Physics 434, 2012: Homework 3

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Back to Physics 434, 2012: Information Processing in Biology.

1. Calculate the cumulant generating function of a normal distribution with the mean $\mu$ and the standard deviation $\sigma$.
2. Calculate the drift and the diffusion constants for a random walk on a lattice with a lattice spacing of $a$, where particles hop between the sites every $\Delta t$ units of time, and they have a probability of $p$ going left, and $q=1-p$ going right. What are these quantities if the particle goes to the left with the probability $p$, to the right with $q$, and stays in place with the probability $r=1-p-q$?
3. Verify by histogramming that the distribution of positions of E. coli after many (100 or more) tumbles, as simulated by the script ecoli.m is, indeed, a gaussian. Plot the mean of the simulated distribution as a function of time. Does is scale linearly with the time? Plot the standard deviation. Does it scale as a square root of time? You will need to run the function I provided many times for a fixed duration of the "experiment" to estimate the distribution of the positions. You will then need to repeat this for many different durations. Please submit your code as well.
4. We have discussed that the mean squared displacement in a random walk scales as $<x^{2}>\propto Dt$. Does this mean that signals propagating due to diffusion are always bound to propagate slowly? Consider the following model, which may be considered as a model describing establishment of morphogen gradients, as we will see later in the class. A 1-dimensional lattice has a particle sitting at each lattice node. However, none of the particles can random walk until they are activated. Activation happens by a contact with another particle. That is, if a particle moves into a site which has an inactive particle, the former activates the latter, both now become motile, and both can activate new particles when they get to them. Originally, only one particle sitting at 0 is motile. Write an Octave code to estimate the growth with time of the average size of the region around 0 where all particles are activated. Does the size of the region go as a square root of time? Linearly with time? Or some other scaling? Explain your results with a qualitative physical argument. This is not an easy code to write, and I encourage you to work together on it.
5. For Graduate Students: Consider a model of a transcription factor moving on a DNA sequence, modeling this as a discrete time random walk. For a single factor bound to the DNA, it will move with $<x^{2}>\propto Dt$. However, if there are multiple transcription factors, they will hinder each other's diffusion -- two or more cannot be bound to the same DNA site. This is called the excluded volume interaction. Estimate numerically using Octave the scaling of $<x^{2}>$ as a function of the fraction of the number of DNA sites occupied by bound transcription factors. Since two transcription factors cannot exchange their position on the 1-d DNA, the diffusion constant should go to zero in the limit when almost every DNA site is bound by some transcription factor. Suppose now we are talking about excluded volume diffusion of receptors on a cellular membrane instead of transcription factors on a DNA. Now particles can go around each other. What is the scaling of $<x^{2}>$ as a function of $t$ in this case?