×

Fluctuating-rate model with multiple gene states. (English) Zbl 1448.92150

Summary: Multiple phenotypic states of single cells often co-exist in the presence of positive feedbacks. Stochastic gene-state switchings and low copy numbers of proteins in single cells cause considerable fluctuations. The chemical master equation (CME) is a powerful tool that describes the dynamics of single cells, but it may be overly complicated. Among many simplified models, a fluctuating-rate (FR) model has been proposed recently to approximate the full CME model in the realistic intermediate region of gene-state switchings. However, only the scenario with two gene states has been carefully analysed. In this paper, we generalise the FR model to the case with multiple gene states, in which the mathematical derivation becomes more complicated. The leading order of fluctuations around each phenotypic state, as well as the transition rates between phenotypic states, in the intermediate gene-state switching region is characterized by the rate function of the stationary distribution of the FR model in the Freidlin-Wentzell-type large deviation principle (LDP). Under certain reasonable assumptions, we show that the derivative of the rate function is equal to the unique nontrivial solution of a dominant generalised eigenvalue problem, leading to a new numerical algorithm for obtaining the LDP rate function directly. Furthermore, we prove the Lyapunov property of the rate function for the corresponding deterministic mean-field dynamics. Finally, through a tristable example, we show that the local fluctuations (the asymptotic variance of the stationary distribution at each phenotypic state) in the intermediate and rapid regions of gene-state switchings are different. Finally, a tri-stable example is constructed to illustrate the validity of our theory.

MSC:

92D10 Genetics and epigenetics
15B48 Positive matrices and their generalizations; cones of matrices
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Acar, M.; Mettetal, JT; van Oudenaarden, A., Stochastic switching as a survival strategy in fluctuating environments, Nat Genet, 40, 471-475 (2008)
[2] Artyomov, MN; Das, J.; Kardar, M.; Chakraborty, AK, Purely stochastic binary decisions in cell signaling models without underlying deterministic bistabilities, Proc Natl Acad Sci USA, 104, 48, 18958-18963 (2007)
[3] Babloyantz, A.; Sanglier, M., Chemical instabilities of “all-or-none” type in beta—galactosidase induction and active transport, FEBS Lett, 23, 3, 364-366 (1972)
[4] Berg, OG, A model for the statistical fluctuations of protein numbers in a microbial population, J Theor Biol, 71, 587-603 (1978)
[5] Bressloff, PC, Path-integral methods for analyzing the effects of fluctuations in stochastic hybrid neural networks, J Math Neurosci, 5, 4, 1-33 (2015) · Zbl 1361.92005
[6] Bressloff, PC; Faugeras, O., On the Hamiltonian structure of large deviations in stochastic hybrid systems, J Stat Mech Theory Exp, 2017, 033206 (2017) · Zbl 1457.82190
[7] Bressloff, PC; Newby, JM, Path integrals and large deviations in stochastic hybrid systems, Phys Rev E, 89, 42, 701 (2014)
[8] Bressloff, PC; Newby, JM, Stochastic hybrid model of spontaneous dendritic NMDA spikes, Phys Biol, 11, 1, 016006 (2014)
[9] Choi, PJ; Cai, L.; Frieda, K.; Xie, XS, A stochastic single-molecule event triggers phenotype switching of a bacterial cell, Science, 322, 5900, 442-446 (2008)
[10] Chu, KWE, Exclusion theorems and the perturbation analysis of the generalized eigenvalue problem, SIAM J Numer Anal, 24, 5, 1114-1125 (1987) · Zbl 0636.15009
[11] Crudu, A.; Debussche, A.; Muller, A.; Radulescu, O., Convergence of stochastic gene networks to hybrid piecewise deterministic processes, Ann Appl Probab, 22, 5, 1822-1859 (2012) · Zbl 1261.60073
[12] Davis, MHA, Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models, J R Stat Soc Ser B (Methodol), 46, 3, 353-388 (1984) · Zbl 0565.60070
[13] Davis, MHA, Markov models and optimization, monographs on statistics and applied probability (1993), London: Chapman and Hall, London · Zbl 0780.60002
[14] Delbrück, M., Statistical fluctuations in autocatalytic reactions, J Chem Phys, 8, 1, 120 (1940)
[15] Deutsch, E.; Neumann, M., Derivatives of the Perron root at an essentially nonnegative matrix and the group inverse of an \(M\)-matrix, J Math Anal Appl, 102, 1, 1-29 (1984) · Zbl 0545.15008
[16] Dürrenberger, P.; Gupta, A.; Khammash, M., A finite state projection method for steady-state sensitivity analysis of stochastic reaction networks, J Chem Phys, 150, 134, 101 (2019)
[17] Dykman, MI; Mori, E.; Ross, J.; Hunt, PM, Large fluctuations and optimal paths in chemical kinetics, J Chem Phys, 100, 5735 (1994)
[18] Eldar, A.; Elowitz, MB, Functional roles for noise in genetic circuits, Nature, 467, 167-173 (2010)
[19] Faggionato, A.; Gabrielli, D.; Crivellari, MR, Non-equilibrium thermodynamics of piecewise deterministic Markov processes, J Stat Phys, 137, 259-304 (2009) · Zbl 1179.82108
[20] Faggionato, A.; Gabrielli, D.; Crivellari, MR, Averaging and large deviation principles for fully-coupled piecewise deterministic Markov processes and applications to molecular motors, Markov Process Relat Fields, 16, 3, 497-548 (2010) · Zbl 1266.60048
[21] Feng, H.; Han, B.; Wang, J., Adiabatic and non-adiabatic non-equilibrium stochastic dynamics of single regulating genes, J Phys Chem, 115, 5, 1254-1261 (2011)
[22] Feng, J.; Kurtz, TG, Large deviations for stochastic processes, mathematical surveys and monographs (2015), Providence: American Mathematical Society, Providence
[23] Frauenfelder, H.; Sligar, SG; Wolynes, PG, The energy landscapes and motions of proteins, Science, 254, 5038, 1598-1603 (1991)
[24] Freidlin, MI; Wentzell, AD, Random perturbations of dynamical systems, Grundlehren der mathematischen Wissenschaften (2014), Berlin: Spinger, Berlin
[25] Frobenius G (1912) Ueber matrizen aus nicht negativen elementen. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, pp 456-477 · JFM 43.0204.09
[26] Ge, H.; Qian, H., Thermodynamic limit of a nonequilibrium steady state: Maxwell-type construction for a bistable biochemical system, Phys Rev Lett, 103, 148, 103 (2009)
[27] Ge, H.; Qian, H.; Xie, XS, Stochastic phenotype transition of a single cell in an intermediate region of gene state switching, Phys Rev Lett, 114, 78, 101 (2015)
[28] Ge, H.; Wu, P.; Qian, H.; Xie, SX, Relatively slow stochastic gene-state switching in the presence of positive feedback significantly broadens the region of bimodality through stabilizing the uninduced phenotypic state, PLoS Comput Biol, 14, 3, e1006051 (2018)
[29] Gershgorin, SA, über die abgrenzung der eigenwerte einer matrix, Bull l’Acad Sci l’URSS Classe Sci Math, 6, 749-754 (1931) · Zbl 0003.00102
[30] Gillespie, DT, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J Comput Phys, 22, 4, 403-434 (1976)
[31] Gillespie, DT, Exact stochastic simulation of coupled chemical reactions, J Phys Chem, 81, 25, 2340-2361 (1977)
[32] Grima, R.; Schmidt, DR; Newman, TJ, Steady-state fluctuations of a genetic feedback loop: An exact solution, J Chem Phys, 137, 3, 035104 (2012)
[33] Gupta, PB; Fillmore, CM; Jiang, G.; Shapira, SD; Tao, K.; Kuperwasser, C.; Lander, ES, Stochastic state transitions give rise to phenotypic equilibrium in populations of cancer cells, Cell, 146, 4, 633-644 (2011)
[34] Gupta, A.; Mikelson, J.; Khammash, M., A finite state projection algorithm for the stationary solution of the chemical master equation, J Chem Phys, 147, 154, 101 (2017)
[35] Hanggi, P.; Grabert, H.; Talkner, P.; Thomas, H., Bistable systems: master equation versus Fokker-Planck modeling, Phys Rev A, 29, 371-378 (1984)
[36] Hasenauer, J.; Wolf, V.; Kazeroonian, A.; Theis, FJ, Method of conditional moments (MCM) for the chemical master equation, J Math Biol, 69, 687-735 (2014) · Zbl 1302.92070
[37] Hegland, M.; Hellander, A.; Lötstedt, P., Sparse grids and hybrid methods for the chemical master equation, BIT Numer Math, 48, 265-283 (2008) · Zbl 1155.65304
[38] Hornos, JEM; Schultz, D.; Innocentini, GCP; Wang, J.; Walczak, AM; Onuchic, JN; Wolynes, PG, Self-regulating gene: an exact solution, Phys Rev E, 72, 51, 907 (2005)
[39] Hufton, PG; Lin, YT; Galla, T.; McKane, AJ, Intrinsic noise in systems with switching environments, Phys Rev E, 93, 52, 119 (2016)
[40] Hufton, PG; Lin, YT; Galla, T., Phenotypic switching of populations of cells in a stochastic environment, J Stat Mech Theory Exp, 023, 501 (2018)
[41] Hufton, PG; Lin, YT; Galla, T., Classical stochastic systems with fast-switching environments: reduced master equations, their interpretation, and limits of validity, Phys Rev E, 99, 32, 121 (2019)
[42] Hufton, PG; Lin, YT; Galla, T., Model reduction methods for population dynamics with fast-switching environments: reduced master equations, stochastic differential equations, and applications, Phys Rev E, 99, 32, 122 (2019)
[43] Ikramov, KD, Matrix pencils: theory, applications, and numerical methods, J Sov Math, 64, 783-853 (1993) · Zbl 0783.15004
[44] Jia, C.; Qian, H.; Chen, M.; Zhang, MQ, Relaxation rates of gene expression kinetics reveal the feedback signs of autoregulatory gene networks, J Chem Phys, 148, 9, 095102 (2018)
[45] Kang, HW; Kurtz, TG, Separation of time-scales and model reduction for stochastic reaction networks, Ann Appl Probab, 23, 2, 529-583 (2013) · Zbl 1377.60076
[46] Karmakar, R.; Bose, I., Graded and binary responses in stochastic gene expression, Phys Biol, 1, 4, 197 (2004)
[47] Kazeev, V.; adn Michael Nip, MK; Schwab, C., Direct solution of the chemical master equation using quantized tensor trains, PLOS Comput Biol, 10, 3, e1003359 (2014)
[48] Keener, JP; Newby, JM, Perturbation analysis of spontaneous action potential initiation by stochastic ion channels, Phys Rev E, 84, 11, 918 (2011)
[49] Kepler, TB; Elston, TC, Stochasticity in transcriptional regulation: origins, consequences, and mathematical representations, Biophys J, 81, 6, 3116-3136 (2001)
[50] Kifer Y (2009) Large deviations and adiabatic transitions for dynamical systems and Markov processes in fully coupled averaging. Mem Am Math Soc 201(944) · Zbl 1222.37002
[51] Knessl, C.; Matkowsky, BJ; Schuss, Z.; Tier, C., An asymptotic theory of large deviations for Markov jump processes, SIAM J Appl Math, 45, 6, 1006-1028 (1985) · Zbl 0582.60081
[52] Kussell, E.; Leibler, S., Phenotypic diversity, population growth, and information in fluctuating environments, Science, 309, 5743, 2075-2078 (2005)
[53] Li, GW; Xie, XS, Central dogma at the single-molecule level in living cells, Nature, 475, 308-315 (2011)
[54] Lin, YT; Doering, CR, Gene expression dynamics with stochastic bursts: construction and exact results for a coarse-grained model, Phys Rev E, 93, 22, 409 (2016)
[55] Lu, M.; Onuchic, J.; Ben-Jacob, E., Construction of an effective landscape for multistate genetic switches, Phys Rev Lett, 113, 78, 102 (2014)
[56] MacNamara, S.; Burrage, K.; Sidje, RB, Multiscale modeling of chemical kinetics via the master equation, Multiscale Model Simul, 6, 4, 1146-1168 (2008) · Zbl 1153.60370
[57] Mateescu, M.; Wolf, V.; Didier, F.; Henzinger, TA, Fast adaptive uniformisation of the chemical master equation, IET Syst Biol, 4, 6, 441-452 (2010)
[58] Munskya, B.; Khammashb, M., The finite state projection algorithm for the solution of the chemical master equation, J Chem Phys, 124, 4, 044104 (2006)
[59] Newby, JM, Isolating intrinsic noise sources in a stochastic genetic switch, Phys Biol, 9, 26, 002 (2012)
[60] Newby, J., Bistable switching asymptotics for the self regulating gene, J Phys A Math Theor, 48, 18, 185001 (2015) · Zbl 1314.92119
[61] Newby, J.; Chapman, J., Metastable behavior in Markov processes with internal states, J Math Biol, 69, 941-976 (2014) · Zbl 1321.60169
[62] Newby, JM; Keener, JP, An asymptotic analysis of the spatially inhomogeneous velocity-jump process, Multiscale Model Simul, 9, 2, 735-765 (2011) · Zbl 1236.60079
[63] Newby, JM; Bressloff, PC; Keener, JP, Breakdown of fast-slow analysis in an excitable system with channel noise, Phys Rev Lett, 111, 128, 101 (2013)
[64] Ochab-Marcinek, A.; Tabaka, M., Bimodal gene expression in noncooperative regulatory systems, Proc Natl Acad Sci USA, 107, 51, 22096-22101 (2010)
[65] Olivieri, E.; Vares, ME, Large deviations and metastability, encyclopedia of mathematics and its applications (2005), Cambridge: Cambridge University Press, Cambridge
[66] Onuchic, JN; Luthey-Schulten, Z.; Wolynes, PG, Theory of protein folding: the energy landscape perspective, Ann Rev Phys Chem, 48, 545-600 (1997)
[67] Ozbudak, EM; Thattai, M.; Lim, HN; Shraiman, BI; van Oudenaarden, A., Multistability in the lactose utilization network of Escherichia coli, Nature, 427, 737-740 (2004)
[68] Paulsson, J., Models of stochastic gene expression, Phys Life Rev, 2, 2, 157-175 (2005)
[69] Peleš, S.; Munsky, B.; Khammash, M., Reduction and solution of the chemical master equation using time scale separation and finite state projection, J Chem Phys, 125, 20, 204104 (2006)
[70] Qian, H., Fitness and entropy production in a cell population dynamics with epigenetic phenotype switching, Quant Biol, 2, 1, 47-53 (2014)
[71] Qian, H.; Shia, PZ; Xing, J., Stochastic bifurcation, slow fluctuations, and bistability as an origin of biochemical complexity, Phys Chem Chem Phys, 24, 11, 4861-4870 (2009)
[72] Ramos, AF; Innocentini, GCP; Hornos, JEM, Exact time-dependent solutions for a self-regulating gene, Phys Rev E, 83, 62, 902 (2011)
[73] Redner, S., A guide to first-passage processes (2007), Cambridge: Cambridge University Press, Cambridge · Zbl 1128.60002
[74] Samad, HE; Khammash, M.; Petzold, L.; Gillespie, D., Stochastic modelling of gene regulatory networks, Int J Robust Nonlinear Control, 15, 15, 691-711 (2005) · Zbl 1090.93002
[75] Santillán, M., Bistable behavior in a model of the lac operon in Escherichia coli with variable growth rate, Biophys J, 94, 6, 2065-2081 (2008)
[76] Taniguchi, Y.; Choi, PJ; Li, GW; Chen, H.; Babu, M.; Hearn, J.; Emili, A.; Xie, XS, Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells, Science, 329, 5991, 533-538 (2010)
[77] Thattai, M.; van Oudenaarden, A., Intrinsic noise in gene regulatory networks, Proc Natl Acad Sci USA, 98, 15, 8614-8619 (2001)
[78] To, TL; Maheshri, N., Noise can induce bimodality in positive transcriptional feedback loops without bistability, Science, 327, 5969, 1142-1145 (2010)
[79] Touchette, H., The large deviation approach to statistical mechanics, Phys Rep, 478, 1-69 (2009)
[80] Vellela, M.; Qian, H., Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlogl model revisited, J R Soc Interface, 6, 39, 925-940 (2008)
[81] Wang, J.; Xu, L.; Wang, E.; Huang, S., The potential landscape of genetic circuits imposes the arrow of time in stem cell differentiation, Biophys J, 99, 1, 29-39 (2010)
[82] Zhou, JX; Aliyu, MDS; Aurell, E.; Huang, S., Quasi-potential landscape in complex multi-stable systems, J R Soc Interface, 9, 77, 3539-3553 (2012)
[83] Zhu, Z.; Shendure, J.; Church, GM, Discovering functional transcription-factor combinations in the human cell cycle, Genome Res, 15, 6, 848-855 (2005)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.