Fig 1.
Functioning of CRISPR-Cas system.
Three spacers are colored according to their age from the time of their acquisition, from dark green marking the youngest (the most recently acquired) spacer to yellow marking the oldest one (which was acquired the earliest). Phages carry protospacers colored similarly to their matching spacers; mutated protospacers are colored white. There are more mutated protospacers among protospacers matching older spacers than among protospacers matching younger ones. Inside the cell, bean-shaped objects are CRISPR effector complexes armed with individual crRNAs. Complexes with crRNA of younger spacers are more abundant than those with older ones. Viral DNA is shown to be simultaneously assessed by two CRISPR effector complexes: the dark green CRISPR spacer matches the non-mutated corresponding protospacer while the protospacer corresponding to the yellow spacer has mutated. The former interaction results in destruction of viral DNA while the latter leaves it intact.
Fig 2.
A cell with S = 3 CRISPR spacers encounters viruses as a Poisson process with an average rate rN. During each encounter there is either a successful interference with probability I or the cell dies with probability 1 − I. We evaluate the probability E(t) of the cell to survive till time t as the measure of performance of its CRISPR-Cas system.
Fig 3.
Typical survival probability profile.
(A) Plot of survival probability E(t) vs. the crRNA decay coefficient δ and the number of spacers in CRISPR array S. Other parameters are: β = 1, χ = 1.4, μ = 0.9, and rNt = 5. (B) Six curves of E(t) vs. S for various values of δ and same β, χ, m, and rNt as in the panel A.
Fig 4.
Effects of mutation rate and binding efficiency.
A set of 25 panels illustrating how the survival probability depends on S and δ for various values of protospacer mutation probability 1 − μ and binding efficiency of effectors β. The δ and S axes in each small panel have the same range as in the panel A in Fig 3, while the scale of the heat-map varies and is indicated to the right of each panel. The external axes describe the variation of mutation probability 1 − μ and effector binding efficiency β. In all panels χ = 1.4 and rNt = 5.
Fig 5.
Effect of parameters on the optimal number of spacers and the maximal survival probability.
The optimal number of spacers and corresponding survival probability as functions of one of the array-unrelated parameters: (A) As function of mutation probability 1 − μ, other parameters are β = 1 and χ = 1.4. (B) As function of binding efficiency β, other parameters are μ = 0.9 and χ = 1.4. (C) As function of interference efficiency χ, other parameters μ = 0.9 and β = 1. The average number of viral infections was rNt = 5 in all panels.
Fig 6.
The optimal number of spacers and maximal cell survival probability.
The optimal number of spacers (A) and the maximal cell survival probability (B) are shown vs. a range of binding efficiencies β and mutation probabilities 1 − μ for rNt = 5 and χ = 1.4.
Fig 7.
CRISPR performance for two virus species.
Plot of the survival probability E(t) as a function of crRNA decay coefficient δ and the number of spacers S of a cell confronting two different viruses with equal population sizes, ν1 = ν2 = 0.5. The binding efficiency is β = 1 and the interference efficiency is χ = 1.4. Viral mutation probability 1 − μ is equal to 0.1 and rNt = 5.
Fig 8.
Survival probability vs diversity of the virus pool.
Plots of the optimized over δ and S cell survival probability and the number of spacers vs the number of viral species and the composition of a two-virus pool for β = 1, χ = 1.4, μ = 0.9 and rNt = 5. (A) Maximal survival probability E(t) (outer plot) and optimal number of spacers Sopt (inner plot) as a function of the number of virus species n. The abundance of virions belonging to different species in the viral pool are the same for all species, ν1 = … = νn = 1/n. (B) The maximal survival probability vs the relative abundance of one of the viruses in a two-virus pool.