Mathematics of the Genome

This work gives a mathematical foundation for bifurcation from a stable equilibrium in the genome. We construct idealized dynamics associated with the genome. For this dynamics, we investigate the two main bifurcations from a stable equilibrium. Finally, we give mathematical proofs of existence and points of bifurcation for the repressilator and the toggle gene circuits.

[1]  J. Monod,et al.  Teleonomic mechanisms in cellular metabolism, growth, and differentiation. , 1961, Cold Spring Harbor symposia on quantitative biology.

[2]  J. Mallet-Paret,et al.  The Poincare-Bendixson theorem for monotone cyclic feedback systems , 1990 .

[3]  Alexey Kuznetsov,et al.  Existence of Limit Cycles in the Repressilator Equations , 2009, Int. J. Bifurc. Chaos.

[4]  M. Elowitz,et al.  A synthetic oscillatory network of transcriptional regulators , 2000, Nature.

[5]  M. Elowitz,et al.  Modeling a synthetic multicellular clock: repressilators coupled by quorum sensing. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[7]  Mark Groudine,et al.  On emerging nuclear order , 2011, The Journal of cell biology.

[8]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[9]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[10]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[11]  L. Glass.  Classification of biological networks by their qualitative dynamics. , 1975, Journal of theoretical biology.

[12]  Konstantin Mischaikow,et al.  Structure of the global attractor of cyclic feedback systems , 1995 .

[13]  James A. Yorke,et al.  Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation , 1982 .

[14]  U. Alon.  An introduction to systems biology : design principles of biological circuits , 2019 .

[15]  J. Alexander,et al.  GLOBAL BIFURCATIONS OF PERIODIC ORBITS. , 1978 .

[16]  J. A. Kuznecov.  Elements of applied bifurcation theory , 1998 .

[17]  G. Sell,et al.  The Hopf Bifurcation and Its Applications , 1976 .

[18]  Leon Glass,et al.  Dynamics in Genetic Networks , 2014, Am. Math. Mon..

[19]  J. Monod,et al.  General Conclusions: Teleonomic Mechanisms in Cellular Metabolism, Growth, and Differentiation , 1978 .

[20]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[21]  S. Kauffman.  Metabolic stability and epigenesis in randomly constructed genetic nets. , 1969, Journal of theoretical biology.

[22]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[23]  B. Dundas,et al.  DIFFERENTIAL TOPOLOGY , 2002 .

[24]  J. Collins,et al.  Construction of a genetic toggle switch in Escherichia coli , 2000, Nature.

[25]  Leon O. Chua,et al.  The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems , 1979 .

[26]  José Carlos Goulart de Siqueira,et al.  Differential Equations , 1919, Nature.

[27]  Felipe Cucker,et al.  Condition - The Geometry of Numerical Algorithms , 2013, Grundlehren der mathematischen Wissenschaften.

[28]  L. Glass,et al.  Stable oscillations in mathematical models of biological control systems , 1978 .

[29]  John J. Tyson,et al.  Existence of periodic solutions for negative feedback cellular control systems , 1977 .