1211 |
+ |
the explicit solvent simulations for this model system. |
| 1212 |
|
|
| 1213 |
< |
\subsection{Summary} |
| 1214 |
< |
According to our simulations, the langevin dynamics is a reliable |
| 1215 |
< |
theory to apply to replace the explicit solvents, especially for the |
| 1216 |
< |
translation properties. For large molecules, the rotation properties |
| 1217 |
< |
are also mimiced reasonablly well. |
| 1213 |
> |
\subsection{Summary of comparisons with explicit solvent simulations} |
| 1214 |
> |
The Langevin rigid-body integrator we have developed is a reliable way |
| 1215 |
> |
to replace explicit solvent simulations in cases where the detailed |
| 1216 |
> |
solute-solvent interactions do not greatly impact the behavior of the |
| 1217 |
> |
solute. As such, it has the potential to greatly increase the length |
| 1218 |
> |
and time scales of coarse grained simulations of large solvated |
| 1219 |
> |
molecules. In cases where the dielectric screening of the solvent, or |
| 1220 |
> |
specific solute-solvent interactions become important for structural |
| 1221 |
> |
or dynamic features of the solute molecule, this integrator may be |
| 1222 |
> |
less useful. However, for the kinds of coarse-grained modeling that |
| 1223 |
> |
have become popular in recent years (ellipsoidal side chains, rigid |
| 1224 |
> |
bodies, and molecular-scale models), this integrator may prove itself |
| 1225 |
> |
to be quite valuable. |
| 1226 |
|
|
| 1227 |
+ |
\begin{figure} |
| 1228 |
+ |
\centering |
| 1229 |
+ |
\includegraphics[width=\linewidth]{graph} |
| 1230 |
+ |
\caption[The mean-squared displacements ($\langle r^2(t) \rangle$) and |
| 1231 |
+ |
orientational correlation functions ($C_2(t)$) for each of the model |
| 1232 |
+ |
rigid bodies studied. The circles are the results for microcanonical |
| 1233 |
+ |
simulations with explicit solvent molecules, while the other data sets |
| 1234 |
+ |
are results for Langevin dynamics using the different hydrodynamic |
| 1235 |
+ |
tensor approximations. The Perrin model for the ellipsoids is |
| 1236 |
+ |
considered the ``exact'' hydrodynamic behavior (this can also be said |
| 1237 |
+ |
for the translational motion of the dumbbell operating under the bead |
| 1238 |
+ |
model). In most cases, the various hydrodynamics models reproduce each |
| 1239 |
+ |
other quantitatively]{} |
| 1240 |
+ |
\label{fig:results} |
| 1241 |
+ |
\end{figure} |
| 1242 |
+ |
|
| 1243 |
|
\begin{table*} |
| 1244 |
|
\begin{minipage}{\linewidth} |
| 1245 |
|
\begin{center} |
| 1246 |
|
\caption{Translational diffusion constants (D) for the model systems |
| 1247 |
|
calculated using microcanonical simulations (with explicit solvent), |
| 1248 |
|
theoretical predictions, and Langevin simulations (with implicit solvent). |
| 1249 |
< |
Analytical solutions for the exactly-solved hydrodynamics models are |
| 1250 |
< |
from Refs. \citen{Einstein05} (sphere), \citen{Perrin1934} and \citen{Perrin1936} |
| 1249 |
> |
Analytical solutions for the exactly-solved hydrodynamics models are obtained |
| 1250 |
> |
from: Stokes' law (sphere), and Refs. \citen{Perrin1934} and \citen{Perrin1936} |
| 1251 |
|
(ellipsoid), \citen{Stimson:1926qy} and \citen{Davis:1969uq} |
| 1252 |
|
(dumbbell). The other model systems have no known analytic solution. |
| 1253 |
< |
All diffusion constants are reported in units of $10^{-3}$ cm$^2$ / ps (= |
| 1253 |
> |
All diffusion constants are reported in units of $10^{-3}$ cm$^2$ / ps (= |
| 1254 |
|
$10^{-4}$ \AA$^2$ / fs). } |
| 1255 |
|
\begin{tabular}{lccccccc} |
| 1256 |
|
\hline |
| 1258 |
|
\cline{2-3} \cline{5-7} |
| 1259 |
|
model & $\eta$ (centipoise) & D & & Analytical & method & Hydrodynamics & simulation \\ |
| 1260 |
|
\hline |
| 1261 |
< |
sphere & 0.261 & ? & & 2.59 & exact & 2.59 & 2.56 \\ |
| 1261 |
> |
sphere & 0.279 & 3.06 & & 2.42 & exact & 2.42 & 2.33 \\ |
| 1262 |
|
ellipsoid & 0.255 & 2.44 & & 2.34 & exact & 2.34 & 2.37 \\ |
| 1263 |
|
& 0.255 & 2.44 & & 2.34 & rough shell & 2.36 & 2.28 \\ |
| 1264 |
< |
dumbbell & 0.322 & ? & & 1.57 & bead model & 1.57 & 1.57 \\ |
| 1265 |
< |
& 0.322 & ? & & 1.57 & rough shell & ? & ? \\ |
| 1264 |
> |
dumbbell & 0.308 & 2.06 & & 1.64 & bead model & 1.65 & 1.62 \\ |
| 1265 |
> |
& 0.308 & 2.06 & & 1.64 & rough shell & 1.59 & 1.62 \\ |
| 1266 |
|
banana & 0.298 & 1.53 & & & rough shell & 1.56 & 1.55 \\ |
| 1267 |
< |
lipid & 0.349 & 0.96 & & & rough shell & 1.33 & 1.32 \\ |
| 1267 |
> |
lipid & 0.349 & 1.41 & & & rough shell & 1.33 & 1.32 \\ |
| 1268 |
|
\end{tabular} |
| 1269 |
|
\label{tab:translation} |
| 1270 |
|
\end{center} |
| 1287 |
|
\cline{2-3} \cline{5-7} |
| 1288 |
|
model & $\eta$ (centipoise) & $\tau$ & & Perrin & method & Hydrodynamic & simulation \\ |
| 1289 |
|
\hline |
| 1290 |
< |
sphere & 0.261 & & & 9.06 & exact & 9.06 & 9.11 \\ |
| 1290 |
> |
sphere & 0.279 & & & 9.69 & exact & 9.69 & 9.64 \\ |
| 1291 |
|
ellipsoid & 0.255 & 46.7 & & 22.0 & exact & 22.0 & 22.2 \\ |
| 1292 |
|
& 0.255 & 46.7 & & 22.0 & rough shell & 22.6 & 22.2 \\ |
| 1293 |
< |
dumbbell & 0.322 & 14.0 & & & bead model & 52.3 & 52.8 \\ |
| 1294 |
< |
& 0.322 & 14.0 & & & rough shell & ? & ? \\ |
| 1293 |
> |
dumbbell & 0.308 & 14.1 & & & bead model & 50.0 & 50.1 \\ |
| 1294 |
> |
& 0.308 & 14.1 & & & rough shell & 41.5 & 41.3 \\ |
| 1295 |
|
banana & 0.298 & 63.8 & & & rough shell & 70.9 & 70.9 \\ |
| 1296 |
|
lipid & 0.349 & 78.0 & & & rough shell & 76.9 & 77.9 \\ |
| 1297 |
|
\hline |
| 1303 |
|
|
| 1304 |
|
\section{Application: A rigid-body lipid bilayer} |
| 1305 |
|
|
| 1306 |
< |
The Langevin dynamics integrator was applied to study the formation of |
| 1307 |
< |
corrugated structures emerging from simulations of the coarse grained |
| 1308 |
< |
lipid molecular models presented above. The initial configuration is |
| 1309 |
< |
taken from our molecular dynamics studies on lipid bilayers with |
| 1310 |
< |
lennard-Jones sphere solvents. The solvent molecules were excluded |
| 1311 |
< |
from the system, and the experimental value for the viscosity of water |
| 1312 |
< |
at 20C ($\eta = 1.00$ cp) was used to mimic the hydrodynamic effects |
| 1313 |
< |
of the solvent. The absence of explicit solvent molecules and the |
| 1314 |
< |
stability of the integrator allowed us to take timesteps of 50 fs. A |
| 1315 |
< |
total simulation run time of 100 ns was sampled. |
| 1316 |
< |
Fig. \ref{fig:bilayer} shows the configuration of the system after 100 |
| 1317 |
< |
ns, and the ripple structure remains stable during the entire |
| 1318 |
< |
trajectory. Compared with using explicit bead-model solvent |
| 1319 |
< |
molecules, the efficiency of the simulation has increased by an order |
| 1320 |
< |
of magnitude. |
| 1306 |
> |
To test the accuracy and efficiency of the new integrator, we applied |
| 1307 |
> |
it to study the formation of corrugated structures emerging from |
| 1308 |
> |
simulations of the coarse grained lipid molecular models presented |
| 1309 |
> |
above. The initial configuration is taken from earlier molecular |
| 1310 |
> |
dynamics studies on lipid bilayers which had used spherical |
| 1311 |
> |
(Lennard-Jones) solvent particles and moderate (480 solvated lipid |
| 1312 |
> |
molecules) system sizes.\cite{SunX._jp0762020} the solvent molecules |
| 1313 |
> |
were excluded from the system and the box was replicated three times |
| 1314 |
> |
in the x- and y- axes (giving a single simulation cell containing 4320 |
| 1315 |
> |
lipids). The experimental value for the viscosity of water at 20C |
| 1316 |
> |
($\eta = 1.00$ cp) was used with the Langevin integrator to mimic the |
| 1317 |
> |
hydrodynamic effects of the solvent. The absence of explicit solvent |
| 1318 |
> |
molecules and the stability of the integrator allowed us to take |
| 1319 |
> |
timesteps of 50 fs. A simulation run time of 30 ns was sampled to |
| 1320 |
> |
calculate structural properties. Fig. \ref{fig:bilayer} shows the |
| 1321 |
> |
configuration of the system after 30 ns. Structural properties of the |
| 1322 |
> |
bilayer (e.g. the head and body $P_2$ order parameters) are nearly |
| 1323 |
> |
identical to those obtained via solvated molecular dynamics. The |
| 1324 |
> |
ripple structure remained stable during the entire trajectory. |
| 1325 |
> |
Compared with using explicit bead-model solvent molecules, the 30 ns |
| 1326 |
> |
trajectory for 4320 lipids with the Langevin integrator is now {\it |
| 1327 |
> |
faster} on the same hardware than the same length trajectory was for |
| 1328 |
> |
the 480-lipid system previously studied. |
| 1329 |
|
|
| 1330 |
|
\begin{figure} |
| 1331 |
|
\centering |
| 1332 |
|
\includegraphics[width=\linewidth]{bilayer} |
| 1333 |
< |
\caption[Snapshot of a bilayer of rigid-body models for lipids]{A |
| 1334 |
< |
snapshot of a bilayer composed of rigid-body models for lipid |
| 1335 |
< |
molecules evolving using the Langevin integrator described in this |
| 1160 |
< |
work.} \label{fig:bilayer} |
| 1333 |
> |
\caption[A snapshot of a bilayer composed of 4320 rigid-body models |
| 1334 |
> |
for lipid molecules evolving using the Langevin integrator described |
| 1335 |
> |
in this work]{} \label{fig:bilayer} |
| 1336 |
|
\end{figure} |
| 1337 |
|
|
| 1338 |
|
\section{Conclusions} |
| 1339 |
|
|
| 1340 |
< |
We have presented a new Langevin algorithm by incorporating the |
| 1341 |
< |
hydrodynamics properties of arbitrary shaped molecules into an |
| 1342 |
< |
advanced symplectic integration scheme. Further studies in systems |
| 1343 |
< |
involving banana shaped molecules illustrated that the dynamic |
| 1344 |
< |
properties could be preserved by using this new algorithm as an |
| 1345 |
< |
implicit solvent model. |
| 1340 |
> |
We have presented a new algorithm for carrying out Langevin dynamics |
| 1341 |
> |
simulations on complex rigid bodies by incorporating the hydrodynamic |
| 1342 |
> |
resistance tensors for arbitrary shapes into a stable and efficient |
| 1343 |
> |
integration scheme. The integrator gives quantitative agreement with |
| 1344 |
> |
both analytic and approximate hydrodynamic theories, and works |
| 1345 |
> |
reasonably well at reproducing the solute dynamical properties |
| 1346 |
> |
(diffusion constants, and orientational relaxation times) from |
| 1347 |
> |
explicitly-solvated simulations. For the cases where there are |
| 1348 |
> |
discrepancies between our Langevin integrator and the explicit solvent |
| 1349 |
> |
simulations, two features of molecular simulations help explain the |
| 1350 |
> |
differences. |
| 1351 |
|
|
| 1352 |
+ |
First, the use of ``stick'' boundary conditions for molecular-sized |
| 1353 |
+ |
solutes in a sea of similarly-sized solvent particles may be |
| 1354 |
+ |
problematic. We are certainly not the first group to notice this |
| 1355 |
+ |
difference between hydrodynamic theories and explicitly-solvated |
| 1356 |
+ |
molecular |
| 1357 |
+ |
simulations.\cite{Schmidt:2004fj,Schmidt:2003kx,Ravichandran:1999fk,TANG:1993lr} |
| 1358 |
+ |
The problem becomes particularly noticable in both the translational |
| 1359 |
+ |
diffusion of the spherical particles and the rotational diffusion of |
| 1360 |
+ |
the ellipsoids. In both of these cases it is clear that the |
| 1361 |
+ |
approximations that go into hydrodynamics are the source of the error, |
| 1362 |
+ |
and not the integrator itself. |
| 1363 |
|
|
| 1364 |
+ |
Second, in the case of structures which have substantial surface area |
| 1365 |
+ |
that is inaccessible to solvent particles, the hydrodynamic theories |
| 1366 |
+ |
(and the Langevin integrator) may overestimate the effects of solvent |
| 1367 |
+ |
friction because they overestimate the exposed surface area of the |
| 1368 |
+ |
rigid body. This is particularly noticable in the rotational |
| 1369 |
+ |
diffusion of the dumbbell model. We believe that given a solvent of |
| 1370 |
+ |
known radius, it may be possible to modify the rough shell approach to |
| 1371 |
+ |
place beads on solvent-accessible surface, instead of on the geometric |
| 1372 |
+ |
surface defined by the van der Waals radii of the components of the |
| 1373 |
+ |
rigid body. Further work to confirm the behavior of this new |
| 1374 |
+ |
approximation is ongoing. |
| 1375 |
+ |
|
| 1376 |
|
\section{Acknowledgments} |
| 1377 |
|
Support for this project was provided by the National Science |
| 1378 |
|
Foundation under grant CHE-0134881. T.L. also acknowledges the |
| 1379 |
< |
financial support from center of applied mathematics at University |
| 1380 |
< |
of Notre Dame. |
| 1379 |
> |
financial support from Center of Applied Mathematics at University of |
| 1380 |
> |
Notre Dame. |
| 1381 |
> |
|
| 1382 |
> |
\end{doublespace} |
| 1383 |
|
\newpage |
| 1384 |
|
|
| 1385 |
< |
\bibliographystyle{jcp} |
| 1385 |
> |
\bibliographystyle{jcp2} |
| 1386 |
|
\bibliography{langevin} |
| 1182 |
– |
|
| 1387 |
|
\end{document} |
