Problem 1:

(a) E.coli swims at about 20 p.111/S by rotating a bundle of helical flagella. If the motors were to turn 10 times faster than normal, what would their swimming speed be? If their fluid environment were made 10 times more viscous, but the motors were to turn at the same rate, what would the swimming speed be? How does the power output of the motor change in these two hypothetical situations?

(b) Two micron-sized spheres, one made of silver and the other gold, sediment (that is, fall under gravity) in a viscous fluid. The silver sphere has twice the radius of the gold one. Which sediments faster?

(c) The left ventricle of the human heart expels about 50 cm3 of blood per heartbeat. Assuming a pulse rate of 1 heartbeat per second and a diameter of the aorta of about 2 cm, what is the mean velocity of blood in the aorta? What is the Reynolds number?

(d) What is the Reynolds number of a swimming bacterium? A tadpole? A blue whale? (Adapted from a problem courtesy of H. C. Berg and D. Nelson.)

Problem 2:

Show that the Boltzmann factor for a system in a state with degeneracies _i∑e^-E_i/k_BT

where i is a sum over degenerate states of the same energy can be written as e^-F_i/k_BT. In the same way, when dealing with constant pressure, the Boltzmann factor becomes e^-G_i/k_BT . This is what you need in problem no. 7.

Problem 3:

Particles in suspension in water have a mean height distribution of z_i *= 2 cm. To produce a colloidal suspension that does not settle at the bottom of a jar of 10cm height, the particles need to be broken up into smaller pieces. In how many equal fragments should these particles be broken so that z₂* = 10 cm?

Problem 4:

A protein has two conformations, compact (C) and expanded (EX) with an energy difference of 3k_bT_R, the lower energy one C has a degeneracy g_c = 2. The higher one EX has a degeneracy g_Ex.

Determine the value of gEx such that at room temperature C is ten times as likely as EX.

This protein is placed in a lipid bilayer, where its configuration is sensitive to the tension in the bilayer. The two states have areas A_c = 10 nm² and AEx = 20 nm² respectively. At what tension will the probability of occupancy of the two states be equal, recalling that under a constant tension the Gibbs free energy will be minimized G = U - TS - TA (T is an applied tension)? Take g_Ex = 5 for this part.

Problem 5:

To add to your insight on the partition function, show that Z = Ωexp(-β), where ,Ω is the number of states and< E> the average energy for a particular choice of thermodynamic variables.

Problem 6:

Helix-coil transition (from Nelson no. 9.5 p. 396)

We will explore a simple model of the helix-coil transition in an artificial polypeptide (see Zimm et al PNAS 45:1601-1607 (1959)). The helix is an ordered molecule which has higher energy than the random coil version, but still as the temperature is raised the transition goes from random coil to helix. The following simple model of "total cooperativity" provides a simple explanation. "total cooperativity" simply means that the each molecule is either a random coil or a helix and not segments of each of the two phases.

(i) In the optical rotation experiment shown above, we see a continuous variation of the rotation angle of polarized light and not a jump from the value associated with the random coil to the value associated with the helix. Explain qualitatively why.

(ii) Now calculate the proportion of molecules in the helix phase as a function of temperature.