Antagonistic coexistence between microorganisms and viruses preying on them often displays rich, complex population dynamics. One of the adaptation mechanisms employed by hosts to curb viruses is the CRISPR-Cas system, a recently discovered adaptive immunity system that is present in the great majority of Archaea and many bacteria [1], [2]. Population studies and agent-based modeling [3] reveal complex behavior of the actual populations that displays long-term persistence of both the host and the virus lineages.

Here we describe and study 3-dimensional Lotka-Volterra type mathematical model of interaction and coevolution of viruses and microbial host that possess CRISPR-Cas hereditary adaptive immunity systems.  A bifurcation analysis of the Malthusian and logistic versions of the model display complex, and in particular quasi-chaotic oscillation regimes. The key factors for the appearance of quasi-chaotic oscillations are three-dimensionality of the model, which divides the hosts into the immune and susceptible populations, and the non-linear dependence of the immunity on the amount of viruses. Quasi-chaotic regimes of virus-host coevolution are likely to be biologically relevant given the extreme evolutionary instability of the CRISPR-Cas loci revealed by comparative genomics [1].

Acknowledgement: this research was supported by the Intramural Research Program of the NIH, NCBI.

References

1. Koonin E.V., Makarova K.S. Evolution of an RNA-based adaptive immunity system in prokaryotes. RNA Biol., 2013, 10(5):679-86.
2. Sorek R., Lawrence C.M., Wiedenheft B. CRISPR-mediated adaptive immune systems in bacteria and archaea. Annu.Rev. Biochem 2013, 82:237-66.
3. Iranzo J., Lobkovsky A.E., Wolf Y.I., Koonin E.V. Evolutionary dynamics of the prokaryotic adaptive immunity system CRISPR-Cas in an explicit ecological context. J. Bacteriol. 2013, 195(17):3834-44.