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A bacteriophage model based on CRISPR/Cas immune system in a chemostat

. Key Laboratory of Eco-environments in Three Gorges Reservoir Region, School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Clustered regularly interspaced short palindromic repeats (CRISPRs) along with Cas proteins are a widespread immune system across bacteria and archaea. In this paper, a mathematical model in a chemostat is proposed to investigate the effect of CRISPR/Cas on the bacteriophage dynamics. It is shown that the introduction of CRISPR/Cas can induce a backward bifurcation and transcritical bifurcation. Numerical simulations reveal the coexistence of a stable infection-free equilibrium with an infection equilibrium, or a stable infection-free equilibrium with a stable periodic solution.

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[1] L. J. Allen,S. W. Vidurupola, Impact of variability in stochastic models of bacteria-phage dynamics applicable to phage therapy, Stochastic Analysis and Applications, 32 (2014): 427-449.

[2] I. Aviram,A. Rabinovitch, Bactria and lytic phage coexistence in a chemostat with periodic nutrient supply, Bulletin of Mathematical Biology, 76 (2014): 225-244.

[3] E. Beretta,Y. Kuang, Modeling and analysis of a marine bacteriophage infection, Mathematical Biosciences, 149 (1998): 57-76.

[4] E. Beretta,Y. Kuang, Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear Analysis: Real World Applications, 2 (2001): 35-74.

[5] B. J. Bohannan,R. E. Lenski, Linking genetic change to community evolution: Insights from studies of bacteria and bacteriophage, Ecology Letters, 3 (2000): 362-377.

[6] S. J. Brouns,M. M. Jore,M. Lundgren, Small CRISPR RNAs guide antiviral defense in prokaryotes, Science, 321 (2008): 960-964.

[7] A. Buckling,M. Brockhurst, Bacteria-virus coevolution, Evolutionary Systems Biology, 751 (2012): 347-370.

[8] J. J. Bull, C. S. Vegge and M. Schmerer, Phenotypic resistance and the dynamics of bacterial escape from phage control PloS One, 9 (2014), e94690.

[9] B. J. Cairns, A. R. Timms and V. Jansen, Quantitative models of vitro bacteriophage-host dynamics and their application to phage therapy PLoS Pathog, 5 (2009), e1000253.

[10] A. Calsina,J. J. Rivaud, A size structured model for bacteria-phages interaction, Nonlinear Analysis: Real World Applications, 15 (2014): 100-117.

[11] A. Campbell, Conditions for existence of bacteriophages, Evolution, 15 (1961): 153-165.

[12] C. L. Carrillo,R. J. Atterbury,A. El-Shibiny, Bacteriophage therapy to reduce Campylobacter jejuni colonization of broiler chickens, Applied and Environmental Microbiology, 71 (2005): 6554-6563.

[13] J. J. Dennehy, What can phages tell us about host-pathogen coevolution? International Journal of Evolutionary Biology, 2012 (2012), Article ID 396165, 12 pages.

[14] H. Deveau,J. E. Garneau,S. Moineau, CRISPR/Cas system and its role in phage-bacteria interactions, Annual Review of Microbiology, 64 (2010): 475-493.

[15] A. Dhooge,W. Govaerts,Y. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans Math Software, 29 (2003): 141-164.

[16] P. van den Driessche,J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosciences, 180 (2002): 29-48.

[17] D. H. Duckworth, Who discovered bacteriophage?, Bacteriological Reviews, 40 (1976): 793-802.

[18] P. C. Fineran,E. Charpentier, Memory of viral infections by CRISPR-Cas adaptive immune systems: Acquisition of new information, Virology, 434 (2012): 202-209.

[19] J. E. Garneau,M. Dupuis, The CRISPR/Cas bacterial immune system cleaves bacteriophage and plasmid DNA, Nature, 468 (2010): 67-71.

[20] P. Gómez,A. Buckling, Bacteria-phage antagonistic coevolution in soil, Science, 332 (2011): 106-109.

[21] S. A. Gourley,Y. Kuang, A delay reaction-diffusion model of the spread of bacteriophage infection, SIAM Journal of Applied Mathematics, 65 (2004): 550-566.

[22] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1993.

[23] Z. Han,H. L. Smith, Bacteriophage-resistant and bacteriophage-sensitive bacteria in a chemostat, Mathematical Biosciences and Engineering, 9 (2012): 737-765.

[24] S. B. Hsu,S. Hubbell,P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, Applied Mathematics, 32 (1977): 366-383.

[25] S. B. Hsu, Limiting behavior for competing species, SIAM Journal on Applied Mathematics, 34 (1978): 760-763.

[26] P. Horvath,R. Barrangou, CRISPR/Cas the immune system of bacteria and archaea, Science, 327 (2010): 167-170.

[27] J. Iranzo,A. E. Lobkovsky,Y. I. Wolf, Evolutionary dynamics of the prokaryotic adaptive immunity system CRISPR-Cas in an explicit ecological context, Journal of Bacteriology, 195 (2013): 3834-3844.

[28] B. R. Levin,F. M. Stewart,L. Chao, Resource-limited growth, competition, and predation: A model, and experimental studies with bacteria and bacteriophage, American Naturalist, 111 (1977): 3-24.

[29] B. R. Levin, Nasty viruses, costly plasmids, population dynamics, and the conditions for establishing and maintaining CRISPR-mediated adaptive immunity in bacteria PLoS Genet, 6 (2010), e1001171.

[30] B. R. Levin, S. Moineau and M. Bushman, The population and evolutionary dynamics of phage and bacteria with CRISPR-mediated immunity PLoS Genet, 9 (2013), e1003312.

[31] T. Li, Analysis of bacterial immune system-A review, Acta Microbiologica Ssinica, 51 (2011): 1297-1303.

[32] M. Lin,H. F. Huo,Y. N. Li, A competitive model in a chemostat with nutrient recycling and antibiotic treatment, Nonlinear Analysis: Real World Applications, 13 (2012): 2540-2555.

[33] Y. Ma,H. Chang, A review on immune system of the bacteria and its self versus non-self discrimination, Chinese Veterinary Science, 42 (2012): 657-660.

[34] L. A. Marraffini,E. J. Sontheimer, Self versus non-self discrimination during CRISPR RNA-directed immunity, Nature, 463 (2010): 568-571.

[35] S. Matsuzaki,M. Rashel,J. Uchiyama, Bacteriophage therapy: a revitalized therapy against bacterial infectious diseases, Journal of Infection and Chemotherapy, 11 (2005): 211-219.

[36] K. Northcott,M. Imran, Competition in the presence of a virus in an aquatic system: an SIS model in the chemostat, Journal of Mathematical Biology, 64 (2012): 1043-1086.

[37] L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1993.

[38] L. M. Proctor,J. A. Fuhrman, Viral mortality of marine bacteria and cyanobacteria, Nature, 343 (1990): 60-62.

[39] J. Reeks,J. H. Naismith,M. F. White, CRISPR interference: A structural perspective, Biochemical Journal, 453 (2013): 155-166.

[40] G. Robledo,F. Grognard,J. L. Gouzé, Global stability for a model of competition in the chemostat with microbial inputs, Nonlinear Analysis: Real World Applications, 13 (2012): 582-598.

[41] K. D. Seed,D. W. Lazinski, A bacteriophage encodes its own CRISPR/Cas adaptive response to evade host innate immunity, Nature, 494 (2013): 489-491.

[42] H. L. Smith,H. R. Thieme, Persistence of bacteria and phages in a chemostat, Journal of Mathematical Biology, 64 (2012): 951-979.

[43] H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992): 755-763.

[44] H. R. Thieme, Persistence under relaxed point-dissipativity with application to an endemic model, SIAM J. Math. Anal., 24 (1993): 407-435.

[45] R. A. Usmani, Applied Linear Algebra, Marcel Dekker, New York, 1987.

[46] W. Wang,X. Zhao, An epidemic model in a patchy environment, Mathematical Biosciences, 190 (2004): 97-112.

[47] R. J. Weld,C. Butts,J. A. Heinemann, Models of phage growth and their applicability to phage therapy, Journal of Theoretical Biology, 227 (2004): 1-11.

[48] X. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, Springer-Verlag, London, 2003.

Copyright Info: © 2017, Wendi Wang, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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