| 43 |
|
torques\cite{Mielke2004}. Membrane fusion is another key biological |
| 44 |
|
process which controls a variety of physiological functions, such as |
| 45 |
|
release of neurotransmitters \textit{etc}. A typical fusion event |
| 46 |
< |
happens on the time scale of millisecond, which is impractical to |
| 46 |
> |
happens on the time scale of a millisecond, which is impractical to |
| 47 |
|
study using atomistic models with newtonian mechanics. With the help |
| 48 |
|
of coarse-grained rigid body model and stochastic dynamics, the |
| 49 |
|
fusion pathways were explored by many |
| 172 |
|
one can write down the translational and rotational resistance |
| 173 |
|
tensors |
| 174 |
|
\begin{eqnarray*} |
| 175 |
< |
\Xi _a^{tt} = 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
| 176 |
< |
\Xi _b^{tt} = \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
| 175 |
> |
\Xi _a^{tt} & = & 16\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - b^2 )S - 2a}}. \\ |
| 176 |
> |
\Xi _b^{tt} & = & \Xi _c^{tt} = 32\pi \eta \frac{{a^2 - b^2 }}{{(2a^2 - 3b^2 )S + |
| 177 |
|
2a}}, |
| 178 |
|
\end{eqnarray*} |
| 179 |
|
and |
| 180 |
|
\begin{eqnarray*} |
| 181 |
< |
\Xi _a^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
| 182 |
< |
\Xi _b^{rr} = \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}}. |
| 181 |
> |
\Xi _a^{rr} & = & \frac{{32\pi }}{3}\eta \frac{{(a^2 - b^2 )b^2 }}{{2a - b^2 S}}, \\ |
| 182 |
> |
\Xi _b^{rr} & = & \Xi _c^{rr} = \frac{{32\pi }}{3}\eta \frac{{(a^4 - b^4 )}}{{(2a^2 - b^2 )S - 2a}}. |
| 183 |
|
\end{eqnarray*} |
| 184 |
|
|
| 185 |
|
\subsubsection{\label{introSection:resistanceTensorRegularArbitrary}\textbf{The Resistance Tensor for Arbitrary Shapes}} |
| 278 |
|
bead $i$ and origin $O$, the elements of resistance tensor at |
| 279 |
|
arbitrary origin $O$ can be written as |
| 280 |
|
\begin{eqnarray} |
| 281 |
< |
\Xi _{}^{tt} = \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\ |
| 282 |
< |
\Xi _{}^{tr} = \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 283 |
< |
\Xi _{}^{rr} = - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\ |
| 281 |
> |
\Xi _{}^{tt} & = & \sum\limits_i {\sum\limits_j {C_{ij} } } \notag , \\ |
| 282 |
> |
\Xi _{}^{tr} & = & \Xi _{}^{rt} = \sum\limits_i {\sum\limits_j {U_i C_{ij} } } , \\ |
| 283 |
> |
\Xi _{}^{rr} & = & - \sum\limits_i {\sum\limits_j {U_i C_{ij} } } U_j. \notag \\ |
| 284 |
|
\label{introEquation:ResistanceTensorArbitraryOrigin} |
| 285 |
|
\end{eqnarray} |
| 286 |
|
The resistance tensor depends on the origin to which they refer. The |
| 528 |
|
\end{center} |
| 529 |
|
\end{table} |
| 530 |
|
|
| 531 |
< |
Another set of calculation were performed to study the efficiency of |
| 531 |
> |
Another set of calculations were performed to study the efficiency of |
| 532 |
|
temperature control using different temperature coupling schemes. |
| 533 |
|
The starting configuration is cooled to 173~K and evolved using NVE, |
| 534 |
|
NVT, and Langevin dynamic with time step of 2 fs. |
| 538 |
|
Nos\'e-Hoover temperature scaling scheme with thermostat of 5 ps |
| 539 |
|
which gives reasonable tight coupling, while the blue one from |
| 540 |
|
Langevin dynamics with viscosity of 0.1 poise demonstrates a faster |
| 541 |
< |
scaling to the desire temperature. In extremely lower friction |
| 542 |
< |
regime (when $ \eta \approx 0$), Langevin dynamics becomes normal |
| 541 |
> |
scaling to the desire temperature. When $ \eta = 0$, Langevin dynamics becomes normal |
| 542 |
|
NVE (see orange curve in Fig.~\ref{langevin:temperature}) which |
| 543 |
|
loses the temperature control ability. |
| 544 |
|
|
| 564 |
|
made of 2266 small identical beads with size of 0.3 \AA on the |
| 565 |
|
surface. Applying the procedure described in |
| 566 |
|
Sec.~\ref{introEquation:ResistanceTensorArbitraryOrigin}, we |
| 567 |
< |
identified the center of resistance at $(0 \AA, 0.7482 \AA, |
| 568 |
< |
-0.1988 \AA)$, as well as the resistance tensor, |
| 567 |
> |
identified the center of resistance at (0 $\rm{\AA}$, 0.7482 $\rm{\AA}$, |
| 568 |
> |
-0.1988 $\rm{\AA}$), as well as the resistance tensor, |
| 569 |
|
\[ |
| 570 |
|
\left( {\begin{array}{*{20}c} |
| 571 |
|
0.9261 & 0 & 0&0&0.08585&0.2057\\ |
| 572 |
|
0& 0.9270&-0.007063& 0.08585&0&0\\ |
| 573 |
< |
0&0.007063&0.7494&0.2057&0&0\\ |
| 574 |
< |
0&0.0858&0.2057& 58.64& 0&-8.5736\\ |
| 573 |
> |
0&-0.007063&0.7494&0.2057&0&0\\ |
| 574 |
> |
0&0.0858&0.2057& 58.64& 0&0\\ |
| 575 |
|
0.08585&0&0&0&48.30&3.219&\\ |
| 576 |
|
0.2057&0&0&0&3.219&10.7373\\ |
| 577 |
|
\end{array}} \right). |
