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

Mengshi Shu; Rui Fu; Wendi Wang

doi:10.3934/mbe.2017070

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December 2017, 14(5&6): 1361-1377. doi: 10.3934/mbe.2017070

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

Mengshi Shu , Rui Fu and Wendi Wang ^,

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

* Corresponding author: Wendi Wang

Received May 31, 2016 Accepted November 21, 2016 Published May 2017

Fund Project: Supported in part by the NSF of China (11571284).

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.

Keywords: Bacteriophage, bifurcation, bistability, backward, evolution.

Mathematics Subject Classification: Primary: 92D30; Secondary: 34D20.

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

References:

[1]	L. J. Allen and 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. doi: 10.1080/07362994.2014.889922.
[2]	I. Aviram and A. Rabinovitch, Bactria and lytic phage coexistence in a chemostat with periodic nutrient supply, Bulletin of Mathematical Biology, 76 (2014), 225-244. doi: 10.1007/s11538-013-9917-3.
[3]	E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection, Mathematical Biosciences, 149 (1998), 57-76. doi: 10.1016/S0025-5564(97)10015-3.
[4]	E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear Analysis: Real World Applications, 2 (2001), 35-74. doi: 10.1016/S0362-546X(99)00285-0.
[5]	B. J. Bohannan and R. E. Lenski, Linking genetic change to community evolution: Insights from studies of bacteria and bacteriophage, Ecology Letters, 3 (2000), 362-377. doi: 10.1046/j.1461-0248.2000.00161.x.
[6]	S. J. Brouns, M. M. Jore and M. Lundgren, Small CRISPR RNAs guide antiviral defense in prokaryotes, Science, 321 (2008), 960-964. doi: 10.1126/science.1159689.
[7]	A. Buckling and M. Brockhurst, Bacteria-virus coevolution, Evolutionary Systems Biology, 751 (2012), 347-370. doi: 10.1007/978-1-4614-3567-9_16.
[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. doi: 10.1371/journal.pone.0094690.
[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. doi: 10.1371/journal.ppat.1000253.
[10]	A. Calsina and J. J. Rivaud, A size structured model for bacteria-phages interaction, Nonlinear Analysis: Real World Applications, 15 (2014), 100-117. doi: 10.1016/j.nonrwa.2013.06.004.
[11]	A. Campbell, Conditions for existence of bacteriophages, Evolution, 15 (1961), 153-165. doi: 10.2307/2406076.
[12]	C. L. Carrillo, R. J. Atterbury and A. El-Shibiny, Bacteriophage therapy to reduce Campylobacter jejuni colonization of broiler chickens, Applied and Environmental Microbiology, 71 (2005), 6554-6563. doi: 10.1128/AEM.71.11.6554-6563.2005.
[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. doi: 10.1155/2012/396165.
[14]	H. Deveau, J. E. Garneau and S. Moineau, CRISPR/Cas system and its role in phage-bacteria interactions, Annual Review of Microbiology, 64 (2010), 475-493. doi: 10.1146/annurev.micro.112408.134123.
[15]	A. Dhooge, W. Govaerts and Y. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans Math Software, 29 (2003), 141-164. doi: 10.1145/779359.779362.
[16]	P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
[17]	D. H. Duckworth, Who discovered bacteriophage?, Bacteriological Reviews, 40 (1976), 793-802.
[18]	P. C. Fineran and E. Charpentier, Memory of viral infections by CRISPR-Cas adaptive immune systems: Acquisition of new information, Virology, 434 (2012), 202-209. doi: 10.1016/j.virol.2012.10.003.
[19]	J. E. Garneau and M. Dupuis, The CRISPR/Cas bacterial immune system cleaves bacteriophage and plasmid DNA, Nature, 468 (2010), 67-71. doi: 10.1038/nature09523.
[20]	P. Gómez and A. Buckling, Bacteria-phage antagonistic coevolution in soil, Science, 332 (2011), 106-109.
[21]	S. A. Gourley and Y. Kuang, A delay reaction-diffusion model of the spread of bacteriophage infection, SIAM Journal of Applied Mathematics, 65 (2004), 550-566. doi: 10.1137/S0036139903436613.
[22]	J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.
[23]	Z. Han and H. L. Smith, Bacteriophage-resistant and bacteriophage-sensitive bacteria in a chemostat, Mathematical Biosciences and Engineering, 9 (2012), 737-765. doi: 10.3934/mbe.2012.9.737.
[24]	S. B. Hsu, S. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, Applied Mathematics, 32 (1977), 366-383. doi: 10.1137/0132030.
[25]	S. B. Hsu, Limiting behavior for competing species, SIAM Journal on Applied Mathematics, 34 (1978), 760-763. doi: 10.1137/0134064.
[26]	P. Horvath and R. Barrangou, CRISPR/Cas the immune system of bacteria and archaea, Science, 327 (2010), 167-170. doi: 10.1126/science.1179555.
[27]	J. Iranzo, A. E. Lobkovsky and 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. doi: 10.1128/JB.00412-13.
[28]	B. R. Levin, F. M. Stewart and L. Chao, Resource-limited growth, competition, and predation: A model, and experimental studies with bacteria and bacteriophage, American Naturalist, 111 (1977), 3-24. doi: 10.1086/283134.
[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. doi: 10.1371/journal.pgen.1001171.
[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. doi: 10.1371/journal.pgen.1003312.
[31]	T. Li, Analysis of bacterial immune system-A review, Acta Microbiologica Ssinica, 51 (2011), 1297-1303.
[32]	M. Lin, H. F. Huo and Y. N. Li, A competitive model in a chemostat with nutrient recycling and antibiotic treatment, Nonlinear Analysis: Real World Applications, 13 (2012), 2540-2555. doi: 10.1016/j.nonrwa.2012.02.016.
[33]	Y. Ma and 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 and E. J. Sontheimer, Self versus non-self discrimination during CRISPR RNA-directed immunity, Nature, 463 (2010), 568-571. doi: 10.1038/nature08703.
[35]	S. Matsuzaki, M. Rashel and J. Uchiyama, Bacteriophage therapy: a revitalized therapy against bacterial infectious diseases, Journal of Infection and Chemotherapy, 11 (2005), 211-219. doi: 10.1007/s10156-005-0408-9.
[36]	K. Northcott and 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. doi: 10.1007/s00285-011-0439-z.
[37]	L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1993.
[38]	L. M. Proctor and J. A. Fuhrman, Viral mortality of marine bacteria and cyanobacteria, Nature, 343 (1990), 60-62. doi: 10.1038/343060a0.
[39]	J. Reeks, J. H. Naismith and M. F. White, CRISPR interference: A structural perspective, Biochemical Journal, 453 (2013), 155-166. doi: 10.1042/BJ20130316.
[40]	G. Robledo, F. Grognard and 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. doi: 10.1016/j.nonrwa.2011.07.049.
[41]	K. D. Seed and D. W. Lazinski, A bacteriophage encodes its own CRISPR/Cas adaptive response to evade host innate immunity, Nature, 494 (2013), 489-491. doi: 10.1038/nature11927.
[42]	H. L. Smith and H. R. Thieme, Persistence of bacteria and phages in a chemostat, Journal of Mathematical Biology, 64 (2012), 951-979. doi: 10.1007/s00285-011-0434-4.
[43]	H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992), 755-763. doi: 10.1007/BF00173267.
[44]	H. R. Thieme, Persistence under relaxed point-dissipativity with application to an endemic model, SIAM J. Math. Anal., 24 (1993), 407-435. doi: 10.1137/0524026.
[45]	R. A. Usmani, Applied Linear Algebra, Marcel Dekker, New York, 1987.
[46]	W. Wang and X. Zhao, An epidemic model in a patchy environment, Mathematical Biosciences, 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001.
[47]	R. J. Weld, C. Butts and J. A. Heinemann, Models of phage growth and their applicability to phage therapy, Journal of Theoretical Biology, 227 (2004), 1-11. doi: 10.1016/S0022-5193(03)00262-5.
[48]	X. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, Springer-Verlag, London, 2003. doi: 10.1007/978-0-387-21761-1.

show all references

References:

[1]	L. J. Allen and 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. doi: 10.1080/07362994.2014.889922.
[2]	I. Aviram and A. Rabinovitch, Bactria and lytic phage coexistence in a chemostat with periodic nutrient supply, Bulletin of Mathematical Biology, 76 (2014), 225-244. doi: 10.1007/s11538-013-9917-3.
[3]	E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection, Mathematical Biosciences, 149 (1998), 57-76. doi: 10.1016/S0025-5564(97)10015-3.
[4]	E. Beretta and Y. Kuang, Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear Analysis: Real World Applications, 2 (2001), 35-74. doi: 10.1016/S0362-546X(99)00285-0.
[5]	B. J. Bohannan and R. E. Lenski, Linking genetic change to community evolution: Insights from studies of bacteria and bacteriophage, Ecology Letters, 3 (2000), 362-377. doi: 10.1046/j.1461-0248.2000.00161.x.
[6]	S. J. Brouns, M. M. Jore and M. Lundgren, Small CRISPR RNAs guide antiviral defense in prokaryotes, Science, 321 (2008), 960-964. doi: 10.1126/science.1159689.
[7]	A. Buckling and M. Brockhurst, Bacteria-virus coevolution, Evolutionary Systems Biology, 751 (2012), 347-370. doi: 10.1007/978-1-4614-3567-9_16.
[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. doi: 10.1371/journal.pone.0094690.
[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. doi: 10.1371/journal.ppat.1000253.
[10]	A. Calsina and J. J. Rivaud, A size structured model for bacteria-phages interaction, Nonlinear Analysis: Real World Applications, 15 (2014), 100-117. doi: 10.1016/j.nonrwa.2013.06.004.
[11]	A. Campbell, Conditions for existence of bacteriophages, Evolution, 15 (1961), 153-165. doi: 10.2307/2406076.
[12]	C. L. Carrillo, R. J. Atterbury and A. El-Shibiny, Bacteriophage therapy to reduce Campylobacter jejuni colonization of broiler chickens, Applied and Environmental Microbiology, 71 (2005), 6554-6563. doi: 10.1128/AEM.71.11.6554-6563.2005.
[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. doi: 10.1155/2012/396165.
[14]	H. Deveau, J. E. Garneau and S. Moineau, CRISPR/Cas system and its role in phage-bacteria interactions, Annual Review of Microbiology, 64 (2010), 475-493. doi: 10.1146/annurev.micro.112408.134123.
[15]	A. Dhooge, W. Govaerts and Y. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans Math Software, 29 (2003), 141-164. doi: 10.1145/779359.779362.
[16]	P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.
[17]	D. H. Duckworth, Who discovered bacteriophage?, Bacteriological Reviews, 40 (1976), 793-802.
[18]	P. C. Fineran and E. Charpentier, Memory of viral infections by CRISPR-Cas adaptive immune systems: Acquisition of new information, Virology, 434 (2012), 202-209. doi: 10.1016/j.virol.2012.10.003.
[19]	J. E. Garneau and M. Dupuis, The CRISPR/Cas bacterial immune system cleaves bacteriophage and plasmid DNA, Nature, 468 (2010), 67-71. doi: 10.1038/nature09523.
[20]	P. Gómez and A. Buckling, Bacteria-phage antagonistic coevolution in soil, Science, 332 (2011), 106-109.
[21]	S. A. Gourley and Y. Kuang, A delay reaction-diffusion model of the spread of bacteriophage infection, SIAM Journal of Applied Mathematics, 65 (2004), 550-566. doi: 10.1137/S0036139903436613.
[22]	J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.
[23]	Z. Han and H. L. Smith, Bacteriophage-resistant and bacteriophage-sensitive bacteria in a chemostat, Mathematical Biosciences and Engineering, 9 (2012), 737-765. doi: 10.3934/mbe.2012.9.737.
[24]	S. B. Hsu, S. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms, Applied Mathematics, 32 (1977), 366-383. doi: 10.1137/0132030.
[25]	S. B. Hsu, Limiting behavior for competing species, SIAM Journal on Applied Mathematics, 34 (1978), 760-763. doi: 10.1137/0134064.
[26]	P. Horvath and R. Barrangou, CRISPR/Cas the immune system of bacteria and archaea, Science, 327 (2010), 167-170. doi: 10.1126/science.1179555.
[27]	J. Iranzo, A. E. Lobkovsky and 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. doi: 10.1128/JB.00412-13.
[28]	B. R. Levin, F. M. Stewart and L. Chao, Resource-limited growth, competition, and predation: A model, and experimental studies with bacteria and bacteriophage, American Naturalist, 111 (1977), 3-24. doi: 10.1086/283134.
[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. doi: 10.1371/journal.pgen.1001171.
[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. doi: 10.1371/journal.pgen.1003312.
[31]	T. Li, Analysis of bacterial immune system-A review, Acta Microbiologica Ssinica, 51 (2011), 1297-1303.
[32]	M. Lin, H. F. Huo and Y. N. Li, A competitive model in a chemostat with nutrient recycling and antibiotic treatment, Nonlinear Analysis: Real World Applications, 13 (2012), 2540-2555. doi: 10.1016/j.nonrwa.2012.02.016.
[33]	Y. Ma and 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 and E. J. Sontheimer, Self versus non-self discrimination during CRISPR RNA-directed immunity, Nature, 463 (2010), 568-571. doi: 10.1038/nature08703.
[35]	S. Matsuzaki, M. Rashel and J. Uchiyama, Bacteriophage therapy: a revitalized therapy against bacterial infectious diseases, Journal of Infection and Chemotherapy, 11 (2005), 211-219. doi: 10.1007/s10156-005-0408-9.
[36]	K. Northcott and 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. doi: 10.1007/s00285-011-0439-z.
[37]	L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1993.
[38]	L. M. Proctor and J. A. Fuhrman, Viral mortality of marine bacteria and cyanobacteria, Nature, 343 (1990), 60-62. doi: 10.1038/343060a0.
[39]	J. Reeks, J. H. Naismith and M. F. White, CRISPR interference: A structural perspective, Biochemical Journal, 453 (2013), 155-166. doi: 10.1042/BJ20130316.
[40]	G. Robledo, F. Grognard and 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. doi: 10.1016/j.nonrwa.2011.07.049.
[41]	K. D. Seed and D. W. Lazinski, A bacteriophage encodes its own CRISPR/Cas adaptive response to evade host innate immunity, Nature, 494 (2013), 489-491. doi: 10.1038/nature11927.
[42]	H. L. Smith and H. R. Thieme, Persistence of bacteria and phages in a chemostat, Journal of Mathematical Biology, 64 (2012), 951-979. doi: 10.1007/s00285-011-0434-4.
[43]	H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology, 30 (1992), 755-763. doi: 10.1007/BF00173267.
[44]	H. R. Thieme, Persistence under relaxed point-dissipativity with application to an endemic model, SIAM J. Math. Anal., 24 (1993), 407-435. doi: 10.1137/0524026.
[45]	R. A. Usmani, Applied Linear Algebra, Marcel Dekker, New York, 1987.
[46]	W. Wang and X. Zhao, An epidemic model in a patchy environment, Mathematical Biosciences, 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001.
[47]	R. J. Weld, C. Butts and J. A. Heinemann, Models of phage growth and their applicability to phage therapy, Journal of Theoretical Biology, 227 (2004), 1-11. doi: 10.1016/S0022-5193(03)00262-5.
[48]	X. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, Springer-Verlag, London, 2003. doi: 10.1007/978-0-387-21761-1.

Figure 1. Bifurcation graphs for $\kappa\geq\mu$ in case $(C1)$. Panel (a) shows the forward bifurcation with $\varepsilon=0.01$. Panel (b) indicates the backward bifurcation with $\varepsilon=0.8$. H denotes a Hopf bifurcation point, LP means a fold bifurcation point and BP represents the branch point $E_1$

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Figure 2. Bifurcation graphs for case $(C2)$. Panel (a) shows the forward bifurcation with $\varepsilon=0.3$ and panel (b) demonstrates the backward bifurcation with $\varepsilon=0.8$. H denotes a Hopf bifurcation point, LP means a fold bifurcation point and BP represents a branch point. The solid lines represent stable branches and dashed lines mean unstable branches

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Figure 3. Bifurcation graphs for case $(C3)$. Panel (a) shows the forward bifurcation at $E_1$ and a transcritical bifurcation at $E_3$ when $\varepsilon=0.1$ and panel (b) demonstrates the bistable phenomena between $E_1, ~E_3$ and a transcritical bifurcation at $E_4$ when $\varepsilon=0.8$. H denotes a Hopf bifurcation point, LP means a fold bifurcation point and BP represents a branch point. The solid lines represent stable branches and dashed lines mean unstable branches

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Figure 4. Graphs of bistable behaviors in case $(C1)$ with $\kappa\geq\mu$. Panel (a) shows the bistability of the infection-free equilibrium and an coexist equilibrium where $\varepsilon=0.01, ~b=130$. Panel (b) indicates the bistable coexistence of the infection-free equilibria with a stable periodic solution where $\varepsilon=0.9, ~b=260$

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

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

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