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

Mengshi Shu; Rui Fu; Wendi Wang; Mengshi Shu; Rui Fu; Wendi Wang

doi:10.3934/mbe.2017070

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 5&6: 1361-1377. doi: 10.3934/mbe.2017070

Previous Article Next Article

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

Received: 31 May 2016 Accepted: 21 November 2016 Published: 01 October 2017
MSC : Primary: 92D30; Secondary: 34D20

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.
- Bacteriophage,
- bifurcation,
- bistability,
- backward,
- evolution
Citation: Mengshi Shu, Rui Fu, Wendi Wang. A bacteriophage model based on CRISPR/Cas immune system in a chemostat[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1361-1377. doi: 10.3934/mbe.2017070

Related Papers:

Abstract

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.

References

[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.

Reader Comments

Your name:*

Email:*
© 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)