Dynamics of adaptive immunity against phage in bacterial populations

Abstract

The CRISPR (clustered regularly interspaced short palindromic repeats) mechanism allows bacteria to adaptively defend against phages by acquiring short genomic sequences (spacers) that target specific sequences in the viral genome. We propose a population dynamical model where immunity can be both acquired and lost. The model predicts regimes where bacterial and phage populations can co-exist, others where the populations exhibit damped oscillations, and still others where one population is driven to extinction. Our model considers two key parameters: (1) ease of acquisition and (2) spacer effectiveness in conferring immunity. Analytical calculations and numerical simulations show that if spacers differ mainly in ease of acquisition, or if the probability of acquiring them is sufficiently high, bacteria develop a diverse population of spacers. On the other hand, if spacers differ mainly in their effectiveness, their final distribution will be highly peaked, akin to a "winner-take-all" scenario, leading to a specialized spacer distribution. Bacteria can interpolate between these limiting behaviors by actively tuning their overall acquisition probability.

Fig 1. Schematic of the CRISPR acquisition and interference mechanism.

PAM stands for protospacer adjacent motif, a short sequence necessary for protospacer recognition by the Cas proteins.

Fig 2. Model of bacteria and phage dynamics.

Bacteria are either wild type or spacer enhanced, grow at different rates f₀ and f₁ and can be infected by phage with rates g and ηg. Spacers can be acquired during infection with a probability α and spacers are lost at a rate κ.

Fig 3. Model of bacteria with a single spacer in the presence of lytic phage.

(Panel a) shows the dynamics of the bacterial concentration in units of the carrying capacity K = 10⁵ and (Panel b) shows the dynamics of the phage population. In both panels, time is shown in units of the inverse growth rate of wild type bacteria (1/f₀) on a logarithmic scale. Parameters are chosen to illustrate the coexistence phase and damped oscillations in the viral population: the acquisition probability is α = 10⁻⁴, the burst size upon lysis is b = 100. All rates are measured in units of the wild type growth rate f₀: the adsorption rate is g/f₀ = 10⁻⁵, the lysis rate of infected bacteria is μ/f₀ = 1, and the spacer loss rate is κ/f₀ = 2 × 10⁻³. The spacer failure probability (η) and growth rate ratio r = f₁/f₀ are as shown in the legend. The initial bacterial population was all wild type, with a size n(0) = 1000, while the initial viral population was v(0) = 10000. The bacterial population has a bottleneck after lysis of the bacteria infected by the initial injection of phage, and then recovers due to CRISPR immunity. Accordingly, the viral population reaches a peak when the first bacteria burst, and drops after immunity is acquired. A higher failure probability η allows a higher steady state phage population, but oscillations can arise because bacteria can lose spacers (see also S1 File). (Panel c) shows the fraction of unused capacity at steady state (Eq 6) as a function of the product of failure probability and burst size (ηb) for a variety of acquisition probabilities (α). In the plots, the burst size upon lysis is b = 100, the growth rate ratio is f₁/f₀ = 1, and the spacer loss rate is κ/f₀ = 10⁻². We see that the fraction of unused capacity diverges as the failure probability approaches the critical value η_c ≈ 1/b (Eq 7) where CRISPR immunity becomes ineffective. The fraction of unused capacity decreases linearly with the acquisition probability following (Eq 6).

Fig 4. The distribution of bacteria with 20 spacer types.

In these simulations, 100 phage are released upon lysis (burst size b = 100) and the carrying capacity for bacteria is K = 10⁵. All rates are measured in units of the bacterial growth rate f: the lysis rate is μ/f = 1, the phage adsorption rate is g/f = 10⁻⁴, the spacer loss rate is κ/f = 10⁻². (Panel a) Distribution of spacers as a function of acquisition probability α_i given a constant failure probability η_i = η. (Eq 10) shows that the abundance depends linearly on the acquisition probability: n_i/n ∝ α_i/α. Horizontal lines give the reference population fraction of all spacers if they all have the same acquisition probability with the indicated failure probability η. (Panel b) Distribution of bacteria with different spacers as a function of failure probability η_i given a constant acquisition probability α_i = α/20. For small α, the distribution is highly peaked around the best spacer while for large α it becomes more uniform. (Panel c) The distribution of spacers when both the acquisition probability α_i and the failure probability η_i vary. The three curves have the same overall acquisition rate α = ∑_i α_i = .0972. The color of the dots indicates the acquisition probability and the x-axis indicates the failure probability of each spacer. When the acquisition probability is constant (green curve i.e. α_i = α/20) the population fraction of a spacer is determined by its failure probability. If the acquisition probability is anti-correlated with the failure probability (blue curve), effective spacers are also more likely to be acquired and this skews the distribution of spacers even further. If the acquisition probability is positively correlated with the failure probability (red curve), more effective spacers are less likely to be acquired. Despite this we see that the most effective spacer still dominates in the population.