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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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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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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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systems\cite{Palacios1998,Berkov2002,Denisov2003}. For instance, the |
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oscillation power spectrum of nanoparticles from Langevin dynamics |
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simulation 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}. In an attempt to reduce the |
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computational cost of simulation, multiple time stepping (MTS) |
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methods have been introduced and have been of great interest to |
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macromolecule and protein community\cite{Tuckerman1992}. Relying on |
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the observation that forces between distant atoms generally |
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demonstrate slower fluctuations than forces between close atoms, MTS |
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method are generally implemented by evaluating the slowly |
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fluctuating forces less frequently than the fast ones. |
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Unfortunately, nonlinear instability resulting from increasing |
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timestep in MTS simulation have became a critical obstruction |
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preventing the long time simulation. Due to the coupling to the heat |
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bath, Langevin dynamics has been shown to be able to damp out the |
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resonance artifact more efficiently\cite{Sandu1999}. |
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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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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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estimation of friction tensor from hydrodynamics theory into the |
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sophisticated rigid body dynamics. |
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\section{Computational methods{\label{methodSec}}} |
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\section{Computational Methods{\label{methodSec}}} |
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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 become |
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impractical when the system become more and more complicate. |
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Instead, various approaches based on hydrodynamics have been |
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developed to calculate the friction coefficients. In general, |
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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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\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shape}} |
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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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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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exactly. In 1936, Perrin extended Stokes's law to general |
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ellipsoids, also called a triaxial ellipsoid, which is given in |
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Cartesian coordinates 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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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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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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: |
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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 : |
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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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tensors |
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\[ |
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\begin{array}{l} |
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\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}} \\ |
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\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + 2a}} \\ |
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\end{array}, |
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\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
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\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
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2a}}, |
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\end{array} |
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\] |
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and |
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\[ |
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\begin{array}{l} |
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\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}} \\ |
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\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
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\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}} \\ |
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\end{array}. |
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\] |
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|
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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}} |
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Unlike spherical and other regular shaped molecules, there is not |
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analytical solution for friction tensor of any arbitrary shaped |
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Unlike spherical and other simply shaped molecules, there is no |
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analytical solution for the friction tensor for arbitrarily shaped |
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rigid molecules. The ellipsoid of revolution model and general |
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triaxial ellipsoid model have been used to approximate the |
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hydrodynamic properties of rigid bodies. However, since the mapping |
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from all possible ellipsoidal space, $r$-space, to all possible |
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combination of rotational diffusion coefficients, $D$-space is not |
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from all possible ellipsoidal spaces, $r$-space, to all possible |
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combination of rotational diffusion coefficients, $D$-space, is not |
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unique\cite{Wegener1979} as well as the intrinsic coupling between |
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translational and rotational motion of rigid body, general ellipsoid |
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is not always suitable for modeling arbitrarily shaped rigid |
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molecule. A number of studies have been devoted to determine the |
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friction tensor for irregularly shaped rigid bodies using more |
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translational and rotational motion of rigid body, general |
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ellipsoids are not always suitable for modeling arbitrarily shaped |
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rigid molecule. A number of studies have been devoted to determine |
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the friction tensor for irregularly shaped rigid bodies using more |
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advanced method where the molecule of interest was modeled by |
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combinations of spheres(beads)\cite{Carrasco1999} and the |
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hydrodynamics properties of the molecule can be calculated using the |
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B_{ij} = \delta _{ij} \frac{I}{{6\pi \eta R}} + (1 - \delta _{ij} |
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)T_{ij} |
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\] |
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where $\delta _{ij}$ is Kronecker delta function. Inverting matrix |
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$B$, we obtain |
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where $\delta _{ij}$ is Kronecker delta function. Inverting the $B$ |
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matrix, we obtain |
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\[ |
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C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
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The resistance tensor depends on the origin to which they refer. The |
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proper location for applying friction force is the center of |
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resistance (reaction), at which the trace of rotational resistance |
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tensor, $ \Xi ^{rr}$ reaches minimum. Mathematically, the center of |
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resistance is defined as an unique point of the rigid body at which |
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the translation-rotation coupling tensor are symmetric, |
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resistance (or center of reaction), at which the trace of rotational |
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resistance tensor, $ \Xi ^{rr}$ reaches a minimum value. |
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Mathematically, the center of resistance is defined as an unique |
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point of the rigid body at which the translation-rotation coupling |
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tensor are symmetric, |
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\begin{equation} |
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\Xi^{tr} = \left( {\Xi^{tr} } \right)^T |
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\label{introEquation:definitionCR} |
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\end{equation} |
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Form Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
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From Equation \ref{introEquation:ResistanceTensorArbitraryOrigin}, |
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we can easily find out that the translational resistance tensor is |
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origin independent, while the rotational resistance tensor and |
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translation-rotation coupling resistance tensor depend on the |
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origin. Given resistance tensor at an arbitrary origin $O$, and a |
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vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
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origin. Given the resistance tensor at an arbitrary origin $O$, and |
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a vector ,$r_{OP}(x_{OP}, y_{OP}, z_{OP})$, from $O$ to $P$, we can |
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obtain the resistance tensor at $P$ by |
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\begin{equation} |
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\begin{array}{l} |
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where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
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joining center of resistance $R$ and origin $O$. |
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|
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\subsection{Langevin dynamics for rigid particles of arbitrary shape\label{LDRB}} |
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\subsection{Langevin Dynamics for Rigid Particles of Arbitrary Shape\label{LDRB}} |
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|
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Consider a Langevin equation of motions in generalized coordinates |
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\begin{equation} |
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+ \frac{h}{2} {\bf \tau}^b(t + h) . |
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\end{align*} |
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|
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\section{Results and discussion} |
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\section{Results and Discussion} |
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|
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The Langevin algorithm described in previous section has been |
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implemented in {\sc oopse}\cite{Meineke2005} and applied to the |
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studies of kinetics and thermodynamic properties in several systems. |
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|
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\subsection{Temperature control} |
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\subsection{Temperature Control} |
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|
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As shown in Eq.~\ref{randomForce}, random collisions associated with |
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the solvent's thermal motions is controlled by the external |
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\begin{center} |
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\begin{tabular}{|l|l|l|} |
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\hline |
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\eta & ${\rm{T}}_{{\rm{avg}}}$ & ${\rm{T}}_{{\rm{rms}}}$ \\ |
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$\eta$ & $\text{T}_{\text{avg}}$ & $\text{T}_{\text{rms}}$ \\ |
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1 & 300.47 & 10.99 \\ |
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0.1 & 301.19 & 11.136 \\ |
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0.01 & 303.04 & 11.796 \\ |
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NVE (see green curve in Fig.~\ref{langevin:temperature}) which loses |
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the temperature control ability. |
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|
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– |
|
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{temperature.eps} |
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temperature fluctuation versus time.} \label{langevin:temperature} |
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\end{figure} |
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|
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\subsection{Langevin dynamics of banana-shaped molecule} |
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\subsection{Langevin Dynamics of Banana Shaped Molecule} |
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|
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In order to verify that Langevin dynamics can mimic the dynamics of |
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the systems absent of explicit solvents, we carried out two sets of |
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simulations and compare their dynamic properties. |
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|
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Fig.~\ref{langevin:twoBanana} shows a snapshot of the simulation |
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made of 256 pentane molecules and two banana shaped molecules at |
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273~K. It has an equivalent implicit solvent system containing only |
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two banana shaped molecules with viscosity of 0.289 center poise. To |
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calculate the hydrodynamic properties of the banana shaped molecule, |
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we create a rough shell model (see Fig.~\ref{langevin:roughShell}), |
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in which the banana shaped molecule is represented as a ``shell'' |
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made of 2266 small identical beads with size of 0.3 $\AA$ on the |
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surface. Applying the procedure described in |
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Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
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identified the center of resistance at $(0, 0.7482, -0.1988)$, as |
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well as the resistance tensor, |
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\[ |
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\left( {\begin{array}{*{20}c} |
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0.9261 & 1.310e-14 & -7.292e-15&5.067e-14&0.08585&0.2057\\ |
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3.968e-14& 0.9270&-0.007063& 0.08585&6.764e-14&4.846e-14\\ |
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-6.561e-16&-0.007063&0.7494&0.2057&4.846e-14&1.5036e-14\\ |
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5.067e-14&0.0858&0.2057& 58.64& 8.563e-13&-8.5736\\ |
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0.08585&6.764e-14&4.846e-14&1.555e-12&48.30&3.219&\\ |
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0.2057&4.846e-14&1.5036e-14&-3.904e-13&3.219&10.7373\\ |
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\end{array}} \right). |
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\] |
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+ |
|
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+ |
|
| 612 |
+ |
|
| 613 |
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\begin{figure} |
| 614 |
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\centering |
| 615 |
< |
\includegraphics[width=\linewidth]{one_banana.eps} |
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\caption[]{.} \label{langevin:banana} |
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> |
\includegraphics[width=\linewidth]{roughShell.eps} |
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\caption[Rough shell model for banana shaped molecule]{Rough shell |
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model for banana shaped molecule.} \label{langevin:roughShell} |
| 618 |
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\end{figure} |
| 619 |
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|
| 620 |
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\begin{figure} |
| 621 |
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\centering |
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\includegraphics[width=\linewidth]{roughShell.eps} |
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\caption[Rough Shell]{Rough Shell.} \label{langevin:roughShell} |
| 622 |
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\includegraphics[width=\linewidth]{twoBanana.eps} |
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\caption[Snapshot from Simulation of Two Banana Shaped Molecules and |
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256 Pentane Molecules]{Snapshot from simulation of two Banana shaped |
| 625 |
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molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
| 626 |
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\end{figure} |
| 627 |
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|
| 628 |
|
\begin{figure} |
| 629 |
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\centering |
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\includegraphics[width=\linewidth]{twoBanana.eps} |
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\caption[Two Banana Shaped Molecules]{Two Banana Shaped Molecules.} |
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\label{langevin:twoBanana} |
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\includegraphics[width=\linewidth]{vacf.eps} |
| 631 |
> |
\caption[Plots of Velocity Auto-correlation functions]{Velocity |
| 632 |
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Auto-correlation function of NVE (blue) and Langevin dynamics |
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(red).} \label{langevin:twoBanana} |
| 634 |
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\end{figure} |
| 635 |
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|
| 636 |
+ |
\begin{figure} |
| 637 |
+ |
\centering |
| 638 |
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\includegraphics[width=\linewidth]{uacf.eps} |
| 639 |
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\caption[Snapshot from Simulation of Two Banana Shaped Molecules and |
| 640 |
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256 Pentane Molecules]{Snapshot from simulation of two Banana shaped |
| 641 |
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molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
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\end{figure} |
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|
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\section{Conclusions} |
