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HW_2_2011_final

# 2 lost in translation in class i briefly touched on

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2. Lost in Translation In class I briefly touched on the idea of molecular recognition based strictly on binding reactions. In this problem, you will work out a simple estimate using the Boltzmann distribution of the rate of errors in translation. Read section 18.4.1 of PBoC that considers biological fidelity. Consider the HP world in which there are only two kinds of amino acids, hydrophobic and polar, and explore the lowest error rates that can be achieved by simple thermodynamic binding reactions. Make sure you explain your simplified picture of the translation process in this world and how you are going to invoke equilibrium statistical mechanics to evaluate its fidelity. The essence 3

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of our analysis is the claim that the error rate is given by error rate = rp err rp err + rp corr , (8) where r is the synthesis rate and p err and p corr are the probabilities of the binding reaction being the incorrect or the correct tRNA. Give a heuristic explanation of the equation given above - what assumptions are being made about the rates of binding and the rate of incorporation to arrive at this equation? Use the fact that the rate of addition of a given species is dN err dt = rp err (9) and dN corr dt = rp corr , (10) where N err and N corr are the number of incorrect and correct incorporations. As we did in chap. 18, take the binding energies to be err and corr , respec- tively. Using what you know about the translation process, make an estimate for Δ = corr - err and then obtain the result that error rate p err p corr (11) and make a plot of this error rate as a function of the energy difference. Make sure you explain why this definition of the error rate is the right idea. Fur- ther, explain why this result demonstrates that there has to be something more to molecular recognition in processes such as translation than mere thermodynamic discrimination. NOTE: In this problem, think of yourself as writing an explanation that could be read by someone else to learn about thermodynamic discrimination. Do not simply copy our way of doing it in the book and make sure you fill in missing steps and that you provide ap- propriate explanatory text. 3. Random Walks, Polymer “Size” and Genome Length The goal in this problem is to estimate the length of the E. coli genome in basepairs by examining an electron microscopy image of its exploded DNA. 4
Chapter 8 of PBoC might be useful for carrying out these estimates and cal- culations. (a) In class I made the claim that the size of the genome is given by size q h R 2 i = a N. (12) State all of the assumptions that go into deriving this expression and then derive it for yourselves. In particular, use the fact that the i th link of the polymer is characterized by the vector a e i to write R = N i =1 a e i . Show that h R i = 0 and explain why that makes sense. Then, use the fact that we can

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