# Physics 434, 2014: Homework 5

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1. Diffusion in steady state: During development, different parts of an embryo make a decision to convert into cells of a different type based on the local concentrations of various signaling molecules that they measure. These molecules are called morphogens. For example, in drosophila development, the mother fly deposits mRNA molecules coding for a certain protein, called bicoid, at one of the ends of the egg. These mRNAs are translated locally into proteins at a constant rate, and the proteins diffuse through the egg and act as a morphogen that controls the activation of expression of subsequent developmental genes. During the diffusion, the proteins get degraded with a fixed rate per molecule and disappears.
• Write down the diffusion equation with degradation that is satisfied by the concentration of bicoid away from the end of the egg where it is produced. In the steady state, the concentration is not a function of time, and so all time derivatives vanish.
• What is the equation obeyed by the bicoid gradient at steady state? Solve it. What is the shape of the concentration of the bicoid protein established in the egg through this mechanism?
• Now consider the problem discussed in Gregor et al. 2005, which measured these gradients. Different flies have sizes different by up to a factor of about 10. How can they establish the same body plans then? Which parameter (protein production rate, diffusion constant, or degradation rate) do you think is changing the most in flies of a different size to develop the same body plan?
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 ${\displaystyle V_{0}<0}$. Every time step ${\displaystyle \Delta t}$ it's voltage can either go up by ${\displaystyle \Delta V}$ or down by ${\displaystyle -\Delta V}$ with probabilities ${\displaystyle p}$ and ${\displaystyle 1-p}$, respectively. Voltage goes down due to the leak current across the membrane, and it increases due to random synaptic inputs. If the voltage reaches ${\displaystyle V_{0}}$, then it cannot be lowered anymore. If it reaches ${\displaystyle V_{\rm {threshold}}}$, then the neuron releases an action potential (that is, it fires), and the voltage is reset back to ${\displaystyle V_{0}}$.
• Write a program to simulate this random walk for arbitrary ${\displaystyle p,\,V_{0},\,V_{\rm {threshold}},\,\Delta V,\,\Delta t}$ 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 ${\displaystyle \Delta t=1ms}$, ${\displaystyle \Delta V=5mV}$, ${\displaystyle V_{0}=-65mV}$, ${\displaystyle V_{\rm {threshold}}=-35mV}$, and for value of ${\displaystyle p=[0.2:0.05:1]}$. Plot the curve of the coefficient of variation, ${\displaystyle \sigma /\mu }$ of the firing time as a function of ${\displaystyle p}$.
• Extra credit question: Now assume that ${\displaystyle p\to 1}$ or ${\displaystyle p\to 0}$. 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 ${\displaystyle p}$.
3. Graduate Students: Let's make the problem above a bit more realistic. The neuron relaxes linearly, in proportion to how far away it is from ${\displaystyle V_{0}}$, and the synaptic input to the neuron is white. That is, the neuronal potential satisfies ${\displaystyle dV/dt=-(V-V_{0})/\tau +\eta }$, where ${\displaystyle \langle \eta \rangle =c}$ and ${\displaystyle \langle \eta (t)\eta (t')\rangle =\Omega \delta (t-t')}$. The neuron fires whenever ${\displaystyle V=V_{\rm {threshold}}}$. Derive analytically the distribution of spikes under the approximation that they are rare. Simulate the problem and confirm your theoretical prediction.
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, which I reproduce below.
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? Polya 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 [above], you could have guessed Polya’s result.