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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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|
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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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|
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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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|
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It is very important to develop stable and efficient methods to |
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integrate the equations of motion of orientational degrees of |
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freedom. Euler angles are the nature choice to describe the |
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rotational degrees of freedom. However, due to its singularity, the |
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numerical integration of corresponding equations of motion is very |
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inefficient and inaccurate. Although an alternative integrator using |
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different sets of Euler angles can overcome this difficulty\cite{}, |
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the computational penalty and the lost of angular momentum |
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conservation still remain. In 1977, a singularity free |
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representation utilizing quaternions was developed by |
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Evans\cite{Evans1977}. Unfortunately, this approach suffer from the |
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nonseparable Hamiltonian resulted from quaternion representation, |
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which prevents the symplectic algorithm to be utilized. Another |
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different approach is to apply holonomic constraints to the atoms |
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belonging to the rigid body\cite{}. Each atom moves independently |
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under the normal forces deriving from potential energy and |
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constraint forces which are used to guarantee the rigidness. |
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However, due to their iterative nature, SHAKE and Rattle algorithm |
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converge very slowly when the number of constraint increases. |
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|
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The break through in geometric literature suggests that, in order to |
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develop a long-term integration scheme, one should preserve the |
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geometric structure of the flow. Matubayasi and Nakahara developed a |
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time-reversible integrator for rigid bodies in quaternion |
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representation. Although it is not symplectic, this integrator still |
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demonstrates a better long-time energy conservation than traditional |
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methods because of the time-reversible nature. Extending |
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Trotter-Suzuki to general system with a flat phase space, Miller and |
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his colleagues devised an novel symplectic, time-reversible and |
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volume-preserving integrator in quaternion representation. However, |
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all of the integrators in quaternion representation suffer from the |
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computational penalty of constructing a rotation matrix from |
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quaternions to evolve coordinates and velocities at every time step. |
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An alternative integration scheme utilizing rotation matrix directly |
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is RSHAKE , in which a conjugate momentum to rotation matrix is |
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introduced to re-formulate the Hamiltonian's equation and the |
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Hamiltonian is evolved in a constraint manifold by iteratively |
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satisfying the orthogonality constraint. However, RSHAKE is |
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inefficient because of the iterative procedure. An extremely |
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efficient integration scheme in rotation matrix representation, |
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which also preserves the same structural properties of the |
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Hamiltonian flow as Miller's integrator, is proposed by Dullweber, |
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Leimkuhler and McLachlan (DLM). |
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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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estimation of friction tensor from hydrodynamics theory into the |
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sophisticated rigid body dynamics. |
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\section{Method{\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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\subsection{\label{introSection:frictionTensor}Friction Tensor} |
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Theoretically, the friction kernel can be determined using the |
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velocity autocorrelation function. However, this approach 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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|
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\subsubsection{\label{introSection:resistanceTensorRegular}\textbf{The Resistance Tensor for Regular Shape}} |
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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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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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\[ |
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\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 |
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}} = 1 |
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\end{array}. |
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\] |
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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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\[ |
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C = B^{ - 1} = \left( {\begin{array}{*{20}c} |
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|
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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} |
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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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\end{equation} |
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where $M_i$ is a $6\times6$ generalized diagonal mass (include mass |
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and moment of inertial) matrix and $V_i$ is a generalized velocity, |
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$V_i = V_i(v_i,\omega _i)$. The right side of Eq. |
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$V_i = V_i(v_i,\omega _i)$. The right side of Eq.~ |
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(\ref{LDGeneralizedForm}) consists of three generalized forces in |
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lab-fixed frame, systematic force $F_{s,i}$, dissipative force |
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$F_{f,i}$ and stochastic force $F_{r,i}$. While the evolution of the |
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\begin{equation} |
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\left\langle {F_{r,i}^l (t)(F_{r,i}^l (t'))^T } \right\rangle = |
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\left\langle {F_{r,i}^b (t)(F_{r,i}^b (t'))^T } \right\rangle = |
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2k_B T\Xi _R \delta (t - t'). |
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2k_B T\Xi _R \delta (t - t'). \label{randomForce} |
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\end{equation} |
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|
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The equation of motion for $v_i$ can be written as |
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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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|
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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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temperature. The capability to maintain the temperature of the whole |
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system was usually used to measure the stability and efficiency of |
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the algorithm. In order to verify the stability of this new |
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algorithm, a series of simulations are performed on system |
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consisiting of 256 SSD water molecules with different viscosities. |
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Initial configuration for the simulations is taken from a 1ns NVT |
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simulation with a cubic box of 19.7166~\AA. All simulation are |
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carried out with cutoff radius of 9~\AA and 2 fs time step for 1 ns |
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with reference temperature at 300~K. Average temperature as a |
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function of $\eta$ is shown in Table \ref{langevin:viscosity} where |
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the temperatures range from 303.04~K to 300.47~K for $\eta = 0.01 - |
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1$ poise. The better temperature control at higher viscosity can be |
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explained by the finite size effect and relative slow relaxation |
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rate at lower viscosity regime. |
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\begin{table} |
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\caption{Average temperatures from Langevin dynamics simulations of |
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SSD water molecules with reference temperature at 300~K.} |
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\label{langevin:viscosity} |
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\begin{center} |
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\begin{tabular}{|l|l|l|} |
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\hline |
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$\eta$ & $\text{T}_{\text{avg}}$ & $\text{T}_{\text{rms}}$ \\ |
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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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\hline |
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\end{tabular} |
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\end{center} |
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\end{table} |
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|
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Another set of calculation were performed to study the efficiency of |
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temperature control using different temperature coupling schemes. |
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The starting configuration is cooled to 173~K and evolved using NVE, |
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NVT, and Langevin dynamic with time step of 2 fs. |
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Fig.~\ref{langevin:temperature} shows the heating curve obtained as |
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the systems reach equilibrium. The orange curve in |
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Fig.~\ref{langevin:temperature} represents the simulation using |
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Nos\'e-Hoover temperature scaling scheme with thermostat of 5 ps |
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which gives reasonable tight coupling, while the blue one from |
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Langevin dynamics with viscosity of 0.1 poise demonstrates a faster |
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scaling to the desire temperature. In extremely lower friction |
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regime (when $ \eta \approx 0$), Langevin dynamics becomes normal |
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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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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{temperature.eps} |
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\caption[Plot of Temperature Fluctuation Versus Time]{Plot of |
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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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|
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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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\subsubsection{Simulations Contain One Banana Shaped Molecule} |
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|
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Fig.~\ref{langevin:oneBanana} shows a snapshot of the simulation |
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made of 256 pentane molecules and one banana shaped molecule at |
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273~K. It has an equivalent implicit solvent system containing only |
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one banana shaped molecule 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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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{roughShell.eps} |
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\caption[Rough shell model for banana shaped molecule]{Rough shell |
616 |
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model for banana shaped molecule.} \label{langevin:roughShell} |
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\end{figure} |
618 |
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|
619 |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{oneBanana.eps} |
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\caption[Snapshot from Simulation of One Banana Shaped Molecules and |
623 |
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256 Pentane Molecules]{Snapshot from simulation of one Banana shaped |
624 |
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molecules and 256 pentane molecules.} \label{langevin:oneBanana} |
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\end{figure} |
626 |
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|
627 |
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\subsubsection{Simulations Contain Two Banana Shaped Molecules} |
628 |
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|
629 |
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\begin{figure} |
630 |
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\centering |
631 |
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\includegraphics[width=\linewidth]{twoBanana.eps} |
632 |
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\caption[Snapshot from Simulation of Two Banana Shaped Molecules and |
633 |
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256 Pentane Molecules]{Snapshot from simulation of two Banana shaped |
634 |
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molecules and 256 pentane molecules.} \label{langevin:twoBanana} |
635 |
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\end{figure} |
636 |
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|
637 |
|
\section{Conclusions} |