Fitness and entropy production in a cell population dynamics with epigenetic phenotype switching

Abstract

Motivated by recent understandings in the stochastic natures of gene expression, biochemical signaling, and spontaneous reversible epigenetic switchings, we study a simple deterministic cell population dynamics in which subpopulations grow with different rates and individual cells can bi-directionally switch between a small number of different epigenetic phenotypes. Two theories in the past, the population dynamics and thermodynamics of master equations, separately defined two important concepts in mathematical terms: the fitness in the former and the (non-adiabatic) entropy production in the latter. Both of them play important roles in the evolution of the cell population dynamics. The switching sustains the variations among the subpopulation growth, thus sustains continuous natural selection. As a form of Price’s equation, the fitness increases with (i) natural selection through variations and (ii) a positive covariance between the per capita growth and switching, which represents a Lamarchian-like behavior. A negative covariance balances the natural selection in a fitness steady state --- “the red queen” scenario. At the same time the growth keeps the proportions of subpopulations away from the “intrinsic” switching equilibrium of individual cells, thus leads to a continuous entropy production. A covariance, between the per capita growth rate and the “chemical potential” of subpopulation, counteracts the entropy production. Analytical results are obtained for the limiting cases of growth dominating switching and vice versa.

Graphical abstract

Keywords

population dynamics / fundamental theorem of natural selection / diversity

References

[1] Haldane, J. B. S. (1924) A mathematical theory of natural and artificial selection. Trans. Cambridge Philos. Soc, 23, 19–41.

[2] Fisher, R. A. (1930) The Genetical Theory of Natural Selection. Oxford: Clarendon Press.

[3] Wright, S. (1932) The roles of mutation, inbreeding, crossbreeding and selection in evolution. Proc. Sixth Intern. Congress Genet. 1, 356–366.

[4] Crow, J. F. and Kimura, M. (1970) An Introduction to Population Genetics Theory. Caldwell: Harper & Row.

[5] Roughgarden, J. (1979) Theory of Population Genetics and Evolutionary Ecology: An Introduction. New York: Macmillan Publishers.

[6] Felsenstein, J. (2011) Theoretical Evolutionary Genetics. Online Books, http: //evolution.gs.washington.edu/pgbook/pgbook.pdf.

[7] Ewens, W. J. (2013) Mathematics, genetics and evolution. Quant. Biol, 1, 9–31

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

[9] Qian, H., Qian, M. and Tang, X. (2002) Thermodynamics of the general diffusion process: Time-reversibility and entropy production. J. Stat. Phys., 107, 1129–1141

[10] Ge, H. and Qian, H. (2010) Physical origins of entropy production, free energy dissipation, and their mathematical representations. Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 81, 051133

[11] Esposito, M. and Van den Broeck, C. (2010) Three detailed fluctuation theorems. Phys. Rev. Lett., 104, 090601

[12] Seifert, U. (2012) Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys., 75, 126001

[13] Qian, H. and Kou, S. C. (2014) Statistics and related topics in single-molecule biophysics. Ann. Rev. Stat. Appl, 1, 465–492.

[14] Zheng, Q. (1999) Progress of a half century in the study of the Luria-Delbrück distribution. Math. Biosci., 162, 1–32

[15] Chang, H. H., Hemberg, M., Barahona, M., Ingber, D. E. and Huang, S. (2008) Transcriptome-wide noise controls lineage choice in mammalian progenitor cells. Nature, 453, 544–547

[16] 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

[17] Huang, S. (2012) The molecular and mathematical basis of Waddington’s epigenetic landscape: a framework for post-Darwinian biology? BioEssays, 34, 149–157

[18] Wang, J., Zhang, K., Xu, L. and Wang, E. (2011) Quantifying the Waddington landscape and biological paths for development and differentiation. Proc. Natl. Acad. Sci. U.S.A., 108, 8257–8262

[19] Huang, S. (2012) Tumor progression: chance and necessity in Darwinian and Lamarckian somatic (mutationless) evolution. Prog. Biophys. Mol. Biol., 110, 69–86

[20] Wang, G., Zhu, X., Hood, L. and Ao, P. (2013) From phage lambda to human cancer: endogenous molecular-cellular network hypothesis. Quant. Biol, 1, 32–49

[21] 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

[22] Wang, J., Xu, L. and Wang, E. (2008) Potential landscape and flux framework of nonequilibrium networks: robustness, dissipation, and coherence of biochemical oscillations. Proc. Natl. Acad. Sci. U.S.A., 105, 12271–12276

[23] Ge, H., Wu, P., Qian, H. and Xie, X. S. (2013) A gene regulation theory for spontaneous transition between cellular phenotypes under nonequilibrium steady-state conditions. (Manuscript in preparation).

[24] Zhou, D., Wu, D.-M., Li, Z., Qian, M.-P. and Zhang, M. Q. (2013) Population dynamics of cancer cells with cell state conversions. Quant. Biol, 1, 201–208

[25] Eigen, M. and Schuster, P. (1979) The Hypercycle: A Principle of Natural Self-Organization. Berlin: Springer-Verlag.

[26] van Nimwegen, E., Crutchfield, J. P. and Huynen, M. (1999) Neutral evolution of mutational robustness. Proc. Natl. Acad. Sci. U.S.A., 96, 9716–9720

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

[28] Mustonen, V. and Lässig, M. (2010) Fitness flux and ubiquity of adaptive evolution. Proc. Natl. Acad. Sci. U.S.A., 107, 4248–4253

[29] Price, G. R. (1972) Fisher’s ‘fundamental theorem’ made clear. Ann. Hum. Genet., 36, 129–140

[30] Edwards, A. W. F. (1994) The fundamental theorem of natural selection. Biol. Rev. Camb. Philos. Soc., 69, 443–474

[31] Qian, H. (2014) The zeroth law of thermodynamics and volume-preserving conservative system in equilibrium with stochastic damping. Phys. Lett. A, 378, 609–616

[32] Ao, P. (2008) Emerging of stochastic dynamical equalities and steady state thermodynamics from Darwinian dynamics. Commun. Commun Theor Phys, 49, 1073–1090

[33] Zhang, F., Xu, L., Zhang, K., Wang, E. and Wang, J. (2012) The potential and flux landscape theory of evolution. J. Chem. Phys., 137, 065102

[34] Iwasa, Y. (1988) Free fitness that always increases in evolution. J. Theor. Biol., 135, 265–281

[35] Maddox, J. (1991) Is Darwinism a thermodynamic necessity? Nature, 350, 653

[36] Varadhan, S. R. S. (2012) Book review: Large deviations for stochastic processes. Am. Math. Soc, 49, 597–601

[37] Voigt, J. (1981) Stochastic operators, information, and entropy. Commun. Math. Phys., 81, 31–38

[38] Mackey, M. C. (1989) The dynamic origin of increasing entropy. Rev. Mod. Phys., 61, 981–1015

[39] Qian, H. (2001) Relative entropy: free energy associated with equilibrium fluctuations and nonequilibrium deviations. Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 63, 042103

[40] Cover, T. M. and Thomas, J. A. (1991) Elements of Information Theory. New York: John Wiley & Sons.

[41] Onsager, L. (1931) Reciprocal relations in irreversible processes. I. Phys. Rev., 37, 405–426

[42] Price, G. R. (1972) Extension of covariance selection mathematics. Ann. Hum. Genet., 35, 485–490

[43] Zhang, X.-J., Qian, H. and Qian, M. (2012) Stochastic theory of nonequilibrium steady states and its applications (Part I). Phys. Rep., 510, 1–86

[44] Ge, H., Qian, M. and Qian, H. (2012) Stochastic theory of nonequilibrium steady states (Part II): Applications in chemical biophysics. Phys. Rep., 510, 87–118

[45] van Valen, L. (1973) A new evolutionary law. Evolu. Th., 1, 1–30.

[46] Ao, P. (2005) Laws in Darwinian evolutionary theory. Phys. Life Rev., 2, 117–156

[47] Schrödinger, E. (1944) What is Life? London: Cambridge University Press.

[48] Qian, H. (2007) Phosphorylation energy hypothesis: open chemical systems and their biological functions. Annu. Rev. Phys. Chem., 58, 113–142

[49] Qian, H. (2013) A decomposition of irreversible diffusion processes without detailed balance. J. Math. Phys., 54, 053302

[50] Lotka, A. J. (1922) Contribution to the energetics of evolution. Proc. Natl. Acad. Sci. U.S.A., 8, 147–151

[51] Lotka, A. J. (1922) Natural selection as a physical principle. Proc. Natl. Acad. Sci. U.S.A., 8, 151–154

[52] Kirschner, M. W. and Gerhart, J. C. (2005) The Plausibility of Life: Resolving Darwin’s Dilemma. New Haven: Yale University Press.

[53] Kauffman, S. A. (1993) The Origins of Order: Self-Organization and Selection in Evolution. New York: Oxford University Press.

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg