Physics 434, 2015: Homework 5

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Back to Physics 434, 2015: Physical Biology.

Please turn on the assignment either as a PDF file to me by email, or as a printout/writeup to my mailbox in physics. Detailed derivations (with explanations) and calculations must be present for the problems for full credit. Submit your code if a problem requires you to program.

Undergraduates

1. Calculate the drift and the diffusion constants for a discrete time random walk on a lattice with a lattice spacing of $a$, where particles go to the left with the probability $p$, to the right with $q$, and stays in place with the probability $r=1-p-q$.
2. 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 Matlab/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. Do you observe this? Can you guess the functional form of the dependence of $D$ on the fraction of occupied sites.
3. Write a Matlab code that would generate random E. coli trajectories, similar to what we saw during class in some movies. Assume that E. coli runs in a straight line with a velocity 'v' for an exponentially distributed time with mean $\tau$. It then tumbles, reorients randomly, and repeats. Focus on just two dimensions. Verify by histogramming that the distribution of positions of E. coli after many (100 or more) tumbles, as simulated by your script, is, indeed, a gaussian. Plot the standard deviation. Does it scale as a square root of time? You will need to run your function many times for different durations to estimate the distribution of the positions. Do the simulations for different parameter values and see how does the variance of the position depend on the velocity and the mean waiting time. Explain your observations analytically.

Graduate Students

1. Calculate the drift and the diffusion constants for a random walk on a lattice with a lattice spacing of $a$, where particles go to the left with the probability $p$, to the right with $q$, and stays in place with the probability $r=1-p-q$, but the time is continuous, with the distribution of steps being exponential, with the mean waiting time of $\tau$.
2. 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 Matlab/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. Do you observe this? Can you guess the functional form of the dependence of $D$ on the fraction of occupied sites.
3. Write a Matlab code that would generate random E. coli trajectories, similar to what we saw during class in some movies. Assume that E. coli runs in a straight line with a velocity 'v' for an exponentially distributed time with mean $\tau$. It then tumbles, reorients randomly, and repeats. Focus on just two dimensions. Verify by histogramming that the distribution of positions of E. coli after many (100 or more) tumbles, as simulated by your script, is, indeed, a gaussian. Plot the standard deviation. Does it scale as a square root of time? You will need to run your function many times for different durations to estimate the distribution of the positions. Do the simulations for different parameter values and see how does the variance of the position depend on the velocity and the mean waiting time. Explain your observations analytically.