Translocation of a linear macromolecule through a nanopore

There are many processes in biology that involve the translocation of an entire molecule through a pore in a membrane (Nelson P. 408). The physics of these are still not entirely understood. Translocation is also being studied as a way of sequencing RNA or DNA, as in the case of alpha-hemolysin and this is being actively studied at ucsc. It's now advanced to a commercial stage where it sequences DNA using a hand held device Links to an external site..

In this problem you can run hw6/pore.py to study the translocation of a molecule as a function of many system parameters.

The polymer is modeled as a freely hinged chain (this could easily be altered to include stiffness as with other problems in this homework set.) The pore is modeled as a potential that is cylindrical in nature between two planes at $y = \pm y_{0}$ $y = \pm y_0$ . The potential is small outside the membrane and within a radius r_0 and large within the membrane, having a value of $V_{0} = 100$ $V_0=100$ . Because the dynamics modeled are continuous the potential should be a smooth function of position. The smoothed out pore potential can be written as

$V_{p} (r, y) = \frac{V_{0}}{1 + \exp ((y^{2} - y_{0}^{2}) / λ^{2})} \frac{1}{1 + \exp (- (r^{2} - r_{0}^{2}) / λ^{2})}$ $V_p(r,y) = \frac{V_0}{1 + \exp((y^2-y_0^2)/\lambda^2)}\frac{1}{1+\exp(-(r^2-r_0^2)/\lambda^2)}$

$y_{0}$ $y_0$ was chosen to be 10.

An external force $f_{e x t}$ $f_{ext}$ is applied to all points on the polymer.

The model for the polymer is that of a freely hinged chain with Langevin dynamics. In addition, a periodic potential has been added in the y-direction to the pore to mimic periodic molecular interactions,

$V_{c} (y) = \frac{a m p l \cos (k y)}{1 + \exp ((y^{2} - y_{0}^{2}) / λ^{2})}$ $V_c(y) = \frac{ampl \cos(k y)}{1 + \exp((y^2-y_0^2)/\lambda^2)}$

The denominator is added to the usual sine term to prevent the potential from leaking out of the membrane.

1. Run the program as it stands now, ampl = 5.0 and T= 1.0. Observer the motion (note

(Note that under some software platforms, only the top and the bottom of the pore are shown (in blue) because that version of visual python used can't handle transparency. )

2. Lower T and observe what happens. Explain this difference in behavior.

3. Plot the center of mass versus time for different temperatures.

4. Set ampl = 0 and try changing other parameters such as r_0 and T.

Try to come up with a qualitative description of the dynamics in these different cases.

5. How well do you think the analysis of Lubensky and Nelson Links to an external site. applies to the parameters studied here?

6. Give biological examples of processes involving translocation of linear macromolecules.

7. Give a mechanism that is used to guarantee that only specific proteins are translocated. Why is this mechanism necessary for proper functioning of a cell?