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\chapter{\label{chapt:methodology}LANGEVIN DYNAMICS FOR RIGID BODIES OF ARBITRARY SHAPE} |
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\chapter{\label{chapt:langevin}LANGEVIN DYNAMICS FOR RIGID BODIES OF ARBITRARY SHAPE} |
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\section{Introduction} |
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%applications of langevin dynamics |
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As an excellent alternative to newtonian dynamics, Langevin |
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dynamics, which mimics a simple heat bath with stochastic and |
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dissipative forces, has been applied in a variety of studies. The |
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stochastic treatment of the solvent enables us to carry out |
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substantially longer time simulation. Implicit solvent Langevin |
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dynamics simulation of met-enkephalin not only outperforms explicit |
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solvent simulation on computation efficiency, but also agrees very |
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well with explicit solvent simulation on dynamics |
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properties\cite{Shen2002}. Recently, applying Langevin dynamics with |
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UNRES model, Liow and his coworkers suggest that protein folding |
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pathways can be possibly exploited within a reasonable amount of |
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time\cite{Liwo2005}. The stochastic nature of the Langevin dynamics |
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also enhances the sampling of the system and increases the |
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probability of crossing energy barrier\cite{Banerjee2004, Cui2003}. |
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Combining Langevin dynamics with Kramers's theory, Klimov and |
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Thirumalai identified the free-energy barrier by studying the |
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viscosity dependence of the protein folding rates\cite{Klimov1997}. |
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In order to account for solvent induced interactions missing from |
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implicit solvent model, Kaya incorporated desolvation free energy |
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barrier into implicit coarse-grained solvent model in protein |
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folding/unfolding study and discovered a higher free energy barrier |
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between the native and denatured states. Because of its stability |
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against noise, Langevin dynamics is very suitable for studying |
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remagnetization processes in various |
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systems\cite{Garcia-Palacios1998,Berkov2002,Denisov2003}. For |
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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, |
9 |
> |
has been applied in a variety of studies. The stochastic treatment |
10 |
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of the solvent enables us to carry out substantially longer time |
11 |
> |
simulations. Implicit solvent Langevin dynamics simulations of |
12 |
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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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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}. In an |
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attempt to reduce the computational cost of simulation, multiple |
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time stepping (MTS) methods have been introduced and have been of |
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great interest to macromolecule and protein |
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community\cite{Tuckerman1992}. Relying on the observation that |
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forces between distant atoms generally demonstrate slower |
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fluctuations than forces between close atoms, MTS method are |
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generally implemented by evaluating the slowly fluctuating forces |
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less frequently than the fast ones. Unfortunately, nonlinear |
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instability resulting from increasing timestep in MTS simulation |
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have became a critical obstruction preventing the long time |
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simulation. Due to the coupling to the heat bath, Langevin dynamics |
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has been shown to be able to damp out the resonance artifact more |
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efficiently\cite{Sandu1999}. |
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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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|
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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 the slow processes in biomolecular systems. Modeling |
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the DNA as a chain of rigid spheres beads, which subject to harmonic |
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potentials as well as excluded volume potentials, Mielke and his |
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coworkers discover rapid superhelical stress generations from the |
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stochastic simulation of twin supercoiling DNA with response to |
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induced torques\cite{Mielke2004}. Membrane fusion is another key |
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biological process which controls a variety of physiological |
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functions, such as release of neurotransmitters \textit{etc}. A |
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typical fusion event happens on the time scale of millisecond, which |
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is impracticable to study using all atomistic model with newtonian |
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mechanics. With the help of coarse-grained rigid body model and |
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stochastic dynamics, the fusion pathways were exploited by many |
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one can study slow processes in biomolecular systems. Modeling DNA |
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as a chain of rigid beads, which are subject to harmonic potentials |
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as well as excluded volume potentials, Mielke and his coworkers |
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discovered rapid superhelical stress generations from the stochastic |
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simulation of twin supercoiling DNA with response to induced |
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torques\cite{Mielke2004}. Membrane fusion is another key biological |
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process which controls a variety of physiological functions, such as |
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release of neurotransmitters \textit{etc}. A typical fusion event |
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happens on the time scale of millisecond, which is impractical to |
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study using atomistic models with newtonian mechanics. With the help |
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of coarse-grained rigid body model and stochastic dynamics, the |
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fusion pathways were explored by many |
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researchers\cite{Noguchi2001,Noguchi2002,Shillcock2005}. Due to the |
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difficulty of numerical integration of anisotropy rotation, most of |
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the rigid body models are simply modeled by sphere, cylinder, |
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ellipsoid or other regular shapes in stochastic simulations. In an |
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effort to account for the diffusion anisotropy of the arbitrary |
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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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error and bias are introduced into the system due to the arbitrary |
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order of applying the noncommuting rotation |
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errors and biases are introduced into the system due to the |
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arbitrary order of applying the noncommuting rotation |
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operators\cite{Beard2003}. Based on the observation the momentum |
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relaxation time is much less than the time step, one may ignore the |
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inertia in Brownian dynamics. However, assumption of the zero |
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inertia in Brownian dynamics. However, the assumption of zero |
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average acceleration is not always true for cooperative motion which |
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is common in protein motion. An inertial Brownian dynamics (IBD) was |
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proposed to address this issue by adding an inertial correction |
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term\cite{Beard2001}. As a complement to IBD which has a lower bound |
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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{Beard2001}. LTID can also deal with the rotational |
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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 cylinder and ellipsoid are inadequate to |
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represent most complex macromolecule assemblies. These intricate |
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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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hydrodynamics properties can be calculated using variant |
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hydrodynamic interaction tensors. |
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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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|
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The goal of the present work is to develop a Langevin dynamics |
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algorithm for arbitrary rigid particles by integrating the accurate |
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estimation of friction tensor from hydrodynamics theory into the |
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sophisticated rigid body 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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|
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\section{Computational methods{\label{methodSec}}} |
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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 velocity |
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autocorrelation function. However, this approach become impractical |
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when the system become more and more complicate. Instead, various |
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approaches based on hydrodynamics have been developed to calculate |
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the friction coefficients. The friction effect is isotropic in |
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Equation, $\zeta$ can be taken as a scalar. In general, friction |
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tensor $\Xi$ is a $6\times 6$ matrix given by |
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Theoretically, the friction kernel can be determined using the |
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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 |
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\[ |
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\Xi = \left( {\begin{array}{*{20}c} |
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{\Xi _{}^{tt} } & {\Xi _{}^{rt} } \\ |
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{\Xi _{}^{tr} } & {\Xi _{}^{rr} } \\ |
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\end{array}} \right). |
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\] |
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Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are translational friction |
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tensor and rotational resistance (friction) tensor respectively, |
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while ${\Xi^{tr} }$ is translation-rotation coupling tensor and $ |
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{\Xi^{rt} }$ is rotation-translation coupling tensor. When a |
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particle moves in a fluid, it may experience friction force or |
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torque along the opposite direction of the velocity or angular |
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velocity, |
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Here, $ {\Xi^{tt} }$ and $ {\Xi^{rr} }$ are $3 \times 3$ |
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translational friction tensor and rotational resistance (friction) |
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tensor respectively, while ${\Xi^{tr} }$ is translation-rotation |
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coupling tensor and $ {\Xi^{rt} }$ is rotation-translation coupling |
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tensor. When a particle moves in a fluid, it may experience friction |
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force or torque along the opposite direction of the velocity or |
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angular velocity, |
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\[ |
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\left( \begin{array}{l} |
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F_R \\ |
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\end{array} \right) |
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\] |
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where $F_r$ is the friction force and $\tau _R$ is the friction |
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toque. |
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torque. |
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|
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\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shape}} |
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\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shapes}} |
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|
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For a spherical particle, the translational and rotational friction |
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constant can be calculated from Stoke's law, |
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For a spherical particle with slip boundary conditions, the |
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translational and rotational friction constant can be calculated |
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from Stoke's law, |
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\[ |
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\Xi ^{tt} = \left( {\begin{array}{*{20}c} |
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{6\pi \eta R} & 0 & 0 \\ |
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\end{array}} \right) |
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\] |
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where $\eta$ is the viscosity of the solvent and $R$ is the |
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hydrodynamics radius. |
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hydrodynamic radius. |
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|
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Other non-spherical shape, such as cylinder and ellipsoid |
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\textit{etc}, are widely used as reference for developing new |
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hydrodynamics theory, because their properties can be calculated |
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exactly. In 1936, Perrin extended Stokes's law to general ellipsoid, |
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also called a triaxial ellipsoid, which is given in Cartesian |
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coordinates by\cite{Perrin1934, Perrin1936} |
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Other non-spherical shapes, such as cylinders and ellipsoids, are |
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widely used as references for developing new hydrodynamics theory, |
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because their properties can be calculated exactly. In 1936, Perrin |
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extended Stokes's law to general ellipsoids, also called a triaxial |
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ellipsoid, which is given in Cartesian coordinates |
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by\cite{Perrin1934, Perrin1936} |
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\[ |
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\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
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}} = 1 |
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\] |
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where the semi-axes are of lengths $a$, $b$, and $c$. Unfortunately, |
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due to the complexity of the elliptic integral, only the ellipsoid |
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with the restriction of two axes having to be equal, \textit{i.e.} |
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with the restriction of two axes being equal, \textit{i.e.} |
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prolate($ a \ge b = c$) and oblate ($ a < b = c $), can be solved |
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exactly. Introducing an elliptic integral parameter $S$ for prolate, |
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exactly. Introducing an elliptic integral parameter $S$ for prolate |
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ellipsoids : |
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\[ |
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S = \frac{2}{{\sqrt {a^2 - b^2 } }}\ln \frac{{a + \sqrt {a^2 - b^2 |
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} }}{b}, |
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\] |
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and oblate, |
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and oblate ellipsoids: |
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\[ |
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S = \frac{2}{{\sqrt {b^2 - a^2 } }}arctg\frac{{\sqrt {b^2 - a^2 } |
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}}{a} |
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\], |
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}}{a}, |
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\] |
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one can write down the translational and rotational resistance |
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tensors |
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\[ |
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\begin{array}{l} |
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\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
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\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
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\end{array}, |
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\] |
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\begin{eqnarray*} |
175 |
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\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
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\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
177 |
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2a}}, |
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\end{eqnarray*} |
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and |
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\[ |
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\begin{array}{l} |
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\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
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\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
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\end{array}. |
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\] |
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\begin{eqnarray*} |
181 |
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\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
182 |
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\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}}. |
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\end{eqnarray*} |
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|
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\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shape}} |
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\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}} |
186 |
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|
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Unlike spherical and other regular shaped molecules, there is not |
188 |
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analytical solution for friction tensor of any arbitrary shaped |
187 |
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Unlike spherical and other simply shaped molecules, there is no |
188 |
> |
analytical solution for the friction tensor for arbitrarily shaped |
189 |
|
rigid molecules. The ellipsoid of revolution model and general |
190 |
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triaxial ellipsoid model have been used to approximate the |
191 |
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hydrodynamic properties of rigid bodies. However, since the mapping |
192 |
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from all possible ellipsoidal space, $r$-space, to all possible |
193 |
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combination of rotational diffusion coefficients, $D$-space is not |
192 |
> |
from all possible ellipsoidal spaces, $r$-space, to all possible |
193 |
> |
combination of rotational diffusion coefficients, $D$-space, is not |
194 |
|
unique\cite{Wegener1979} as well as the intrinsic coupling between |
195 |
< |
translational and rotational motion of rigid body, general ellipsoid |
196 |
< |
is not always suitable for modeling arbitrarily shaped rigid |
197 |
< |
molecule. A number of studies have been devoted to determine the |
198 |
< |
friction tensor for irregularly shaped rigid bodies using more |
199 |
< |
advanced method where the molecule of interest was modeled by |
200 |
< |
combinations of spheres(beads)\cite{Carrasco1999} and the |
195 |
> |
translational and rotational motion of rigid bodies, general |
196 |
> |
ellipsoids are not always suitable for modeling arbitrarily shaped |
197 |
> |
rigid molecules. A number of studies have been devoted to |
198 |
> |
determining the friction tensor for irregularly shaped rigid bodies |
199 |
> |
using more advanced methods where the molecule of interest was |
200 |
> |
modeled by a combinations of spheres\cite{Carrasco1999} and the |
201 |
|
hydrodynamics properties of the molecule can be calculated using the |
202 |
|
hydrodynamic interaction tensor. Let us consider a rigid assembly of |
203 |
< |
$N$ beads immersed in a continuous medium. Due to hydrodynamics |
203 |
> |
$N$ beads immersed in a continuous medium. Due to hydrodynamic |
204 |
|
interaction, the ``net'' velocity of $i$th bead, $v'_i$ is different |
205 |
|
than its unperturbed velocity $v_i$, |
206 |
|
\[ |
214 |
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\label{introEquation:tensorExpression} |
215 |
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\end{equation} |
216 |
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This equation is the basis for deriving the hydrodynamic tensor. In |
217 |
< |
1930, Oseen and Burgers gave a simple solution to Equation |
218 |
< |
\ref{introEquation:tensorExpression} |
217 |
> |
1930, Oseen and Burgers gave a simple solution to |
218 |
> |
Eq.~\ref{introEquation:tensorExpression} |
219 |
|
\begin{equation} |
220 |
|
T_{ij} = \frac{1}{{8\pi \eta r_{ij} }}\left( {I + \frac{{R_{ij} |
221 |
|
R_{ij}^T }}{{R_{ij}^2 }}} \right). \label{introEquation:oseenTensor} |
231 |
|
\frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right)} \right]. |
232 |
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\label{introEquation:RPTensorNonOverlapped} |
233 |
|
\end{equation} |
234 |
< |
Both of the Equation \ref{introEquation:oseenTensor} and Equation |
235 |
< |
\ref{introEquation:RPTensorNonOverlapped} have an assumption $R_{ij} |
236 |
< |
\ge \sigma _i + \sigma _j$. An alternative expression for |
234 |
> |
Both of the Eq.~\ref{introEquation:oseenTensor} and |
235 |
> |
Eq.~\ref{introEquation:RPTensorNonOverlapped} have an assumption |
236 |
> |
$R_{ij} \ge \sigma _i + \sigma _j$. An alternative expression for |
237 |
|
overlapping beads with the same radius, $\sigma$, is given by |
238 |
|
\begin{equation} |
239 |
|
T_{ij} = \frac{1}{{6\pi \eta R_{ij} }}\left[ {\left( {1 - |
241 |
|
\frac{2}{{32}}\frac{{R_{ij} R_{ij}^T }}{{R_{ij} \sigma }}} \right] |
242 |
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\label{introEquation:RPTensorOverlapped} |
243 |
|
\end{equation} |
260 |
– |
|
244 |
|
To calculate the resistance tensor at an arbitrary origin $O$, we |
245 |
|
construct a $3N \times 3N$ matrix consisting of $N \times N$ |
246 |
|
$B_{ij}$ blocks |
256 |
|
B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
257 |
|
)T_{ij} |
258 |
|
\] |
259 |
< |
where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
260 |
< |
$B$, we obtain |
278 |
< |
|
259 |
> |
where $\delta _{ij}$ is the Kronecker delta function. Inverting the |
260 |
> |
$B$ matrix, we obtain |
261 |
|
\[ |
262 |
|
C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
263 |
|
{C_{11} } & \ldots & {C_{1N} } \\ |
264 |
|
\vdots & \ddots & \vdots \\ |
265 |
|
{C_{N1} } & \cdots & {C_{NN} } \\ |
266 |
< |
\end{array}} \right) |
266 |
> |
\end{array}} \right), |
267 |
|
\] |
268 |
< |
, which can be partitioned into $N \times N$ $3 \times 3$ block |
269 |
< |
$C_{ij}$. With the help of $C_{ij}$ and skew matrix $U_i$ |
268 |
> |
which can be partitioned into $N \times N$ $3 \times 3$ block |
269 |
> |
$C_{ij}$. With the help of $C_{ij}$ and the skew matrix $U_i$ |
270 |
|
\[ |
271 |
|
U_i = \left( {\begin{array}{*{20}c} |
272 |
|
0 & { - z_i } & {y_i } \\ |
275 |
|
\end{array}} \right) |
276 |
|
\] |
277 |
|
where $x_i$, $y_i$, $z_i$ are the components of the vector joining |
278 |
< |
bead $i$ and origin $O$. Hence, the elements of resistance tensor at |
278 |
> |
bead $i$ and origin $O$, the elements of resistance tensor at |
279 |
|
arbitrary origin $O$ can be written as |
280 |
< |
\begin{equation} |
281 |
< |
\begin{array}{l} |
300 |
< |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } , \\ |
280 |
> |
\begin{eqnarray} |
281 |
> |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\ |
282 |
|
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
283 |
< |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j \\ |
303 |
< |
\end{array} |
283 |
> |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\ |
284 |
|
\label{introEquation:ResistanceTensorArbitraryOrigin} |
285 |
< |
\end{equation} |
306 |
< |
|
285 |
> |
\end{eqnarray} |
286 |
|
The resistance tensor depends on the origin to which they refer. The |
287 |
< |
proper location for applying friction force is the center of |
288 |
< |
resistance (reaction), at which the trace of rotational resistance |
289 |
< |
tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
290 |
< |
resistance is defined as an unique point of the rigid body at which |
291 |
< |
the translation-rotation coupling tensor are symmetric, |
287 |
> |
proper location for applying the friction force is the center of |
288 |
> |
resistance (or center of reaction), at which the trace of rotational |
289 |
> |
resistance tensor, $ \Xi ^{rr}$ reaches a minimum value. |
290 |
> |
Mathematically, the center of resistance is defined as an unique |
291 |
> |
point of the rigid body at which the translation-rotation coupling |
292 |
> |
tensors are symmetric, |
293 |
|
\begin{equation} |
294 |
|
\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
295 |
|
\label{introEquation:definitionCR} |
296 |
|
\end{equation} |
297 |
< |
Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
298 |
< |
we can easily find out that the translational resistance tensor is |
297 |
> |
From Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
298 |
> |
we can easily derive that the translational resistance tensor is |
299 |
|
origin independent, while the rotational resistance tensor and |
300 |
|
translation-rotation coupling resistance tensor depend on the |
301 |
< |
origin. Given resistance tensor at an arbitrary origin $O$, and a |
302 |
< |
vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
301 |
> |
origin. Given the resistance tensor at an arbitrary origin $O$, and |
302 |
> |
a vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
303 |
|
obtain the resistance tensor at $P$ by |
304 |
|
\begin{equation} |
305 |
|
\begin{array}{l} |
317 |
|
{ - y_{OP} } & {x_{OP} } & 0 \\ |
318 |
|
\end{array}} \right) |
319 |
|
\] |
320 |
< |
Using Equations \ref{introEquation:definitionCR} and |
321 |
< |
\ref{introEquation:resistanceTensorTransformation}, one can locate |
322 |
< |
the position of center of resistance, |
320 |
> |
Using Eq.~\ref{introEquation:definitionCR} and |
321 |
> |
Eq.~\ref{introEquation:resistanceTensorTransformation}, one can |
322 |
> |
locate the position of center of resistance, |
323 |
|
\begin{eqnarray*} |
324 |
|
\left( \begin{array}{l} |
325 |
|
x_{OR} \\ |
336 |
|
(\Xi _O^{tr} )_{xy} - (\Xi _O^{tr} )_{yx} \\ |
337 |
|
\end{array} \right) \\ |
338 |
|
\end{eqnarray*} |
359 |
– |
|
339 |
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
340 |
|
joining center of resistance $R$ and origin $O$. |
341 |
|
|
342 |
< |
\subsection{Langevin dynamics for rigid particles of arbitrary shape\label{LDRB}} |
342 |
> |
\subsection{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}} |
343 |
|
|
344 |
< |
Consider a Langevin equation of motions in generalized coordinates |
344 |
> |
Consider the Langevin equations of motion in generalized coordinates |
345 |
|
\begin{equation} |
346 |
|
M_i \dot V_i (t) = F_{s,i} (t) + F_{f,i(t)} + F_{r,i} (t) |
347 |
|
\label{LDGeneralizedForm} |
348 |
|
\end{equation} |
349 |
|
where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
350 |
|
and moment of inertial) matrix and $V_i$ is a generalized velocity, |
351 |
< |
$V_i = V_i(v_i,\omega _i)$. The right side of Eq.~ |
352 |
< |
(\ref{LDGeneralizedForm}) consists of three generalized forces in |
351 |
> |
$V_i = V_i(v_i,\omega _i)$. The right side of |
352 |
> |
Eq.~\ref{LDGeneralizedForm} consists of three generalized forces in |
353 |
|
lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
354 |
|
$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
355 |
|
system in Newtownian mechanics typically refers to lab-fixed frame, |
358 |
|
in body-fixed frame and converted back to lab-fixed frame by: |
359 |
|
\[ |
360 |
|
\begin{array}{l} |
361 |
< |
F_{f,i}^l (t) = A^T F_{f,i}^b (t), \\ |
362 |
< |
F_{r,i}^l (t) = A^T F_{r,i}^b (t) \\ |
363 |
< |
\end{array}. |
361 |
> |
F_{f,i}^l (t) = Q^T F_{f,i}^b (t), \\ |
362 |
> |
F_{r,i}^l (t) = Q^T F_{r,i}^b (t). \\ |
363 |
> |
\end{array} |
364 |
|
\] |
365 |
|
Here, the body-fixed friction force $F_{r,i}^b$ is proportional to |
366 |
|
the body-fixed velocity at center of resistance $v_{R,i}^b$ and |
367 |
< |
angular velocity $\omega _i$, |
367 |
> |
angular velocity $\omega _i$ |
368 |
|
\begin{equation} |
369 |
|
F_{r,i}^b (t) = \left( \begin{array}{l} |
370 |
|
f_{r,i}^b (t) \\ |
384 |
|
\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
385 |
|
2k_B T\Xi _R \delta (t - t'). \label{randomForce} |
386 |
|
\end{equation} |
408 |
– |
|
387 |
|
The equation of motion for $v_i$ can be written as |
388 |
|
\begin{equation} |
389 |
|
m\dot v_i (t) = f_{t,i} (t) = f_{s,i} (t) + f_{f,i}^l (t) + |
398 |
|
\tau _{r,i}^b = \tau _{r,i}^b +r_{MR} \times f_{r,i}^b |
399 |
|
\end{equation} |
400 |
|
where $r_{MR}$ is the vector from the center of mass to the center |
401 |
< |
of the resistance. Instead of integrating angular velocity in |
402 |
< |
lab-fixed frame, we consider the equation of motion of angular |
403 |
< |
momentum in body-fixed frame |
401 |
> |
of the resistance. Instead of integrating the angular velocity in |
402 |
> |
lab-fixed frame, we consider the equation of angular momentum in |
403 |
> |
body-fixed frame |
404 |
|
\begin{equation} |
405 |
|
\dot j_i (t) = \tau _{t,i} (t) = \tau _{s,i} (t) + \tau _{f,i}^b (t) |
406 |
|
+ \tau _{r,i}^b(t) |
407 |
|
\end{equation} |
430 |
– |
|
408 |
|
Embedding the friction terms into force and torque, one can |
409 |
|
integrate the langevin equations of motion for rigid body of |
410 |
|
arbitrary shape in a velocity-Verlet style 2-part algorithm, where |
421 |
|
{\bf j}\left(t + h / 2 \right) &\leftarrow {\bf j}(t) |
422 |
|
+ \frac{h}{2} {\bf \tau}^b(t), \\ |
423 |
|
% |
424 |
< |
\mathsf{A}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
424 |
> |
\mathsf{Q}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j} |
425 |
|
(t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right). |
426 |
|
\end{align*} |
450 |
– |
|
427 |
|
In this context, the $\mathrm{rotate}$ function is the reversible |
428 |
|
product of the three body-fixed rotations, |
429 |
|
\begin{equation} |
432 |
|
/ 2) \cdot \mathsf{G}_x(a_x /2), |
433 |
|
\end{equation} |
434 |
|
where each rotational propagator, $\mathsf{G}_\alpha(\theta)$, |
435 |
< |
rotates both the rotation matrix ($\mathsf{A}$) and the body-fixed |
435 |
> |
rotates both the rotation matrix ($\mathsf{Q}$) and the body-fixed |
436 |
|
angular momentum (${\bf j}$) by an angle $\theta$ around body-fixed |
437 |
|
axis $\alpha$, |
438 |
|
\begin{equation} |
439 |
|
\mathsf{G}_\alpha( \theta ) = \left\{ |
440 |
|
\begin{array}{lcl} |
441 |
< |
\mathsf{A}(t) & \leftarrow & \mathsf{A}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
441 |
> |
\mathsf{Q}(t) & \leftarrow & \mathsf{Q}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\ |
442 |
|
{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf |
443 |
|
j}(0). |
444 |
|
\end{array} |
458 |
|
\end{array} |
459 |
|
\right). |
460 |
|
\end{equation} |
461 |
< |
All other rotations follow in a straightforward manner. |
461 |
> |
All other rotations follow in a straightforward manner. After the |
462 |
> |
first part of the propagation, the forces and body-fixed torques are |
463 |
> |
calculated at the new positions and orientations |
464 |
|
|
487 |
– |
After the first part of the propagation, the forces and body-fixed |
488 |
– |
torques are calculated at the new positions and orientations |
489 |
– |
|
465 |
|
{\tt doForces:} |
466 |
|
\begin{align*} |
467 |
|
{\bf f}(t + h) &\leftarrow |
470 |
|
{\bf \tau}^{s}(t + h) &\leftarrow {\bf u}(t + h) |
471 |
|
\times \frac{\partial V}{\partial {\bf u}}, \\ |
472 |
|
% |
473 |
< |
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{A}(t + h) |
473 |
> |
{\bf \tau}^{b}(t + h) &\leftarrow \mathsf{Q}(t + h) |
474 |
|
\cdot {\bf \tau}^s(t + h). |
475 |
|
\end{align*} |
476 |
+ |
Once the forces and torques have been obtained at the new time step, |
477 |
+ |
the velocities can be advanced to the same time value. |
478 |
|
|
502 |
– |
{\sc oopse} automatically updates ${\bf u}$ when the rotation matrix |
503 |
– |
$\mathsf{A}$ is calculated in {\tt moveA}. Once the forces and |
504 |
– |
torques have been obtained at the new time step, the velocities can |
505 |
– |
be advanced to the same time value. |
506 |
– |
|
479 |
|
{\tt moveB:} |
480 |
|
\begin{align*} |
481 |
|
{\bf v}\left(t + h \right) &\leftarrow {\bf v}\left(t + h / 2 |
487 |
|
+ \frac{h}{2} {\bf \tau}^b(t + h) . |
488 |
|
\end{align*} |
489 |
|
|
490 |
< |
\section{Results and discussion} |
490 |
> |
\section{Results and Discussion} |
491 |
|
|
492 |
|
The Langevin algorithm described in previous section has been |
493 |
< |
implemented in {\sc oopse}\cite{Meineke2005} and applied to the |
494 |
< |
studies of kinetics and thermodynamic properties in several systems. |
493 |
> |
implemented in {\sc oopse}\cite{Meineke2005} and applied to studies |
494 |
> |
of the static and dynamic properties in several systems. |
495 |
|
|
496 |
< |
\subsection{Temperature control} |
496 |
> |
\subsection{Temperature Control} |
497 |
|
|
498 |
|
As shown in Eq.~\ref{randomForce}, random collisions associated with |
499 |
|
the solvent's thermal motions is controlled by the external |
502 |
|
the algorithm. In order to verify the stability of this new |
503 |
|
algorithm, a series of simulations are performed on system |
504 |
|
consisiting of 256 SSD water molecules with different viscosities. |
505 |
< |
Initial configuration for the simulations is taken from a 1ns NVT |
506 |
< |
simulation with a cubic box of 19.7166~\AA. All simulation are |
505 |
> |
The initial configuration for the simulations is taken from a 1ns |
506 |
> |
NVT simulation with a cubic box of 19.7166~\AA. All simulation are |
507 |
|
carried out with cutoff radius of 9~\AA and 2 fs time step for 1 ns |
508 |
< |
with reference temperature at 300~K. Average temperature as a |
508 |
> |
with reference temperature at 300~K. The average temperature as a |
509 |
|
function of $\eta$ is shown in Table \ref{langevin:viscosity} where |
510 |
|
the temperatures range from 303.04~K to 300.47~K for $\eta = 0.01 - |
511 |
|
1$ poise. The better temperature control at higher viscosity can be |
512 |
|
explained by the finite size effect and relative slow relaxation |
513 |
|
rate at lower viscosity regime. |
514 |
|
\begin{table} |
515 |
< |
\caption{Average temperatures from Langevin dynamics simulations of |
516 |
< |
SSD water molecules with reference temperature at 300~K.} |
515 |
> |
\caption{AVERAGE TEMPERATURES FROM LANGEVIN DYNAMICS SIMULATIONS OF |
516 |
> |
SSD WATER MOLECULES WITH REFERENCE TEMPERATURE AT 300~K.} |
517 |
|
\label{langevin:viscosity} |
518 |
|
\begin{center} |
519 |
< |
\begin{tabular}{|l|l|l|} |
519 |
> |
\begin{tabular}{lll} |
520 |
|
\hline |
521 |
< |
\eta & $\text{T}_{\text{avg}}$ & $\text{T}_{\text{rms}}$ \\ |
521 |
> |
$\eta$ & $\text{T}_{\text{avg}}$ & $\text{T}_{\text{rms}}$ \\ |
522 |
> |
\hline |
523 |
|
1 & 300.47 & 10.99 \\ |
524 |
|
0.1 & 301.19 & 11.136 \\ |
525 |
|
0.01 & 303.04 & 11.796 \\ |
540 |
|
Langevin dynamics with viscosity of 0.1 poise demonstrates a faster |
541 |
|
scaling to the desire temperature. In extremely lower friction |
542 |
|
regime (when $ \eta \approx 0$), Langevin dynamics becomes normal |
543 |
< |
NVE (see green curve in Fig.~\ref{langevin:temperature}) which loses |
544 |
< |
the temperature control ability. |
543 |
> |
NVE (see orange curve in Fig.~\ref{langevin:temperature}) which |
544 |
> |
loses the temperature control ability. |
545 |
|
|
573 |
– |
|
546 |
|
\begin{figure} |
547 |
|
\centering |
548 |
|
\includegraphics[width=\linewidth]{temperature.eps} |
550 |
|
temperature fluctuation versus time.} \label{langevin:temperature} |
551 |
|
\end{figure} |
552 |
|
|
553 |
< |
\subsection{Langevin dynamics of banana-shaped molecule} |
553 |
> |
\subsection{Langevin Dynamics of Banana Shaped Molecules} |
554 |
|
|
555 |
+ |
In order to verify that Langevin dynamics can mimic the dynamics of |
556 |
+ |
the systems absent of explicit solvents, we carried out two sets of |
557 |
+ |
simulations and compare their dynamic properties. |
558 |
+ |
Fig.~\ref{langevin:twoBanana} shows a snapshot of the simulation |
559 |
+ |
made of 256 pentane molecules and two banana shaped molecules at |
560 |
+ |
273~K. It has an equivalent implicit solvent system containing only |
561 |
+ |
two banana shaped molecules with viscosity of 0.289 center poise. To |
562 |
+ |
calculate the hydrodynamic properties of the banana shaped molecule, |
563 |
+ |
we created a rough shell model (see Fig.~\ref{langevin:roughShell}), |
564 |
+ |
in which the banana shaped molecule is represented as a ``shell'' |
565 |
+ |
made of 2266 small identical beads with size of 0.3 \AA on the |
566 |
+ |
surface. Applying the procedure described in |
567 |
+ |
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
568 |
+ |
identified the center of resistance at $(0 \AA, 0.7482 \AA, |
569 |
+ |
-0.1988 \AA)$, as well as the resistance tensor, |
570 |
+ |
\[ |
571 |
+ |
\left( {\begin{array}{*{20}c} |
572 |
+ |
0.9261 & 0 & 0&0&0.08585&0.2057\\ |
573 |
+ |
0& 0.9270&-0.007063& 0.08585&0&0\\ |
574 |
+ |
0&0.007063&0.7494&0.2057&0&0\\ |
575 |
+ |
0&0.0858&0.2057& 58.64& 0&-8.5736\\ |
576 |
+ |
0.08585&0&0&0&48.30&3.219&\\ |
577 |
+ |
0.2057&0&0&0&3.219&10.7373\\ |
578 |
+ |
\end{array}} \right). |
579 |
+ |
\] |
580 |
+ |
%\[ |
581 |
+ |
%\left( {\begin{array}{*{20}c} |
582 |
+ |
%0.9261 & 1.310e-14 & -7.292e-15&5.067e-14&0.08585&0.2057\\ |
583 |
+ |
%3.968e-14& 0.9270&-0.007063& 0.08585&6.764e-14&4.846e-14\\ |
584 |
+ |
%-6.561e-16&-0.007063&0.7494&0.2057&4.846e-14&1.5036e-14\\ |
585 |
+ |
%5.067e-14&0.0858&0.2057& 58.64& 8.563e-13&-8.5736\\ |
586 |
+ |
%0.08585&6.764e-14&4.846e-14&1.555e-12&48.30&3.219&\\ |
587 |
+ |
%0.2057&4.846e-14&1.5036e-14&-3.904e-13&3.219&10.7373\\ |
588 |
+ |
%\end{array}} \right). |
589 |
+ |
%\] |
590 |
+ |
|
591 |
+ |
Curves of the velocity auto-correlation functions in |
592 |
+ |
Fig.~\ref{langevin:vacf} were shown to match each other very well. |
593 |
+ |
However, because of the stochastic nature, simulation using Langevin |
594 |
+ |
dynamics was shown to decay slightly faster than MD. In order to |
595 |
+ |
study the rotational motion of the molecules, we also calculated the |
596 |
+ |
auto-correlation function of the principle axis of the second GB |
597 |
+ |
particle, $u$. The discrepancy shown in Fig.~\ref{langevin:uacf} was |
598 |
+ |
probably due to the reason that the viscosity using in the |
599 |
+ |
simulations only partially preserved the dynamics of the system. |
600 |
+ |
|
601 |
|
\begin{figure} |
602 |
|
\centering |
603 |
< |
\includegraphics[width=\linewidth]{one_banana.eps} |
604 |
< |
\caption[]{.} \label{langevin:banana} |
603 |
> |
\includegraphics[width=\linewidth]{roughShell.eps} |
604 |
> |
\caption[Rough shell model for banana shaped molecule]{Rough shell |
605 |
> |
model for banana shaped molecule.} \label{langevin:roughShell} |
606 |
|
\end{figure} |
607 |
|
|
608 |
|
\begin{figure} |
609 |
|
\centering |
610 |
< |
\includegraphics[width=\linewidth]{roughShell.eps} |
611 |
< |
\caption[Rough Shell]{Rough Shell.} \label{langevin:roughShell} |
610 |
> |
\includegraphics[width=\linewidth]{twoBanana.eps} |
611 |
> |
\caption[Snapshot from Simulation of Two Banana Shaped Molecules and |
612 |
> |
256 Pentane Molecules]{Snapshot from simulation of two Banana shaped |
613 |
> |
molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
614 |
|
\end{figure} |
615 |
|
|
616 |
|
\begin{figure} |
617 |
|
\centering |
618 |
< |
\includegraphics[width=\linewidth]{twoBanana.eps} |
619 |
< |
\caption[Two Banana Shaped Molecules]{Two Banana Shaped Molecules.} |
620 |
< |
\label{langevin:twoBanana} |
618 |
> |
\includegraphics[width=\linewidth]{vacf.eps} |
619 |
> |
\caption[Plots of Velocity Auto-correlation Functions]{Velocity |
620 |
> |
auto-correlation functions of NVE (explicit solvent) in blue) and |
621 |
> |
Langevin dynamics (implicit solvent) in red.} \label{langevin:vacf} |
622 |
|
\end{figure} |
623 |
|
|
624 |
+ |
\begin{figure} |
625 |
+ |
\centering |
626 |
+ |
\includegraphics[width=\linewidth]{uacf.eps} |
627 |
+ |
\caption[Auto-correlation functions of the principle axis of the |
628 |
+ |
middle GB particle]{Auto-correlation functions of the principle axis |
629 |
+ |
of the middle GB particle of NVE (blue) and Langevin dynamics |
630 |
+ |
(red).} \label{langevin:uacf} |
631 |
+ |
\end{figure} |
632 |
+ |
|
633 |
|
\section{Conclusions} |
634 |
+ |
|
635 |
+ |
We have presented a new Langevin algorithm by incorporating the |
636 |
+ |
hydrodynamics properties of arbitrary shaped molecules into an |
637 |
+ |
advanced symplectic integration scheme. The temperature control |
638 |
+ |
ability of this algorithm was demonstrated by a set of simulations |
639 |
+ |
with different viscosities. It was also shown to have significant |
640 |
+ |
advantage of producing rapid thermal equilibration over |
641 |
+ |
Nos\'{e}-Hoover method. Further studies in systems involving banana |
642 |
+ |
shaped molecules illustrated that the dynamic properties could be |
643 |
+ |
preserved by using this new algorithm as an implicit solvent model. |