On some multidimensional models of gene network functioning

Akinshin, Andrey; Golubyatnikov, V. P.; Golubyatnikov, I. V.

doi:10.1134/s1990478913030022

Cited by 5 publications

(7 citation statements)

References 3 publications

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“…This article continues the study started in [1][2][3] of the phase portraits of the nonlinear dynamical systems of a special kind:…”

Section: Introductionsupporting

confidence: 55%

“…The following is proved like its analogs in [2,9,10]: (6), the trajectories of systems (4) and (5) are rectilinear and their extensions intersect at one point U B .…”

Section: Construction Of Diagramsmentioning

confidence: 99%

“…The article [2] described the construction of a cycle C of system (4) traversing diagram (7) and symmetric with respect to the cyclic change of variables ζ and established that the phase portrait of (4) contains the two stable equilibrium points:…”

Section: Construction Of Diagramsmentioning

confidence: 99%

“…Construct a three-dimensional piecewise linear hypersurface M 3 ⊂ Q 4 separating the attraction domains of U 0 ∈ {0101} and U 1 ∈ {1010} and containing the cycle C of the system constructed in [2] and the two-dimensional invariant surface M 2 ⊂ Π 4 "spanned" by this cycle. Such a separatrix must traverse E. For the convenience of computations, introduce the coordinate system (X 1 , X 2 , X 3 , X 4 ) in which X i = x i − 1 and E is the origin.…”

Section: Geometry Of the Phase Portrait Of (4) Construction Of A Sepmentioning

confidence: 99%

“…Since in {1100} all trajectories of (4) are described by homotheties with center Z 1 = (A − 1, −1, −1, A − 1) (see [2]), the coordinates of Z 1 must satisfy (10), and thus β 3 = (A − 1) · (α 1 + α 3 ) − α 2 . On the face F 1 = {1100}∩{1101}, where X 4 = 0, the plane P 3 1 cuts out the plane P 2 1 with the equation…”

Section: Geometry Of the Phase Portrait Of (4) Construction Of A Sepmentioning

confidence: 99%

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On two classes of nonlinear dynamical systems: The four-dimensional case

Ayupova

Golubyatnikov

2015

Sib Math J

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We consider two four-dimensional piecewise linear dynamical systems of chemical kinetics. For one of them, we give an explicit construction of a hypersurface that separates the attraction basins of two stable equilibrium points and contains an unstable cycle of this system. For the other system, we prove the existence of a trajectory not contained in the attraction basin of the stable cycle of this system described earlier by Glass and Pasternack. The homotopy types of the phase portraits of these two systems are compared.

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“…This article continues the study started in [1][2][3] of the phase portraits of the nonlinear dynamical systems of a special kind:…”

Section: Introductionsupporting

confidence: 55%

“…The following is proved like its analogs in [2,9,10]: (6), the trajectories of systems (4) and (5) are rectilinear and their extensions intersect at one point U B .…”

Section: Construction Of Diagramsmentioning

confidence: 99%

Section: Construction Of Diagramsmentioning

confidence: 99%

Section: Geometry Of the Phase Portrait Of (4) Construction Of A Sepmentioning

confidence: 99%

Section: Geometry Of the Phase Portrait Of (4) Construction Of A Sepmentioning

confidence: 99%

See 3 more Smart Citations

On two classes of nonlinear dynamical systems: The four-dimensional case

Ayupova

Golubyatnikov

2015

Sib Math J

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On a Cycle in a $$5$$-Dimensional Circular Gene Network Model

Ayupova

Golubyatnikov

2021

J. Appl. Ind. Math.

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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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On some multidimensional models of gene network functioning

Cited by 5 publications

References 3 publications

On two classes of nonlinear dynamical systems: The four-dimensional case

On two classes of nonlinear dynamical systems: The four-dimensional case

On a Cycle in a $$5$$-Dimensional Circular Gene Network Model

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

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