BioE41, Final Exam (take home)

Due: Friday, outside E350B (3rd floor Clark Center)

This is a 6hr take home final. You can choose anytime in the next 48hrs tocomplete the exam. Please complete the exam in a single sitting and indicate clearlyon the top of the exam what time you started working on the same and what timeyou stopped. You can use your regular textbooks and any class notes as reference andthe exam is open book. Please do not use any other sources (internet/papers etc).The work submitted for the final exam should be done strictly independently (unlikeproblem sets where we encourage discussions). You are not allowed to discuss theproblems with your class mates during this 48hr period and we will take any cases ofStanford honor code violation very seriously.

If you give me two free parameters, I can describe an elephant. If you give methree, I can make him wiggle his tail - Eugene Wigner, 1902-1995

Problem 1 Radiometer toy and maxwell’s demon, 10 points

A radiometer was a device invented in 1872 by William Crooke to measure the energyemitted by a light source, but today it is a novelty or a toy sold in science shops.Inside a sealed, partially evacuated glass bulb, four vertical metal vanes are attachedto a metal hub that can rotate around a vertical needle. The vanes have the samearrangement of colors: white on one side and black on the other side. When thedevice is mounted near a light source, the vanes and the hub rotate around thevertical needle, rotating faster for brighter light.(a) What causes this rotation at the first place? What is the direction (does the blackside of the vane lead or the white side of the vane)?(b) Did we break any laws of thermodynamics by converting useless energy to usefulenergy?(b) Could you reverse this rotation? How?

Problem 2 Transition between B to S-form DNA, 15 points

DNA when subjected to stretching force exceeding 60pN of force undergoes a struc-tural transition from the usual B form to the so called S form (”S” for stretch). Here

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Figure 1: A Solar Radiometer with four vanes colored black and white on the sides.The vanes are white on one side, black on the other with the same arrangement goingaround the needle. If you have never seen this in action; you can watch the youtubevideo at this link: https://www.youtube.com/watch?v=s2yXly2xNVM (This is theonly time you can use the computer - to watch this single video).

we examine a simple model of this transition based on the freely jointed chain modelof DNA and compare the model to experimental data.

Figure 2: (A) Random-walk model for B to S DNA transition. The B DNA linkshave a length b and the S DNA links have length a. (B) Force-extension curve fordouble stranded DNA. (B, from C. Bustamante et al., Nature 421:423, 2003.)

(A) Consider the freely jointed chain model in one dimension. Each link of thepolymer points in the +x or the −x direction. There is a force f in the direction +xapplied at one of the ends (see figure 2A). To account for the B-to-S transition, weassume that links are of length b (B state) or a (S state), with a > b. Furthermore,there is an energy penalty ε of transforming the link from a B state to an S state.(This is the energy, presumably, for unstacking the base pairs in DNA). Write downthe expression for (a) energy and the Boltzman factor for each of the four states of asingle link and (b) the partition function.

(B) Compute the average end-to-end distance for one link. The average end-to-enddistance for a chain of N links is N times as large.

(C ) Plot the average end-to-end distance normalized by Nb (that is, the relativeextension or fractional extension) as a function of the force using the numbers ap-

BioE41, Final Exam 2013 Final Exam

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Figure 3: Toy model of protein folding showing four different conformations.

propriate for DNA, namely, b = 100nm, and, a = 190nm. To estimate ε , take the”energy per base pair” for transforming B-DNA to S-DNA to be 5kBT (the length ofone base pair is approximately 1/3 nm for B DNA). (Hint: You can use a computeror a calculator to qualitatively plot the fractional extension as a function of force).How does your model compare with experimental data in figure 2(B)?

Problem 3 Statistical mechanics of an optical trap, 10 points

Lasers are often used to trap micron-sized beads (as described in chapter 5, p.207,PBOC). The dynamics of such a bead can be thought of as the Brownian motion ofa particle in a quadratic energy well. Compute the mean-squared excursion < x2 >of such a bead in a one-dimensional quadratic well with a potential-energy profileU(x) = 1

2kx2 and show that we can then determine the trap stiffness as k = kBT

<x2>.

Problem 4 Toy model of protein folding, 10 points

A four-residue protein can take on the four different conformations shown in figurebelow. Three conformations are open and have energy ε (where ε > 0) and one thatis compact, and has energy zero.(a) At temperature T , what is the probability, po, of finding the molecule in an openconformation? What is the probability, pc, that it is compact?(b) What happens to the probability pc, calculated in (a), in the limit of very largeand very low temperatures.(c) What is the average energy of the molecule at temperature T?

Problem 5 Rubber band thermodynamics, 10 points

(a) Quickly stretch a rubber band while holding it against your upper lip. Why doesthe rubber band become warm enough for your lip to sense the same?

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(b) With the rubber band still stretched, hold it away from your lip for a minute, andput it back on your lip, and then let it quickly contract. Why does it become cool?c) Can you imagine building a rubber band heat engine? If yes, provide details (A heatengine is a machine that converts heat energy into mechanical motion, say spinningof a disc).

Best of luck.Manu

BioE41, Final Exam 2013 Final Exam