Molecular distributions in gene regulatory dynamics. (English) Zbl 1331.92046
Summary: Extending the work of N. Friedman et al. [“Linking stochastic dynamics to population distribution: an analytical framework of gene expression”, Phys. Rev. Lett. 97, No. 16, Article ID 168302, 4 p. (2006; doi:10.1103/PhysRevLett.97.168302)], we study the stationary density of the distribution of molecular constituents in the presence of noise arising from either bursting transcription or translation, or noise in degradation rates. We examine both the global stability of the stationary density as well as its bifurcation structure. We have compared our results with an analysis of the same model systems (either inducible or repressible operons) in the absence of any stochastic effects, and shown the correspondence between behaviour in the deterministic system and the stochastic analogs. We have identified key dimensionless parameters that control the appearance of one or two stable steady states in the deterministic case, or unimodal and bimodal densities in the stochastic systems, and detailed the analytic requirements for the occurrence of different behaviours. This approach provides, in some situations, an alternative to computationally intensive stochastic simulations. Our results indicate that, within the context of the simple models we have examined, bursting and degradation noise cannot be distinguished analytically when present alone.
MSC:
| 92C40 | Biochemistry, molecular biology |
| 92C45 | Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) |
References:
| [1] | Acar, M.; Becskei, A.; van Oudenaarden, A., Enhancement of cellular memory by reducing stochastic transitions, Nature, 435, 228-232 (2005) |
| [2] | Arnold, L., Random dynamical systems, Springer Monographs in Mathematics (1998), Springer-Verlag: Springer-Verlag Berlin · Zbl 0906.34001 |
| [3] | Berg, O. G., A model for the statistical fluctuations of protein numbers in a microbial population, J. Theor. Biol., 71, 4, 587-603 (1978) |
| [4] | Blake, W.; Balázsi, G.; Kohanski, M.; Issacs, F.; Murphy, K.; Kuang, Y.; Cantor, C.; Walt, D.; Collins, J., Phenotypic consequences of promoter-mediated transcriptional noise, Mol. Cell, 24, 853-865 (2006) |
| [5] | Blake, W.; Kaern, M.; Cantor, C.; Collins, J., Noise in eukaryotic gene expression, Nature, 422, 633-637 (2003) |
| [6] | Bobrowski, A.; Lipniacki, T.; Pichór, K.; Rudnicki, R., Asymptotic behavior of distributions of mRNA and protein levels in a model of stochastic gene expression, J. Math. Anal. Appl., 333, 2, 753-769 (2007) · Zbl 1115.92018 |
| [7] | Cai, L.; Friedman, N.; Xie, X., Stochastic protein expression in individual cells at the single molecule level, Nature, 440, 358-362 (2006) |
| [8] | Chubb, J.; Trcek, T.; Shenoy, S.; Singer, R., Transcriptional pulsing of a developmental gene, Curr. Biol., 16, 1018-1025 (2006) |
| [9] | Elowitz, M.; Levine, A.; Siggia, E.; Swain, P., Stochastic gene expression in a single cell, Science, 297, 1183-1186 (2002) |
| [10] | Feistel, R.; Ebeling, W., Evolution of Complex Systems (1989), VEB Deutscher Verlag der Wissenschaften: VEB Deutscher Verlag der Wissenschaften Berlin · Zbl 0774.92012 |
| [11] | Fraser, H.; Hirsh, A.; Glaever, G.; Kumm, J.; Eisen, M., Noise minimization in eukaryotic gene expression, PLoS Biol., 2 (2004), 8343-838 |
| [12] | Friedman, N.; Cai, L.; Xie, X. S., Linking stochastic dynamics to population distribution: an analytical framework of gene expression, Phys. Rev. Lett., 97 (2006), 168302-1-4 |
| [13] | Gardiner, C., Handbook of Stochastic Methods (1983), Springer Verlag: Springer Verlag Berlin, Heidelberg |
| [14] | Gardner, T.; Cantor, C.; Collins, J., Construction of a genetic toggle switch in Escherichia coli, Nature, 403, 339-342 (2000) |
| [15] | Gillespie, D., Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81, 2340-2361 (1977) |
| [16] | Golding, I.; Paulsson, J.; Zawilski, S.; Cox, E., Real-time kinetics of gene activity in individual bacteria, Cell, 123, 1025-1036 (2005) |
| [17] | Goodwin, B. C., Oscillatory behavior in enzymatic control processes, Adv. Enzyme Regul., 3, 425-428 (1965), IN1-IN2, 429-430, IN3-IN6, 431-437 |
| [18] | Griffith, J., Mathematics of cellular control processes. I. Negative feedback to one gene, J. Theor. Biol., 20, 202-208 (1968) |
| [19] | Griffith, J., Mathematics of cellular control processes. II. Positive feedback to one gene, J. Theor. Biol., 20, 209-216 (1968) |
| [20] | Haken, H., 1983. Synergetics: An Introduction, third ed. Springer Series in Synergetics, vol. 1. Springer-Verlag, Berlin.; Haken, H., 1983. Synergetics: An Introduction, third ed. Springer Series in Synergetics, vol. 1. Springer-Verlag, Berlin. · Zbl 0523.93001 |
| [21] | Hawkins, K.; Smolke, C., The regulatory roles of the galactose permease and kinase in the induction response of the GAL network in Saccharomyces cerevisiae, J. Biol. Chem., 281, 13485-13492 (2006) |
| [22] | Hierro, J.; Dopazo, C., Singular boundaries in the forward Chapman-Kolmogorov differential equation, J. Stat. phys., 137, 305-329 (2009) · Zbl 1187.60051 |
| [23] | Horsthemke, W.; Lefever, R., Noise Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology (1984), Springer-Verlag: Springer-Verlag Berlin, New York, Heidelberg · Zbl 0529.60085 |
| [24] | Jacob, F.; Perrin, D.; Snchez, C.; Monod, J., L’opéron: groupe de gènes à expression coordonnée par un opérateur, C. R. Acad. Sci. Paris, 250, 1727-1729 (1960) |
| [25] | Kaern, M.; Elston, T.; Blake, W.; Collins, J., Stochasticity in gene expression: from theories to phenotypes, Nature Rev. Genet., 6, 451-464 (2005) |
| [26] | Kepler, T.; Elston, T., Stochasticity in transcriptional regulation: origins, consequences, and mathematical representations, Biophys. J., 81, 3116-3136 (2001) |
| [27] | Lasota, A.; Mackey, M., Chaos, Fractals, and Noise. Applied Mathematical Sciences, vol. 97 (1994), Springer-Verlag: Springer-Verlag New York · Zbl 0784.58005 |
| [28] | Lipniacki, T.; Paszek, P.; Marciniak-Czochra, A.; Brasier, A.; Kimmel, M., Transcriptional stochasticity in gene expression, J. Theor. Biol., 238, 2, 348-367 (2006) · Zbl 1445.92113 |
| [29] | Lipshtat, A.; Loinger, A.; Balaban, N. Q.; Biham, O., Genetic toggle switch without cooperative binding, Phys. Rev. Lett., 96, 18, 188101 (2006) |
| [30] | Mackey, M. C.; Tyran-Kamińska, M., Dynamics and density evolution in piecewise deterministic growth processes, Ann. Polon. Math., 94, 111-129 (2008) · Zbl 1183.47038 |
| [31] | Mariani, L.; Schulz, E.; Lexberg, M.; Helmstetter, C.; Radbruch, A.; Löhning, M.; Höfer, T., Short-term memory in gene induction reveals the regulatory principle behind stochastic IL-4 expression, Mol. Sys. Biol., 6, 359 (2010) |
| [32] | Ochab-Marcinek, A., Predicting the asymmetric response of a genetic switch to noise, J. Theor. Biol., 254, 37-44 (2008) · Zbl 1400.92231 |
| [33] | Ochab-Marcinek, A., Extrinsic noise passing through a Michaelis-Menten reaction: a universal response of a genetic switch, J. Theor. Biol., 263, 510-520 (2010) · Zbl 1406.92274 |
| [34] | Oppenheim, I.; Schuler, K.; Weiss, G., Stochastic and deterministic formulation of chemical rate equations, J. Chem. Phys., 50, 460-466 (1969) |
| [35] | Othmer, H., The qualitative dynamics of a class of biochemical control circuits, J. Math. Biol., 3, 53-78 (1976) · Zbl 0334.92001 |
| [36] | Pichór, K.; Rudnicki, R., Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249, 668-685 (2000) · Zbl 0965.47026 |
| [37] | Polynikis, A.; Hogan, S.; di Bernardo, M., Comparing different ODE modelling approaches for gene regulatory networks, J. Theor. Biol., 261, 511-530 (2009) · Zbl 1403.92095 |
| [38] | Raj, A.; Peskin, C.; Tranchina, D.; Vargas, D.; Tyagi, S., Stochastic mRNA synthesis in mammalian cells, PLoS Biol., 4, 1707-1719 (2006) |
| [39] | Raj, A.; van Oudenaarden, A., Nature, nurture, or chance: stochastic gene expression and its consequences, Cell, 135, 216-226 (2008) |
| [40] | Raser, J.; O’Shea, E., Control of stochasticity in eukaryotic gene expression, Science, 304, 1811-1814 (2004) |
| [41] | Scott, M.; Ingallls, B.; Kærn, M., Estimations of intrinsic and extrinsic noise in models of nonlinear genetic networks, Chaos, 16 (2006), 026107-1-15 · Zbl 1152.92339 |
| [42] | Selgrade, J., Mathematical analysis of a cellular control process with positive feedback, SIAM J. Appl. Math., 36, 219-229 (1979) · Zbl 0426.34037 |
| [43] | Shahrezaei, V.; Ollivier, J.; Swain, P., Colored extrinsic fluctuations and stochastic gene expression, Mol. Syst. Biol., 4, 196-205 (2008) |
| [44] | Shahrezaei, V.; Swain, P., Analytic distributions for stochastic gene expression, Proc. Nat. Acad. Sci, 105, 17256-17261 (2008) |
| [45] | Shahrezaei, V.; Swain, P., The stochastic nature of biochemical networks, Cur. Opinion Biotech., 19, 369-374 (2008) |
| [46] | Sigal, A.; Milo, R.; Cohen, A.; Geva-Zatorsky, N.; Klein, Y.; Liron, Y.; Rosenfeld, N.; Danon, T.; Perzov, N.; Alon, U., Variability and memory of protein levels in human cells, Nature, 444, 643-646 (2006) |
| [47] | Smith, H., Monotone Dynamical Systems Mathematical Surveys and Monographs, vol. 41 (1995), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0821.34003 |
| [48] | Song, C.; Phenix, H.; Abedi, V.; Scott, M.; Ingalls, B.; Perkins, M. K.T., Estimating the stochastic bifurcation structure of cellular networks, PLos Comp. Biol., 6 (2010), e1000699/1-11 |
| [49] | Stratonovich, R.L., 1963. Topics in the theory of random noise. Vol. I: General theory of random processes. Nonlinear transformations of signals and noise. Revised English edition. Translated from the Russian by Richard A. Silverman. Gordon and Breach Science Publishers, New York.; Stratonovich, R.L., 1963. Topics in the theory of random noise. Vol. I: General theory of random processes. Nonlinear transformations of signals and noise. Revised English edition. Translated from the Russian by Richard A. Silverman. Gordon and Breach Science Publishers, New York. |
| [50] | Swain, P.; Elowitz, M.; Siggia, E., Intrinsic and extrinsic contributions to stochasticity in gene expression, Proc. Nat. Acad. Sci., 99, 12795-12800 (2002) |
| [51] | Titular, U., A systematic solution procedure for the Fokker-Planck equation of a Brownian particle in the high-friction case, Physica, 91A, 321-344 (1978) |
| [52] | Wilemski, G., On the derivation of Smoluchowski equations with corrections in the classical theory of Brownian motion, J. Stat. Phys., 14, 153-169 (1976) |
| [53] | Yildirim, N.; Santillán, M.; Horike, D.; Mackey, M. C., Dynamics and bistability in a reduced model of the lac operon, Chaos, 14, 279-292 (2004) · Zbl 1080.92031 |
| [54] | Yu, J.; Xiao, J.; Ren, X.; Lao, K.; Xie, X., Probing gene expression in live cells, one protein molecule at a time, Science, 311, 1600-1603 (2006) |
| [55] | Zacharioudakis, I.; Gligoris, T.; Tzamarias, D., A yeast catabolic enzyme controls transcriptional memory, Curr. Biol., 17, 2041-2046 (2007) |
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.
