This homework addresses some aspects of electrophoresis, Brownian Ratchets, and life at low Reynolds number. You may find some of the assigned reading helpful, such as Bader and Purcell. This assignment is due in my mailbox (in mailroom, 2nd floor of NPB) before 5:00 pm, Thursday April 11, 2002. You may work together on the homework, but you should hand in your own solution, of course.

1. Consider the following liquids: pentane, glycerol, crude oil, and liquid helium (but not superfluid helium).

(a) How fast would a bacterium of diameter 1 &mum have to swim through each of these liquids, in order to attain a Reynolds number of Re = 1?

(b) Suppose that a molecular motor is designed to provide a force of 300 pN. Explain very briefly how the performance of that motor might be affected by immersion in each of the above liquids.

(c) Consider again the bacterium of diameter 1 &mum, and traveling at a speed of 10 &mum/s through each of the above liquids. If it suddenly stops swimming, how far will it travel (in each liquid) before it stops? (You may assume the bacterium is mostly made of water, regardless of what medium it is swimming in.)

2. We have repeatedly mentioned the spreading Gaussian behavior of diffusing particles, P(x,t) = Po exp(-x^2/4Dt) / sqrt(4 &pi Dt).

(a) If we combine Fick's Law (which we discussed in class), with the additional requirement that the total number of diffusing particles does not change over time (even though they spread out), we can obtain the diffusion equation, dP/dt = D d^2 P/dx^2. This means that P(x,t) must be a solution to the diffusion equation. Show that the spreading Gaussian P(x,t) does satisfy the diffusion equation.

(b) Use your favorite plotting software to show P(x,t) on the interval -3a < x < 3a for different times, specifically Dt/a^2 = .01, .1, .2, .5, 1, and 2. Can you verify that the total number of particles in the system (i.e. the area under the curve) doesn't change with time?

3. Consider an ideal Brownian Ratchet device for the separation of DNA. The device employs multiple steps of length d. Recall that, in this case, the average speed of a molecule characterized by a diffusion constant D is given by v = 2D/d. Thus, a particle moves an average distance x = 2Dt/d in a time t.

(a) The expression for v implies that, as d gets smaller, the speed of the molecules in the ratchet grows without limit. What is wrong with this argument? What should be the real upper limit vmax to the speed of the ratcheting particle? [Note: Yes, I know that the particles cannot exceed the speed of light c, but that is not an important limit here.]

(b) What is the physical significance of dmax, where vmax = 2D/dmax? In other words, how would you know what kind of values for d cause the velocity formula to break down?

(c) The Brownian Ratchet seems like a 'free lunch' -- a violation of the 2nd Law of thermodynamics -- although in fact the ratchet is doing work. Estimate the power (= work per time = force times velocity) required to ratchet a particle with diffusion constant D. There are several reasonable ways to do this.

(d) We wish to use the ratchet to separate two lengths of double-stranded DNA, with lengths (in base pairs) N and N' respectively. Obviously, all the particles of the same size do not move at precisely the same speed, so the band of particles moving down the ratchet tends to get a little blurry with time. Therefore, there is an average spread in x(t). That is, let's suppose the square of the standard deviation in the position x is sigma^2 = Dt. Draw a sketch of the concentration versus position on the ratchet, for several different times. The idea is to show how the distribution in position evolves over time.

(e) Find a general formula for the amount of time t that you would need to move the mixture of particles across the ratchet, in order to cleanly separate the N from the N' particles. You may assume that N is not very different from N'.

(f) This problem will require some research. You may be familiar with the binomial distribution, in which a randomly walking particle has a certain probability p for moving to the right, and a probability 1-p for moving to the left, at each step. This gives rise to a certain standard deviation in the position of the particles. Since the particle in our Brownian ratchet cannot move to the left, the ratchet is actually better described by a different distribution, in which the particle has a probability p of moving to the right, and a probability 1-p of not moving at all with any step. This gives rise to a different standard deviation. What is the name of the distribution that is associated with this process? What is the standard deviation of the position of the particles after n steps?

4. Consider the electrophoresis of DNA, in a case where the randomly coiled DNA is pulled towards the edge of an unusual gel by an electric field E. Recall that DNA of length L carries negative charge per unit length (= -Q/L). This gel is made of long thin tubes that run parallel to the electric field, so if one end of the DNA enters one of those tubes, the random coil will slowly be pulled into the tube and into the gel (see figure).
   

   (a) What is the electrostatic energy U of the molecule as a function of the distance x to which it has entered the gel? You may find it useful to assume that the radius of gyration of the randomly coiled region is far smaller than x.

   (b) Now consider the entropy of the molecule. What is the entropy S of the randomly coiled portion (with length = L - x) of the molecule? You may wish to recall our simple calculation of the entropy of freely jointed chain, which has a number of configurations (let's call it gamma) available to each free link. You should have an expression for S in terms of L, p, gamma, x, and any other basic physical parameters.

   (c) Combining your results above, write down the expression for the free energy F = U - TS of the molecule, as a function of x. Sketch this function, and show how the expected behavior of the molecule is different depending on the temperature and the electric field.

   (d) At what value of the electric field does the expected behavior change?

   (e) The derivative of the free energy -dF/dx represents an effective force that pulls this molecule through the gel. Derive an expression for the time required for the molecule to be pulled into the gel by an electric field E. Assume the solvent is water with a viscosity eta.

   (f) Suppose that the electric field is suddenly turned off just as the tail end of the DNA is entering the gel. Derive an expression for the time required for the molecule to leave the gel again.

   (g) Does this suggest a mechanism by which DNA molecules can be sorted according to length? Explain.

   A device of this type is described by Turner, Craighead, and colleagues in this article