Regulated CRISPR Modules Exploit a Dual Defense Strategy of Restriction and Abortive Infection in a Model of Prokaryote-Phage Coevolution
(A) p, q and v denote densities of uninfected, infected cells, and phage respectively. q undergoes autoimmunity at a rate of γq→ϕ, while p undergoes autoimmunity at a suppressed rate determined by δγp→ϕ. The second figure shows the bifurcation behavior of the free cell densities with respect to the control parameter δ, beyond a certain critical value of which one of the steady states vanishes. (B,C) Two-parameter bifurcation diagram revealing coexistence (C) and host extinction (E). Each plot instance is denoted by a tuple <AB> where A and B can indicate low (L) or high (H) values or extinction (E) of prokaryote (free—solid, infected—dotted) and phage (dashed) respectively. GP→ϕ and GQ→ϕ denote the rescaled free cell autoimmunity rate and infected cell autoimmunity (abortive infection) rates respectively. High values of GQ→ϕ lead to complete phage evasion. Parameter values αp = 1 hr−1, Φ = 108 cells ml−1, γv = 5 hr−1, αv = 50, αq = 5×10−9 ml phage−1hr−1.