Nasty viruses, costly plasmids, population dynamics, and the conditions for establishing and maintaining CRISPR-mediated adaptive immunity in bacteria

Abstract

Clustered, Regularly Interspaced Short Palindromic Repeats (CRISPR) abound in the genomes of almost all archaebacteria and nearly half the eubacteria sequenced. Through a genetic interference mechanism, bacteria with CRISPR regions carrying copies of the DNA of previously encountered phage and plasmids abort the replication of phage and plasmids with these sequences. Thus it would seem that protection against infecting phage and plasmids is the selection pressure responsible for establishing and maintaining CRISPR in bacterial populations. But is it? To address this question and provide a framework and hypotheses for the experimental study of the ecology and evolution of CRISPR, I use mathematical models of the population dynamics of CRISPR-encoding bacteria with lytic phage and conjugative plasmids. The results of the numerical (computer simulation) analysis of the properties of these models with parameters in the ranges estimated for Escherichia coli and its phage and conjugative plasmids indicate: (1) In the presence of lytic phage there are broad conditions where bacteria with CRISPR-mediated immunity will have an advantage in competition with non-CRISPR bacteria with otherwise higher Malthusian fitness. (2) These conditions for the existence of CRISPR are narrower when there is envelope resistance to the phage. (3) While there are situations where CRISPR-mediated immunity can provide bacteria an advantage in competition with higher Malthusian fitness bacteria bearing deleterious conjugative plasmids, the conditions for this to obtain are relatively narrow and the intensity of selection favoring CRISPR weak. The parameters of these models can be independently estimated, the assumption behind their construction validated, and the hypotheses generated from the analysis of their properties tested in experimental populations of bacteria with lytic phage and conjugative plasmids. I suggest protocols for estimating these parameters and outline the design of experiments to evaluate the validity of these models and test these hypotheses.

Figure 1. Adsorption rate as a function of the multiplicity of infection (MOI), δ_MIN = 10⁻¹⁴, δ_MAX = 5×10⁻⁹, x = 0.5, or x = 0.2 q = 10², and n = 2.

Figure 2. Model of the population dynamics of lytic phage with CRISPR-mediated adaptive immunity and envelope resistance in continuous culture: P – phage, N – phage sensitive non–CRISPR bacteria, N_R – envelope resistant, non–CRISPR bacteria C - phage sensitive CRISPR bacteria, C_R - phage immune CRISPR bacteria.

The δs are the adsorption rate constants, m is the fraction of C to which phage are adsorbed that enter the immune state, ν is the rate at which immune CRISPR cells lose their immunity, and μ is the rate of mutation to envelope resistance. While the phage adsorb to immune CRISPR cells at the maximum rate and are removed from the phage population, their replication on CRISPR cells and the rate of mortality of immune CRISPR is either 0 or a monotonically increasing function of the multiplicity of infection (equation (1)). The bacteria reproduce at a rate proportional to the concentration of a limiting resource and their maximum rates of replication. Phage replication is through the killing of adsorbed bacteria and their burst size, β, on that cell line. The limiting resource in the reservoir is at concentration A µg/ml and enters the vessel at a rate, w, which is the same rate at which the phage and bacterial populations and excess resource, R, are removed from the vessel. For more details see the text.

Figure 3. Model of the population dynamics of a conjugative plasmid with CRISPR-mediated adaptive immunity in continuous culture.

N - plasmid-free non–CRISPR, N_P - plasmid-bearing non–CRISPR, C - plasmid-free CRISPR, C _P - plasmid-bearing CRISPR, C_X - immune CRISPR. The γs are the rate constants of plasmid transfer, m is the fraction of C_P that enter the immune state C_X upon receiving the plasmid from an N_P or C_P, ν is the rate at which immune CRISPR cells lose their immunity and z the rate at which the CRISPR cells lose the CRISPR element and become N or N_P. The bacteria reproduce at a rate proportional to the concentration of a limiting resource and their maximum rates of replication. The limiting resource in the reservoir is at concentration A µg/ml and enters the vessel at the rate, w, which is the same as the rate at which the phage and bacterial populations and excess resource, R, are removed from the vessel. For more details see the text.

Figure 4. Population dynamics of lytic phage, P, with sensitive non–CRISPR bacteria, N, non-immune and immune CRISPR-encoding cells, C and C_R, respectively.

Changes in the densities of the bacterial and phage populations and the concentration of the limiting resource, R. In this and the other simulations, A = 50 µg/m, w = 0.2 per hour, e = 5×10⁻⁷µg, k = 0.25 µg. In these phage simulations, β_N = β_C = β_CP. (a) The dynamics of sensitive bacteria and phage in the absence of CRISPR, V_N = 1.0 hr⁻¹, δ_N = 5×10⁻⁹. (b) Invasion of CRISPR in the presence of phage, no MOI effect (x = 0), V_N = 1.0. V_C = 0.95, V_CR = 0.90, δ_N = δ_C = 5×10⁻⁹, δ_CP = δ_MIN = 10⁻¹⁴ (δ_MAX = 5×10⁻⁹) (c) Invasion of CRISPR with presence of phage V_N = 1.0. V_C = 0.95, V_CR = 0.90, δ_N = δ_C = 5×10⁻⁹, Strong MOI effect (x = 0.5, n = 2.0, q = 10², δ_MIN = 10⁻¹⁴, δ_MAX = 5×10⁻⁹). (d) Invasion of CRISPR with presence of phage, V_N = 1.0. V_C = 0.95, V_CR = 0.90, δ_N = δ_C = 5×10⁻⁹, Modest MOI effect (x = 0.2, n = 2.0, q = 10², δ_MIN = 10⁻¹⁴, δ_MAX = 5×10⁻⁹).

Figure 5. Population dynamics of lytic phage, P, with sensitive and resistant non–CRISPR bacteria, N and N_R, non-immune and immune CRISPR-encoding cells, C and C_R, respectively.

Changes in the densities of the bacterial and phage populations and the concentration of the limiting resource, R. Unless otherwise noted, the parameter values used are those in Figure 4B. (a) Invasion of C and N_R into a population with phage, modest cost of resistance, V_NR = 0.85. (b) Invasion of C and N_R into a population with phage, with a greater cost of resistance, V_NR = 0.70.

Figure 6. Population dynamics of a conjugative plasmid with non–CRISPR, N and N_P and CRISPR, C, C_P and C_X populations; changes in the densities of the bacterial populations.

Unless otherwise noted all of the rate constants of plasmid transfer, the γ_ijs = 10⁻⁹ , the segregation rates, τ_N and τ_C = 10⁻³, the rate of loss of immunity ν = 10⁻³, upon receiving the plasmid the rate of conversion of C_P to C_X = 0.2, and the rate of conversion of CRISPR cells to N or N_P, z = 10⁻⁸. (a) No CRISPR – Just N and N_P 1 - Deleterious plasmid V_N = 1, V_NP = 0.95; 2 - a beneficial plasmid V_N = 1, V_NP = 1.2 and 3- deleterious plasmid V_N = 1, V_NP = 0.95, γ_NN = 10⁻¹¹. (b) Invasion of bacteria carrying a deleterious plasmid into a lower fitness CRISPR, C, population, V_N = 1, V_NP = 0.95, V_C = 0.97, V_CP = 0.88, V_x = 0.96, (c) Invasion of CRISPR X into a equilibrium population of plasmid-bearing and plasmid free cells, N-NP with a deleterious plasmid (parameters the same as b). (d) Invasion of cells carrying a higher fitness plasmid, NP, into a C population, V_N = 1, V_NP = 1.2 V_C = 0.97, V_CP = 1.1, V_x = 0.96.