--- trunk/tengDissertation/Langevin.tex	2006/06/29 23:00:35	2909
+++ trunk/tengDissertation/Langevin.tex	2006/07/18 05:47:33	2944
@@ -11,27 +11,27 @@ solvent simulations for dynamical properties\cite{Shen
 simulations. Implicit solvent Langevin dynamics simulations of
 met-enkephalin not only outperform explicit solvent simulations for
 computational efficiency, but also agrees very well with explicit
-solvent simulations for dynamical properties\cite{Shen2002}.
+solvent simulations for dynamical properties.\cite{Shen2002}
 Recently, applying Langevin dynamics with the UNRES model, Liow and
 his coworkers suggest that protein folding pathways can be possibly
-explored within a reasonable amount of time\cite{Liwo2005}. The
+explored within a reasonable amount of time.\cite{Liwo2005} The
 stochastic nature of the Langevin dynamics also enhances the
 sampling of the system and increases the probability of crossing
-energy barriers\cite{Banerjee2004, Cui2003}. Combining Langevin
+energy barriers.\cite{Banerjee2004, Cui2003} Combining Langevin
 dynamics with Kramers's theory, Klimov and Thirumalai identified
 free-energy barriers by studying the viscosity dependence of the
-protein folding rates\cite{Klimov1997}. In order to account for
+protein folding rates.\cite{Klimov1997} In order to account for
 solvent induced interactions missing from implicit solvent model,
 Kaya incorporated desolvation free energy barrier into implicit
 coarse-grained solvent model in protein folding/unfolding studies
 and discovered a higher free energy barrier between the native and
 denatured states. Because of its stability against noise, Langevin
 dynamics is very suitable for studying remagnetization processes in
-various systems\cite{Palacios1998,Berkov2002,Denisov2003}. For
+various systems.\cite{Palacios1998,Berkov2002,Denisov2003} For
 instance, the oscillation power spectrum of nanoparticles from
 Langevin dynamics simulation has the same peak frequencies for
 different wave vectors, which recovers the property of magnetic
-excitations in small finite structures\cite{Berkov2005a}.
+excitations in small finite structures.\cite{Berkov2005a}
 
 %review langevin/browninan dynamics for arbitrarily shaped rigid body
 Combining Langevin or Brownian dynamics with rigid body dynamics,
@@ -40,14 +40,14 @@ torques\cite{Mielke2004}. Membrane fusion is another k
 as well as excluded volume potentials, Mielke and his coworkers
 discovered rapid superhelical stress generations from the stochastic
 simulation of twin supercoiling DNA with response to induced
-torques\cite{Mielke2004}. Membrane fusion is another key biological
+torques.\cite{Mielke2004} Membrane fusion is another key biological
 process which controls a variety of physiological functions, such as
 release of neurotransmitters \textit{etc}. A typical fusion event
-happens on the time scale of millisecond, which is impractical to
+happens on the time scale of a millisecond, which is impractical to
 study using atomistic models with newtonian mechanics. With the help
 of coarse-grained rigid body model and stochastic dynamics, the
 fusion pathways were explored by many
-researchers\cite{Noguchi2001,Noguchi2002,Shillcock2005}. Due to the
+researchers.\cite{Noguchi2001,Noguchi2002,Shillcock2005} Due to the
 difficulty of numerical integration of anisotropic rotation, most of
 the rigid body models are simply modeled using spheres, cylinders,
 ellipsoids or other regular shapes in stochastic simulations. In an
@@ -56,20 +56,20 @@ consecutive rotations\cite{Fernandes2002}. Unfortunate
 dynamics simulation algorithm\cite{Ermak1978,Allison1991} by
 incorporating a generalized $6\times6$ diffusion tensor and
 introducing a simple rotation evolution scheme consisting of three
-consecutive rotations\cite{Fernandes2002}. Unfortunately, unexpected
+consecutive rotations.\cite{Fernandes2002} Unfortunately, unexpected
 errors and biases are introduced into the system due to the
 arbitrary order of applying the noncommuting rotation
-operators\cite{Beard2003}. Based on the observation the momentum
+operators.\cite{Beard2003} Based on the observation the momentum
 relaxation time is much less than the time step, one may ignore the
 inertia in Brownian dynamics. However, the assumption of zero
 average acceleration is not always true for cooperative motion which
 is common in protein motion. An inertial Brownian dynamics (IBD) was
 proposed to address this issue by adding an inertial correction
-term\cite{Beard2000}. As a complement to IBD which has a lower bound
+term.\cite{Beard2000} As a complement to IBD which has a lower bound
 in time step because of the inertial relaxation time, long-time-step
 inertial dynamics (LTID) can be used to investigate the inertial
 behavior of the polymer segments in low friction
-regime\cite{Beard2000}. LTID can also deal with the rotational
+regime.\cite{Beard2000} LTID can also deal with the rotational
 dynamics for nonskew bodies without translation-rotation coupling by
 separating the translation and rotation motion and taking advantage
 of the analytical solution of hydrodynamics properties. However,
@@ -172,15 +172,15 @@ tensors
 one can write down the translational and rotational resistance
 tensors
 \begin{eqnarray*}
- \Xi _a^{tt}  = 16\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - b^2 )S - 2a}}. \\
- \Xi _b^{tt}  = \Xi _c^{tt}  = 32\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - 3b^2 )S +
+ \Xi _a^{tt}  & = & 16\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - b^2 )S - 2a}}. \\
+ \Xi _b^{tt}  & = & \Xi _c^{tt}  = 32\pi \eta \frac{{a^2  - b^2 }}{{(2a^2  - 3b^2 )S +
  2a}},
 \end{eqnarray*}
 and
 \begin{eqnarray*}
- \Xi _a^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^2  - b^2 )b^2 }}{{2a - b^2 S}}, \\
- \Xi _b^{rr}  = \Xi _c^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^4  - b^4 )}}{{(2a^2  - b^2 )S - 2a}}.
-\end{eqnarray}
+ \Xi _a^{rr} & = & \frac{{32\pi }}{3}\eta \frac{{(a^2  - b^2 )b^2 }}{{2a - b^2 S}}, \\
+ \Xi _b^{rr} & = & \Xi _c^{rr}  = \frac{{32\pi }}{3}\eta \frac{{(a^4  - b^4 )}}{{(2a^2  - b^2 )S - 2a}}.
+\end{eqnarray*}
 
 \subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}}
 
@@ -223,7 +223,7 @@ Garc\'{i}a de la Torre and Bloomfield\cite{Torre1977},
 Here $R_{ij}$ is the distance vector between bead $i$ and bead $j$.
 A second order expression for element of different size was
 introduced by Rotne and Prager\cite{Rotne1969} and improved by
-Garc\'{i}a de la Torre and Bloomfield\cite{Torre1977},
+Garc\'{i}a de la Torre and Bloomfield,\cite{Torre1977}
 \begin{equation}
 T_{ij}  = \frac{1}{{8\pi \eta R_{ij} }}\left[ {\left( {I +
 \frac{{R_{ij} R_{ij}^T }}{{R_{ij}^2 }}} \right) + R\frac{{\sigma
@@ -278,9 +278,9 @@ arbitrary origin $O$ can be written as
 bead $i$ and origin $O$, the elements of resistance tensor at
 arbitrary origin $O$ can be written as
 \begin{eqnarray}
- \Xi _{}^{tt}  = \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\
- \Xi _{}^{tr}  = \Xi _{}^{rt}  = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\
- \Xi _{}^{rr}  =  - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\
+ \Xi _{}^{tt}  & = & \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\
+ \Xi _{}^{tr}  & = & \Xi _{}^{rt}  = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\
+ \Xi _{}^{rr}  & = &  - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\
 \label{introEquation:ResistanceTensorArbitraryOrigin}
 \end{eqnarray}
 The resistance tensor depends on the origin to which they refer. The
@@ -528,7 +528,7 @@ Another set of calculation were performed to study the
 \end{center}
 \end{table}
 
-Another set of calculation were performed to study the efficiency of
+Another set of calculations were performed to study the efficiency of
 temperature control using different temperature coupling schemes.
 The starting configuration is cooled to 173~K and evolved using NVE,
 NVT, and Langevin dynamic with time step of 2 fs.
@@ -538,8 +538,7 @@ scaling to the desire temperature. In extremely lower 
 Nos\'e-Hoover temperature scaling scheme with thermostat of 5 ps
 which gives reasonable tight coupling, while the blue one from
 Langevin dynamics with viscosity of 0.1 poise demonstrates a faster
-scaling to the desire temperature. In extremely lower friction
-regime (when $ \eta  \approx 0$), Langevin dynamics becomes normal
+scaling to the desire temperature. When $ \eta = 0$, Langevin dynamics becomes normal
 NVE (see orange curve in Fig.~\ref{langevin:temperature}) which
 loses the temperature control ability.
 
@@ -565,38 +564,28 @@ identified the center of resistance at $(0\AA, 0.7482\
 made of 2266 small identical beads with size of 0.3 \AA on the
 surface. Applying the procedure described in
 Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we
-identified the center of resistance at $(0\AA, 0.7482\AA,
--0.1988\AA)$, as well as the resistance tensor,
+identified the center of resistance at (0 $\rm{\AA}$, 0.7482 $\rm{\AA}$,
+-0.1988 $\rm{\AA}$), as well as the resistance tensor,
 \[
 \left( {\begin{array}{*{20}c}
 0.9261 & 0 & 0&0&0.08585&0.2057\\
 0& 0.9270&-0.007063& 0.08585&0&0\\
-0&0.007063&0.7494&0.2057&0&0\\
-0&0.0858&0.2057& 58.64& 0&-8.5736\\
+0&-0.007063&0.7494&0.2057&0&0\\
+0&0.0858&0.2057& 58.64& 0&0\\
 0.08585&0&0&0&48.30&3.219&\\
 0.2057&0&0&0&3.219&10.7373\\
 \end{array}} \right).
 \]
-%\[
-%\left( {\begin{array}{*{20}c}
-%0.9261 & 1.310e-14 & -7.292e-15&5.067e-14&0.08585&0.2057\\
-%3.968e-14& 0.9270&-0.007063& 0.08585&6.764e-14&4.846e-14\\
-%-6.561e-16&-0.007063&0.7494&0.2057&4.846e-14&1.5036e-14\\
-%5.067e-14&0.0858&0.2057& 58.64& 8.563e-13&-8.5736\\
-%0.08585&6.764e-14&4.846e-14&1.555e-12&48.30&3.219&\\
-%0.2057&4.846e-14&1.5036e-14&-3.904e-13&3.219&10.7373\\
-%\end{array}} \right).
-%\]
+where the units for translational, translation-rotation coupling and rotational tensors are $\frac{kcal \cdot fs}{mol \cdot \rm{\AA}^2}$, $\frac{kcal \cdot fs}{mol \cdot \rm{\AA} \cdot rad}$ and $\frac{kcal \cdot fs}{mol \cdot rad^2}$ respectively.
 
 Curves of the velocity auto-correlation functions in
 Fig.~\ref{langevin:vacf} were shown to match each other very well.
 However, because of the stochastic nature, simulation using Langevin
 dynamics was shown to decay slightly faster than MD. In order to
 study the rotational motion of the molecules, we also calculated the
-auto- correlation function of the principle axis of the second GB
+auto-correlation function of the principle axis of the second GB
 particle, $u$. The discrepancy shown in Fig.~\ref{langevin:uacf} was
-probably due to the reason that the viscosity using in the
-simulations only partially preserved the dynamics of the system.
+probably due to the reason that we used the experimental viscosity directly instead of calculating bulk viscosity from simulation.
 
 \begin{figure}
 \centering
@@ -617,7 +606,7 @@ auto-correlation functions of NVE (explicit solvent) i
 \centering
 \includegraphics[width=\linewidth]{vacf.eps}
 \caption[Plots of Velocity Auto-correlation Functions]{Velocity
-auto-correlation functions of NVE (explicit solvent) in blue) and
+auto-correlation functions of NVE (explicit solvent) in blue and
 Langevin dynamics (implicit solvent) in red.} \label{langevin:vacf}
 \end{figure}