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Bounded noises as a natural tool to model extrinsic fluctuations in biomolecular networks

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Abstract

In the first part of this invited paper we review the role of both extrinsic and intrinsic stochasticity in shaping the dynamics of biomolecular networks. In particular, we stress the use of bounded stochastic processes as a model of extrinsic random perturbations. In the second part, we propose three examples of molecular circuits under the influence of external fluctuations modeled by means of bounded noises. The first two examples involve the linear decay of a protein with, respectively, large or low number of molecules, so to stress different modeling approaches. The third example concerns the spatio-temporal dynamics of proteins determining the chemotaxis-driven polarization of a cell. In these examples one can observe phenomena that are dependent on the specific class of employed bounded noise.

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References

  • Alberts B, Johnson A, Lewis J, Raff M, Roberts K, Walter P (2009) Molecular biology of the cell, 5th ed. Garland, New York

  • Alon U (2006) An introduction to systems biology: design principles of biological circuits. Chapman & Hall/CRC Mathematical & Computational Biology, Boca Raton, FL

    Google Scholar 

  • Angeli D, Ferrell JE Jr, Sontag ED (2004) Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems. PNAS 101(7):1822–1827

    Article  Google Scholar 

  • Arnold L (1989) Random dynamical systems. Springer, Berlin

    Google Scholar 

  • Arnold L (1998) Random dynamical systems. Springer, Berlin

    Book  MATH  Google Scholar 

  • Becskei A, Serrano L (2000) Engineering stability in gene networks by autoregulation. Nature 405:590–593

    Article  Google Scholar 

  • Becskei A, Kaufmann BB, van Oudenaarden AE (2000) Contributions of low molecule number and chromosomal positioning to stochastic gene expression. Nat Genet 37:937–944

    Article  Google Scholar 

  • Bobryk RV, Chrzeszczyk A (2005) Transitions induced by bounded noise. Phys A 358(2–4):263

    Article  Google Scholar 

  • Bobryk RV, Chrzeszczyk A (2008) Transitions in a duffing oscillator excited by random noise. Nonlinear Dyn 51:541

    Article  MATH  MathSciNet  Google Scholar 

  • Borland L (1998) Ito–Langevin equations within generalized thermostatistics. Phys Lett A f245(1–2):67

    Article  MathSciNet  Google Scholar 

  • Cai CQ, Lin YK (1996) Generation of non-gaussian stationary stochastic processes. Phys Rev E 54:299

    Article  Google Scholar 

  • Cai L, Friedman N, Xie XS (2006) Stochastic protein expression in individual cells at the single molecule level. Nature 440:358–362

    Article  Google Scholar 

  • Caravagna G, Mauri G, d’Onofrio A (2013) The interplay of intrinsic and extrinsic bounded noises in biomolecular networks. PLoS One 8:e51174

    Article  Google Scholar 

  • Caravagna G, Mauri G, d’Onofrio A (2013) Bounded extrinsic noises affecting biochemical networks with low molecule numbers. Chapter in d’Onofrio A (ed) Bounded noises in physics, biology and engineering, Birkauser, Verlag. ISBN 978-1-4614-7348-8

  • Caravagna G, Mauri G, d’Onofrio A (2013) NoisySIM: exact simulation of stochastic chemically reacting systems with extrinsic noises. In proceedings of the Symposium on theory of modeling and simulation vol 12, pp 1–6. Society for Computer Simulation International, San Diego, CA

  • Chang HH, Oh PY, Ingber DE, Huang S (2006) Stochastic approaches for systems biology. BMC Cell Biol 7:11–23

    Article  Google Scholar 

  • Cinquin O, Demongeot J (2005) High-dimensional switches and the modelling of cellular differentiation. J Theor Biol 233:391–411

    Article  Google Scholar 

  • deFranciscis S, d’Onofrio A (2012) Spatiotemporal bounded noises and transitions induced by them in solutions of the real Ginzburg–Landau model. Phys Rev E 86:021118

    Article  Google Scholar 

  • deFranciscis S, d’Onofrio A (2013) Cellular polarization: interaction between extrinsic bounded noises and the wave-pinning mechanism. Phys Rev E 88:032709

    Article  Google Scholar 

  • deFranciscis S, d’Onofrio A (2013) Spatio-temporal Sine–Wiener bounded noise and its effect on Ginzburg–Landau model. Nonlinear Dyn 74:607

    Article  MathSciNet  Google Scholar 

  • Detwiler PB, Ramanathan S, Sengupta A, Shraiman BI (2000) Engineering aspects of enzymatic signal transduction: photoreceptors in the retina. Biophys J 79:2801–2817

    Article  Google Scholar 

  • Dimentberg M (1988) Statistical dynamics of nonlinear and time-varying systems. Research Studies Press, Baldock

    MATH  Google Scholar 

  • d’Onofrio A (2013) Multifaceted aspects of the kinetics of immunoevasion from tumor dormancy. In: Enderling H, Almog N and Hlatky L (eds.) Systems biology of tumor dormancy, advances in experimental medicine and biology, vol 734, Springer, Berlin, p 111

  • d’Onofrio A (ed) (2013) Bounded noises in physics, biology and engineering, Birkauser, Basel-Boston. ISBN 978-1-4614-7348-8.

  • d’Onofrio A (2010) Bounded-noise-induced transitions in a tumor-immune system interplay. Phys Rev E 81:021923

    Article  MathSciNet  Google Scholar 

  • d’Onofrio A, Gandolfi A (2010) Resistance to antitumor chemotherapy due to bounded-noise-induced transitions. Phys Rev E 82:061901

    Article  MathSciNet  Google Scholar 

  • Eldar A, Elowitz MB (2010) Functional role for noise in genetic circuits. Nature 467:167–173

    Article  Google Scholar 

  • Elowitz MB, Levine AJ, Siggia ED, Swain PS (2002) Stochastic gene expression in a single cell. Science 298:1183–1186

    Article  Google Scholar 

  • Gamba A, de Candia A, Di Talia S, A Coniglio A, Bussolino F, Serini G (2005) Diffusion-limited phase separation in eukaryotic chemotaxis. PNAS 102(47):16927

    Article  Google Scholar 

  • García-Ojalvo J, Sancho JM, Ramírez-Piscina L (1992) Generation of spatiotemporal colored noise. Phys Rev A 46:4670

    Article  Google Scholar 

  • García-Ojalvo J, Sancho JM, Ramírez-Piscina L (1992) A nonequilibrium phase transition with colored noise. Phys Lett A 168(1):35–39

    Article  Google Scholar 

  • Garcia-Ojalvo J, Sancho JM, Ramirez-Piscina I (1992) Generation of spatiotemporal colored noise. Phys Rev E 46:4670

    Google Scholar 

  • Gardiner CW (1985) Handbook of stochastic methods, 2nd edn. Springer, Berlin

    Google Scholar 

  • Gardner TR, Cantor CR, Collins JJ (2000) Construction of a genetic toggle switch in escherichiacoli. Nature 403:339–342

    Article  Google Scholar 

  • Ghaemmaghami S, Huh W, Bower K, Howson RW, Belle A, Dephoure N, O’Shea EK, Weissman JS (2003) Global analysis of protein expression in yeast. Nature 425:737–743

    Article  Google Scholar 

  • Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12:30–39

    Article  Google Scholar 

  • Gillespie DT (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comp Phys 22(4):403–434

    Article  MathSciNet  Google Scholar 

  • Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81:2340–2361

    Google Scholar 

  • Gillespie DT (1980) Approximating the master equation by Fokker–Planck-type equations for single-variable chemical systems. J Phys Chem 72:5363–5371

    Google Scholar 

  • Gillespie DT (2000) The chemical Langevin equation. J Phys Chem 113:297–306

    Google Scholar 

  • Glass L, Kauffman SA (1968) Logical analysis of systems comprising feedback loops. J Theor Biol 39:103–129

    Article  Google Scholar 

  • Grabert H, Hänggi P, Oppenheim I (1983) Fluctuations in reversible chemical reactions. Phys A 117:300–316

    Article  MathSciNet  Google Scholar 

  • Graudenzi A, Caravagna G, De Matteis G, Antoniotti M (2014) Investigating the relation between stochastic differentiation and homeostasis in intestinal crypts via multiscale modeling. bioRxv, http://biorxiv.org/content/early/2013/11/25/000927

  • Griffith JS (1968) Mathematics of cellular control processes ii positive feedback to one gene. J Theor Biol 20:209–216

    Article  Google Scholar 

  • Guo W, Du LC, Mei DC (2012) Transitions induced by time delays and cross-correlated Sine–Wiener noises in a tumor-immune system interplay. Phys A 391:1270–1280

    Article  Google Scholar 

  • Hasty J, Pradines J, Dolnik M, Collins JJ (2000) Noise-based switches and amplifiers for gene expression. PNAS 97(5):2075–2080

    Article  Google Scholar 

  • Homburg AJ, Young TR, Gharaei M (2013) Bifurcations of random differential equations with bounded noise. In d’Onofrio A (ed) Bounded noises in physics, biology and engineering, Birkauser, Verlag. ISBN 978-1-4614-7348-8

  • Horsthemke W, Lefever R (2006) Noise-induced transitions: theory and applications in physics, chemistry, and biology, Series in Synergetics Springer, Berlin

    Google Scholar 

  • Iglesias PA, Devreotes PN (2008) Navigating through models of chemotaxis. Curr Opin Cell Biol 20:35–40

    Article  Google Scholar 

  • Iglesias PA, Ingalls PB (2010) Control theory and systems biology. MIT Press, Cambridge

    MATH  Google Scholar 

  • Jung P, Hänggi P (1987) Dynamical systems: a unified colored-noise approximation. Phys Rev A 35:4464

    Article  Google Scholar 

  • Kauffman SA (1969) Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol 22:437–467

    Article  MathSciNet  Google Scholar 

  • Kramer BP, Fussenegger M (2005) Hysteresis in a synthetic mammalian gene network. PNAS 102:9517–9522

    Article  Google Scholar 

  • Lestas I, Vinnicombe G, Paulsson J (2010) Fundamental limits on the suppression of molecular fluctuations. Nature 467:174–178

    Article  Google Scholar 

  • Losick R, Desplan C (2008) Stochasticity and cell fate. Science 320:65–68

    Article  Google Scholar 

  • Macieira-Coelho A (2007) Asymmetric cell division. Springer, Berlin

    Book  Google Scholar 

  • Mandelbrot BB (1963) The variation of certain speculative prices. J Bus (Chicago) 36:394–419

    Article  Google Scholar 

  • Markevich NI, Hoek JB, Kholodenko BN (2004) Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades. J Cell Biol 164:353–359

    Article  Google Scholar 

  • Meinhardt H (1999) Orientation of chemotactic cells and growth cones: models and mechanisms. J Cell Sci 17:2867

    Google Scholar 

  • Mori Y, Jilkine A, Edelstein-Keshet L (2008) Wave-pinning and cell polarity from a bistable reaction-diffusion system. Biophys J 94:3684

    Article  Google Scholar 

  • Murray JD (2002) Mathematical biology. Springer, New York

    MATH  Google Scholar 

  • Onsum MD, Rao CV (2009) Calling heads from tails: the role of mathematical modeling in understanding cell polarization. Curr Opin Cell Biol 21(1):74

    Article  Google Scholar 

  • Paulsson BO (2011) Systems biology simulation of dynamic network states. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Rao CV, Wolf D, Arkin AP (2002) Control, exploitation and tolerance of intracellular noise. Nature 420:231–237

    Article  Google Scholar 

  • Ridolfi L, D’Odorico P, Laio F (2011) Noise-induced phenomena in the environmental sciences. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Rigney DR, Schieve WC (1977) Stochastic model of linear, continuous protein-synthesis in bacterial populations. J Theor Biol 69:761–766

    Article  Google Scholar 

  • Samoilov M, Plyasunov S, Arkin AP (2005) Stochastic amplification and signaling in enzymatic futile cycles through noise-induced bistability with oscillations. PNAS 102(7):2310–2315

    Article  Google Scholar 

  • Sanft KR, Gillespie DT, Petzold LR (2011) Legitimacy of the stochastic michaelis-menten approximation. IET Sys Bio 5(1):58–69

    Article  Google Scholar 

  • Semplice M, Veglio A, Naldi G, Serini G, Gamba A (2012) A bistable model of cell polarity. PLoS One 7:e30977

    Article  Google Scholar 

  • Shahrezaei V, Ollivier JF, Swain PS (2008) Colored extrinsic fluctuations and stochastic gene expression. Mol Sys Biol 4:196

    Google Scholar 

  • Siegal-Gaskins D, Grotewold E, Smith GD (2009) The capacity for multistability in small gene regulatory networks. BMC Sys Biol 3:96

    Article  Google Scholar 

  • Simon Z (1965) Multi-steady-state model for cell differentiation. J Theor Biol 8:258–263

    Article  Google Scholar 

  • Sugita M (1964) Functional analysis of chemical systems in vivo using a logical circuit equivalent ii the idea of a molecular automaton. J Theor Biol 4:437–467

    Google Scholar 

  • Thattai M, Van Oudenaarden A (2001) Attenuation of noise in ultrasensitive signaling cascades. Biophys J 82:2943–2950

    Article  Google Scholar 

  • Thattai M, Van Oudenaarden A (2001) Intrisic noise in gene regulatory networks. PNAS 98:8614–8619

    Article  Google Scholar 

  • Thomas R (1978) Logical analysis of systems comprising feedback loops. J Theor Biol 73:631–656

    Article  Google Scholar 

  • Tomas R, d’Ari R (1990) Biological feedbacks. Chapman & Hall/CRC Mathematical & Computational Biology, Boca Raton, FL

    Google Scholar 

  • Turing AM (1952) The chemical basis of morphogenesis. Phil Trans R Soc Lond B 237(641):37–72

    Article  Google Scholar 

  • Tze-Leung T, Mahesci N (2010) Stochasticity and cell fate. Science 327:1142–1145

    Article  Google Scholar 

  • Ullah M, Wolkhenauer O (2011) Stochastic approaches for systems biology. Springer, Berlin

    Book  MATH  Google Scholar 

  • Walther GR, Marée AF, Edelstein-Keshet L, Grieneisen VA (2012) Deterministic versus stochastic cell polarisation through wave-pinning. Bull Math Biol 74:2570

    MATH  MathSciNet  Google Scholar 

  • Wang L, Walker BL, Iannaccone S, Bhatt D, Kennedy PJ, Tse WT (2009) Bistable switches control memory and plasticity in cellular differentiation. PNAS 106(16):6638–6643

    Article  Google Scholar 

  • Wilkinson DJ (2006) Stochastic modelling for systems biology. Chapman & Hall/CRC Mathematical & Computational Biology, Boca Raton, FL

    MATH  Google Scholar 

  • Wio HS, Lindenberg K (2003) Modern challenges in statistical mechanics. In proceedings of the AIP conference vol 658(1)

  • Wio Hs, Deza RR (2013) Noise-induced phenomena: effects of noises based on Tsallis statistics. In d’Onofrio A (ed.) Bounded noises in physics, biology and engineering, Birkauser, Verlag. ISBN 978-1-4614-7348-8

  • Wio H, Toral R (2004) Effect of non-Gaussian noises in a noise induced transition. Phys D 193:161

    Article  MATH  Google Scholar 

  • Xiong W, Ferrell JE Jr (2003) A positive-feedback-based bistable ’memory module’ that governs a cell fate decision. Nature 426:460–465

    Article  Google Scholar 

  • Yamada T, Bork P (2009) Evolution of biomolecular networks: lessons from metabolic and protein interactions. Nat Rev Mol Cell Bio 10:791–803

    Article  Google Scholar 

  • Zhdanov VP (2011) Interplay of bistable kinetics of gene expression during cellular growth. Phys A 390:57

    Article  Google Scholar 

  • Zhdanov VP (2012) Periodic perturbation of genetic oscillations. Chaos, Solitons & Fractals 45:577–587

    Article  Google Scholar 

  • Zhu WQ, Cai GQ (2013) On Bounded stochastic processes. In d’Onofrio A (ed) bounded noises in physics, biology and engineering, Birkauser, Verlag. ISBN 978-1-4614-7348-8

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Acknowledgments

The authors would like to thank the anonymous referees for invaluable comments which improved the paper. AdO wishes to thank the organizers of Wivace (Italian Workshop on Artificial Life and Evolutionary Computation) for the invitation to present these topics to the 2013 edition of the workshop and in this special issue.

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Correspondence to Alberto d’Onofrio.

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The research of A. d’Onofrio and S. de Franciscis has been done in the framework of the Integrated Project “P-medicine—from data sharing and integration via VPH models to personalized medicine” (Project ID: 270089), which is partially funded by the European Commission under the 7th framework program.

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de Franciscis, S., Caravagna, G. & d’Onofrio, A. Bounded noises as a natural tool to model extrinsic fluctuations in biomolecular networks. Nat Comput 13, 297–307 (2014). https://doi.org/10.1007/s11047-014-9424-y

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