Two genes repress each other with a Hill coefficient (three regulator molecules bind…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Biophysics & Medical Physics · Genetic circuits, regulation and biological information

Forces, Machines and Signals, Ch Genetic circuits and biological pattern

Problem

Two genes repress each other with a Hill coefficient n=3n=3 (three regulator molecules bind together at each promoter). In scaled, dimensionless units the mutual-repression circuit obeys u˙=α1+v3−u,v˙=α1+u3−v,\dot u = \frac{\alpha}{1+v^3} - u, \qquad \dot v = \frac{\alpha}{1+u^3} - v, where uu and vv are the scaled protein levels of the two genes and α\alpha is the scaled maximal production rate. ((a)) Write the equation satisfied by the symmetric fixed point u=v=u∗u=v=u^\ast, and derive the condition on α\alpha for this symmetric state to be unstable to a perturbation that raises uu and lowers vv (or vice versa). Express the threshold as a critical value αc\alpha_c obtained from the corresponding u∗u^\ast. ((b)) Evaluate αc\alpha_c for n=3n=3, to two significant figures. ((c)) Take α=2.5\alpha=2.5. Find u∗u^\ast by trial to two significant figures, evaluate the antisymmetric eigenvalue, and state whether the circuit is bistable at this α\alpha.

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