Phys 139: Problem set 7

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Phys 139: Problem set 7

DUE BY 5pm ON FRIDAY, MAY 31

May 22, 2024

1. Do some literature research to answer the following: what are the names of the RNA polymerase(s) in prokaryotes, and in eukaryotes? How fast do they typically travel when transcribing a gene? What is the average gene length in prokaryotes vs eukaryotes, and thus how long does RNA production typically take?

2. We discussed in class the multimeric nature of many transcription factors (TFs), and the way this changes their interaction with induc-ers (small molecules, i.e. ligands, whose binding modulates the TFs DNA-binding activity). The classic model system of ligand interac-tions with a multimeric receptor is that of hemoglobin, a tetrameric protein within red blood cells that is responsible for binding to oxy-gen (O2) and delivering it to tissues. Each hemoglobin can bind four oxygen molecules.

(a) First assume that each binding event is independent of the others, and that each bound O2 has an energy E relative to that of free O2 in solution. Assume that the free O2 has a chemical potential µ, and thus an “ activity” λ ≡ exp[µ/T]. Use a grand canonical ensemble approach to sketch i) the probability that one and only one O2 is bound to hemoglobin versus λ, and ii) the probability that exactly four O2 are bound to hemoglobin. In order to draw each sketch with some accuracy, you should approximate the de-pendency of probability on λ in both the limits of large and small λ, and you should indicate those dependencies on your sketch.

(b) Now make the Hill approximation, and assume binding is fully cooperative, i.e. with ‘all or none’ dynamics, but with the four-bound-O2 state still having total energy 4E. Sketch (with approx-imations, as above) the probability of binding four O2 versus λ. Compare to the results of the previous part, and comment on the comparison.

3. Sensitivity of negative autoregulation. A protein acts as its own repressor, with autorepression dictated by a Hill function with coeffi-cient n. The dynamics of the protein concentration, X, can thus be described by

with binding constant K, maximal production rate β, and degradation parameter α.

(a) In the limit of strong autorepression, (X/K) n >> 1, find the tra-jectory of concentration, X(t), in terms of the steady-state con-centration Xs, and assuming X(0) = 0. Calculate the response time of the system as T1/2 , where X(T1/2 ) = Xs/2. How does this response time correspond to that of a system with simple production dynamics, ˙X = β − αX?

(b) The parameter sensitivity coefficient, S, for a variable A that depends on a parameter B, is defined as the relative change in A given a small relative change ∆B/B in B. Specifically,

For this (strong) autorepressor, analyze the sensitivity of the steady state concentration to the maximal production rate, i.e. what is S(Xs, β)? Why might a biological system evolve to have a Hill parameter n > 1?

4. Double positive feedback. Consider a system of two proteins, X, Y , that act as activators for each other.

Such a system will generally obey the rate equations

where α is the degradation rate (assumed the same for both), and f(Z) is the production rate. Assume the following form for the production rate:

where the parameters β and K are, for simplicity, taken to be the same for both proteins.

(a) Write a few words indicating what motivates this functional form of the production rate.

(b) Analysis of the rate equations, and some algebra, show that the following equation defines the steady state, Xs:

with an identical equation defining the steady state Ys. Show that this system can be bistable, and find the critical point that controls the appearance of bistability. What are the relative con-centrations of the two species at the fixed points?

(c) Select relevant parameters to make a phase portrait showing the streamlines and nullclines of the system, for the situations on both sides of the critical point. Mathematica provides relatively straightforward functions for plotting streamlines (e.g. ‘Stream-Plot’), though you can find web-based plotters online. If possible, overlay the nullcline plot with the streamline plot. Comment on the stability of the various fixed points.