1 Consider a series of 8 ips of a fair coin a Calculate the

1. Consider a series of 8 ips of a fair coin.

(a) Calculate the probabilities for obtaining 0-8 heads. We will consider each of thesenine outcomes to be macrostates of the system.

(b) Calculate the entropy for each of these macrostates.

(c) Sketch plots for the probabilities and entropies of the macrostates. Use a ruler to draw your axes and make sure your scales are evenly spaced.

2. A transcription factor is a molecule that binds to a specic DNA sequence in order tocontrol the expression of the nearby genetic material. In this problem we will look at theprocess by which the factor nds the correct sequence to bind to. Assume that factorcan bind loosely to a DNA sequence that does not match the target sequence. Afterevery 10 ns the molecule will move 1 base pair with a 50/50 chance that the move is tothe left or to the right.

(a) After 100 ns, what is the probability that the transcription factor is 6 base pairs tothe left of where it started? (hint: treat each step like the coin ip example fromlecture)

(b) Where is the most likely place for the factor to be after 100 ns? What is theprobability that it ends up at the most likely place?

(c) Does your answer from part (b) mean that the factor will never nd its targetsequence? Why or why not?

3. Ethane (C2H6) can rotate around the central C–C bond. When the two –CH3 groupsare aligned with each other there is a repulsion between the Hydrogen atoms that resultsin an energy 2.0 × 1020 Joules higher than when the Hydrogens are staggered. If theethane molecules are at room temperature (25 C), what is the ratio of molecules thatare perfectly aligned to molecules that are perfectly staggered?

4. The density of the atmosphere can be estimated using the Boltzmann distribution andthe gravitational potential energy of the gas molecules. Assume that the air temperatureis (20 C) and use the molecular weight of diatomic nitrogen (N2). Calculate the followingratios:

(a) The probability that a molecule is at sea level to the probability that it is at 1600meters (Denver).

(b) The probability that a molecule is at sea level to the probability that it is at 3200meters (tall peak in Rockies).

(c) The probability that a molecule is at sea level to the probability that it is at 9000meters (cruising altitude of commercial airplane).

Solution

3)

what is the ratio of molecules thatare perfectly aligned to molecules that are perfectly staggered?

1.25

4)

a) 0.245

b) 0.1215

c) 0.568