Abstract
In this paper, we study the stability of an Ordinary Differential Equation (ODE) usually referred to as Cyclic Feedback Loop, which typically models a biological network of \(d\) molecules where each molecule regulates its successor in a cycle (\(A_{1}\rightarrow A_{2}\rightarrow \cdots \rightarrow A_{d-1} \rightarrow A_{d} \rightarrow A_{1}\)). Regulations, which can be either positive or negative, are modelled by increasing or decreasing functions. We make an analysis of this model for a wide range of functions (including affine and Hill functions) by determining the parameters for which bistability and oscillatory behaviours arise. These results encompass previous theoretical studies of gene regulatory networks, which are particular cases of this model.
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Notes
It is in fact sufficient to assume that these two functions are convex, and that at least one of them is strictly convex.
In the Appendix of [12] we show that many other usual sigmoid functions are strictly \(\gamma ^{1/2}-\)convex.
References
Goodwin, B.C., et al.: Temporal organization in cells. A dynamic theory of cellular control processes. In Temporal organization in cells. A dynamic theory of cellular control processes (1963)
Goodwin, B.C.: Oscillatory behavior in enzymatic control processes. Adv. Enzyme Regul. 3, 425–437 (1965)
Angeli, D., Ferrell Jr., J.E.F., Sontag, E.D.: Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems. Proc. Natl. Acad. Sci. 101(7), 1822–1827 (2004)
Gardner, T.S., Cantor, C.R., Collins, J.J.: Construction of a genetic toggle switch in escherichia coli. Nature 403(6767), 339–342 (2000)
Banks, H.T., Mahaffy, J.M.: Stability of cyclic gene models for systems involving repression. J. Theor. Biol. 74(2), 323–334 (1978)
Müller, S., Hofbauer, J., Endler, L., Flamm, C., Widder, S., Schuster, P.: A generalized model of the repressilator. J. Math. Biol. 53, 905–937 (2006)
Selgrade, J.F.: Asymptotic behavior of solutions to single loop positive feedback systems. J. Differ. Equ. 38(1), 80–103 (1980)
Smith, H.: Oscillations and multiple steady states in a cyclic gene model with repression. J. Math. Biol. 25(2), 169–190 (1987)
Smith, H.L.: Systems of ordinary differential equations which generate an order preserving flow. A survey of results. SIAM Rev. 30(1), 87–113 (1988)
Widder, S., Schicho, J., Schuster, P.: Dynamic patterns of gene regulation I: simple two-gene systems. J. Theor. Biol. 246(3), 395–419 (2007)
Cherry, J.L., Adler, F.R.: How to make a biological switch. J. Theor. Biol. 203(2), 117–133 (2000)
Guilberteau, J., Pouchol, C., Duteil, N.P.: Monostability and bistability of biological switches. J. Math. Biol. 83(6–7), 65 (2021)
Perko, L.: Differential Equations and Dynamical Systems, vol. 7. Springer, Berlin (2013)
Mallet-Paret, J., Smith, H.: The Poincaré-Bendixson theorem for monotone cyclic feedback systems. J. Dyn. Differ. Equ. 2(4), 367–421 (1990)
Hirsch, M.W.: Systems of differential equations which are competitive or cooperative: I. Limit sets. SIAM J. Math. Anal. 13(2), 167–179 (1982)
Hirsch, M.W.: The dynamical systems approach to differential equations. Bull. Am. Math. Soc. 11(1), 1–64 (1984)
Hirsch, M.W.: Differential equations and convergence almost everywhere in strongly monotone semiflows. Contemp. Math. 17, 267–285 (1983)
Buse, O., Pérez, R., Kuznetsov, A.: Dynamical properties of the repressilator model. Phys. Rev. E 81(6), 066206 (2010)
Eren Ahsen, M., Özbay, H., Niculescu, S.-I.: Analysis of deterministic cyclic gene regulatory network models with delays. In: SpringerBriefs in Control, Automation and Robotics, pp. 1–92. Springer, Berlin (2015)
Hurewicz, W., Wallman, H.: Dimension Theory (PMS-4), Volume 4, vol. 63. Princeton University Press, Princeton (2015)
Jia, D., Kumar Jolly, M., Harrison, W., Boareto, M., Ben-Jacob, E., Levine, H.: Operating principles of tristable circuits regulating cellular differentiation. Phys. Biol. 14(3), 035007 (2017)
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The author thanks Nastassia Pouradier Duteil and Camille Pouchol for their proofreading and their guidance throughout the writing of this paper.
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Guilberteau, J. Bistability and Oscillatory Behaviours of Cyclic Feedback Loops. Acta Appl Math 188, 6 (2023). https://doi.org/10.1007/s10440-023-00618-x
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DOI: https://doi.org/10.1007/s10440-023-00618-x