2015
Journal article
Open Access
On the impact of discreteness and abstractions on modelling noise in gene regulatory networks
Bodei C., Bortolussi L., Chiarugi D., Guerriero M. L., Policriti A., Romanel A.
Gene regulatory networks Stochastic noise Computational Mathematics Structural Biology Quasi-steady state approximation Discrete modeling Biochemistry Hybrid system Organic Chemistry
In this paper, we explore the impact of different forms of model abstraction and the role of discreteness on the dynamical behaviour of a simple model of gene regulation where a transcriptional repressor negatively regulates its own expression. We first investigate the relation between a minimal set of parameters and the system dynamics in a purely discrete stochastic framework, with the twofold purpose of providing an intuitive explanation of the different behavioural patterns exhibited and of identifying the main sources of noise. Then, we explore the effect of combining hybrid approaches and quasi-steady state approximations on model behaviour (and simulation time), to understand to what extent dynamics and quantitative features such as noise intensity can be preserved.
Source: Computational biology and chemistry (Print) 56 (2015): 98–108. doi:10.1016/j.compbiolchem.2015.04.004
Publisher: Elsevier,, Oxford , Regno Unito
Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., Walter, P., 2002. Mole. Biol. Cell. Garland Science.
Balázsi, G., van Oudenaarden, A., Collins, J.J., 2011. Cellular decision making and biological noise: from microbes to mammals. Cell 144 (6), 910-925.
Bartholomay, A.F., 1958. Stochastic models for chemical reactions. Bull. Math. Biophys. 20 (3), 175-190.
Becksei, A., Séraphin, B., Serrano, L., 2001. Positive feedback in eukaryotic gene networks: cell differentiation by graded binary response conversion. EMBO J. 20 (10), 2528-2535.
Bortolussi, L., Policriti, A., 2009. Hybrid semantics of stochastic programs with dynamic reconfiguration. In: Proceedings of CompMod'09, pp. 63-76.
Bortolussi, L., Policriti, A., 2013. (Hybrid) automata and (stochastic) programs: the hybrid automata lattice of a stochastic program. J. Logic Comput. 23 (4), 761-798.
Burden, R.L., Faires, J.D., 2005. Numerical Analysis. Thomson Brooks/Cole.
Cao, Y., Petzold, L., 2006. Accuracy limitations and the measurement of errors in the stochastic simulation of chemically reacting systems. J. Comput. Phys. 212 (1), 6-24.
Cao, Y., Gillespie, D.T., Petzold, L.R., 2005. The slow-scale stochastic simulation algorithm. J. Chem. Phys. 122 (1), 014116.
Chabot, J.R., Pedraza, J.M., Luitel, P., van Oudenaarden, A., 2007. Stochastic gene expression out-of-steady-state in the cyanobacterial circadian clock. Nature 450 (7173), 1249-1252.
Costa, O.L.V., Dufour, F., 2008. Stability and Ergodicity of Piecewise Deterministic Markov Processes. SIAM J. Control Optim. 47, 1053-1077.
Crudu, A., Debussche, A., Radulescu, O., 2009. Hybrid stochastic simplifications for multiscale gene networks. BMC Syst. Biol. 3, 89.
Dauxois, T., Di Patti, F., Fanelli, D., McKane, A.J., 2009. Enhanced stochastic oscillations in autocatalytic reactions. Phys. Rev. E: Stat. Nonlinear Soft Matter Phys. 79, 036112.
Davis, M.H.A., 1993. Markov Models and Optimization. Chapman & Hall.
Elowitz, M.B., Levine, A.J., Siggia, E.D., Swain, P.S., 2002. Stochastic gene expression in a single cell. Science 297 (5584), 1183-1186.
Gibson, M., Bruck, J., 2000. Efficient exact simulation of chemical systems with many species and many channels. J. Phys. Chem. 104 (9), 1876-1889.
Gillespie, D.T., Petzold, L., 2006. Numerical simulation for biochemical kinetics. In: Ch. System Modelling in Cellular Biology. MIT Press, pp. 1-29.
Gillespie, D., 1977. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81 (25), 2340-2361.
Gillespie, D.T., 2000. The chemical Langevin equation. J. Chem. Phys. 113 (1), 297-306.
Golding, I., Paulsson, J., Zawilski, S.M., Cox, E.C., 2005. Real-time kinetics of gene activity in individual bacteria. Cell 123 (6), 1025-1036.
Hoops, S., Sahle, S., Gauges, R., Lee, C., Pahle, J., Simus, N., Singhal, M., Xu, L., Mendes, P., Kummer, U., 2006. COPASI: a COmplex PAthway SImulator. Bioinformatics 22 (24), 3067-3074.
Hume, D.A., 2000. Probability in transcriptional regulation and its implications for leukocyte differentiation and inducible gene expression. Blood 96 (7), 2323-2328.
Lillacci, G., Khammash, M., 2010. Parameter estimation and model selection in computational biology. PLoS Comput. Biol. 6 (3), e1000696.
Mann, H.B., Whitney, D.R., 1947. On a test of whether one of two random variables is stochastically larger than the other. Ann. Math. Stat. 18 (1), 50-60.
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 (12), 4271-4288.
Marquez Lago, T.T., Stelling, J., 2010. Counter-intuitive stochastic behavior of simple gene circuits with negative feedback. Biophys. J. 98 (9), 1742-1750.
W.E., Liu, D., Vanden-Eijnden, E., 2005. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys. 123, 194107.
W.E., Liu, D., Vanden-Eijnden, E., 2007. Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Comput. Phys. 221 (1), 158-180.
Mastny, E.A., Haseltine, E.L., Rawlings, J.B., 2007. Two classes of quasi-steady-state model reductions for stochastic kinetics. J. Chem. Phys. 127 (9), 094106.
McQuarrie, D., 1967. Stochastic approach to chemical kinetics. J. Appl. Probab. 4 (3), 413-478.
Murphy, K.F., Adams, R.M., Wang, X., Balázsi, G., Collins, J.J., 2010. Tuning and controlling gene expression noise in synthetic gene networks. Nucleic Acids Res. 38 (8), 2712-2726.
Novic, A., Weiner, M., 1957. Enzyme induction as an all-or-none phenomenon. Proc. Natl. Acad. Sci. U. S. A. 43 (7), 553-566.
Pahle, J., 2009. Biochemical simulations: stochastic, approximate stochastic and hybrid approaches. Brief Bioinform. 10 (1), 53-64.
Peccoud, J., Ycart, B., 1995. Markovian modelling of gene-product synthesis. Theor. Popul. Biol. 48 (2), 222-234.
Raj, A., van Oudenaarden, A., 2008. Nature, nurture, or chance: stochastic gene expression and its consequences. Cell 135 (2), 216-226.
Raj, A., van Oudenaarden, A., 2009. Single-molecule approaches to stochastic gene expression. Annu. Rev. Biophys. 38, 255-270.
Raj, A., Peskin, C.S., Tranchina, D., Vargas, D.Y., Tyagi, S., 2006. Stochastic mRNA synthesis in mammalian cells. PLoS Biol. 4 (10), e309.
Raj, A., Rifkin, S.A., Andersen, E., van Oudenaarden, A., 2010. Variability in gene expression underlies incomplete penetrance. Nature 463 (7283), 913-918.
Ramaswamy, R., Sbalzarini, I., 2010. A partial-propensity variant of the compositionrejection stochastic simulation algorithm for chemical reaction networks. J. Chem. Phys. 132 (4), 044102.
Rao, C.V., Arkin, A.P., 2003. Stochastic chemical kinetics and the quasi-steady state assumption: application to the Gillespie algorithm. J. Chem. Phys. 118 (11), 4999-5010.
Rathinam, M., Petzold, L.R., Cao, Y., Gillespie, D.T., 2003. Stiffness in stochastic chemically reacting systems: the implicit tau-leaping method. Journal of Chemical Physics 119 (24), 12784-12794.
Resat, H., Petzold, L., Pettigrew, M.F., 2009. Kinetic modeling of biological systems. Methods Mol. Biol. 541 (10), 311-335.
Ross, I.L., Browne, C.M., Hume, D.A., 1994. Transcription of individual genes in eukaryotic cells occurs randomly and infrequently. Immunol. Cell Biol. 72 (2), 177-185.
Salis, H., Kaznessis, Y., 2005. Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions. J. Chem. Phys. 122 (5), 54103.
Salis, H., Sotiropoulos, V., Kaznessis, Y.N., 2006. Multiscale Hy3S: hybrid stochastic simulation for supercomputers. BMC Bioinform. 7, 93.
Sheskin, D.J., 2004. Handbook of Parametric and Nonparametric Statistical Procedures, 3rd ed. Chapman & Hall.
Stamatakis, M., Zygourakis, K., 2010. A mathematical and computational approach for integrating the major sources of cell population heterogeneity. J. Theor. Biol. 266 (1), 41-61.
Stamatakis, M., Zygourakis, K., 2011. Deterministic and stochastic population level simulations of an artificial lac operon genetic network. BMC Bioinform. 12, 301.
Stekel, D.J., Jenkins, D.J., 2008. Strong negative self regulation of prokaryotic transcription factors increases the intrinsic noise of protein expression. BMC Syst. Biol. 2, 6.
Trivedi, K.S., Kulkarni, V.G., 1993. FSPNs: fluid stochastic Petri nets. In: Application and Theory of Petri Nets., pp. 24-31.
Vogel, U., Jensen, K.F., 1994. The RNA chain elongation rate in Escherichia coli depends on the growth rate. J. Bacteriol. 176 (10), 2807-2813.
Walker, J.J., Terry, J.R., Tsaneva-Atanasova, K., Armstrong, S.P., McArdle, C.A., Lightman, S.L., 2010. Encoding and decoding mechanisms of pulsatile hormone secretion. J. Neuroendocrinol. 22 (12), 1226-1238.
Zhou, G.-Q., Zhong, W.-Z., 1982. Diffusion-controlled reactions of enzymes. Eur. J. Biochem. 128 (2-3), 383-387.
Balázsi, G., van Oudenaarden, A., Collins, J.J., 2011. Cellular decision making and biological noise: from microbes to mammals. Cell 144 (6), 910-925.
Bartholomay, A.F., 1958. Stochastic models for chemical reactions. Bull. Math. Biophys. 20 (3), 175-190.
Becksei, A., Séraphin, B., Serrano, L., 2001. Positive feedback in eukaryotic gene networks: cell differentiation by graded binary response conversion. EMBO J. 20 (10), 2528-2535.
Bortolussi, L., Policriti, A., 2009. Hybrid semantics of stochastic programs with dynamic reconfiguration. In: Proceedings of CompMod'09, pp. 63-76.
Bortolussi, L., Policriti, A., 2013. (Hybrid) automata and (stochastic) programs: the hybrid automata lattice of a stochastic program. J. Logic Comput. 23 (4), 761-798.
Burden, R.L., Faires, J.D., 2005. Numerical Analysis. Thomson Brooks/Cole.
Cao, Y., Petzold, L., 2006. Accuracy limitations and the measurement of errors in the stochastic simulation of chemically reacting systems. J. Comput. Phys. 212 (1), 6-24.
Cao, Y., Gillespie, D.T., Petzold, L.R., 2005. The slow-scale stochastic simulation algorithm. J. Chem. Phys. 122 (1), 014116.
Chabot, J.R., Pedraza, J.M., Luitel, P., van Oudenaarden, A., 2007. Stochastic gene expression out-of-steady-state in the cyanobacterial circadian clock. Nature 450 (7173), 1249-1252.
Costa, O.L.V., Dufour, F., 2008. Stability and Ergodicity of Piecewise Deterministic Markov Processes. SIAM J. Control Optim. 47, 1053-1077.
Crudu, A., Debussche, A., Radulescu, O., 2009. Hybrid stochastic simplifications for multiscale gene networks. BMC Syst. Biol. 3, 89.
Dauxois, T., Di Patti, F., Fanelli, D., McKane, A.J., 2009. Enhanced stochastic oscillations in autocatalytic reactions. Phys. Rev. E: Stat. Nonlinear Soft Matter Phys. 79, 036112.
Davis, M.H.A., 1993. Markov Models and Optimization. Chapman & Hall.
Elowitz, M.B., Levine, A.J., Siggia, E.D., Swain, P.S., 2002. Stochastic gene expression in a single cell. Science 297 (5584), 1183-1186.
Gibson, M., Bruck, J., 2000. Efficient exact simulation of chemical systems with many species and many channels. J. Phys. Chem. 104 (9), 1876-1889.
Gillespie, D.T., Petzold, L., 2006. Numerical simulation for biochemical kinetics. In: Ch. System Modelling in Cellular Biology. MIT Press, pp. 1-29.
Gillespie, D., 1977. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81 (25), 2340-2361.
Gillespie, D.T., 2000. The chemical Langevin equation. J. Chem. Phys. 113 (1), 297-306.
Golding, I., Paulsson, J., Zawilski, S.M., Cox, E.C., 2005. Real-time kinetics of gene activity in individual bacteria. Cell 123 (6), 1025-1036.
Hoops, S., Sahle, S., Gauges, R., Lee, C., Pahle, J., Simus, N., Singhal, M., Xu, L., Mendes, P., Kummer, U., 2006. COPASI: a COmplex PAthway SImulator. Bioinformatics 22 (24), 3067-3074.
Hume, D.A., 2000. Probability in transcriptional regulation and its implications for leukocyte differentiation and inducible gene expression. Blood 96 (7), 2323-2328.
Lillacci, G., Khammash, M., 2010. Parameter estimation and model selection in computational biology. PLoS Comput. Biol. 6 (3), e1000696.
Mann, H.B., Whitney, D.R., 1947. On a test of whether one of two random variables is stochastically larger than the other. Ann. Math. Stat. 18 (1), 50-60.
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 (12), 4271-4288.
Marquez Lago, T.T., Stelling, J., 2010. Counter-intuitive stochastic behavior of simple gene circuits with negative feedback. Biophys. J. 98 (9), 1742-1750.
W.E., Liu, D., Vanden-Eijnden, E., 2005. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys. 123, 194107.
W.E., Liu, D., Vanden-Eijnden, E., 2007. Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Comput. Phys. 221 (1), 158-180.
Mastny, E.A., Haseltine, E.L., Rawlings, J.B., 2007. Two classes of quasi-steady-state model reductions for stochastic kinetics. J. Chem. Phys. 127 (9), 094106.
McQuarrie, D., 1967. Stochastic approach to chemical kinetics. J. Appl. Probab. 4 (3), 413-478.
Murphy, K.F., Adams, R.M., Wang, X., Balázsi, G., Collins, J.J., 2010. Tuning and controlling gene expression noise in synthetic gene networks. Nucleic Acids Res. 38 (8), 2712-2726.
Novic, A., Weiner, M., 1957. Enzyme induction as an all-or-none phenomenon. Proc. Natl. Acad. Sci. U. S. A. 43 (7), 553-566.
Pahle, J., 2009. Biochemical simulations: stochastic, approximate stochastic and hybrid approaches. Brief Bioinform. 10 (1), 53-64.
Peccoud, J., Ycart, B., 1995. Markovian modelling of gene-product synthesis. Theor. Popul. Biol. 48 (2), 222-234.
Raj, A., van Oudenaarden, A., 2008. Nature, nurture, or chance: stochastic gene expression and its consequences. Cell 135 (2), 216-226.
Raj, A., van Oudenaarden, A., 2009. Single-molecule approaches to stochastic gene expression. Annu. Rev. Biophys. 38, 255-270.
Raj, A., Peskin, C.S., Tranchina, D., Vargas, D.Y., Tyagi, S., 2006. Stochastic mRNA synthesis in mammalian cells. PLoS Biol. 4 (10), e309.
Raj, A., Rifkin, S.A., Andersen, E., van Oudenaarden, A., 2010. Variability in gene expression underlies incomplete penetrance. Nature 463 (7283), 913-918.
Ramaswamy, R., Sbalzarini, I., 2010. A partial-propensity variant of the compositionrejection stochastic simulation algorithm for chemical reaction networks. J. Chem. Phys. 132 (4), 044102.
Rao, C.V., Arkin, A.P., 2003. Stochastic chemical kinetics and the quasi-steady state assumption: application to the Gillespie algorithm. J. Chem. Phys. 118 (11), 4999-5010.
Rathinam, M., Petzold, L.R., Cao, Y., Gillespie, D.T., 2003. Stiffness in stochastic chemically reacting systems: the implicit tau-leaping method. Journal of Chemical Physics 119 (24), 12784-12794.
Resat, H., Petzold, L., Pettigrew, M.F., 2009. Kinetic modeling of biological systems. Methods Mol. Biol. 541 (10), 311-335.
Ross, I.L., Browne, C.M., Hume, D.A., 1994. Transcription of individual genes in eukaryotic cells occurs randomly and infrequently. Immunol. Cell Biol. 72 (2), 177-185.
Salis, H., Kaznessis, Y., 2005. Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions. J. Chem. Phys. 122 (5), 54103.
Salis, H., Sotiropoulos, V., Kaznessis, Y.N., 2006. Multiscale Hy3S: hybrid stochastic simulation for supercomputers. BMC Bioinform. 7, 93.
Sheskin, D.J., 2004. Handbook of Parametric and Nonparametric Statistical Procedures, 3rd ed. Chapman & Hall.
Stamatakis, M., Zygourakis, K., 2010. A mathematical and computational approach for integrating the major sources of cell population heterogeneity. J. Theor. Biol. 266 (1), 41-61.
Stamatakis, M., Zygourakis, K., 2011. Deterministic and stochastic population level simulations of an artificial lac operon genetic network. BMC Bioinform. 12, 301.
Stekel, D.J., Jenkins, D.J., 2008. Strong negative self regulation of prokaryotic transcription factors increases the intrinsic noise of protein expression. BMC Syst. Biol. 2, 6.
Trivedi, K.S., Kulkarni, V.G., 1993. FSPNs: fluid stochastic Petri nets. In: Application and Theory of Petri Nets., pp. 24-31.
Vogel, U., Jensen, K.F., 1994. The RNA chain elongation rate in Escherichia coli depends on the growth rate. J. Bacteriol. 176 (10), 2807-2813.
Walker, J.J., Terry, J.R., Tsaneva-Atanasova, K., Armstrong, S.P., McArdle, C.A., Lightman, S.L., 2010. Encoding and decoding mechanisms of pulsatile hormone secretion. J. Neuroendocrinol. 22 (12), 1226-1238.
Zhou, G.-Q., Zhong, W.-Z., 1982. Diffusion-controlled reactions of enzymes. Eur. J. Biochem. 128 (2-3), 383-387.
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BibTeX entry
@article{oai:it.cnr:prodotti:424148,
title = {On the impact of discreteness and abstractions on modelling noise in gene regulatory networks},
author = {Bodei C. and Bortolussi L. and Chiarugi D. and Guerriero M. L. and Policriti A. and Romanel A.},
publisher = {Elsevier,, Oxford , Regno Unito},
doi = {10.1016/j.compbiolchem.2015.04.004},
journal = {Computational biology and chemistry (Print)},
volume = {56},
pages = {98–108},
year = {2015}
}
0000-0001-8874-4001
