Modeling stochastic phenotype switching and bet-hedging in bacteria: stochastic nonlinear dynamics and critical state identification

Abstract

Fluctuating environments pose tremendous challenges to bacterial populations. It is observed in numerous bacterial species that individual cells can stochastically switch among multiple phenotypes for the population to survive in rapidly changing environments. This kind of phenotypic heterogeneity with stochastic phenotype switching is generally understood to be an adaptive bet-hedging strategy. Mathematical models are essential to gain a deeper insight into the principle behind bet-hedging and the pattern behind experimental data. Traditional deterministic models cannot provide a correct description of stochastic phenotype switching and bet-hedging, and traditional Markov chain models at the cellular level fail to explain their underlying molecular mechanisms. In this paper, we propose a nonlinear stochastic model of multistable bacterial systems at the molecular level. It turns out that our model not only provides a clear description of stochastic phenotype switching and bet-hedging within isogenic bacterial populations, but also provides a deeper insight into the analysis of multidimensional experimental data. Moreover, we use some deep mathematical theories to show that our stochastic model and traditional Markov chain models are essentially consistent and reflect the dynamic behavior of the bacterial system at two different time scales. In addition, we provide a quantitative characterization of the critical state of multistable bacterial systems and develop an effective data-driven method to identify the critical state without resorting to specific mathematical models.

Graphical abstract

Keywords

phenotypic heterogeneity / phenotypic variation / multistability / gene network / stochastic gene expression

References

[1] Kussell, E. and Leibler, S. (2005) Phenotypic diversity, population growth, and information in fluctuating environments. Science, 309, 2075-2078

[2] Smits, W. K., Kuipers, O. P. and Veening, J. W. (2006) Phenotypic variation in bacteria: the role of feedback regulation. Nat. Rev. Microbiol., 4, 259-271

[3] Dubnau, D. and Losick, R. (2006) Bistability in bacteria. Mol. Microbiol., 61, 564-572

[4] Avery, S. V. (2006) Microbial cell individuality and the underlying sources of heterogeneity. Nat. Rev. Microbiol., 4, 577-587

[5] Dhar, N. and McKinney, J. D. (2007) Microbial phenotypic heterogeneity and antibiotic tolerance. Curr. Opin. Microbiol., 10, 30-38

[6] Lu, T., Shen, T., Bennett, M. R., Wolynes, P. G. and Hasty, J. (2007) Phenotypic variability of growing cellular populations. Proc. Natl. Acad. Sci. USA, 104, 18982-18987

[7] Veening, J. W., Smits, W. K. and Kuipers, O. P. (2008) Bistability, epigenetics, and bet-hedging in bacteria. Annu. Rev. Microbiol., 62, 193-210

[8] Fraser, D. and Kaern, M. (2009) A chance at survival: gene expression noise and phenotypic diversification strategies. Mol. Microbiol., 71, 1333-1340

[9] Jablonka, E. and Raz, G. (2009) Transgenerational epigenetic inheritance: prevalence, mechanisms, and implications for the study of heredity and evolution. Q. Rev. Biol., 84, 131-176

[10] Snijder, B. and Pelkmans, L. (2011) Origins of regulated cell-to-cell variability. Nat. Rev. Mol. Cell Biol., 12, 119-125

[11] Mao, J., Blanchard, A. E. and Lu, T. (2014) Slow and steady wins the race: A bacterial exploitative competition strategy in fluctuating environments. ACS Synth. Biol.

[12] Rulands, S., Jahn, D. and Frey, E. (2014) Specialization and bet hedging in heterogeneous populations. Phys. Rev. Lett., 113, 108102

[13] Acar, M., Mettetal, J. T. and van Oudenaarden, A. (2008) Stochastic switching as a survival strategy in fluctuating environments. Nat. Genet., 40, 471-475

[14] Salathé, M., Van Cleve, J. and Feldman, M. W. (2009) Evolution of stochastic switching rates in asymmetric fitness landscapes. Genetics, 182, 1159-1164

[15] Leisner, M., Stingl, K., Frey, E. and Maier, B. (2008) Stochastic switching to competence. Curr. Opin. Microbiol., 11, 553-559

[16] Gaál, B., Pitchford, J. W. and Wood, A. J. (2010) Exact results for the evolution of stochastic switching in variable asymmetric environments. Genetics, 184, 1113-1119

[17] Libby E, Rainey PB (2011) Exclusion rules, bottlenecks and the evolution of stochastic phenotype switching. P. Roy. Soc. B-Biol. Sci., 278: 3574-3583

[18] Rainey, P. B., Beaumont, H. J., Ferguson, G. C., Gallie, J., Kost, C., Libby, E. and Zhang, X. X. (2011) The evolutionary emergence of stochastic phenotype switching in bacteria. Microb. Cell Fact., 10, S14

[19] Ozbudak, E. M., Thattai, M., Lim, H. N., Shraiman, B. I. and Van Oudenaarden, A. (2004) Multistability in the lactose utilization network of Escherichia coli. Nature, 427, 737-740

[20] Süel, G. M., Garcia-Ojalvo, J., Liberman, L. M. and Elowitz, M. B. (2006) An excitable gene regulatory circuit induces transient cellular differentiation. Nature, 440, 545-550

[21] Tsang, J. and Van Oudenaarden, A. (2006) Exciting fluctuations: monitoring competence induction dynamics at the single-cell level. Mol. Syst. Biol., 2, 0025

[22] Schultz, D., Ben Jacob, E., Onuchic, J. N. and Wolynes, P. G. (2007) Molecular level stochastic model for competence cycles in Bacillus subtilis. Proc. Natl. Acad. Sci. USA, 104, 17582-17587

[23] Sonenshein, A.L., Hoch, J.A. and Losick R. (2002) Bacillus subtilis and its closest relatives: from genes to cells Washington: American Society for Microbiology.

[24] Errington, J. (2003) Regulation of endospore formation in Bacillus subtilis. Nat. Rev. Microbiol., 1, 117-126

[25] Morohashi, M., Ohashi, Y., Tani, S., Ishii, K., Itaya, M., Nanamiya, H., Kawamura, F., Tomita, M. and Soga, T. (2007) Model-based definition of population heterogeneity and its effects on metabolism in sporulating Bacillus subtilis. J. Biochem., 142, 183-191

[26] de Jong, I. G., Veening, J. W. and Kuipers, O. P. (2010) Heterochronic phosphorelay gene expression as a source of heterogeneity in Bacillus subtilis spore formation. J. Bacteriol., 192, 2053-2067

[27] Sureka K., Ghosh, B., Dasugpta, A., Basu J., Kundu, M. and Bose Z., (2008) Positive feedback and noise activate the stringent response regulator rel in mycobacteria. PLoS One3: e1771.

[28] Gefen, O. and Balaban, N. Q. (2009) The importance of being persistent: heterogeneity of bacterial populations under antibiotic stress. FEMS Microbiol. Rev., 33, 704-717

[29] Ghosh, S., Sureka, K., Ghosh, B., Bose, I., Basu, J. and Kundu, M. (2011) Phenotypic heterogeneity in mycobacterial stringent response. BMC Syst. Biol., 5, 18

[30] Veening, J. W., Stewart, E. J., Berngruber, T. W., Taddei, F., Kuipers, O. P. and Hamoen, L. W. (2008) Bet-hedging and epigenetic inheritance in bacterial cell development. Proc. Natl. Acad. Sci. USA., 105, 4393-4398

[31] Beaumont, H. J., Gallie, J., Kost, C., Ferguson, G. C. and Rainey, P. B. (2009) Experimental evolution of bet hedging. Nature, 462, 90-93

[32] Wolf, D. M., Vazirani, V. V. and Arkin, A. P. (2005) Diversity in times of adversity: probabilistic strategies in microbial survival games. J. Theor. Biol., 234, 227-253

[33] Gupta, P. B., Fillmore, C. M., Jiang, G., Shapira, S. D., Tao, K., Kuperwasser, C. and Lander, E. S. (2011) Stochastic state transitions give rise to phenotypic equilibrium in populations of cancer cells. Cell, 146, 633-644

[34] Zhou D, Wu D, Li Z, Qian M, Zhang MQ (2013) Population dynamics of cancer cells with cell state conversions. Quant. Biol. 1, 1-8.

[35] Karmakar, R. and Bose, I. (2007) Positive feedback, stochasticity and genetic competence. Phys. Biol., 4, 29-37

[36] Mitrophanov, A. Y. and Groisman, E. A. (2008) Positive feedback in cellular control systems. BioEssays, 30, 542-555

[37] Mantzaris, N. V. (2007) From single-cell genetic architecture to cell population dynamics: quantitatively decomposing the effects of different population heterogeneity sources for a genetic network with positive feedback architecture. Biophys. J., 92, 4271-4288

[38] Vellela, M. and Qian, H. (2009) Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited. J. R. Soc. Interface, 6, 925-940

[39] Qian, H. and Bishop, L. M. (2010) The chemical master equation approach to nonequilibrium steady-state of open biochemical systems: linear single-molecule enzyme kinetics and nonlinear biochemical reaction networks. Int. J. Mol. Sci., 11, 3472-3500

[40] Qian, H. (2011) Nonlinear stochastic dynamics of mesoscopic homogeneous biochemical reaction systemsłan analytical theory. Nonlinearity, 24, R19-R49

[41] Ge, H. and Qian, H. (2011) Non-equilibrium phase transition in mesoscopic biochemical systems: from stochastic to nonlinear dynamics and beyond. J. R. Soc. Interface, 8, 107-116

[42] Qian, H. (2012) Cooperativity in cellular biochemical processes: noise-enhanced sensitivity, fluctuating enzyme, bistability with nonlinear feedback, and other mechanisms for sigmoidal responses. Annu. Rev. Biophys., 41, 179-204

[43] Qian, H. and Ge, H. (2012) Mesoscopic biochemical basis of isogenetic inheritance and canalization: stochasticity, nonlinearity, and emergent landscape. Mol Cell Biomech, 9, 1-30

[44] McAdams, H. H. and Arkin, A. (1997) Stochastic mechanisms in gene expression. Proc. Natl. Acad. Sci. USA., 94, 814-819

[45] Elowitz, M. B., Levine, A. J., Siggia, E. D. and Swain, P. S. (2002) Stochastic gene expression in a single cell. Science, 297, 1183-1186

[46] Ozbudak, E. M., Thattai, M., Kurtser, I., Grossman, A. D. and van Oudenaarden, A. (2002) Regulation of noise in the expression of a single gene. Nat. Genet., 31, 69-73

[47] Paulsson, J. (2004) Summing up the noise in gene networks. Nature, 427, 415-418

[48] Kaern, M., Elston, T. C., Blake, W. J. and Collins, J. J. (2005) Stochasticity in gene expression: from theories to phenotypes. Nat. Rev. Genet., 6, 451-464

[49] Raser, J. M. and O’Shea, E. K. (2005) Noise in gene expression: origins, consequences, and control. Science, 309, 2010-2013

[50] Cai, L., Friedman, N. and Xie, X. S. (2006) Stochastic protein expression in individual cells at the single molecule level. Nature, 440, 358-362

[51] Yu, J., Xiao, J., Ren, X., Lao, K. and Xie, X. S. (2006) Probing gene expression in live cells, one protein molecule at a time. Science, 311, 1600-1603

[52] Xie, X. S., Choi, P. J., Li, G. W., Lee, N. K. and Lia, G. (2008) Single-molecule approach to molecular biology in living bacterial cells. Annu. Rev. Biophys., 37, 417-444

[53] Sanchez, A., Choubey, S. and Kondev, J. (2013) Regulation of noise in gene expression. Annu. Rev. Biophys., 42, 469-491

[54] Freidlin, M. I., Szücs, J. and Wentzell, A. D. (2012) Random perturbations of dynamical systems, New York: Springer-Verlag

[55] Kim, D., Rath, O., Kolch, W. and Cho, K. H. (2007) A hidden oncogenic positive feedback loop caused by crosstalk between Wnt and ERK pathways. Oncogene, 26, 4571-4579

[56] Kellershohn, N. and Laurent, M. (2001) Prion diseases: dynamics of the infection and properties of the bistable transition. Biophys. J., 81, 2517-2529

[57] Olivieri, E. and Vares, M. E. (2005) Large deviations and metastability. UK: Cambridge University Press.

[58] Liu R, Li M.Y., Liu Z.P., Wu J.R., Chen L.N. and Aihara K. (2012) Identifying critical transitions and their leading biomolecular networks in complex diseases. Sci. Rep.,

[59] Dai, L., Vorselen, D., Korolev, K. S. and Gore, J. (2012) Generic indicators for loss of resilience before a tipping point leading to population collapse. Science, 336, 1175-1177

[60] Angeli, D., Ferrell, J. E. Jr and Sontag, E. D. (2004) Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems. Proc. Natl. Acad. Sci. USA., 101, 1822-1827

[61] Gouzé, J. L. (1998) Positive and negative circuits in dynamical systems. J. Biol. Syst., 6, 11-15

[62] Snoussi, E. H. (1998) Necessary conditions for multistationarity and stable periodicity. J. Biol. Syst., 6, 3-9

[63] Cinquin, O. and Demongeot, J. (2002) Positive and negative feedback: striking a balance between necessary antagonists. J. Theor. Biol., 216, 229-241

[64] Gillespie, D. T. (1992) A rigorous derivation of the chemical master equation. Physica A, 188, 404-425

[65] Kurtz, T. G. (1970) Solutions of ordinary differential equations as limits of pure jump markov processes. J. Appl. Probab., 7, 49-58

[66] Kurtz, T. G. (1971) Limit theorems for sequences of jump markov processes approximating ordinary differential processes. J. Appl. Probab., 8, 344-356

[67] Kurtz, T. G. (1972) The relationship between stochastic and deterministic models for chemical reactions. J. Chem. Phys., 57, 2976-2978

[68] Gillespie, D. T. (2000) The chemical langevin equation. J. Chem. Phys., 113, 297-306

[69] Rao, C. V., Wolf, D. M. and Arkin, A. P. (2002) Control, exploitation and tolerance of intracellular noise. Nature, 420, 231-237

[70] Wilkinson, D. J. (2009) Stochastic modelling for quantitative description of heterogeneous biological systems. Nat. Rev. Genet., 10, 122-133

[71] Meister A, Du C, Li YH, Wong WH (2014) Modeling stochastic noise in gene regulatory systems. Quant. Biol.: 1-29

[72] Yao, G., Lee, T. J., Mori, S., Nevins, J. R. and You, L. (2008) A bistable Rb-E2F switch underlies the restriction point. Nat. Cell Biol., 10, 476-482

[73] Ferrell, J. E. Jr and Machleder, E. M. (1998) The biochemical basis of an all-or-none cell fate switch in Xenopus oocytes. Science, 280, 895-898

[74] Xiong, W. and Ferrell, J. E. Jr. (2003) A positive-feedback-based bistable ‘memory module’ that governs a cell fate decision. Nature, 426, 460-465

[75] Corson, F. and Siggia, E. D. (2012) Geometry, epistasis, and developmental patterning. Proc. Natl. Acad. Sci. USA., 109, 5568-5575

[76] Yan, L., Yang, M., Guo, H., Yang, L., Wu, J., Li, R., Liu, P., Lian, Y., Zheng, X., Yan, J., (2013) Single-cell RNA-Seq profiling of human preimplantation embryos and embryonic stem cells. Nat. Struct. Mol. Biol., 20, 1131-1139

[77] Eissing, T., Conzelmann, H., Gilles, E. D., Allgöwer, F., Bullinger, E. and Scheurich, P. (2004) Bistability analyses of a caspase activation model for receptor-induced apoptosis. J. Biol. Chem., 279, 36892-36897

[78] Spencer, S. L., Gaudet, S., Albeck, J. G., Burke, J. M. and Sorger, P. K. (2009) Non-genetic origins of cell-to-cell variability in TRAIL-induced apoptosis. Nature, 459, 428-432

ACKNOWLEDGMENTS

The authors gratefully acknowledge Prof. Michael S.Waterman at the University of Southern California and Prof. Michael Q. Zhang at Tsinghua University for stimulating discussions. This work is supported by NSFC ( Nos. 11271029 and 11171024).

COMPLIANCE WITH ETHICS GUIDELINES

The authors Chen Jia, Minping Qian, Yu Kang and Daquan Jiang declare that they have no conflict of interests.

This article does not contain any studies with human or animal subjects performed by any of the authors.

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg