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\chapter{\label{chapt:methodology}Langevin Dynamics for Rigid Bodies of Arbitrary Shape} |
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\chapter{\label{chapt:methodology}LANGEVIN DYNAMICS FOR RIGID BODIES OF ARBITRARY SHAPE} |
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\section{Introduction} |
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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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%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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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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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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%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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\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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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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\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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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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The equation of motion for $v_i$ can be written as |
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\section{Results and discussion} |
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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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\subsection{Temperature control} |
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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 & ${\rm{T}}_{{\rm{avg}}}$ & ${\rm{T}}_{{\rm{rms}}}$ \\ |
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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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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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\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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\subsection{Langevin dynamics of banana-shaped molecule} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{one_banana.eps} |
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\caption[]{.} \label{langevin:banana} |
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\end{figure} |
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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]{Rough Shell.} \label{langevin:roughShell} |
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\end{figure} |
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\begin{figure} |
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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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\end{figure} |
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\section{Conclusions} |