On the existence and stability of cycles in five-dimensional models of gene networks

Golubyatnikov, V. P.; Golubyatnikov, I. V.; Лихошвай, В. А.

doi:10.1134/s199542391004004x

Cited by 8 publications

(4 citation statements)

References 5 publications

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“…where L i are the step functions: L i ( x )= β i α i for 0≤ x ≤1; L i ( x )=0 for x >1. Similar systems were studied in [ 41 – 43 ] in analysis of circular gene networks, i =2,3,…, n .…”

Section: Resultsmentioning

confidence: 98%

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Limit cycles in models of circular gene networks regulated by negative feedback loops

2020

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Background The regulatory feedback loops that present in structural and functional organization of molecular-genetic systems and the phenomenon of the regulatory signal delay, a time period between the moment of signal reception and its implementation, provide natural conditions for complicated dynamic regimes in these systems. The delay phenomenon at the intracellular level is a consequence of the matrix principle of data transmission, implemented through the rather complex processes of transcription and translation.However, the rules of the influence of system structure on system dynamics are not clearly understood. Knowledge of these rules is particularly important for construction of synthetic gene networks with predetermined properties. Results We study dynamical properties of models of simplest circular gene networks regulated by negative feedback mechanisms. We have shown existence and stability of oscillating trajectories (cycles) in these models. Two algorithms of construction and localization of these cycles have been proposed. For one of these models, we have solved an inverse problem of parameters identification. Conclusions The modeling results demonstrate that non-stationary dynamics in the models of circular gene networks with negative feedback loops is achieved by a high degree of non-linearity of the mechanism of the autorepressor influence on its own expression, by the presence of regulatory signal delay, the value of which must exceed a certain critical value, and transcription/translation should be initiated from a sufficiently strong promoter/Shine-Dalgarno site. We believe that the identified patterns are key elements of the oscillating construction design.

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Section: Resultsmentioning

confidence: 98%

“…For its piece-wise version, the sufficient and necessary conditions of existence of a limit cycle were obtained in [42]. For similar 5-dimensional system, sufficient condition of existence of a cycle was obtained in [43], where some results on its stability were presented as well.…”

Section: Limit Cycles Of Gene Network Models For N >mentioning

confidence: 99%

Limit cycles in models of circular gene networks regulated by negative feedback loops

2020

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“…In the even-dimensional dynamical systems, the sum (1) of these indices with the above-indicated behavior of trajectories on the boundary of Q equals +1. Similar arguments concerning the topological characteristics of the vector fields were used previously for modeling some other gene networks [15,16].…”

Section: Construction Of An Invariant Region Of the Systemmentioning

confidence: 94%

Mathematical modeling of the first phase of morphogenesis of mechanoreceptors in D. melanogaster

Бухарина

Golubyatnikov

et al. 2012

J. Appl. Ind. Math.

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The mechanoreceptors (in particular, macrochaetes) of drosophila pass three stages in its development (morphogenesis) whose genetic support is described in terms of gene networks.The key object of the gene networks of macrochaete morphogenesis is the achaete-scute (AS-C) complex of genes. Each mechanoreceptor develops from one parent cell distinguished by high concentration of the AS-C protein. The activity of this complex, which ensures the protein concentration critical for initiating morphogenesis, is determined by the central regulatory contour which includes a system of interactions among certain objects of the networks (genes and their products).

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“…In the Sobolev Institute of Mathematics SB RAS is studied the qualitative theory of dynamical systems describing various gene networks that are regulated by feedbacks. Golubyatnikov and his colleagues have studied in their works (Gaidov, Golubyatnikov, 2007;Golubyatnikov et al, 2010;Akinshin, Golubyatnikov, 2012;Golubyatnikov, Kazantsev, 2016; Golu-…”

Section: Application Of Gene Network In the Modeling Of Oscillatory Systemsmentioning

confidence: 99%

The finding and researching algorithm for potentially oscillating enzymatic systems

et al. 2021

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Many processes in living organisms are subject to periodic oscillations at different hierarchical levels of their organization: from molecular-genetic to population and ecological. Oscillatory processes are responsible for cell cycles in both prokaryotes and eukaryotes, for circadian rhythms, for synchronous coupling of respiration with cardiac contractions, etc. Fluctuations in the numbers of organisms in natural populations can be caused by the populations’ own properties, their age structure, and ecological relationships with other species. Along with experimental approaches, mathematical and computer modeling is widely used to study oscillating biological systems. This paper presents classical mathematical models that describe oscillatory behavior in biological systems. Methods for the search for oscillatory molecular-genetic systems are presented by the example of their special case – oscillatory enzymatic systems. Factors influencing the cyclic dynamics in living systems, typical not only of the molecular-genetic level, but of higher levels of organization as well, are considered. Application of different ways to describe gene networks for modeling oscillatory molecular-genetic systems is considered, where the most important factor for the emergence of cyclic behavior is the presence of feedback. Techniques for finding potentially oscillatory enzymatic systems are presented. Using the method described in the article, we present and analyze, in a step-by-step manner, first the structural models (graphs) of gene networks and then the reconstruction of the mathematical models and computational experiments with them. Structural models are ideally suited for the tasks of an automatic search for potential oscillating contours (linked subgraphs), whose structure can correspond to the mathematical model of the molecular-genetic system that demonstrates oscillatory behavior in dynamics. At the same time, it is the numerical study of mathematical models for the selected contours that makes it possible to confirm the presence of stable limit cycles in them. As an example of application of the technology, a network of 300 metabolic reactions of the bacterium Escherichia coli was analyzed using mathematical and computer modeling tools. In particular, oscillatory behavior was shown for a loop whose reactions are part of the tryptophan biosynthesis pathway.

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On the existence and stability of cycles in five-dimensional models of gene networks

Cited by 8 publications

References 5 publications

Limit cycles in models of circular gene networks regulated by negative feedback loops

Limit cycles in models of circular gene networks regulated by negative feedback loops

Mathematical modeling of the first phase of morphogenesis of mechanoreceptors in D. melanogaster

The finding and researching algorithm for potentially oscillating enzymatic systems

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