635 |
|
\caption[Plot of Temperature Fluctuation Versus Time]{Plot of |
636 |
|
temperature fluctuation versus time.} \label{langevin:temperature} |
637 |
|
\end{figure} |
638 |
+ |
|
639 |
+ |
\subsection{Langevin Dynamics simulation {\it vs} NVE simulations} |
640 |
+ |
|
641 |
+ |
To validate our langevin dynamics simulation. We performed several NVE |
642 |
+ |
simulations with explicit solvents for different shaped |
643 |
+ |
molecules. There are one solute molecule and 1929 solvent molecules in |
644 |
+ |
NVE simulation. The parameters are shown in table |
645 |
+ |
\ref{tab:parameters}. The force field between spheres is standard |
646 |
+ |
Lennard-Jones, and ellipsoids interact with other ellipsoids and |
647 |
+ |
spheres with generalized Gay-Berne potential. All simulations are |
648 |
+ |
carried out at 300 K and 1 Atm. The time step is 25 ns, and a |
649 |
+ |
switching function was applied to all potentials to smoothly turn off |
650 |
+ |
the interactions between a range of $22$ and $25$ \AA. The switching |
651 |
+ |
function was the standard (cubic) function, |
652 |
+ |
\begin{equation} |
653 |
+ |
s(r) = |
654 |
+ |
\begin{cases} |
655 |
+ |
1 & \text{if $r \le r_{\text{sw}}$},\\ |
656 |
+ |
\frac{(r_{\text{cut}} + 2r - 3r_{\text{sw}})(r_{\text{cut}} - r)^2} |
657 |
+ |
{(r_{\text{cut}} - r_{\text{sw}})^3} |
658 |
+ |
& \text{if $r_{\text{sw}} < r \le r_{\text{cut}}$}, \\ |
659 |
+ |
0 & \text{if $r > r_{\text{cut}}$.} |
660 |
+ |
\end{cases} |
661 |
+ |
\label{eq:switchingFunc} |
662 |
+ |
\end{equation} |
663 |
+ |
We have computed translational diffusion constants for lipid molecules |
664 |
+ |
from the mean-square displacement, |
665 |
+ |
\begin{equation} |
666 |
+ |
D = \lim_{t\rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
667 |
+ |
\end{equation} |
668 |
+ |
of the solute molecules. Translational diffusion constants for the |
669 |
+ |
different shaped molecules are shown in table |
670 |
+ |
\ref{tab:translation}. We have also computed orientational correlation |
671 |
+ |
times for different shaped molecules from fits of the second-order |
672 |
+ |
Legendre polynomial correlation function, |
673 |
+ |
\begin{equation} |
674 |
+ |
C_{\ell}(t) = \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
675 |
+ |
\mu}_{i}(0) \right) |
676 |
+ |
\end{equation} |
677 |
+ |
the results are shown in table \ref{tab:rotation}. We used einstein |
678 |
+ |
format of the pressure correlation function, |
679 |
+ |
\begin{equation} |
680 |
+ |
C_{\ell}(t) = \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
681 |
+ |
\mu}_{i}(0) \right) |
682 |
+ |
\end{equation} |
683 |
+ |
to estimate the viscosity of the systems from NVE simulations. The |
684 |
+ |
viscosity can also be calculated by Green-Kubo pressure correlaton |
685 |
+ |
function, |
686 |
+ |
\begin{equation} |
687 |
+ |
C_{\ell}(t) = \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
688 |
+ |
\mu}_{i}(0) \right) |
689 |
+ |
\end{equation} |
690 |
+ |
However, this method converges slowly, and the statistics are not good |
691 |
+ |
enough to give us a very accurate value. The langevin dynamics |
692 |
+ |
simulations for different shaped molecules are performed at the same |
693 |
+ |
conditions as the NVE simulations with viscosity estimated from NVE |
694 |
+ |
simulations. To get better statistics, 1024 non-interacting solute |
695 |
+ |
molecules are put into one simulation box for each langevin |
696 |
+ |
simulation, this is equal to 1024 simulations for single solute |
697 |
+ |
systems. The diffusion constants and rotation relaxation times for |
698 |
+ |
different shaped molecules are shown in table \ref{tab:translation} |
699 |
+ |
and \ref{tab:rotation} to compare to the results calculated from NVE |
700 |
+ |
simulations. The theoretical values for sphere is calculated from the |
701 |
+ |
Stokes-Einstein law, the theoretical values for ellipsoid is |
702 |
+ |
calculated from Perrin's fomula, the theoretical values for dumbbell |
703 |
+ |
is calculated from StinXX and Davis theory. The exact method is |
704 |
+ |
applied to the langevin dynamics simulations for sphere and ellipsoid, |
705 |
+ |
the bead model is applied to the simulation for dumbbell molecule, and |
706 |
+ |
the rough shell model is applied to ellipsoid, dumbbell, banana and |
707 |
+ |
lipid molecules. The results from all the langevin dynamics |
708 |
+ |
simulations, including exact, bead model and rough shell, match the |
709 |
+ |
theoretical values perfectly for all different shaped molecules. This |
710 |
+ |
indicates that our simulation package for langevin dynamics is working |
711 |
+ |
well. The approxiate methods ( bead model and rough shell model) are |
712 |
+ |
accurate enough for the current simulations. The goal of the langevin |
713 |
+ |
dynamics theory is to replace the explicit solvents by the friction |
714 |
+ |
forces. We compared the dynamic properties of different shaped |
715 |
+ |
molecules in langevin dynamics simulations with that in NVE |
716 |
+ |
simulations. The results are reasonable close. Overall, the |
717 |
+ |
translational diffusion constants calculated from langevin dynamics |
718 |
+ |
simulations are very close to the values from the NVE simulation. For |
719 |
+ |
sphere and lipid molecules, the diffusion constants are a little bit |
720 |
+ |
off from the NVE simulation results. One possible reason is that the |
721 |
+ |
calculation of the viscosity is very difficult to be accurate. Another |
722 |
+ |
possible reason is that although we save very frequently during the |
723 |
+ |
NVE simulations and run pretty long time simulations, there is only |
724 |
+ |
one solute molecule in the system which makes the calculation for the |
725 |
+ |
diffusion constant difficult. The sphere molecule behaves as a free |
726 |
+ |
rotor in the solvent, so there is no rotation relaxation time |
727 |
+ |
calculated from NVE simulations. The rotation relaxation time is not |
728 |
+ |
very close to the NVE simulations results. The banana and lipid |
729 |
+ |
molecules match the NVE simulations results pretty well. The mismatch |
730 |
+ |
between langevin dynamics and NVE simulation for ellipsoid is possibly |
731 |
+ |
caused by the slip boundary condition. For dumbbell, the mismatch is |
732 |
+ |
caused by the size of the solvent molecule is pretty large compared to |
733 |
+ |
dumbbell molecule in NVE simulations. |
734 |
|
|
735 |
+ |
According to our simulations, the langevin dynamics is a reliable |
736 |
+ |
theory to apply to replace the explicit solvents, especially for the |
737 |
+ |
translation properties. For large molecules, the rotation properties |
738 |
+ |
are also mimiced reasonablly well. |
739 |
+ |
|
740 |
+ |
\begin{table*} |
741 |
+ |
\begin{minipage}{\linewidth} |
742 |
+ |
\begin{center} |
743 |
+ |
\caption{} |
744 |
+ |
\begin{tabular}{llccccccc} |
745 |
+ |
\hline |
746 |
+ |
& & Sphere & Ellipsoid & Dumbbell(2 spheres) & Banana(3 ellpsoids) & |
747 |
+ |
Lipid(head) & lipid(tail) & Solvent \\ |
748 |
+ |
\hline |
749 |
+ |
$d$ (\AA) & & 6.5 & 4.6 & 6.5 & 4.2 & 6.5 & 4.6 & 4.7 \\ |
750 |
+ |
$l$ (\AA) & & $= d$ & 13.8 & $=d$ & 11.2 & $=d$ & 13.8 & 4.7 \\ |
751 |
+ |
$\epsilon^s$ (kcal/mol) & & 0.8 & 0.8 & 0.8 & 0.8 & 0.185 & 0.8 & 0.8 \\ |
752 |
+ |
$\epsilon_r$ (well-depth aspect ratio)& & 1 & 0.2 & 1 & 0.2 & 1 & 0.2 & 1 \\ |
753 |
+ |
$m$ (amu) & & 190 & 200 & 190 & 240 & 196 & 760 & 72.06 \\ |
754 |
+ |
%$\overleftrightarrow{\mathsf I}$ (amu \AA$^2$) & & & & \\ |
755 |
+ |
%\multicolumn{2}c{$I_{xx}$} & 1125 & 45000 & N/A \\ |
756 |
+ |
%\multicolumn{2}c{$I_{yy}$} & 1125 & 45000 & N/A \\ |
757 |
+ |
%\multicolumn{2}c{$I_{zz}$} & 0 & 9000 & N/A \\ |
758 |
+ |
%$\mu$ (Debye) & & varied & 0 & 0 \\ |
759 |
+ |
\end{tabular} |
760 |
+ |
\label{tab:parameters} |
761 |
+ |
\end{center} |
762 |
+ |
\end{minipage} |
763 |
+ |
\end{table*} |
764 |
+ |
|
765 |
+ |
\begin{table*} |
766 |
+ |
\begin{minipage}{\linewidth} |
767 |
+ |
\begin{center} |
768 |
+ |
\caption{} |
769 |
+ |
\begin{tabular}{lccccc} |
770 |
+ |
\hline |
771 |
+ |
& & & & &Translation \\ |
772 |
+ |
\hline |
773 |
+ |
& NVE & & Theoretical & Langevin & \\ |
774 |
+ |
\hline |
775 |
+ |
& $\eta$ & D & D & method & D \\ |
776 |
+ |
\hline |
777 |
+ |
sphere & 3.480159e-03 & 1.643135e-04 & 1.942779e-04 & exact & 1.982283e-04 \\ |
778 |
+ |
ellipsoid & 2.551262e-03 & 2.437492e-04 & 2.335756e-04 & exact & 2.374905e-04 \\ |
779 |
+ |
& 2.551262e-03 & 2.437492e-04 & 2.335756e-04 & rough shell & 2.284088e-04 \\ |
780 |
+ |
dumbell & 2.41276e-03 & 2.129432e-04 & 2.090239e-04 & bead model & 2.148098e-04 \\ |
781 |
+ |
& 2.41276e-03 & 2.129432e-04 & 2.090239e-04 & rough shell & 2.013219e-04 \\ |
782 |
+ |
banana & 2.9846e-03 & 1.527819e-04 & & rough shell & 1.54807e-04 \\ |
783 |
+ |
lipid & 3.488661e-03 & 0.9562979e-04 & & rough shell & 1.320987e-04 \\ |
784 |
+ |
\end{tabular} |
785 |
+ |
\label{tab:translation} |
786 |
+ |
\end{center} |
787 |
+ |
\end{minipage} |
788 |
+ |
\end{table*} |
789 |
+ |
|
790 |
+ |
\begin{table*} |
791 |
+ |
\begin{minipage}{\linewidth} |
792 |
+ |
\begin{center} |
793 |
+ |
\caption{} |
794 |
+ |
\begin{tabular}{lccccc} |
795 |
+ |
\hline |
796 |
+ |
& & & & &Rotation \\ |
797 |
+ |
\hline |
798 |
+ |
& NVE & & Theoretical & Langevin & \\ |
799 |
+ |
\hline |
800 |
+ |
& $\eta$ & $\tau_0$ & $\tau_0$ & method & $\tau_0$ \\ |
801 |
+ |
\hline |
802 |
+ |
sphere & 3.480159e-03 & & 1.208178e+04 & exact & 1.20628e+04 \\ |
803 |
+ |
ellipsoid & 2.551262e-03 & 4.66806e+04 & 2.198986e+04 & exact & 2.21507e+04 \\ |
804 |
+ |
& 2.551262e-03 & 4.66806e+04 & 2.198986e+04 & rough shell & 2.21714e+04 \\ |
805 |
+ |
dumbell & 2.41276e-03 & 1.42974e+04 & & bead model & 7.12435e+04 \\ |
806 |
+ |
& 2.41276e-03 & 1.42974e+04 & & rough shell & 7.04765e+04 \\ |
807 |
+ |
banana & 2.9846e-03 & 6.38323e+04 & & rough shell & 7.0945e+04 \\ |
808 |
+ |
lipid & 3.488661e-03 & 7.79595e+04 & & rough shell & 7.78886e+04 \\ |
809 |
+ |
\end{tabular} |
810 |
+ |
\label{tab:rotation} |
811 |
+ |
\end{center} |
812 |
+ |
\end{minipage} |
813 |
+ |
\end{table*} |
814 |
+ |
|
815 |
+ |
Langevin dynamics simulations are applied to study the formation of |
816 |
+ |
the ripple phase of lipid membranes. The initial configuration is |
817 |
+ |
taken from our molecular dynamics studies on lipid bilayers with |
818 |
+ |
lennard-Jones sphere solvents. The solvent molecules are excluded from |
819 |
+ |
the system, the experimental value of water viscosity is applied to |
820 |
+ |
mimic the heat bath. Fig. XXX is the snapshot of the stable |
821 |
+ |
configuration of the system, the ripple structure stayed stable after |
822 |
+ |
100 ns run. The efficiency of the simulation is increased by one order |
823 |
+ |
of magnitude. |
824 |
+ |
|
825 |
|
\subsection{Langevin Dynamics of Banana Shaped Molecules} |
826 |
|
|
827 |
|
In order to verify that Langevin dynamics can mimic the dynamics of |