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. 2013 Apr;10(2):025004.
doi: 10.1088/1478-3975/10/2/025004. Epub 2013 Mar 15.

Physical Model of the Immune Response of Bacteria Against Bacteriophage Through the Adaptive CRISPR-Cas Immune System

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Free PMC article

Physical Model of the Immune Response of Bacteria Against Bacteriophage Through the Adaptive CRISPR-Cas Immune System

Pu Han et al. Phys Biol. .
Free PMC article

Abstract

Bacteria and archaea have evolved an adaptive, heritable immune system that recognizes and protects against viruses or plasmids. This system, known as the CRISPR-Cas system, allows the host to recognize and incorporate short foreign DNA or RNA sequences, called 'spacers' into its CRISPR system. Spacers in the CRISPR system provide a record of the history of bacteria and phage coevolution. We use a physical model to study the dynamics of this coevolution as it evolves stochastically over time. We focus on the impact of mutation and recombination on bacteria and phage evolution and evasion. We discuss the effect of different spacer deletion mechanisms on the coevolutionary dynamics. We make predictions about bacteria and phage population growth, spacer diversity within the CRISPR locus, and spacer protection against the phage population.

Figures

Figure 1
Figure 1
a) Typical repeat in S. thermophilus CRISPR1, data taken from [7]. b) RNA repeat. c) Left: secondary structure (hairpin) of RNA repeat. right: pre crRNA. d) Targeting and protecting during CRISPR immunity.
Figure 2
Figure 2
Addition of a new spacer to the CRISPR locus at the leader-proximal end. The protospacer in virus i added to CRISPR. Spacers 1– (N −1) are shifted to the right. Spacer N is deleted. Other deletion mechanisms will be discussed later.
Figure 3
Figure 3
In the Markov process, five categories of transition events change the state of the system. The rates of all of these events are denoted by φi. Processes 2, 3, and 5 all result from phages infecting the bacteria. Processes 4 results from phage infecting some bacteria, which could be the population under study, or a different host population of bacteria. There is an additional category of events, not shown in this figure, which is evolution of the virus due to mutation or recombination.
Figure 4
Figure 4
The solution of the differential equation and the result of the stochastic method. Parameters are c = 0.15, r = 0.045, β = 2 × 10−5, γ = 0.1, vmax = 17500, and xmax = 4500. There are 2 spacers in a CRISPR. The error bars are one standard error. The bacterial growth rate sets the explicit time scale in this model.
Figure 5
Figure 5
When multiple phages infect the same bacteria, two parents may produce a descendant by the polymerase copying along one strand with probability 1−pc and switching to another strand with probability pc. This process leads to recombination between the phage genomes.
Figure 6
Figure 6
Population of bacteria and phages with time. We show results for constant and density-dependent phage growth rates. The parameters are c = 0.15, r = 0.05, β = 2 × 10−5, and γ = 0.1. The mutation rate per sequence per replication is μ = 0.01. The maximum population of phage is vmax = 6000, and the maximum population of bacteria is xmax = 12000. The maximum number of spacers in a CRISPR is 30. When the number of spacers in the CRISPR array is over 30, the oldest spacer is deleted from the leader-distal end. There are 149 phage strains with a logarithmic initial population distribution.
Figure 7
Figure 7
Diversity of spacers at CRISPR position i at different times. The parameters are the same as in Fig. 6.
Figure 8
Figure 8
Protection afforded by spacers at different positions of CRISPR at different times. Protection is defined as the number matches between the spacer and the protospacers in the current phage population. The parameters are the same as in Fig. 6.
Figure 9
Figure 9
Spacer diversity versus location when the probability of deleting spacer i is proportional to i. a) When the number of spacers in the CRISPR array is over 30, one spacer is selected to be deleted with a possibility proportional to its distance to the leader proximal end. b) When the number of spacers in the CRISPR array is over 30, one spacer at a random location is deleted. The parameters are the same as in Fig. 6.
Figure 10
Figure 10
Diversity at different positions of CRISPR at different times for l = 2 with a) mutation only and b) recombination only. In this case, CRISPR recognize phage with zero or one mismatch between the spacer in the bacterium and protospacer in the phage. The recombination rate per sequence per replication is ν = 0.01, and pc = 0.5. The other parameters are the same as in Fig. 6. Spacer diversity is not particularly sensitive to whether the phage evolve by mutation or recombination.
Figure 11
Figure 11
Diversity of the phage for l = 2 with mutation only. The parameters are the same as in Fig. 10.
Figure 12
Figure 12
Protection at different positions of CRISPR at different times for l = 2 with a) mutation only and b) recombination only. The parameters are as in Fig. 10.
Figure 13
Figure 13
Immunity, β Σk Σi,j xi,jvk(δi,k + δj,k), at different mutation rate and recombination rate when l = 1 (a) and l = 2 (b). The other parameters are as in Fig 6. The immunity is averaged over the time range t = 100 to t = 300.

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