Symmetry and Stochastic Gene Regulation

Ramos, Alexandre F.; Hornos, José Eduardo Martinho

doi:10.1103/physrevlett.99.108103

Cited by 21 publications

(40 citation statements)

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“…This is, of course, in agreement with the symmetry analysis performed in Ref. [19] and in light of the adiabatic analysis done formerly [17]. We emphasize that our model gives an effective phenomenological description of the combined transcription-translation processes, and its parameters are selected to reflect globally the results of the chemical reactions responsible for the deactivation of the regulatory protein and the gene state control.…”

mentioning

confidence: 56%

“…Exact solutions for the stationary regime of the single gene spin-boson model were already * alex.ramos@usp.br † innocentini@ursa.ifsc.usp.br ‡ hornos@ifsc.usp.br obtained [17,18]. The underlying symmetries responsible for the integrability of this model were identified [19,20], and several applications of the model have been discussed in the literature [21,22]. In this Brief Report we solve completely the model exhibiting analytical solutions for the time-dependent stochastic process in terms of Heun functions [23][24][25][26].…”

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confidence: 98%

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Exact time-dependent solutions for a self-regulating gene

2011

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The exact time-dependent solution for the stochastic equations governing the behavior of a binary self-regulating gene is presented. Using the generating function technique to rephrase the master equations in terms of partial differential equations, we show that the model is totally integrable and the analytical solutions are the celebrated confluent Heun functions. Self-regulation plays a major role in the control of gene expression, and it is remarkable that such a microscopic model is completely integrable in terms of well-known complex functions.

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confidence: 56%

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confidence: 98%

Exact time-dependent solutions for a self-regulating gene

2011

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“…[14][15][16]. The corrections are presented in Table S1 (Supplemental Material) on a clear and condensed fashion.…”

Section: Discussionmentioning

confidence: 99%

“…7 -8, and the master equations utilized in the works of Hornos and his coworkers [8,[13][14][15][16]. We point out differences between the two sets of equations that are either because of the extra reaction in our model (Eq.…”

Section: Master Equationsmentioning

confidence: 99%

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Self-regulating genes. Exact steady state solution by using Poisson representation

Sugár

Simon

2014

Open Physics

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Abstract:Systems biology studies the structure and behavior of complex gene regulatory networks. One of its aims is to develop a quantitative understanding of the modular components that constitute such networks. The self-regulating gene is a type of auto regulatory genetic modules which appears in over 40% of known transcription factors in E. coli. In this work, using the technique of Poisson Representation, we are able to provide exact steady state solutions for this feedback model. By using the methods of synthetic biology (P.E.M. Purnick and Weiss, R., Nature Reviews, Molecular Cell Biology, 2009, 10: 410-422) one can build the system itself from modules like this.

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A model of gene expression based on random dynamical systems reveals modularity properties of gene regulatory networks

Antoneli

Ferreira

Briones

2016

Mathematical Biosciences

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Here we propose a new approach to modeling gene expression based on the theory of random dynamical systems (RDS) that provides a general coupling prescription between the nodes of any given regulatory network given the dynamics of each node is modeled by a RDS. The main virtues of this approach are the following: (i) it provides a natural way to obtain arbitrarily large networks by coupling together simple basic pieces, thus revealing the modularity of regulatory networks; (ii) the assumptions about the stochastic processes used in the modeling are fairly general, in the sense that the only requirement is stationarity; (iii) there is a well developed mathematical theory, which is a blend of smooth dynamical systems theory, ergodic theory and stochastic analysis that allows one to extract relevant dynamical and statistical information without solving the system; (iv) one may obtain the classical rate equations form the corresponding stochastic version by averaging the dynamic random variables (small noise limit). It is important to emphasize that unlike the deterministic case, where coupling two equations is a trivial matter, coupling two RDS is non-trivial, specially in our case, where the coupling is performed between a state variable of one gene and the switching stochastic process of another gene and, hence, it is not a priori true that the resulting coupled system will satisfy the definition of a random dynamical system. We shall provide the necessary arguments that ensure that our coupling prescription does indeed furnish a coupled regulatory network of random dynamical systems. Finally, the fact that classical rate equations are the small noise limit of our stochastic model ensures that any validation or prediction made on the basis of the classical theory is also a validation or prediction of our model. We illustrate our framework with some simple examples of single-gene system and network motifs.

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Symmetry and Stochastic Gene Regulation

Cited by 21 publications

References 20 publications

Exact time-dependent solutions for a self-regulating gene

Exact time-dependent solutions for a self-regulating gene

Self-regulating genes. Exact steady state solution by using Poisson representation

A model of gene expression based on random dynamical systems reveals modularity properties of gene regulatory networks

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