Problem Set 6

 

1.  You should read the classic paper by Berg and Purcell:

 

Physics of chemoreception.  HC Berg & EM Purcell, Biophys J 20, 93-219 (1977).

 

As much as possible, try to follow the mathematical derivations.  If there are places where you get stuck, make notes and bring these to the discussion on Thursday.

  1. To help you understand some of what is going on in Berg & Purcell, you should do a small simulation of diffusion.  Start in one dimension, and imagine a particle that can sit at (for example) 1000 different points along a line.  At each time step, flip a coin and decide whether the particle moves to the right or the left; to avoid trouble with the particle leaving the region you are studying, you can wrap the line into a ring, so that if the particle at site number 1000 and tries to move right, it goes to site 1.  You should be able to do this in a few lines of MATLAB code.  Before running the code, what is the mean displacement across one time step?  What is the mean square displacement across on time step?  Now run the code, say for tens of thousands of time steps.  The result should be a trajectory x(t).  Plot this result.  To analyze it, collect �data� on how much the particle moves over a time t -> t+ T.  Show that the distribution of displacements starts t look Gaussian when T becomes large, and that the variance of this Gaussian is growing linearly with T, as expected for diffusion.
  2. Generalize this calculation (1 particle with 1000 sites) to have, say 100 particles and 10,000 sites; assume that particles are completely independent of each other, even if they land on the same site.  Suppose that all the particles start at the same site.  As your simulation progresses, they disperse, corresponding to the fact that concentration gradients relax via diffusion.   Suppose you summarize the state of the system at time t by computing the variance in position of all the particles; how does this variance change with time?  What does the distribution of particle positions look like?  Can you show that, although the variance is changing, the shape of the distribution (suitably normalized) is not?  What is this shape? How many time steps do you need before the distribution of particles starts to look uniform along the line?  Does this make sense given what you know about diffusion? Once the distribution looks uniform, suppose you count the number of molecules in a window of 100 sites.  What is the average of this number (you should be able to answer that without looking at the data!)?  What is the variance in this number?  Since you have many samples, you should be able to estimate the full probability distribution for this molecule count.  Plot your result and see if you can fit it to the Poisson distribution.
  3. Go back to having just one particle, but now allow for motion in three dimensions (it takes some thought to see how to set this up). For example, let the particle move in a box that is 10x10x10 sites in size, and at each moment you flip independent coins to decide if the x, y, and z coordinates should increase or decrease by one unit; as before you need to wrap the edges around, so that if you are at position 10 and try to increase the coordinate by one you go to position 1.  Let your one particle start at some random position, and ask how long it takes before it hits one particular site that you choose—the site where some chemical reaction might happen.  Suppose you do this experiment many times, and measure each time how long it takes to hit the chosen site.  What is the distribution of these times?  What happens when you make the box bigger (e.g., 20x20x20 sites)?  What if you make the site you have to hit bigger, say consisting of a 2x2x2 box, such that you declare a �hit� if the particle enters any of these 8 sites?  Explain how these simulations relate to the idea of diffusion limited reaction rates.
  4. If you look at your simulations in (4), you�ll see that the distribution of times required to hit the site is broad, but well behaved.  Repeat the same simulation in one dimension (back to the line of 1000 sites).  What does the distribution look like now?  You might need to do the simulation many times in order to get a convincing answer.  Hopefully you�ll see that one dimension is different from three!

 

As an aside, you will note that Berg and Purcell provide an interesting mixture of intuitive and mathematical arguments, often sliding (perhaps suspiciously!) from one qualitative picture to another.  Some of you might wish that things could be made more precise.  This turns out to be a bit of work, but it is kind of fun, and shows that what Berg and Purcell did was really much more general than one might have thought.

 

Physical limits to biochemical signaling. W Bialek & S Setayeshgar, Proc Nat�l Acad Sci (USA) 102, 10040-10045 (2005).