Extend the repressilator idea to a ring of four repressor genes: gene 1 represses gene 2
PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Genetic circuits, regulation and biological information · Genetic circuits, regulation and biological information
Forces, Machines and Signals, Ch Genetic circuits and biological pattern
Problem
Extend the repressilator idea to a ring of four repressor genes: gene 1 represses gene 2, gene 2 represses gene 3, gene 3 represses gene 4, and gene 4 represses gene 1, each obeying the same scaled kinetics x˙i=f(xi−1)−xi. Linearising about the symmetric steady state, each gene's own decay contributes −1 on the diagonal and the repression from its neighbour contributes an off-diagonal slope g<0 (with ∣g∣ increasing with promoter strength and cooperativity). By the same reasoning used for the ring of three, the eigenvalues of the linearised system are λk=−1+gωk,ωk=e2πik/4,k=0,1,2,3, the fourth roots of unity.
((a)) Evaluate ωk for k=0,1,2,3 and give the four eigenvalues λk in terms of g.
((b)) Identify which eigenvalue(s) can acquire a positive real part as ∣g∣ increases from zero, and find the value of ∣g∣ at which this first happens.
((c)) State whether the mode that goes unstable first is real or part of a complex-conjugate pair, and use that to decide whether a ring of four repressors is expected to oscillate like the ring of three or to show a toggle-like instability like the ring of two.