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A6 : Due 11/14

1. Consider a system consisting of three proteins A, I and G respectively. A and I are constitutively expressed. A associates with itself to form homodimers D, which can also dissociate. Homodimers form active transcription factors that drive the expression of G. On the other hand, A and I associate to make heterodimers, H, which can also dissociate. The heterodimers do not form active transcription factors. Thus, the I proteins are called transcriptional co-repressors. Finally, I degrades at a rate proportional to the concentration of a small molecule u, that can be added exogenously (like an inducer) and diffuses freely throughout the cell. Thus when u is added, the I proteins degrade, allowing the A proteins to form more homodimers, thus driving the expression of G.

a) Design a mass action kinetics model of this system. Assume that the gene g for G is in two states gon and goff and that the homodimers bind and unbind the off gene to make an on gene. Don't forget dilution of all species (except the gene).
b) Derive the differential equation model from the mass action kinetics model. Assume association reactions are 2X faster than dissociation reactions, which are 10X faster than gene expression, which is 10X faster than dilution/degradation. Assume degradation of I by u is about the same rate as dissociation.
c) Simulation the equations with u>0, starting in initial conditions defined by u=0. What happens as the heterodimer association rate is increased?

Note: This problem is designed to help you think about making models of biological systems, and is a bit of a review. Focus on making a good, convincing model in which all of the biology makes sense.

2. Consider the reactions

X → X + Y
Y → Y + X
X → ∅
Y → ∅

where the first two reactions have rates k1 and the second two have rates k2.

a) Using the extended generator, find the equations for the rates at which the means and variances of X and Y change.
b) Discuss the different behaviors obtained with k1 < k2, k1 = k2, and k1 > k2. Simulate the means and variances for each of these cases (you choose the exact values). Start with the deterministic state having 1 X and 2 Y (meaning that the variances are initially zero).
c) [EXTRA CREDIT] Simulate the system using the Gillespie algorithm and overlay the means and one-standard deviation window (as in Lecture 15, slide 4).

3. (a) Find a register machine that determines whether the initial value of of register R0 is even or odd. If it is even, the machine should terminate with a 0 in register R1. Otherwise, it should terminate with a 1 in register R1. (b) List the stochastic chemical reactions for your system. (c) Simulation your system in gro. The easiest thing to do would be to have your cells not grow. You might want to use multiple cells running in parallel to see which get the right answer and which don't. Collect data and plot histograms of the resulting computations.

Turn in A6 here: