Physics 380, 2011: Homework 3

Back to the main Teaching page.

Back to Physics 380, 2011: Information Processing in 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.

1. 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 $r=1-p-q$?
2. In class, we have discussed the first passage time in random walk models. Let's analyze generation of a neural action potential from this perspective. Let's suppose a neuron starts at rest with the voltage $V_{0}<0$. Every time step $\Delta t$ it's voltage can either go up by $\Delta V$ or down by $-\Delta V$ with probabilities $p$ and $1-p$, respectively. Voltage goes down due to the leak current across the membrane, and it increases due to random synaptic inputs. If the voltage reaches $V_{0}$, then it cannot be lowered anymore. If it reaches $V_{\rm {threshold}}$, then the neuron releases an action potential (that is, it fires). Write an Octave program to simulate this random walk for arbitrary $p,\,V_{0},\,V_{\rm {threshold}},\,\Delta V,\,\Delta t$ and to record the time it takes for the neuron to fire. Run this program sufficient number of times to estimate the mean and the standard deviation of the time to firing for $\Delta t=1ms$, $\Delta V=5mV$, $V_{0}=-65mV$, $V_{\rm {threshold}}=-35mV$, and for value of $p=[0.1:0.05:1]$. Plot the curve of the coefficient of variation, $\sigma /\mu$ of the firing time as a function of $p$. Now assume that $p\to 1$ or $p\to 0$. Analytically derive expressions for the coefficient of variation in these cases. You may want to look at Bel et al., 2010, for a derivation of this result for an arbitrary $p$.
3. 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.
4. Let's verify whether what I told you in class about random walk return probabilities is correct. We will solve Problem 1.1.17 in Grinstead and Snell book. Mathematicians have been known to get some of the best ideas while sitting in a cafe, riding on a bus, or strolling in the park. In the early 1900s the famous mathematician George Polya lived in a hotel near the woods in Zurich. He liked to walk in the woods and think about mathematics. Polya describes the following incident:

At the hotel there lived also some students with whom I usually took my meals and had friendly relations. On a certain day one of them expected the visit of his fiancee, what (sic) I knew, but I did not foresee that he and his fiancee would also set out for a stroll in the woods, and then suddenly I met them there. And then I met them the same morning repeatedly, I don’t remember how many times, but certainly much too often and I felt embarrassed: It looked as if I was snooping around which was, I assure you, not the case.

This set him to thinking about whether random walkers were destined to meet. Polya considered random walkers in one, two, and three dimensions. In one dimension, he envisioned the walker on a very long street. At each intersection the walker flips a fair coin to decide which direction to walk next. In two dimensions, the walker is walking on a grid of streets, and at each intersection he chooses one of the four possible directions with equal probability. In three dimensions (we might better speak of a random climber), the walker moves on a three-dimensional grid, and at each intersection there are now six different directions that the walker may choose, each with equal probability.

• Write a program to simulate a random walk in one dimension starting at 0. Have your program print out the lengths of the times between returns to the starting point (returns to 0). See if you can guess from this simulation the answer to the following question: Will the walker always return to his starting point eventually or might he drift away forever?
• The paths of two walkers in two dimensions who meet after n steps can be considered to be a single path that starts at (0, 0) and returns to (0, 0) after 2n steps. This means that the probability that two random walkers in two dimensions meet is the same as the probability that a single walker in two dimensions ever returns to the starting point. Thus the question of whether two walkers are sure to meet is the same as the question of whether a single walker is sure to return to the starting point. Write a program to simulate a random walk in two dimensions and see if you think that the walker is sure to return to (0, 0). If so, Po ́lya would be sure to keep meeting his friends in the park. Perhaps by now you have conjectured the answer to the question: Is a random walker in one or two dimensions sure to return to the starting point? Po ́lya answered this question for dimensions one, two, and three. He established the remarkable result that the answer is yes in one and two dimensions and no in three dimensions.
• Write a program to simulate a random walk in three dimensions and see whether, from this simulation and the results of (a) and (b), you could have guessed Polya’s result.

1. 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? Do a numerical simulation to see this. What is it in 3 dimensions?