Counting single cells and computing their heterogeneity: from phenotypic frequencies to mean value of a quantitative biomarker

Abstract

This tutorial presents a mathematical theory that relates the probability of sample frequencies, of M phenotypes in an isogenic population of N cells, to the probability distribution of the sample mean of a quantitative biomarker, when the N is very large. An analogue to the statistical mechanics of canonical ensemble is discussed.

Keywords

large deviation principle / chemical kinetics / Boltzmann’s law / variational Bayesian method / maximum entropy principle

References

[1] Pence, C. H. (2011) “Describing our whole experience”: the statistical philosophies of W. F. R. Weldon and Karl Pearson. Stud. Hist. Philos. Biol. Biomed. Sci., 42, 475–485

[2] Chibbaro, S., Rondoni, L. and Vulpiani, A. (2014) Reductionism, Emergence and Levels of Reality. New York: Springer

[3] Anderson, P. W. (1972) More is different. Science, 177, 393–396

[4] Qian, H., Ao, P., Tu, Y. and Wang, J. (2016) A framework towards understanding mesoscopic phenomena: Emergent unpredictability, symmetry breaking and dynamics across scales. Chem. Phys. Lett., 665, 153–161

[5] Ge, H. and Qian, H. (2016) Mesoscopic kinetic basis of macroscopic chemical thermodynamics: A mathematical theory. Phys. Rev. E, 94, 052150

[6] Dembo, A. and Zeitouni, O. (1998) Large Deviations Techniques and Applications, 2nd ed. New York: Springer

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

[8] Liggett, T. M. (1985) Interacting Particle Systems. New York: Springer-Verlag

[9] Derrida, B. (1998) An exactly soluble nonequilibrium system: The asymmetric simple exclusion process. Phys. Rep., 301, 65–83

[10] Gang, H. (1986) Lyapunov function and stationary probability distribution. Zeit. Physik B: Cond. Matt. 65, 103–106

[11] Chang, R. and Goldsby, K. A. (2012) Chemistry, 11th ed. New York: McGraw-Hill

[12] Qian, H. (2019) Stochastic population kinetics and its underlying mathematicothermodynamics. In: The Dynamics of Biological Systems, Bianchi, A., Hillen, T., Lewis, M., Yi, Y. eds., pp. 149–188. Springer: New York

[13] Murray, J. D. (2011) Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd ed. New York: Springer

[14] von Bertalanffy, L. (1950) The theory of open systems in physics and biology. Science, 111, 23–29

[15] Huang, K. (1963) Statistical Mechanics. New York: John Wiley & Sons

[16] Sharp, K. and Matschinsky, F. (2015) Translation of Ludwig Boltzmann’s paper “On the relationship between the second fundamental theorem of the mechanical theory of heat and probability calculations regarding the conditions for thermal equilibrium”. Entropy (Basel), 17, 1971–2009

[17] Ghahramani, Z. (2001) An introduction to hidden Markov models and Bayesian networks. Int. J. Pattern Recognit. Artif. Intell., 15, 9–42

[18] Pauli, W. (1973) Pauli Lectures on Physics: Thermodynamics and the Kinetic Theory of Gas. Cambridge: The MIT Press

[19] Jaynes, E. T. (2003) Probability Theory: The Logic of Science. London: Cambridge University Press

[20] Shore, J. E. and Johnson, R. W. (1980) Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. IEEE Trans. Inf. Theory, 26, 26–37

[21] van Campenhout, J. M. and Cover, T. M. (1981) Maximum entropy and conditional probability. IEEE Trans. Inf. Theory, 27, 483–489

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature