Figure 2.

Periodic solutions and travelling waves in the generalized repressilator model. (a) The period of the limit cycle of the deterministic model of odd rings (solid line) increases linearly with the length of the ring. Simulations of the stochastic version of this system using the Gillespie algorithm show that the period (shown as circles) follows the same trend, although they are slightly larger. The period of the quasi-stable solutions found in even rings (deterministic and stochastic) increases linearly with the number of genes but with a slope that is half that of the odd rings. The inset shows representative time traces of the periodic solutions in odd rings (stable) and in even rings (quasi-stable). (b) Time snapshot of the spatial distribution of the concentrations of two successive protein concentrations for the periodic solution in the odd ring with n = 23. The solution has a travelling wave structure with a kink-like perturbation propagating around the ring, indicated by the arrow in the bottom figure. The bottom figure represents the minimum distance |Δp_j| = min(|p_u − p_j|, |p_d − p_j|) between the travelling wave solution and the dimerized solution with an alternating pattern of protein expression given by p_u and p_d. The distance becomes large around the kink in the travelling wave solution. (c) Same as (b) for the quasi-stable periodic solution of the even ring with n = 22. In this case, the travelling wave solution has two kinks that propagate around the ring, as indicated by the arrows.