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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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\author{Teng Lin, Xiuquan Sun and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: |
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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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University of Notre Dame\\ |
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\section{Introduction} |
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|
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%applications of langevin dynamics |
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As alternative to Newtonian dynamics, Langevin dynamics, which |
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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 |
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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 |
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free-energy barriers by studying the viscosity dependence of the |
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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 |
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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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Langevin dynamics, which mimics a simple heat bath with stochastic and |
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dissipative forces, has been applied in a variety of situations as an |
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alternative to molecular dynamics with explicit solvent molecules. |
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The stochastic treatment of the solvent allows the use of simulations |
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with substantially longer time and length scales. In general, the |
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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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Recent examples of the usefulness of Langevin simulations include a |
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study of met-enkephalin in which Langevin simulations predicted |
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dynamical properties that were largely in agreement with explicit |
| 63 |
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solvent simulations.\cite{Shen2002} By applying Langevin dynamics with |
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the UNRES model, Liow and his coworkers suggest that protein folding |
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pathways can be explored within a reasonable amount of |
| 66 |
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time.\cite{Liwo2005} |
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|
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The stochastic nature of Langevin dynamics also enhances the sampling |
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of the system and increases the probability of crossing energy |
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barriers.\cite{Cui2003,Banerjee2004} Combining Langevin dynamics with |
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Kramers's theory, Klimov and Thirumalai identified free-energy |
| 72 |
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barriers by studying the viscosity dependence of the protein folding |
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rates.\cite{Klimov1997} In order to account for solvent induced |
| 74 |
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interactions missing from the implicit solvent model, Kaya |
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incorporated a desolvation free energy barrier into protein |
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folding/unfolding studies and discovered a higher free energy barrier |
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between the native and denatured states.\cite{XXX} |
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|
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Because of its stability against noise, Langevin dynamics has also |
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proven useful for studying remagnetization processes in various |
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systems.\cite{Palacios1998,Berkov2002,Denisov2003} [Check: 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 has the same peak frequencies for different wave |
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vectors, which recovers the property of magnetic excitations in small |
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finite structures.\cite{Berkov2005a}] |
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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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In typical LD simulations, the friction and random forces on |
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individual atoms are taken from the Stokes-Einstein hydrodynamic |
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approximation, |
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\begin{eqnarray} |
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m \dot{v}(t) & = & -\nabla U(x) - \xi m v(t) + R(t) \\ |
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\langle R(t) \rangle & = & 0 \\ |
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\langle R(t) R(t') \rangle & = & 2 k_B T \xi m \delta(t - t') |
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\end{eqnarray} |
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where $\xi \approx 6 \pi \eta a$. Here $\eta$ is the viscosity of the |
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implicit solvent, and $a$ is the hydrodynamic radius of the atom. |
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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. |
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Although an alternative integrator using multiple sets of Euler |
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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} |
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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 |
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deriving from potential energy and constraint forces which are used |
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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 use of rigid substructures,\cite{???} |
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coarse-graining,\cite{Ayton,Sun,Zannoni} and ellipsoidal |
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representations of protein side chains~\cite{Schulten} has made the |
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use of the Stokes-Einstein approximation problematic. A rigid |
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substructure moves as a single unit with orientational as well as |
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translational degrees of freedom. This requires a more general |
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treatment of the hydrodynamics than the spherical approximation |
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provides. The atoms involved in a rigid or coarse-grained structure |
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should properly have solvent-mediated interactions with each |
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other. The theory of interactions {\it between} bodies moving through |
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a fluid has been developed over the past century and has been applied |
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to simulations of Brownian |
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motion.\cite{MarshallNewton,GarciaDeLaTorre} There a need to have a |
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more thorough treatment of hydrodynamics included in the library of |
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methods available for performing Langevin simulations. |
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|
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A break-through in geometric literature suggests that, in order to |
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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 simulations.\cite{Sun2008} |
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Many of the water models in common use are also rigid-body |
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models,\cite{TIPs,SPC/E} although they are typically evolved using |
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constraints 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 |
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degrees of freedom. However, due to $1 \over \sin \theta$ |
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singularities, the numerical integration of corresponding equations of |
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these motion can become inaccurate (and inefficient). Although an |
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alternative integrator using multiple sets of Euler angles can |
| 128 |
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overcome this problem,\cite{Barojas1973} the computational penalty and |
| 129 |
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the loss of angular momentum conservation remain. A singularity-free |
| 130 |
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representation utilizing quaternions was developed by Evans in |
| 131 |
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1977.\cite{Evans1977} Unfortunately, this approach uses a nonseparable |
| 132 |
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Hamiltonian resulting from the quaternion representation, which |
| 133 |
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prevented symplectic algorithms from being utilized until very |
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recently.\cite{Miller2002} Another approach is the application of |
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holonomic constraints to the atoms belonging to the rigid body. Each |
| 136 |
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atom moves independently under the normal forces deriving from |
| 137 |
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potential energy and constraint forces which are used to guarantee the |
| 138 |
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rigidness. However, due to their iterative nature, the SHAKE and |
| 139 |
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Rattle algorithms also converge very slowly when the number of |
| 140 |
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constraints increases.\cite{Ryckaert1977,Andersen1983} |
| 141 |
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|
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A breakthrough 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 |
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proposed to evolve the Hamiltonian system in a constraint manifold |
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by iteratively satisfying the orthogonality constraint $Q^T Q = 1$. |
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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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equation, a symplectic integrator, RSHAKE,\cite{Kol1997} was proposed |
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to evolve the Hamiltonian system in a constraint manifold by |
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iteratively satisfying the orthogonality constraint $Q^T Q = 1$. An |
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alternative method using the quaternion representation was developed |
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by Omelyan.\cite{Omelyan1998} However, both of these methods are |
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iterative and suffer from some related inefficiencies. A symplectic |
| 152 |
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Lie-Poisson integrator for rigid bodies developed by Dullweber {\it et |
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al.}\cite{Dullweber1997} gets around most of the limitations mentioned |
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above and has become the basis for our Langevin integrator. |
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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 |
| 161 |
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discovered rapid superhelical stress generations from the stochastic |
| 162 |
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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 |
| 167 |
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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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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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|
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\subsection{The Hydrodynamic tensor and Brownian dynamics} |
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Combining Brownian dynamics with rigid substructures, one can study |
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slow processes in biomolecular systems. Modeling DNA as a chain of |
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beads which are subject to harmonic potentials as well as excluded |
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volume potentials, Mielke and his coworkers discovered rapid |
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superhelical stress generations from the stochastic simulation of twin |
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supercoiling DNA with response to induced torques.\cite{Mielke2004} |
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Membrane fusion is another key biological process which controls a |
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variety of physiological functions, such as release of |
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neurotransmitters \textit{etc}. A typical fusion event happens on the |
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time scale of a millisecond, which is impractical to study using |
| 168 |
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atomistic models with newtonian mechanics. With the help of |
| 169 |
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coarse-grained rigid body model and stochastic dynamics, the fusion |
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pathways were explored by Noguchi and others.\cite{Noguchi2001,Noguchi2002,Shillcock2005} |
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|
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Due to the difficulty of numerically integrating anisotropic |
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rotational motion, most of the coarse-grained rigid body models are |
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treated as spheres, cylinders, ellipsoids or other regular shapes in |
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stochastic simulations. In an effort to account for the diffusion |
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anisotropy of arbitrarily-shaped particles, Fernandes and Garc\'{i}a |
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de la Torre improved the original Brownian dynamics simulation |
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algorithm~\cite{Ermak1978,Allison1991} by incorporating a generalized |
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$6\times6$ diffusion tensor and introducing a rotational evolution |
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scheme consisting of three consecutive rotations.\cite{Fernandes2002} |
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Unfortunately, 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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inertia in Brownian dynamics. However, the assumption of zero average |
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acceleration is not always true for cooperative motion which is common |
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in proteins. An inertial Brownian dynamics (IBD) was proposed to |
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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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separating the translation and rotation motion and taking advantage of |
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the analytical solution of hydrodynamics properties. However, typical |
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nonskew bodies like cylinders and ellipsoids are inadequate to |
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represent most complex macromolecular assemblies. |
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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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accurate estimation of friction tensor from hydrodynamics theory into |
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a symplectic rigid body dynamics propagator. In the sections below, |
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we review some of the theory of hydrodynamic tensors developed for |
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Brownian simulations of rigid multi-particle systems, we then present |
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our integration method for a set of generalized Langevin equations of |
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motion, and we compare the behavior of the new Langevin integrator to |
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dynamical quantities obtained via explicit solvent molecular dynamics. |
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|
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\section{Computational Methods{\label{methodSec}}} |
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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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\subsection{\label{introSection:frictionTensor}The Friction Tensor} |
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Theoretically, a complete friction kernel can be determined using the |
| 212 |
|
velocity autocorrelation function. However, this approach becomes |
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impractical when the system becomes more and more complicated. |
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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 |
| 217 |
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\[ |
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impractical when the solute becomes complex. Instead, various |
| 214 |
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approaches based on hydrodynamics have been developed to calculate the |
| 215 |
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friction coefficients. In general, the friction tensor $\Xi$ is a |
| 216 |
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$6\times 6$ matrix given by |
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\begin{equation} |
| 218 |
|
\Xi = \left( {\begin{array}{*{20}c} |
| 219 |
|
{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
| 220 |
|
{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
| 221 |
|
\end{array}} \right). |
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\] |
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Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are $3 \times 3$ |
| 224 |
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translational friction tensor and rotational resistance (friction) |
| 225 |
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tensor respectively, while ${\Xi^{tr} }$ is translation-rotation |
| 226 |
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coupling tensor and $ {\Xi^{rt} }$ is rotation-translation coupling |
| 227 |
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tensor. When a particle moves in a fluid, it may experience friction |
| 228 |
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force or torque along the opposite direction of the velocity or |
| 229 |
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angular velocity, |
| 230 |
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\[ |
| 222 |
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\end{equation} |
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Here, $\Xi^{tt}$ and $\Xi^{rr}$ are $3 \times 3$ translational and |
| 224 |
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rotational resistance (friction) tensors respectively, while |
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$\Xi^{tr}$ is translation-rotation coupling tensor and $\Xi^{rt}$ is |
| 226 |
> |
rotation-translation coupling tensor. When a particle moves in a |
| 227 |
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fluid, it may experience friction force ($\mathbf{F}_f$) and torque |
| 228 |
> |
($\mathbf{\tau}_f$) in opposition to the directions of the velocity |
| 229 |
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($\mathbf{v}$) and body-fixed angular velocity ($\mathbf{\omega}$), |
| 230 |
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\begin{equation} |
| 231 |
|
\left( \begin{array}{l} |
| 232 |
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F_R \\ |
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< |
\tau _R \\ |
| 232 |
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\mathbf{F}_f \\ |
| 233 |
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\mathbf{\tau}_f \\ |
| 234 |
|
\end{array} \right) = - \left( {\begin{array}{*{20}c} |
| 235 |
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{\Xi ^{tt} } & {\Xi ^{rt} } \\ |
| 236 |
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{\Xi ^{tr} } & {\Xi ^{rr} } \\ |
| 235 |
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\Xi ^{tt} & \Xi ^{rt} \\ |
| 236 |
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\Xi ^{tr} & \Xi ^{rr} \\ |
| 237 |
|
\end{array}} \right)\left( \begin{array}{l} |
| 238 |
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v \\ |
| 239 |
< |
w \\ |
| 240 |
< |
\end{array} \right) |
| 241 |
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\] |
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where $F_r$ is the friction force and $\tau _R$ is the friction |
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torque. |
| 238 |
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\mathbf{v} \\ |
| 239 |
> |
\mathbf{\omega} \\ |
| 240 |
> |
\end{array} \right). |
| 241 |
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\end{equation} |
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|
|
| 243 |
|
\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shapes}} |
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|
| 245 |
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For a spherical particle with slip boundary conditions, the |
| 246 |
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translational and rotational friction constant can be calculated |
| 247 |
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from Stoke's law, |
| 248 |
< |
\[ |
| 217 |
< |
\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
| 244 |
> |
For a spherical particle under ``stick'' boundary conditions, the |
| 245 |
> |
translational and rotational friction tensors can be calculated from |
| 246 |
> |
Stoke's law, |
| 247 |
> |
\begin{equation} |
| 248 |
> |
\Xi^{tt} = \left( {\begin{array}{*{20}c} |
| 249 |
|
{6\pi \eta R} & 0 & 0 \\ |
| 250 |
|
0 & {6\pi \eta R} & 0 \\ |
| 251 |
|
0 & 0 & {6\pi \eta R} \\ |
| 252 |
|
\end{array}} \right) |
| 253 |
< |
\] |
| 253 |
> |
\end{equation} |
| 254 |
|
and |
| 255 |
< |
\[ |
| 255 |
> |
\begin{equation} |
| 256 |
|
\Xi ^{rr} = \left( {\begin{array}{*{20}c} |
| 257 |
|
{8\pi \eta R^3 } & 0 & 0 \\ |
| 258 |
|
0 & {8\pi \eta R^3 } & 0 \\ |
| 259 |
|
0 & 0 & {8\pi \eta R^3 } \\ |
| 260 |
|
\end{array}} \right) |
| 261 |
< |
\] |
| 261 |
> |
\end{equation} |
| 262 |
|
where $\eta$ is the viscosity of the solvent and $R$ is the |
| 263 |
|
hydrodynamic radius. |
| 264 |
|
|
| 265 |
|
Other non-spherical shapes, such as cylinders and ellipsoids, are |
| 266 |
< |
widely used as references for developing new hydrodynamics theory, |
| 266 |
> |
widely used as references for developing new hydrodynamics theories, |
| 267 |
|
because their properties can be calculated exactly. In 1936, Perrin |
| 268 |
|
extended Stokes's law to general ellipsoids, also called a triaxial |
| 269 |
|
ellipsoid, which is given in Cartesian coordinates |
| 270 |
< |
by\cite{Perrin1934, Perrin1936} |
| 271 |
< |
\[ |
| 272 |
< |
\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
| 273 |
< |
}} = 1 |
| 274 |
< |
\] |
| 275 |
< |
where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
| 276 |
< |
due to the complexity of the elliptic integral, only the ellipsoid |
| 277 |
< |
with the restriction of two axes being equal, \textit{i.e.} |
| 278 |
< |
prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
| 279 |
< |
exactly. Introducing an elliptic integral parameter $S$ for prolate |
| 280 |
< |
ellipsoids : |
| 281 |
< |
\[ |
| 251 |
< |
S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
| 252 |
< |
} }}{b}, |
| 253 |
< |
\] |
| 270 |
> |
by\cite{Perrin1934,Perrin1936} |
| 271 |
> |
\begin{equation} |
| 272 |
> |
\frac{x^2 }{a^2} + \frac{y^2}{b^2} + \frac{z^2 }{c^2} = 1 |
| 273 |
> |
\end{equation} |
| 274 |
> |
where the semi-axes are of lengths $a$, $b$, and $c$. Due to the |
| 275 |
> |
complexity of the elliptic integral, only uniaxial ellipsoids, |
| 276 |
> |
{\it i.e.} prolate ($ a \ge b = c$) and oblate ($ a < b = c $), can |
| 277 |
> |
be solved exactly. Introducing an elliptic integral parameter $S$ for |
| 278 |
> |
prolate ellipsoids : |
| 279 |
> |
\begin{equation} |
| 280 |
> |
S = \frac{2}{\sqrt{a^2 - b^2}} \ln \frac{a + \sqrt{a^2 - b^2}}{b}, |
| 281 |
> |
\end{equation} |
| 282 |
|
and oblate ellipsoids: |
| 283 |
< |
\[ |
| 284 |
< |
S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
| 285 |
< |
}}{a}, |
| 258 |
< |
\] |
| 283 |
> |
\begin{equation} |
| 284 |
> |
S = \frac{2}{\sqrt {b^2 - a^2 }} \arctan \frac{\sqrt {b^2 - a^2}}{a}, |
| 285 |
> |
\end{equation} |
| 286 |
|
one can write down the translational and rotational resistance |
| 287 |
< |
tensors |
| 287 |
> |
tensors for oblate, |
| 288 |
|
\begin{eqnarray*} |
| 289 |
< |
\Xi _a^{tt} & = & 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
| 290 |
< |
\Xi _b^{tt} & = & \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
| 264 |
< |
2a}}, |
| 289 |
> |
\Xi_a^{tt} & = & 16\pi \eta \frac{a^2 - b^2}{(2a^2 - b^2 )S - 2a}. \\ |
| 290 |
> |
\Xi_b^{tt} = \Xi_c^{tt} & = & 32\pi \eta \frac{a^2 - b^2 }{(2a^2 - 3b^2 )S + 2a}, |
| 291 |
|
\end{eqnarray*} |
| 292 |
< |
and |
| 292 |
> |
and prolate, |
| 293 |
|
\begin{eqnarray*} |
| 294 |
< |
\Xi _a^{rr} & = & \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
| 295 |
< |
\Xi _b^{rr} & = & \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}}. |
| 294 |
> |
\Xi_a^{rr} & = & \frac{32\pi}{3} \eta \frac{(a^2 - b^2 )b^2}{2a - b^2 S}, \\ |
| 295 |
> |
\Xi_b^{rr} = \Xi_c^{rr} & = & \frac{32\pi}{3} \eta \frac{(a^4 - b^4)}{(2a^2 - b^2 )S - 2a} |
| 296 |
|
\end{eqnarray*} |
| 297 |
+ |
ellipsoids. For both spherical and ellipsoidal particles, the |
| 298 |
+ |
translation-rotation and rotation-translation coupling tensors are |
| 299 |
+ |
zero. |
| 300 |
|
|
| 301 |
|
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}} |
| 302 |
|
|
| 304 |
|
analytical solution for the friction tensor for arbitrarily shaped |
| 305 |
|
rigid molecules. The ellipsoid of revolution model and general |
| 306 |
|
triaxial ellipsoid model have been used to approximate the |
| 307 |
< |
hydrodynamic properties of rigid bodies. However, since the mapping |
| 308 |
< |
from all possible ellipsoidal spaces, $r$-space, to all possible |
| 309 |
< |
combination of rotational diffusion coefficients, $D$-space, is not |
| 310 |
< |
unique\cite{Wegener1979} as well as the intrinsic coupling between |
| 311 |
< |
translational and rotational motion of rigid bodies, general |
| 312 |
< |
ellipsoids are not always suitable for modeling arbitrarily shaped |
| 313 |
< |
rigid molecules. A number of studies have been devoted to |
| 307 |
> |
hydrodynamic properties of rigid bodies. However, the mapping from all |
| 308 |
> |
possible ellipsoidal spaces, $r$-space, to all possible combination of |
| 309 |
> |
rotational diffusion coefficients, $D$-space, is not |
| 310 |
> |
unique.\cite{Wegener1979} Additionally, because there is intrinsic |
| 311 |
> |
coupling between translational and rotational motion of rigid bodies, |
| 312 |
> |
general ellipsoids are not always suitable for modeling arbitrarily |
| 313 |
> |
shaped rigid molecules. A number of studies have been devoted to |
| 314 |
|
determining the friction tensor for irregularly shaped rigid bodies |
| 315 |
< |
using more advanced methods where the molecule of interest was |
| 316 |
< |
modeled by a combinations of spheres\cite{Carrasco1999} and the |
| 317 |
< |
hydrodynamics properties of the molecule can be calculated using the |
| 318 |
< |
hydrodynamic interaction tensor. Let us consider a rigid assembly of |
| 319 |
< |
$N$ beads immersed in a continuous medium. Due to hydrodynamic |
| 320 |
< |
interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different |
| 321 |
< |
than its unperturbed velocity $v_i$, |
| 315 |
> |
using more advanced methods where the molecule of interest was modeled |
| 316 |
> |
by a combinations of spheres\cite{Carrasco1999} and the hydrodynamics |
| 317 |
> |
properties of the molecule can be calculated using the hydrodynamic |
| 318 |
> |
interaction tensor. Let us consider a rigid assembly of $N$ beads |
| 319 |
> |
immersed in a continuous medium. Due to hydrodynamic interaction, the |
| 320 |
> |
``net'' velocity of $i$th bead, $v'_i$ is different than its |
| 321 |
> |
unperturbed velocity $v_i$, |
| 322 |
|
\[ |
| 323 |
|
v'_i = v_i - \sum\limits_{j \ne i} {T_{ij} F_j } |
| 324 |
|
\] |
| 396 |
|
\begin{eqnarray} |
| 397 |
|
\Xi _{}^{tt} & = & \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\ |
| 398 |
|
\Xi _{}^{tr} & = & \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 399 |
< |
\Xi _{}^{rr} & = & - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\ |
| 399 |
> |
\Xi _{}^{rr} & = & - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } |
| 400 |
> |
U_j + 6 \eta V {\bf I}. \notag |
| 401 |
|
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 402 |
|
\end{eqnarray} |
| 403 |
+ |
The final term in the expression for $\Xi^{rr}$ is correction that |
| 404 |
+ |
accounts for errors in the rotational motion of certain kinds of bead |
| 405 |
+ |
models. The additive correction uses the solvent viscosity ($\eta$) |
| 406 |
+ |
as well as the total volume of the beads that contribute to the |
| 407 |
+ |
hydrodynamic model, |
| 408 |
+ |
\begin{equation} |
| 409 |
+ |
V = \frac{4 \pi}{3} \sum_{i=1}^{N} \sigma_i^3, |
| 410 |
+ |
\end{equation} |
| 411 |
+ |
where $\sigma_i$ is the radius of bead $i$. This correction term was |
| 412 |
+ |
rigorously tested and compared with the analytical results for |
| 413 |
+ |
two-sphere and ellipsoidal systems by Garcia de la Torre and |
| 414 |
+ |
Rodes.\cite{Torre:1983lr} |
| 415 |
+ |
|
| 416 |
+ |
|
| 417 |
|
The resistance tensor depends on the origin to which they refer. The |
| 418 |
|
proper location for applying the friction force is the center of |
| 419 |
|
resistance (or center of reaction), at which the trace of rotational |
| 470 |
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
| 471 |
|
joining center of resistance $R$ and origin $O$. |
| 472 |
|
|
| 429 |
– |
\subsection{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}} |
| 473 |
|
|
| 474 |
+ |
\section{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}} |
| 475 |
|
Consider the Langevin equations of motion in generalized coordinates |
| 476 |
|
\begin{equation} |
| 477 |
|
M_i \dot V_i (t) = F_{s,i} (t) + F_{f,i(t)} + F_{r,i} (t) |
| 618 |
|
+ \frac{h}{2} {\bf \tau}^b(t + h) . |
| 619 |
|
\end{align*} |
| 620 |
|
|
| 621 |
< |
\section{Results and Discussion} |
| 621 |
> |
\section{Validating the Method\label{sec:validating}} |
| 622 |
> |
In order to validate our Langevin integrator for arbitrarily-shaped |
| 623 |
> |
rigid bodies, we implemented the algorithm in {\sc |
| 624 |
> |
oopse}\cite{Meineke2005} and compared the results of this algorithm |
| 625 |
> |
with the known |
| 626 |
> |
hydrodynamic limiting behavior for a few model systems, and to |
| 627 |
> |
microcanonical molecular dynamics simulations for some more |
| 628 |
> |
complicated bodies. The model systems and their analytical behavior |
| 629 |
> |
(if known) are summarized below. Parameters for the primary particles |
| 630 |
> |
comprising our model systems are given in table \ref{tab:parameters}, |
| 631 |
> |
and a sketch of the arrangement of these primary particles into the |
| 632 |
> |
model rigid bodies is shown in figure \ref{fig:models}. In table |
| 633 |
> |
\ref{tab:parameters}, $d$ and $l$ are the physical dimensions of |
| 634 |
> |
ellipsoidal (Gay-Berne) particles. For spherical particles, the value |
| 635 |
> |
of the Lennard-Jones $\sigma$ parameter is the particle diameter |
| 636 |
> |
($d$). Gay-Berne ellipsoids have an energy scaling parameter, |
| 637 |
> |
$\epsilon^s$, which describes the well depth for two identical |
| 638 |
> |
ellipsoids in a {\it side-by-side} configuration. Additionally, a |
| 639 |
> |
well depth aspect ratio, $\epsilon^r = \epsilon^e / \epsilon^s$, |
| 640 |
> |
describes the ratio between the well depths in the {\it end-to-end} |
| 641 |
> |
and side-by-side configurations. For spheres, $\epsilon^r \equiv 1$. |
| 642 |
> |
Moments of inertia are also required to describe the motion of primary |
| 643 |
> |
particles with orientational degrees of freedom. |
| 644 |
|
|
| 645 |
< |
The Langevin algorithm described in previous section has been |
| 646 |
< |
implemented in {\sc oopse}\cite{Meineke2005} and applied to studies |
| 581 |
< |
of the static and dynamic properties in several systems. |
| 582 |
< |
|
| 583 |
< |
\subsection{Temperature Control} |
| 584 |
< |
|
| 585 |
< |
As shown in Eq.~\ref{randomForce}, random collisions associated with |
| 586 |
< |
the solvent's thermal motions is controlled by the external |
| 587 |
< |
temperature. The capability to maintain the temperature of the whole |
| 588 |
< |
system was usually used to measure the stability and efficiency of |
| 589 |
< |
the algorithm. In order to verify the stability of this new |
| 590 |
< |
algorithm, a series of simulations are performed on system |
| 591 |
< |
consisiting of 256 SSD water molecules with different viscosities. |
| 592 |
< |
The initial configuration for the simulations is taken from a 1ns |
| 593 |
< |
NVT simulation with a cubic box of 19.7166~\AA. All simulation are |
| 594 |
< |
carried out with cutoff radius of 9~\AA and 2 fs time step for 1 ns |
| 595 |
< |
with reference temperature at 300~K. The average temperature as a |
| 596 |
< |
function of $\eta$ is shown in Table \ref{langevin:viscosity} where |
| 597 |
< |
the temperatures range from 303.04~K to 300.47~K for $\eta = 0.01 - |
| 598 |
< |
1$ poise. The better temperature control at higher viscosity can be |
| 599 |
< |
explained by the finite size effect and relative slow relaxation |
| 600 |
< |
rate at lower viscosity regime. |
| 601 |
< |
\begin{table} |
| 602 |
< |
\caption{AVERAGE TEMPERATURES FROM LANGEVIN DYNAMICS SIMULATIONS OF |
| 603 |
< |
SSD WATER MOLECULES WITH REFERENCE TEMPERATURE AT 300~K.} |
| 604 |
< |
\label{langevin:viscosity} |
| 645 |
> |
\begin{table*} |
| 646 |
> |
\begin{minipage}{\linewidth} |
| 647 |
|
\begin{center} |
| 648 |
< |
\begin{tabular}{lll} |
| 649 |
< |
\hline |
| 650 |
< |
$\eta$ & $\text{T}_{\text{avg}}$ & $\text{T}_{\text{rms}}$ \\ |
| 651 |
< |
\hline |
| 652 |
< |
1 & 300.47 & 10.99 \\ |
| 653 |
< |
0.1 & 301.19 & 11.136 \\ |
| 654 |
< |
0.01 & 303.04 & 11.796 \\ |
| 655 |
< |
\hline |
| 648 |
> |
\caption{Parameters for the primary particles in use by the rigid body |
| 649 |
> |
models in figure \ref{fig:models}.} |
| 650 |
> |
\begin{tabular}{lrcccccccc} |
| 651 |
> |
\hline |
| 652 |
> |
& & & & & & & \multicolumn{3}c{$\overleftrightarrow{\mathsf I}$ (amu \AA$^2$)} \\ |
| 653 |
> |
& & $d$ (\AA) & $l$ (\AA) & $\epsilon^s$ (kcal/mol) & $\epsilon^r$ & |
| 654 |
> |
$m$ (amu) & $I_{xx}$ & $I_{yy}$ & $I_{zz}$ \\ \hline |
| 655 |
> |
Sphere & & 6.5 & $= d$ & 0.8 & 1 & 190 & 802.75 & 802.75 & 802.75 \\ |
| 656 |
> |
Ellipsoid & & 4.6 & 13.8 & 0.8 & 0.2 & 200 & 2105 & 2105 & 421 \\ |
| 657 |
> |
Dumbbell &(2 identical spheres) & 6.5 & $= d$ & 0.8 & 1 & 190 & 802.75 & 802.75 & 802.75 \\ |
| 658 |
> |
Banana &(3 identical ellipsoids)& 4.2 & 11.2 & 0.8 & 0.2 & 240 & 10000 & 10000 & 0 \\ |
| 659 |
> |
Lipid: & Spherical Head & 6.5 & $= d$ & 0.185 & 1 & 196 & & & \\ |
| 660 |
> |
& Ellipsoidal Tail & 4.6 & 13.8 & 0.8 & 0.2 & 760 & 45000 & 45000 & 9000 \\ |
| 661 |
> |
Solvent & & 4.7 & $= d$ & 0.8 & 1 & 72.06 & & & \\ |
| 662 |
> |
\hline |
| 663 |
|
\end{tabular} |
| 664 |
+ |
\label{tab:parameters} |
| 665 |
|
\end{center} |
| 666 |
< |
\end{table} |
| 666 |
> |
\end{minipage} |
| 667 |
> |
\end{table*} |
| 668 |
|
|
| 618 |
– |
Another set of calculations were performed to study the efficiency of |
| 619 |
– |
temperature control using different temperature coupling schemes. |
| 620 |
– |
The starting configuration is cooled to 173~K and evolved using NVE, |
| 621 |
– |
NVT, and Langevin dynamic with time step of 2 fs. |
| 622 |
– |
Fig.~\ref{langevin:temperature} shows the heating curve obtained as |
| 623 |
– |
the systems reach equilibrium. The orange curve in |
| 624 |
– |
Fig.~\ref{langevin:temperature} represents the simulation using |
| 625 |
– |
Nos\'e-Hoover temperature scaling scheme with thermostat of 5 ps |
| 626 |
– |
which gives reasonable tight coupling, while the blue one from |
| 627 |
– |
Langevin dynamics with viscosity of 0.1 poise demonstrates a faster |
| 628 |
– |
scaling to the desire temperature. When $ \eta = 0$, Langevin dynamics becomes normal |
| 629 |
– |
NVE (see orange curve in Fig.~\ref{langevin:temperature}) which |
| 630 |
– |
loses the temperature control ability. |
| 631 |
– |
|
| 669 |
|
\begin{figure} |
| 670 |
|
\centering |
| 671 |
< |
\includegraphics[width=\linewidth]{temperature.pdf} |
| 672 |
< |
\caption[Plot of Temperature Fluctuation Versus Time]{Plot of |
| 673 |
< |
temperature fluctuation versus time.} \label{langevin:temperature} |
| 671 |
> |
\includegraphics[width=3in]{sketch} |
| 672 |
> |
\caption[Sketch of the model systems]{A sketch of the model systems |
| 673 |
> |
used in evaluating the behavior of the rigid body Langevin |
| 674 |
> |
integrator.} \label{fig:models} |
| 675 |
|
\end{figure} |
| 676 |
|
|
| 677 |
< |
\subsection{Langevin Dynamics simulation {\it vs} NVE simulations} |
| 677 |
> |
\subsection{Simulation Methodology} |
| 678 |
> |
We performed reference microcanonical simulations with explicit |
| 679 |
> |
solvents for each of the different model system. In each case there |
| 680 |
> |
was one solute model and 1929 solvent molecules present in the |
| 681 |
> |
simulation box. All simulations were equilibrated using a |
| 682 |
> |
constant-pressure and temperature integrator with target values of 300 |
| 683 |
> |
K for the temperature and 1 atm for pressure. Following this stage, |
| 684 |
> |
further equilibration and sampling was done in a microcanonical |
| 685 |
> |
ensemble. Since the model bodies are typically quite massive, we were |
| 686 |
> |
able to use a time step of 25 fs. |
| 687 |
|
|
| 688 |
< |
To validate our langevin dynamics simulation. We performed several NVE |
| 689 |
< |
simulations with explicit solvents for different shaped |
| 690 |
< |
molecules. There are one solute molecule and 1929 solvent molecules in |
| 691 |
< |
NVE simulation. The parameters are shown in table |
| 692 |
< |
\ref{tab:parameters}. The force field between spheres is standard |
| 693 |
< |
Lennard-Jones, and ellipsoids interact with other ellipsoids and |
| 694 |
< |
spheres with generalized Gay-Berne potential. All simulations are |
| 695 |
< |
carried out at 300 K and 1 Atm. The time step is 25 ns, and a |
| 696 |
< |
switching function was applied to all potentials to smoothly turn off |
| 697 |
< |
the interactions between a range of $22$ and $25$ \AA. The switching |
| 698 |
< |
function was the standard (cubic) function, |
| 688 |
> |
The model systems studied used both Lennard-Jones spheres as well as |
| 689 |
> |
uniaxial Gay-Berne ellipoids. In its original form, the Gay-Berne |
| 690 |
> |
potential was a single site model for the interactions of rigid |
| 691 |
> |
ellipsoidal molecules.\cite{Gay81} It can be thought of as a |
| 692 |
> |
modification of the Gaussian overlap model originally described by |
| 693 |
> |
Berne and Pechukas.\cite{Berne72} The potential is constructed in the |
| 694 |
> |
familiar form of the Lennard-Jones function using |
| 695 |
> |
orientation-dependent $\sigma$ and $\epsilon$ parameters, |
| 696 |
> |
\begin{equation*} |
| 697 |
> |
V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
| 698 |
> |
r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 699 |
> |
{\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u |
| 700 |
> |
}_i}, |
| 701 |
> |
{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} |
| 702 |
> |
-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 703 |
> |
{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right] |
| 704 |
> |
\label{eq:gb} |
| 705 |
> |
\end{equation*} |
| 706 |
> |
|
| 707 |
> |
The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
| 708 |
> |
\hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
| 709 |
> |
\hat{u}}_{j},{\bf \hat{r}}_{ij}))$ parameters |
| 710 |
> |
are dependent on the relative orientations of the two ellipsoids (${\bf |
| 711 |
> |
\hat{u}}_{i},{\bf \hat{u}}_{j}$) as well as the direction of the |
| 712 |
> |
inter-ellipsoid separation (${\bf \hat{r}}_{ij}$). The shape and |
| 713 |
> |
attractiveness of each ellipsoid is governed by a relatively small set |
| 714 |
> |
of parameters: $l$ and $d$ describe the length and width of each |
| 715 |
> |
uniaxial ellipsoid, while $\epsilon^s$, which describes the well depth |
| 716 |
> |
for two identical ellipsoids in a {\it side-by-side} configuration. |
| 717 |
> |
Additionally, a well depth aspect ratio, $\epsilon^r = \epsilon^e / |
| 718 |
> |
\epsilon^s$, describes the ratio between the well depths in the {\it |
| 719 |
> |
end-to-end} and side-by-side configurations. Details of the potential |
| 720 |
> |
are given elsewhere,\cite{Luckhurst90,Golubkov06,SunGezelter08} and an |
| 721 |
> |
excellent overview of the computational methods that can be used to |
| 722 |
> |
efficiently compute forces and torques for this potential can be found |
| 723 |
> |
in Ref. \citen{Golubkov06} |
| 724 |
> |
|
| 725 |
> |
For the interaction between nonequivalent uniaxial ellipsoids (or |
| 726 |
> |
between spheres and ellipsoids), the spheres are treated as ellipsoids |
| 727 |
> |
with an aspect ratio of 1 ($d = l$) and with an well depth ratio |
| 728 |
> |
($\epsilon^r$) of 1 ($\epsilon^e = \epsilon^s$). The form of the |
| 729 |
> |
Gay-Berne potential we are using was generalized by Cleaver {\it et |
| 730 |
> |
al.} and is appropriate for dissimilar uniaxial |
| 731 |
> |
ellipsoids.\cite{Cleaver96} |
| 732 |
> |
|
| 733 |
> |
A switching function was applied to all potentials to smoothly turn |
| 734 |
> |
off the interactions between a range of $22$ and $25$ \AA. The |
| 735 |
> |
switching function was the standard (cubic) function, |
| 736 |
|
\begin{equation} |
| 737 |
|
s(r) = |
| 738 |
|
\begin{cases} |
| 744 |
|
\end{cases} |
| 745 |
|
\label{eq:switchingFunc} |
| 746 |
|
\end{equation} |
| 747 |
< |
We have computed translational diffusion constants for lipid molecules |
| 748 |
< |
from the mean-square displacement, |
| 747 |
> |
|
| 748 |
> |
To measure shear viscosities from our microcanonical simulations, we |
| 749 |
> |
used the Einstein form of the pressure correlation function,\cite{hess:209} |
| 750 |
|
\begin{equation} |
| 751 |
< |
D = \lim_{t\rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
| 751 |
> |
\eta = \lim_{t->\infty} \frac{V}{2 k_B T} \frac{d}{dt} \left\langle \left( |
| 752 |
> |
\int_{t_0}^{t_0 + t} P_{xz}(t') dt' \right)^2 \right\rangle_{t_0}. |
| 753 |
> |
\label{eq:shear} |
| 754 |
|
\end{equation} |
| 755 |
< |
of the solute molecules. Translational diffusion constants for the |
| 669 |
< |
different shaped molecules are shown in table |
| 670 |
< |
\ref{tab:translation}. We have also computed orientational correlation |
| 671 |
< |
times for different shaped molecules from fits of the second-order |
| 672 |
< |
Legendre polynomial correlation function, |
| 755 |
> |
A similar form exists for the bulk viscosity |
| 756 |
|
\begin{equation} |
| 757 |
< |
C_{\ell}(t) = \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
| 758 |
< |
\mu}_{i}(0) \right) |
| 757 |
> |
\kappa = \lim_{t->\infty} \frac{V}{2 k_B T} \frac{d}{dt} \left\langle \left( |
| 758 |
> |
\int_{t_0}^{t_0 + t} |
| 759 |
> |
\left(P\left(t'\right)-\left\langle P \right\rangle \right)dt' |
| 760 |
> |
\right)^2 \right\rangle_{t_0}. |
| 761 |
|
\end{equation} |
| 762 |
< |
the results are shown in table \ref{tab:rotation}. We used einstein |
| 763 |
< |
format of the pressure correlation function, |
| 762 |
> |
Alternatively, the shear viscosity can also be calculated using a |
| 763 |
> |
Green-Kubo formula with the off-diagonal pressure tensor correlation function, |
| 764 |
|
\begin{equation} |
| 765 |
< |
C_{\ell}(t) = \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
| 766 |
< |
\mu}_{i}(0) \right) |
| 765 |
> |
\eta = \frac{V}{k_B T} \int_0^{\infty} \left\langle P_{xz}(t_0) P_{xz}(t_0 |
| 766 |
> |
+ t) \right\rangle_{t_0} dt, |
| 767 |
|
\end{equation} |
| 768 |
< |
to estimate the viscosity of the systems from NVE simulations. The |
| 769 |
< |
viscosity can also be calculated by Green-Kubo pressure correlaton |
| 770 |
< |
function, |
| 768 |
> |
although this method converges extremely slowly and is not practical |
| 769 |
> |
for obtaining viscosities from molecular dynamics simulations. |
| 770 |
> |
|
| 771 |
> |
The Langevin dynamics for the different model systems were performed |
| 772 |
> |
at the same temperature as the average temperature of the |
| 773 |
> |
microcanonical simulations and with a solvent viscosity taken from |
| 774 |
> |
Eq. (\ref{eq:shear}) applied to these simulations. We used 1024 |
| 775 |
> |
independent solute simulations to obtain statistics on our Langevin |
| 776 |
> |
integrator. |
| 777 |
> |
|
| 778 |
> |
\subsection{Analysis} |
| 779 |
> |
|
| 780 |
> |
The quantities of interest when comparing the Langevin integrator to |
| 781 |
> |
analytic hydrodynamic equations and to molecular dynamics simulations |
| 782 |
> |
are typically translational diffusion constants and orientational |
| 783 |
> |
relaxation times. Translational diffusion constants for point |
| 784 |
> |
particles are computed easily from the long-time slope of the |
| 785 |
> |
mean-square displacement, |
| 786 |
|
\begin{equation} |
| 787 |
< |
C_{\ell}(t) = \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
| 688 |
< |
\mu}_{i}(0) \right) |
| 787 |
> |
D = \lim_{t\rightarrow \infty} \frac{1}{6 t} \left\langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \right\rangle, |
| 788 |
|
\end{equation} |
| 789 |
< |
However, this method converges slowly, and the statistics are not good |
| 790 |
< |
enough to give us a very accurate value. The langevin dynamics |
| 791 |
< |
simulations for different shaped molecules are performed at the same |
| 792 |
< |
conditions as the NVE simulations with viscosity estimated from NVE |
| 793 |
< |
simulations. To get better statistics, 1024 non-interacting solute |
| 794 |
< |
molecules are put into one simulation box for each langevin |
| 795 |
< |
simulation, this is equal to 1024 simulations for single solute |
| 796 |
< |
systems. The diffusion constants and rotation relaxation times for |
| 797 |
< |
different shaped molecules are shown in table \ref{tab:translation} |
| 798 |
< |
and \ref{tab:rotation} to compare to the results calculated from NVE |
| 700 |
< |
simulations. The theoretical values for sphere is calculated from the |
| 701 |
< |
Stokes-Einstein law, the theoretical values for ellipsoid is |
| 702 |
< |
calculated from Perrin's fomula, the theoretical values for dumbbell |
| 703 |
< |
is calculated from StinXX and Davis theory. The exact method is |
| 704 |
< |
applied to the langevin dynamics simulations for sphere and ellipsoid, |
| 705 |
< |
the bead model is applied to the simulation for dumbbell molecule, and |
| 706 |
< |
the rough shell model is applied to ellipsoid, dumbbell, banana and |
| 707 |
< |
lipid molecules. The results from all the langevin dynamics |
| 708 |
< |
simulations, including exact, bead model and rough shell, match the |
| 709 |
< |
theoretical values perfectly for all different shaped molecules. This |
| 710 |
< |
indicates that our simulation package for langevin dynamics is working |
| 711 |
< |
well. The approxiate methods ( bead model and rough shell model) are |
| 712 |
< |
accurate enough for the current simulations. The goal of the langevin |
| 713 |
< |
dynamics theory is to replace the explicit solvents by the friction |
| 714 |
< |
forces. We compared the dynamic properties of different shaped |
| 715 |
< |
molecules in langevin dynamics simulations with that in NVE |
| 716 |
< |
simulations. The results are reasonable close. Overall, the |
| 717 |
< |
translational diffusion constants calculated from langevin dynamics |
| 718 |
< |
simulations are very close to the values from the NVE simulation. For |
| 719 |
< |
sphere and lipid molecules, the diffusion constants are a little bit |
| 720 |
< |
off from the NVE simulation results. One possible reason is that the |
| 721 |
< |
calculation of the viscosity is very difficult to be accurate. Another |
| 722 |
< |
possible reason is that although we save very frequently during the |
| 723 |
< |
NVE simulations and run pretty long time simulations, there is only |
| 724 |
< |
one solute molecule in the system which makes the calculation for the |
| 725 |
< |
diffusion constant difficult. The sphere molecule behaves as a free |
| 726 |
< |
rotor in the solvent, so there is no rotation relaxation time |
| 727 |
< |
calculated from NVE simulations. The rotation relaxation time is not |
| 728 |
< |
very close to the NVE simulations results. The banana and lipid |
| 729 |
< |
molecules match the NVE simulations results pretty well. The mismatch |
| 730 |
< |
between langevin dynamics and NVE simulation for ellipsoid is possibly |
| 731 |
< |
caused by the slip boundary condition. For dumbbell, the mismatch is |
| 732 |
< |
caused by the size of the solvent molecule is pretty large compared to |
| 733 |
< |
dumbbell molecule in NVE simulations. |
| 789 |
> |
of the solute molecules. For models in which the translational |
| 790 |
> |
diffusion tensor (${\bf D}_{tt}$) has non-degenerate eigenvalues |
| 791 |
> |
(i.e. any non-spherically-symmetric rigid body), it is possible to |
| 792 |
> |
compute the diffusive behavior for motion parallel to each body-fixed |
| 793 |
> |
axis by projecting the displacement of the particle onto the |
| 794 |
> |
body-fixed reference frame at $t=0$. With an isotropic solvent, as we |
| 795 |
> |
have used in this study, there are differences between the three |
| 796 |
> |
diffusion constants, but these must converge to the same value at |
| 797 |
> |
longer times. Translational diffusion constants for the different |
| 798 |
> |
shaped models are shown in table \ref{tab:translation}. |
| 799 |
|
|
| 800 |
+ |
In general, the three eigenvalues ($D_1, D_2, D_3$) of the rotational |
| 801 |
+ |
diffusion tensor (${\bf D}_{rr}$) measure the diffusion of an object |
| 802 |
+ |
{\it around} a particular body-fixed axis and {\it not} the diffusion |
| 803 |
+ |
of a vector pointing along the axis. However, these eigenvalues can |
| 804 |
+ |
be combined to find 5 characteristic rotational relaxation |
| 805 |
+ |
times,\cite{PhysRev.119.53,Berne90} |
| 806 |
+ |
\begin{eqnarray} |
| 807 |
+ |
1 / \tau_1 & = & 6 D_r + 2 \Delta \\ |
| 808 |
+ |
1 / \tau_2 & = & 6 D_r - 2 \Delta \\ |
| 809 |
+ |
1 / \tau_3 & = & 3 (D_r + D_1) \\ |
| 810 |
+ |
1 / \tau_4 & = & 3 (D_r + D_2) \\ |
| 811 |
+ |
1 / \tau_5 & = & 3 (D_r + D_3) |
| 812 |
+ |
\end{eqnarray} |
| 813 |
+ |
where |
| 814 |
+ |
\begin{equation} |
| 815 |
+ |
D_r = \frac{1}{3} \left(D_1 + D_2 + D_3 \right) |
| 816 |
+ |
\end{equation} |
| 817 |
+ |
and |
| 818 |
+ |
\begin{equation} |
| 819 |
+ |
\Delta = \left( (D_1 - D_2)^2 + (D_3 - D_1 )(D_3 - D_2)\right)^{1/2} |
| 820 |
+ |
\end{equation} |
| 821 |
+ |
Each of these characteristic times can be used to predict the decay of |
| 822 |
+ |
part of the rotational correlation function when $\ell = 2$, |
| 823 |
+ |
\begin{equation} |
| 824 |
+ |
C_2(t) = \frac{a^2}{N^2} e^{-t/\tau_1} + \frac{b^2}{N^2} e^{-t/\tau_2}. |
| 825 |
+ |
\end{equation} |
| 826 |
+ |
This is the same as the $F^2_{0,0}(t)$ correlation function that |
| 827 |
+ |
appears in Ref. \citen{Berne90}. The amplitudes of the two decay |
| 828 |
+ |
terms are expressed in terms of three dimensionless functions of the |
| 829 |
+ |
eigenvalues: $a = \sqrt{3} (D_1 - D_2)$, $b = (2D_3 - D_1 - D_2 + |
| 830 |
+ |
2\Delta)$, and $N = 2 \sqrt{\Delta b}$. Similar expressions can be |
| 831 |
+ |
obtained for other angular momentum correlation |
| 832 |
+ |
functions.\cite{PhysRev.119.53,Berne90} In all of the model systems we |
| 833 |
+ |
studied, only one of the amplitudes of the two decay terms was |
| 834 |
+ |
non-zero, so it was possible to derive a single relaxation time for |
| 835 |
+ |
each of the hydrodynamic tensors. In many cases, these characteristic |
| 836 |
+ |
times are averaged and reported in the literature as a single relaxation |
| 837 |
+ |
time,\cite{Garcia-de-la-Torre:1997qy} |
| 838 |
+ |
\begin{equation} |
| 839 |
+ |
1 / \tau_0 = \frac{1}{5} \sum_{i=1}^5 \tau_{i}^{-1}, |
| 840 |
+ |
\end{equation} |
| 841 |
+ |
although for the cases reported here, this averaging is not necessary |
| 842 |
+ |
and only one of the five relaxation times is relevant. |
| 843 |
+ |
|
| 844 |
+ |
To test the Langevin integrator's behavior for rotational relaxation, |
| 845 |
+ |
we have compared the analytical orientational relaxation times (if |
| 846 |
+ |
they are known) with the general result from the diffusion tensor and |
| 847 |
+ |
with the results from both the explicitly solvated molecular dynamics |
| 848 |
+ |
and Langevin simulations. Relaxation times from simulations (both |
| 849 |
+ |
microcanonical and Langevin), were computed using Legendre polynomial |
| 850 |
+ |
correlation functions for a unit vector (${\bf u}$) fixed along one or |
| 851 |
+ |
more of the body-fixed axes of the model. |
| 852 |
+ |
\begin{equation} |
| 853 |
+ |
C_{\ell}(t) = \left\langle P_{\ell}\left({\bf u}_{i}(t) \cdot {\bf |
| 854 |
+ |
u}_{i}(0) \right) \right\rangle |
| 855 |
+ |
\end{equation} |
| 856 |
+ |
For simulations in the high-friction limit, orientational correlation |
| 857 |
+ |
times can then be obtained from exponential fits of this function, or by |
| 858 |
+ |
integrating, |
| 859 |
+ |
\begin{equation} |
| 860 |
+ |
\tau = \ell (\ell + 1) \int_0^{\infty} C_{\ell}(t) dt. |
| 861 |
+ |
\end{equation} |
| 862 |
+ |
In lower-friction solvents, the Legendre correlation functions often |
| 863 |
+ |
exhibit non-exponential decay, and may not be characterized by a |
| 864 |
+ |
single decay constant. |
| 865 |
+ |
|
| 866 |
+ |
In table \ref{tab:rotation} we show the characteristic rotational |
| 867 |
+ |
relaxation times (based on the diffusion tensor) for each of the model |
| 868 |
+ |
systems compared with the values obtained via microcanonical and Langevin |
| 869 |
+ |
simulations. |
| 870 |
+ |
|
| 871 |
+ |
\subsection{Spherical particles} |
| 872 |
+ |
Our model system for spherical particles was a Lennard-Jones sphere of |
| 873 |
+ |
diameter ($\sigma$) 6.5 \AA\ in a sea of smaller spheres ($\sigma$ = |
| 874 |
+ |
4.7 \AA). The well depth ($\epsilon$) for both particles was set to |
| 875 |
+ |
an arbitrary value of 0.8 kcal/mol. |
| 876 |
+ |
|
| 877 |
+ |
The Stokes-Einstein behavior of large spherical particles in |
| 878 |
+ |
hydrodynamic flows is well known, giving translational friction |
| 879 |
+ |
coefficients of $6 \pi \eta R$ (stick boundary conditions) and |
| 880 |
+ |
rotational friction coefficients of $8 \pi \eta R^3$. Recently, |
| 881 |
+ |
Schmidt and Skinner have computed the behavior of spherical tag |
| 882 |
+ |
particles in molecular dynamics simulations, and have shown that {\it |
| 883 |
+ |
slip} boundary conditions ($\Xi_{tt} = 4 \pi \eta R$) may be more |
| 884 |
+ |
appropriate for molecule-sized spheres embedded in a sea of spherical |
| 885 |
+ |
solvent particles.\cite{Schmidt:2004fj,Schmidt:2003kx} |
| 886 |
+ |
|
| 887 |
+ |
Our simulation results show similar behavior to the behavior observed |
| 888 |
+ |
by Schmidt and Skinner. The diffusion constant obtained from our |
| 889 |
+ |
microcanonical molecular dynamics simulations lies between the slip |
| 890 |
+ |
and stick boundary condition results obtained via Stokes-Einstein |
| 891 |
+ |
behavior. Since the Langevin integrator assumes Stokes-Einstein stick |
| 892 |
+ |
boundary conditions in calculating the drag and random forces for |
| 893 |
+ |
spherical particles, our Langevin routine obtains nearly quantitative |
| 894 |
+ |
agreement with the hydrodynamic results for spherical particles. One |
| 895 |
+ |
avenue for improvement of the method would be to compute elements of |
| 896 |
+ |
$\Xi_{tt}$ assuming behavior intermediate between the two boundary |
| 897 |
+ |
conditions. |
| 898 |
+ |
|
| 899 |
+ |
In the explicit solvent simulations, both our solute and solvent |
| 900 |
+ |
particles were structureless, exerting no torques upon each other. |
| 901 |
+ |
Therefore, there are not rotational correlation times available for |
| 902 |
+ |
this model system. |
| 903 |
+ |
|
| 904 |
+ |
\subsection{Ellipsoids} |
| 905 |
+ |
For uniaxial ellipsoids ($a > b = c$), Perrin's formulae for both |
| 906 |
+ |
translational and rotational diffusion of each of the body-fixed axes |
| 907 |
+ |
can be combined to give a single translational diffusion |
| 908 |
+ |
constant,\cite{Berne90} |
| 909 |
+ |
\begin{equation} |
| 910 |
+ |
D = \frac{k_B T}{6 \pi \eta a} G(\rho), |
| 911 |
+ |
\label{Dperrin} |
| 912 |
+ |
\end{equation} |
| 913 |
+ |
as well as a single rotational diffusion coefficient, |
| 914 |
+ |
\begin{equation} |
| 915 |
+ |
\Theta = \frac{3 k_B T}{16 \pi \eta a^3} \left\{ \frac{(2 - \rho^2) |
| 916 |
+ |
G(\rho) - 1}{1 - \rho^4} \right\}. |
| 917 |
+ |
\label{ThetaPerrin} |
| 918 |
+ |
\end{equation} |
| 919 |
+ |
In these expressions, $G(\rho)$ is a function of the axial ratio |
| 920 |
+ |
($\rho = b / a$), which for prolate ellipsoids, is |
| 921 |
+ |
\begin{equation} |
| 922 |
+ |
G(\rho) = (1- \rho^2)^{-1/2} \ln \left\{ \frac{1 + (1 - |
| 923 |
+ |
\rho^2)^{1/2}}{\rho} \right\} |
| 924 |
+ |
\label{GPerrin} |
| 925 |
+ |
\end{equation} |
| 926 |
+ |
Again, there is some uncertainty about the correct boundary conditions |
| 927 |
+ |
to use for molecular-scale ellipsoids in a sea of similarly-sized |
| 928 |
+ |
solvent particles. Ravichandran and Bagchi found that {\it slip} |
| 929 |
+ |
boundary conditions most closely resembled the simulation |
| 930 |
+ |
results,\cite{Ravichandran:1999fk} in agreement with earlier work of |
| 931 |
+ |
Tang and Evans.\cite{TANG:1993lr} |
| 932 |
+ |
|
| 933 |
+ |
Even though there are analytic resistance tensors for ellipsoids, we |
| 934 |
+ |
constructed a rough-shell model using 2135 beads (each with a diameter |
| 935 |
+ |
of 0.25 \AA) to approximate the shape of the model ellipsoid. We |
| 936 |
+ |
compared the Langevin dynamics from both the simple ellipsoidal |
| 937 |
+ |
resistance tensor and the rough shell approximation with |
| 938 |
+ |
microcanonical simulations and the predictions of Perrin. As in the |
| 939 |
+ |
case of our spherical model system, the Langevin integrator reproduces |
| 940 |
+ |
almost exactly the behavior of the Perrin formulae (which is |
| 941 |
+ |
unsurprising given that the Perrin formulae were used to derive the |
| 942 |
+ |
drag and random forces applied to the ellipsoid). We obtain |
| 943 |
+ |
translational diffusion constants and rotational correlation times |
| 944 |
+ |
that are within a few percent of the analytic values for both the |
| 945 |
+ |
exact treatment of the diffusion tensor as well as the rough-shell |
| 946 |
+ |
model for the ellipsoid. |
| 947 |
+ |
|
| 948 |
+ |
The translational diffusion constants from the microcanonical simulations |
| 949 |
+ |
agree well with the predictions of the Perrin model, although the rotational |
| 950 |
+ |
correlation times are a factor of 2 shorter than expected from hydrodynamic |
| 951 |
+ |
theory. One explanation for the slower rotation |
| 952 |
+ |
of explicitly-solvated ellipsoids is the possibility that solute-solvent |
| 953 |
+ |
collisions happen at both ends of the solute whenever the principal |
| 954 |
+ |
axis of the ellipsoid is turning. In the upper portion of figure |
| 955 |
+ |
\ref{fig:explanation} we sketch a physical picture of this explanation. |
| 956 |
+ |
Since our Langevin integrator is providing nearly quantitative agreement with |
| 957 |
+ |
the Perrin model, it also predicts orientational diffusion for ellipsoids that |
| 958 |
+ |
exceed explicitly solvated correlation times by a factor of two. |
| 959 |
+ |
|
| 960 |
+ |
\subsection{Rigid dumbbells} |
| 961 |
+ |
Perhaps the only {\it composite} rigid body for which analytic |
| 962 |
+ |
expressions for the hydrodynamic tensor are available is the |
| 963 |
+ |
two-sphere dumbbell model. This model consists of two non-overlapping |
| 964 |
+ |
spheres held by a rigid bond connecting their centers. There are |
| 965 |
+ |
competing expressions for the 6x6 resistance tensor for this |
| 966 |
+ |
model. Equation (\ref{introEquation:oseenTensor}) above gives the |
| 967 |
+ |
original Oseen tensor, while the second order expression introduced by |
| 968 |
+ |
Rotne and Prager,\cite{Rotne1969} and improved by Garc\'{i}a de la |
| 969 |
+ |
Torre and Bloomfield,\cite{Torre1977} is given above as |
| 970 |
+ |
Eq. (\ref{introEquation:RPTensorNonOverlapped}). In our case, we use |
| 971 |
+ |
a model dumbbell in which the two spheres are identical Lennard-Jones |
| 972 |
+ |
particles ($\sigma$ = 6.5 \AA\ , $\epsilon$ = 0.8 kcal / mol) held at |
| 973 |
+ |
a distance of 6.532 \AA. |
| 974 |
+ |
|
| 975 |
+ |
The theoretical values for the translational diffusion constant of the |
| 976 |
+ |
dumbbell are calculated from the work of Stimson and Jeffery, who |
| 977 |
+ |
studied the motion of this system in a flow parallel to the |
| 978 |
+ |
inter-sphere axis,\cite{Stimson:1926qy} and Davis, who studied the |
| 979 |
+ |
motion in a flow {\it perpendicular} to the inter-sphere |
| 980 |
+ |
axis.\cite{Davis:1969uq} We know of no analytic solutions for the {\it |
| 981 |
+ |
orientational} correlation times for this model system (other than |
| 982 |
+ |
those derived from the 6 x 6 tensors mentioned above). |
| 983 |
+ |
|
| 984 |
+ |
The bead model for this model system comprises the two large spheres |
| 985 |
+ |
by themselves, while the rough shell approximation used 3368 separate |
| 986 |
+ |
beads (each with a diameter of 0.25 \AA) to approximate the shape of |
| 987 |
+ |
the rigid body. The hydrodynamics tensors computed from both the bead |
| 988 |
+ |
and rough shell models are remarkably similar. Computing the initial |
| 989 |
+ |
hydrodynamic tensor for a rough shell model can be quite expensive (in |
| 990 |
+ |
this case it requires inverting a 10104 x 10104 matrix), while the |
| 991 |
+ |
bead model is typically easy to compute (in this case requiring |
| 992 |
+ |
inversion of a 6 x 6 matrix). |
| 993 |
+ |
|
| 994 |
+ |
\begin{figure} |
| 995 |
+ |
\centering |
| 996 |
+ |
\includegraphics[width=2in]{RoughShell} |
| 997 |
+ |
\caption[Model rigid bodies and their rough shell approximations]{The |
| 998 |
+ |
model rigid bodies (left column) used to test this algorithm and their |
| 999 |
+ |
rough-shell approximations (right-column) that were used to compute |
| 1000 |
+ |
the hydrodynamic tensors. The top two models (ellipsoid and dumbbell) |
| 1001 |
+ |
have analytic solutions and were used to test the rough shell |
| 1002 |
+ |
approximation. The lower two models (banana and lipid) were compared |
| 1003 |
+ |
with explicitly-solvated molecular dynamics simulations. } |
| 1004 |
+ |
\label{fig:roughShell} |
| 1005 |
+ |
\end{figure} |
| 1006 |
+ |
|
| 1007 |
+ |
|
| 1008 |
+ |
Once the hydrodynamic tensor has been computed, there is no additional |
| 1009 |
+ |
penalty for carrying out a Langevin simulation with either of the two |
| 1010 |
+ |
different hydrodynamics models. Our naive expectation is that since |
| 1011 |
+ |
the rigid body's surface is roughened under the various shell models, |
| 1012 |
+ |
the diffusion constants will be even farther from the ``slip'' |
| 1013 |
+ |
boundary conditions than observed for the bead model (which uses a |
| 1014 |
+ |
Stokes-Einstein model to arrive at the hydrodynamic tensor). For the |
| 1015 |
+ |
dumbbell, this prediction is correct although all of the Langevin |
| 1016 |
+ |
diffusion constants are within 6\% of the diffusion constant predicted |
| 1017 |
+ |
from the fully solvated system. |
| 1018 |
+ |
|
| 1019 |
+ |
For rotational motion, Langevin integration (and the hydrodynamic tensor) |
| 1020 |
+ |
yields rotational correlation times that are substantially shorter than those |
| 1021 |
+ |
obtained from explicitly-solvated simulations. It is likely that this is due |
| 1022 |
+ |
to the large size of the explicit solvent spheres, a feature that prevents |
| 1023 |
+ |
the solvent from coming in contact with a substantial fraction of the surface |
| 1024 |
+ |
area of the dumbbell. Therefore, the explicit solvent only provides drag |
| 1025 |
+ |
over a substantially reduced surface area of this model, while the |
| 1026 |
+ |
hydrodynamic theories utilize the entire surface area for estimating |
| 1027 |
+ |
rotational diffusion. A sketch of the free volume available in the explicit |
| 1028 |
+ |
solvent simulations is shown in figure \ref{fig:explanation}. |
| 1029 |
+ |
|
| 1030 |
+ |
|
| 1031 |
+ |
\begin{figure} |
| 1032 |
+ |
\centering |
| 1033 |
+ |
\includegraphics[width=6in]{explanation} |
| 1034 |
+ |
\caption[Explanations of the differences between orientational |
| 1035 |
+ |
correlation times for explicitly-solvated models and hydrodynamics |
| 1036 |
+ |
predictions]{Explanations of the differences between orientational |
| 1037 |
+ |
correlation times for explicitly-solvated models and hydrodynamic |
| 1038 |
+ |
predictions. For the ellipsoids (upper figures), rotation of the |
| 1039 |
+ |
principal axis can involve correlated collisions at both sides of the |
| 1040 |
+ |
solute. In the rigid dumbbell model (lower figures), the large size |
| 1041 |
+ |
of the explicit solvent spheres prevents them from coming in contact |
| 1042 |
+ |
with a substantial fraction of the surface area of the dumbbell. |
| 1043 |
+ |
Therefore, the explicit solvent only provides drag over a |
| 1044 |
+ |
substantially reduced surface area of this model, where the |
| 1045 |
+ |
hydrodynamic theories utilize the entire surface area for estimating |
| 1046 |
+ |
rotational diffusion. |
| 1047 |
+ |
} \label{fig:explanation} |
| 1048 |
+ |
\end{figure} |
| 1049 |
+ |
|
| 1050 |
+ |
|
| 1051 |
+ |
|
| 1052 |
+ |
\subsection{Composite banana-shaped molecules} |
| 1053 |
+ |
Banana-shaped rigid bodies composed of three Gay-Berne ellipsoids have |
| 1054 |
+ |
been used by Orlandi {\it et al.} to observe mesophases in |
| 1055 |
+ |
coarse-grained models for bent-core liquid crystalline |
| 1056 |
+ |
molecules.\cite{Orlandi:2006fk} We have used the same overlapping |
| 1057 |
+ |
ellipsoids as a way to test the behavior of our algorithm for a |
| 1058 |
+ |
structure of some interest to the materials science community, |
| 1059 |
+ |
although since we are interested in capturing only the hydrodynamic |
| 1060 |
+ |
behavior of this model, we have left out the dipolar interactions of |
| 1061 |
+ |
the original Orlandi model. |
| 1062 |
+ |
|
| 1063 |
+ |
A reference system composed of a single banana rigid body embedded in a |
| 1064 |
+ |
sea of 1929 solvent particles was created and run under standard |
| 1065 |
+ |
(microcanonical) molecular dynamics. The resulting viscosity of this |
| 1066 |
+ |
mixture was 0.298 centipoise (as estimated using Eq. (\ref{eq:shear})). |
| 1067 |
+ |
To calculate the hydrodynamic properties of the banana rigid body model, |
| 1068 |
+ |
we created a rough shell (see Fig.~\ref{fig:roughShell}), in which |
| 1069 |
+ |
the banana is represented as a ``shell'' made of 3321 identical beads |
| 1070 |
+ |
(0.25 \AA\ in diameter) distributed on the surface. Applying the |
| 1071 |
+ |
procedure described in Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
| 1072 |
+ |
identified the center of resistance, ${\bf r} = $(0 \AA, 0.81 \AA, 0 \AA), as |
| 1073 |
+ |
well as the resistance tensor, |
| 1074 |
+ |
\begin{equation*} |
| 1075 |
+ |
\Xi = |
| 1076 |
+ |
\left( {\begin{array}{*{20}c} |
| 1077 |
+ |
0.9261 & 0 & 0&0&0.08585&0.2057\\ |
| 1078 |
+ |
0& 0.9270&-0.007063& 0.08585&0&0\\ |
| 1079 |
+ |
0&-0.007063&0.7494&0.2057&0&0\\ |
| 1080 |
+ |
0&0.0858&0.2057& 58.64& 0&0\\0.08585&0&0&0&48.30&3.219&\\0.2057&0&0&0&3.219&10.7373\\\end{array}} \right), |
| 1081 |
+ |
\end{equation*} |
| 1082 |
+ |
where the units for translational, translation-rotation coupling and |
| 1083 |
+ |
rotational tensors are (kcal fs / mol \AA$^2$), (kcal fs / mol \AA\ rad), |
| 1084 |
+ |
and (kcal fs / mol rad$^2$), respectively. |
| 1085 |
+ |
|
| 1086 |
+ |
The Langevin rigid-body integrator (and the hydrodynamic diffusion tensor) |
| 1087 |
+ |
are essentially quantitative for translational diffusion of this model. |
| 1088 |
+ |
Orientational correlation times under the Langevin rigid-body integrator |
| 1089 |
+ |
are within 11\% of the values obtained from explicit solvent, but these |
| 1090 |
+ |
models also exhibit some solvent inaccessible surface area in the |
| 1091 |
+ |
explicitly-solvated case. |
| 1092 |
+ |
|
| 1093 |
+ |
\subsection{Composite sphero-ellipsoids} |
| 1094 |
+ |
Spherical heads perched on the ends of Gay-Berne ellipsoids have been |
| 1095 |
+ |
used recently as models for lipid |
| 1096 |
+ |
molecules.\cite{SunGezelter08,Ayton01} |
| 1097 |
+ |
MORE DETAILS |
| 1098 |
+ |
|
| 1099 |
+ |
A reference system composed of a single lipid rigid body embedded in a |
| 1100 |
+ |
sea of 1929 solvent particles was created and run under standard |
| 1101 |
+ |
(microcanonical) molecular dynamics. The resulting viscosity of this |
| 1102 |
+ |
mixture was 0.349 centipoise (as estimated using |
| 1103 |
+ |
Eq. (\ref{eq:shear})). To calculate the hydrodynamic properties of |
| 1104 |
+ |
the lipid rigid body model, we created a rough shell (see |
| 1105 |
+ |
Fig.~\ref{fig:roughShell}), in which the lipid is represented as a |
| 1106 |
+ |
``shell'' made of 3550 identical beads (0.25 \AA\ in diameter) |
| 1107 |
+ |
distributed on the surface. Applying the procedure described in |
| 1108 |
+ |
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
| 1109 |
+ |
identified the center of resistance, ${\bf r} = $(0 \AA, 0 \AA, 1.46 |
| 1110 |
+ |
\AA). |
| 1111 |
+ |
|
| 1112 |
+ |
|
| 1113 |
+ |
\subsection{Summary} |
| 1114 |
|
According to our simulations, the langevin dynamics is a reliable |
| 1115 |
|
theory to apply to replace the explicit solvents, especially for the |
| 1116 |
|
translation properties. For large molecules, the rotation properties |
| 1117 |
|
are also mimiced reasonablly well. |
| 1118 |
|
|
| 1119 |
< |
\begin{table*} |
| 1120 |
< |
\begin{minipage}{\linewidth} |
| 1121 |
< |
\begin{center} |
| 1122 |
< |
\caption{} |
| 1123 |
< |
\begin{tabular}{llccccccc} |
| 1124 |
< |
\hline |
| 1125 |
< |
& & Sphere & Ellipsoid & Dumbbell(2 spheres) & Banana(3 ellpsoids) & |
| 1126 |
< |
Lipid(head) & lipid(tail) & Solvent \\ |
| 1127 |
< |
\hline |
| 1128 |
< |
$d$ (\AA) & & 6.5 & 4.6 & 6.5 & 4.2 & 6.5 & 4.6 & 4.7 \\ |
| 1129 |
< |
$l$ (\AA) & & $= d$ & 13.8 & $=d$ & 11.2 & $=d$ & 13.8 & 4.7 \\ |
| 1130 |
< |
$\epsilon^s$ (kcal/mol) & & 0.8 & 0.8 & 0.8 & 0.8 & 0.185 & 0.8 & 0.8 \\ |
| 1131 |
< |
$\epsilon_r$ (well-depth aspect ratio)& & 1 & 0.2 & 1 & 0.2 & 1 & 0.2 & 1 \\ |
| 1132 |
< |
$m$ (amu) & & 190 & 200 & 190 & 240 & 196 & 760 & 72.06 \\ |
| 1133 |
< |
%$\overleftrightarrow{\mathsf I}$ (amu \AA$^2$) & & & & \\ |
| 1134 |
< |
%\multicolumn{2}c{$I_{xx}$} & 1125 & 45000 & N/A \\ |
| 1135 |
< |
%\multicolumn{2}c{$I_{yy}$} & 1125 & 45000 & N/A \\ |
| 757 |
< |
%\multicolumn{2}c{$I_{zz}$} & 0 & 9000 & N/A \\ |
| 758 |
< |
%$\mu$ (Debye) & & varied & 0 & 0 \\ |
| 759 |
< |
\end{tabular} |
| 760 |
< |
\label{tab:parameters} |
| 761 |
< |
\end{center} |
| 762 |
< |
\end{minipage} |
| 763 |
< |
\end{table*} |
| 1119 |
> |
\begin{figure} |
| 1120 |
> |
\centering |
| 1121 |
> |
\includegraphics[width=\linewidth]{graph} |
| 1122 |
> |
\caption[Mean squared displacements and orientational |
| 1123 |
> |
correlation functions for each of the model rigid bodies.]{The |
| 1124 |
> |
mean-squared displacements ($\langle r^2(t) \rangle$) and |
| 1125 |
> |
orientational correlation functions ($C_2(t)$) for each of the model |
| 1126 |
> |
rigid bodies studied. The circles are the results for microcanonical |
| 1127 |
> |
simulations with explicit solvent molecules, while the other data sets |
| 1128 |
> |
are results for Langevin dynamics using the different hydrodynamic |
| 1129 |
> |
tensor approximations. The Perrin model for the ellipsoids is |
| 1130 |
> |
considered the ``exact'' hydrodynamic behavior (this can also be said |
| 1131 |
> |
for the translational motion of the dumbbell operating under the bead |
| 1132 |
> |
model). In most cases, the various hydrodynamics models reproduce |
| 1133 |
> |
each other quantitatively.} |
| 1134 |
> |
\label{fig:results} |
| 1135 |
> |
\end{figure} |
| 1136 |
|
|
| 1137 |
|
\begin{table*} |
| 1138 |
|
\begin{minipage}{\linewidth} |
| 1139 |
|
\begin{center} |
| 1140 |
< |
\caption{} |
| 1141 |
< |
\begin{tabular}{lccccc} |
| 1140 |
> |
\caption{Translational diffusion constants (D) for the model systems |
| 1141 |
> |
calculated using microcanonical simulations (with explicit solvent), |
| 1142 |
> |
theoretical predictions, and Langevin simulations (with implicit solvent). |
| 1143 |
> |
Analytical solutions for the exactly-solved hydrodynamics models are |
| 1144 |
> |
from Refs. \citen{Einstein05} (sphere), \citen{Perrin1934} and \citen{Perrin1936} |
| 1145 |
> |
(ellipsoid), \citen{Stimson:1926qy} and \citen{Davis:1969uq} |
| 1146 |
> |
(dumbbell). The other model systems have no known analytic solution. |
| 1147 |
> |
All diffusion constants are reported in units of $10^{-3}$ cm$^2$ / ps (= |
| 1148 |
> |
$10^{-4}$ \AA$^2$ / fs). } |
| 1149 |
> |
\begin{tabular}{lccccccc} |
| 1150 |
|
\hline |
| 1151 |
< |
& & & & &Translation \\ |
| 1152 |
< |
\hline |
| 1153 |
< |
& NVE & & Theoretical & Langevin & \\ |
| 774 |
< |
\hline |
| 775 |
< |
& $\eta$ & D & D & method & D \\ |
| 1151 |
> |
& \multicolumn{2}c{microcanonical simulation} & & \multicolumn{3}c{Theoretical} & Langevin \\ |
| 1152 |
> |
\cline{2-3} \cline{5-7} |
| 1153 |
> |
model & $\eta$ (centipoise) & D & & Analytical & method & Hydrodynamics & simulation \\ |
| 1154 |
|
\hline |
| 1155 |
< |
sphere & 3.480159e-03 & 1.643135e-04 & 1.942779e-04 & exact & 1.982283e-04 \\ |
| 1156 |
< |
ellipsoid & 2.551262e-03 & 2.437492e-04 & 2.335756e-04 & exact & 2.374905e-04 \\ |
| 1157 |
< |
& 2.551262e-03 & 2.437492e-04 & 2.335756e-04 & rough shell & 2.284088e-04 \\ |
| 1158 |
< |
dumbell & 2.41276e-03 & 2.129432e-04 & 2.090239e-04 & bead model & 2.148098e-04 \\ |
| 1159 |
< |
& 2.41276e-03 & 2.129432e-04 & 2.090239e-04 & rough shell & 2.013219e-04 \\ |
| 1160 |
< |
banana & 2.9846e-03 & 1.527819e-04 & & rough shell & 1.54807e-04 \\ |
| 1161 |
< |
lipid & 3.488661e-03 & 0.9562979e-04 & & rough shell & 1.320987e-04 \\ |
| 1155 |
> |
sphere & 0.279 & 3.06 & & 2.42 & exact & 2.42 & 2.33 \\ |
| 1156 |
> |
ellipsoid & 0.255 & 2.44 & & 2.34 & exact & 2.34 & 2.37 \\ |
| 1157 |
> |
& 0.255 & 2.44 & & 2.34 & rough shell & 2.36 & 2.28 \\ |
| 1158 |
> |
dumbbell & 0.308 & 2.06 & & 1.64 & bead model & 1.65 & 1.62 \\ |
| 1159 |
> |
& 0.308 & 2.06 & & 1.64 & rough shell & 1.59 & 1.62 \\ |
| 1160 |
> |
banana & 0.298 & 1.53 & & & rough shell & 1.56 & 1.55 \\ |
| 1161 |
> |
lipid & 0.349 & 0.96 & & & rough shell & 1.33 & 1.32 \\ |
| 1162 |
|
\end{tabular} |
| 1163 |
|
\label{tab:translation} |
| 1164 |
|
\end{center} |
| 1168 |
|
\begin{table*} |
| 1169 |
|
\begin{minipage}{\linewidth} |
| 1170 |
|
\begin{center} |
| 1171 |
< |
\caption{} |
| 1172 |
< |
\begin{tabular}{lccccc} |
| 1171 |
> |
\caption{Orientational relaxation times ($\tau$) for the model systems using |
| 1172 |
> |
microcanonical simulation (with explicit solvent), theoretical |
| 1173 |
> |
predictions, and Langevin simulations (with implicit solvent). All |
| 1174 |
> |
relaxation times are for the rotational correlation function with |
| 1175 |
> |
$\ell = 2$ and are reported in units of ps. The ellipsoidal model has |
| 1176 |
> |
an exact solution for the orientational correlation time due to |
| 1177 |
> |
Perrin, but the other model systems have no known analytic solution.} |
| 1178 |
> |
\begin{tabular}{lccccccc} |
| 1179 |
|
\hline |
| 1180 |
< |
& & & & &Rotation \\ |
| 1181 |
< |
\hline |
| 1182 |
< |
& NVE & & Theoretical & Langevin & \\ |
| 799 |
< |
\hline |
| 800 |
< |
& $\eta$ & $\tau_0$ & $\tau_0$ & method & $\tau_0$ \\ |
| 1180 |
> |
& \multicolumn{2}c{microcanonical simulation} & & \multicolumn{3}c{Theoretical} & Langevin \\ |
| 1181 |
> |
\cline{2-3} \cline{5-7} |
| 1182 |
> |
model & $\eta$ (centipoise) & $\tau$ & & Perrin & method & Hydrodynamic & simulation \\ |
| 1183 |
|
\hline |
| 1184 |
< |
sphere & 3.480159e-03 & & 1.208178e+04 & exact & 1.20628e+04 \\ |
| 1185 |
< |
ellipsoid & 2.551262e-03 & 4.66806e+04 & 2.198986e+04 & exact & 2.21507e+04 \\ |
| 1186 |
< |
& 2.551262e-03 & 4.66806e+04 & 2.198986e+04 & rough shell & 2.21714e+04 \\ |
| 1187 |
< |
dumbell & 2.41276e-03 & 1.42974e+04 & & bead model & 7.12435e+04 \\ |
| 1188 |
< |
& 2.41276e-03 & 1.42974e+04 & & rough shell & 7.04765e+04 \\ |
| 1189 |
< |
banana & 2.9846e-03 & 6.38323e+04 & & rough shell & 7.0945e+04 \\ |
| 1190 |
< |
lipid & 3.488661e-03 & 7.79595e+04 & & rough shell & 7.78886e+04 \\ |
| 1184 |
> |
sphere & 0.279 & & & 9.69 & exact & 9.69 & 9.64 \\ |
| 1185 |
> |
ellipsoid & 0.255 & 46.7 & & 22.0 & exact & 22.0 & 22.2 \\ |
| 1186 |
> |
& 0.255 & 46.7 & & 22.0 & rough shell & 22.6 & 22.2 \\ |
| 1187 |
> |
dumbbell & 0.308 & 14.1 & & & bead model & 50.0 & 50.1 \\ |
| 1188 |
> |
& 0.308 & 14.1 & & & rough shell & 41.5 & 41.3 \\ |
| 1189 |
> |
banana & 0.298 & 63.8 & & & rough shell & 70.9 & 70.9 \\ |
| 1190 |
> |
lipid & 0.349 & 78.0 & & & rough shell & 76.9 & 77.9 \\ |
| 1191 |
> |
\hline |
| 1192 |
|
\end{tabular} |
| 1193 |
|
\label{tab:rotation} |
| 1194 |
|
\end{center} |
| 1195 |
|
\end{minipage} |
| 1196 |
|
\end{table*} |
| 1197 |
|
|
| 1198 |
< |
Langevin dynamics simulations are applied to study the formation of |
| 1199 |
< |
the ripple phase of lipid membranes. The initial configuration is |
| 1198 |
> |
\section{Application: A rigid-body lipid bilayer} |
| 1199 |
> |
|
| 1200 |
> |
The Langevin dynamics integrator was applied to study the formation of |
| 1201 |
> |
corrugated structures emerging from simulations of the coarse grained |
| 1202 |
> |
lipid molecular models presented above. The initial configuration is |
| 1203 |
|
taken from our molecular dynamics studies on lipid bilayers with |
| 1204 |
< |
lennard-Jones sphere solvents. The solvent molecules are excluded from |
| 1205 |
< |
the system, the experimental value of water viscosity is applied to |
| 1206 |
< |
mimic the heat bath. Fig. XXX is the snapshot of the stable |
| 1207 |
< |
configuration of the system, the ripple structure stayed stable after |
| 1208 |
< |
100 ns run. The efficiency of the simulation is increased by one order |
| 1204 |
> |
lennard-Jones sphere solvents. The solvent molecules were excluded |
| 1205 |
> |
from the system, and the experimental value for the viscosity of water |
| 1206 |
> |
at 20C ($\eta = 1.00$ cp) was used to mimic the hydrodynamic effects |
| 1207 |
> |
of the solvent. The absence of explicit solvent molecules and the |
| 1208 |
> |
stability of the integrator allowed us to take timesteps of 50 fs. A |
| 1209 |
> |
total simulation run time of 100 ns was sampled. |
| 1210 |
> |
Fig. \ref{fig:bilayer} shows the configuration of the system after 100 |
| 1211 |
> |
ns, and the ripple structure remains stable during the entire |
| 1212 |
> |
trajectory. Compared with using explicit bead-model solvent |
| 1213 |
> |
molecules, the efficiency of the simulation has increased by an order |
| 1214 |
|
of magnitude. |
| 1215 |
|
|
| 825 |
– |
\subsection{Langevin Dynamics of Banana Shaped Molecules} |
| 826 |
– |
|
| 827 |
– |
In order to verify that Langevin dynamics can mimic the dynamics of |
| 828 |
– |
the systems absent of explicit solvents, we carried out two sets of |
| 829 |
– |
simulations and compare their dynamic properties. |
| 830 |
– |
Fig.~\ref{langevin:twoBanana} shows a snapshot of the simulation |
| 831 |
– |
made of 256 pentane molecules and two banana shaped molecules at |
| 832 |
– |
273~K. It has an equivalent implicit solvent system containing only |
| 833 |
– |
two banana shaped molecules with viscosity of 0.289 center poise. To |
| 834 |
– |
calculate the hydrodynamic properties of the banana shaped molecule, |
| 835 |
– |
we created a rough shell model (see Fig.~\ref{langevin:roughShell}), |
| 836 |
– |
in which the banana shaped molecule is represented as a ``shell'' |
| 837 |
– |
made of 2266 small identical beads with size of 0.3 \AA on the |
| 838 |
– |
surface. Applying the procedure described in |
| 839 |
– |
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
| 840 |
– |
identified the center of resistance at (0 $\rm{\AA}$, 0.7482 $\rm{\AA}$, |
| 841 |
– |
-0.1988 $\rm{\AA}$), as well as the resistance tensor, |
| 842 |
– |
\[ |
| 843 |
– |
\left( {\begin{array}{*{20}c} |
| 844 |
– |
0.9261 & 0 & 0&0&0.08585&0.2057\\ |
| 845 |
– |
0& 0.9270&-0.007063& 0.08585&0&0\\ |
| 846 |
– |
0&-0.007063&0.7494&0.2057&0&0\\ |
| 847 |
– |
0&0.0858&0.2057& 58.64& 0&0\\ |
| 848 |
– |
0.08585&0&0&0&48.30&3.219&\\ |
| 849 |
– |
0.2057&0&0&0&3.219&10.7373\\ |
| 850 |
– |
\end{array}} \right). |
| 851 |
– |
\] |
| 852 |
– |
where the units for translational, translation-rotation coupling and rotational tensors are $\frac{kcal \cdot fs}{mol \cdot \rm{\AA}^2}$, $\frac{kcal \cdot fs}{mol \cdot \rm{\AA} \cdot rad}$ and $\frac{kcal \cdot fs}{mol \cdot rad^2}$ respectively. |
| 853 |
– |
Curves of the velocity auto-correlation functions in |
| 854 |
– |
Fig.~\ref{langevin:vacf} were shown to match each other very well. |
| 855 |
– |
However, because of the stochastic nature, simulation using Langevin |
| 856 |
– |
dynamics was shown to decay slightly faster than MD. In order to |
| 857 |
– |
study the rotational motion of the molecules, we also calculated the |
| 858 |
– |
auto-correlation function of the principle axis of the second GB |
| 859 |
– |
particle, $u$. The discrepancy shown in Fig.~\ref{langevin:uacf} was |
| 860 |
– |
probably due to the reason that we used the experimental viscosity directly instead of calculating bulk viscosity from simulation. |
| 861 |
– |
|
| 1216 |
|
\begin{figure} |
| 1217 |
|
\centering |
| 1218 |
< |
\includegraphics[width=\linewidth]{roughShell.pdf} |
| 1219 |
< |
\caption[Rough shell model for banana shaped molecule]{Rough shell |
| 1220 |
< |
model for banana shaped molecule.} \label{langevin:roughShell} |
| 1218 |
> |
\includegraphics[width=\linewidth]{bilayer} |
| 1219 |
> |
\caption[Snapshot of a bilayer of rigid-body models for lipids]{A |
| 1220 |
> |
snapshot of a bilayer composed of rigid-body models for lipid |
| 1221 |
> |
molecules evolving using the Langevin integrator described in this |
| 1222 |
> |
work.} \label{fig:bilayer} |
| 1223 |
|
\end{figure} |
| 1224 |
|
|
| 869 |
– |
\begin{figure} |
| 870 |
– |
\centering |
| 871 |
– |
\includegraphics[width=\linewidth]{twoBanana.pdf} |
| 872 |
– |
\caption[Snapshot from Simulation of Two Banana Shaped Molecules and |
| 873 |
– |
256 Pentane Molecules]{Snapshot from simulation of two Banana shaped |
| 874 |
– |
molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
| 875 |
– |
\end{figure} |
| 876 |
– |
|
| 877 |
– |
\begin{figure} |
| 878 |
– |
\centering |
| 879 |
– |
\includegraphics[width=\linewidth]{vacf.pdf} |
| 880 |
– |
\caption[Plots of Velocity Auto-correlation Functions]{Velocity |
| 881 |
– |
auto-correlation functions of NVE (explicit solvent) in blue and |
| 882 |
– |
Langevin dynamics (implicit solvent) in red.} \label{langevin:vacf} |
| 883 |
– |
\end{figure} |
| 884 |
– |
|
| 885 |
– |
\begin{figure} |
| 886 |
– |
\centering |
| 887 |
– |
\includegraphics[width=\linewidth]{uacf.pdf} |
| 888 |
– |
\caption[Auto-correlation functions of the principle axis of the |
| 889 |
– |
middle GB particle]{Auto-correlation functions of the principle axis |
| 890 |
– |
of the middle GB particle of NVE (blue) and Langevin dynamics |
| 891 |
– |
(red).} \label{langevin:uacf} |
| 892 |
– |
\end{figure} |
| 893 |
– |
|
| 1225 |
|
\section{Conclusions} |
| 1226 |
|
|
| 1227 |
|
We have presented a new Langevin algorithm by incorporating the |
| 1228 |
|
hydrodynamics properties of arbitrary shaped molecules into an |
| 1229 |
< |
advanced symplectic integration scheme. The temperature control |
| 1230 |
< |
ability of this algorithm was demonstrated by a set of simulations |
| 1231 |
< |
with different viscosities. It was also shown to have significant |
| 1232 |
< |
advantage of producing rapid thermal equilibration over |
| 902 |
< |
Nos\'{e}-Hoover method. Further studies in systems involving banana |
| 903 |
< |
shaped molecules illustrated that the dynamic properties could be |
| 904 |
< |
preserved by using this new algorithm as an implicit solvent model. |
| 1229 |
> |
advanced symplectic integration scheme. Further studies in systems |
| 1230 |
> |
involving banana shaped molecules illustrated that the dynamic |
| 1231 |
> |
properties could be preserved by using this new algorithm as an |
| 1232 |
> |
implicit solvent model. |
| 1233 |
|
|
| 1234 |
|
|
| 1235 |
|
\section{Acknowledgments} |
| 1239 |
|
of Notre Dame. |
| 1240 |
|
\newpage |
| 1241 |
|
|
| 1242 |
< |
\bibliographystyle{jcp2} |
| 1242 |
> |
\bibliographystyle{jcp} |
| 1243 |
|
\bibliography{langevin} |
| 1244 |
|
|
| 1245 |
|
\end{document} |
