Biological Delay Systems and the Mikhailov Criterion of Stability

Foryś, Urszula

doi:10.1142/s0218339004001014

Cited by 42 publications

(46 citation statements)

References 16 publications

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“…Györi and Ladas 1991;Hale 1997;Foryś 2004). Below τ cr , the stationary point is asymptotically stable but it is approached in an oscillatory manner, see Fig.…”

Section: Deterministic Descriptionmentioning

confidence: 95%

Stochastic Models of Gene Expression with Delayed Degradation

et al. 2011

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In many biochemical reactions occurring in living cells, number of various molecules might be low which results in significant stochastic fluctuations. In addition, most reactions are not instantaneous, there exist natural time delays in the evolution of cell states. It is a challenge to develop a systematic and rigorous treatment of stochastic dynamics with time delays and to investigate combined effects of stochasticity and delays in concrete models.We propose a new methodology to deal with time delays in biological systems and apply it to simple models of gene expression with delayed degradation. We show that time delay of protein degradation does not cause oscillations as it was recently argued. It follows from our rigorous analysis that one should look for different mechanisms responsible for oscillations observed in biological experiments.We develop a systematic analytical treatment of stochastic models of time delays. Specifically we take into account that some reactions, for example degradation, are consuming, that is: once molecules start to degrade they cannot be part in other degradation processes. We introduce an auxiliary stochastic process and calculate analytically the variance and the autocorrelation function of the number of protein molecules in stationary states in basic models of delayed protein degradation.

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“…Györi and Ladas 1991;Hale 1997;Foryś 2004). Below τ cr , the stationary point is asymptotically stable but it is approached in an oscillatory manner, see Fig.…”

Section: Deterministic Descriptionmentioning

confidence: 95%

Stochastic Models of Gene Expression with Delayed Degradation

et al. 2011

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“…For the case with delay we use the Mikhailov Criterion (see [8]). To get stability we require that the change of argument of the vector W(iω) while ω changes from 0 to +∞ is equal to kπ/2, which is impossible because W(0) < 0 and argument of W(iω) tends to kπ/2 as ω → +∞.…”

Section: Definition 24 (Definition 21 In [14]) Let H : D → D Be a Cmentioning

confidence: 99%

“…Let denote it byp. Now, we introduce the following change of variables 8) wherep is a unique positive solution to (3.7). With the change of variables (3.8), and allowing time delay to be distributed one, the system (3.6) readṡ…”

Section: Hes1 Gene Expression Modelmentioning

confidence: 99%

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General model of a cascade of reactions with time delays: Global stability analysis

Bodnar

2015

Journal of Differential Equations

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“…It should be noted that if the critical point is unstable in the case without delay, then it is also unstable for every discrete non-zero delays (it can be shown using, for example, Mikhailov criterion; for details, see [17,23,28]). …”

Section: The Function E(t) Decreases At the Beginning But The Same Imentioning

confidence: 99%

Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models

Foryś

Kheifetz²,

Kogan

2005

Mathematical Biosciences and Engineering

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Abstract. We perform critical-point analysis for three-variable systems that represent essential processes of the growth of the angiogenic tumor, namely, tumor growth, vascularization, and generation of angiogenic factor (protein) as a function of effective vessel density. Two models that describe tumor growth depending on vascular mass and regulation of new vessel formation through a key angiogenic factor are explored. The first model is formulated in terms of ODEs, while the second assumes delays in this regulation, thus leading to a system of DDEs. In both models, the only nontrivial critical point is always unstable, while one of the trivial critical points is always stable. The models predict unlimited growth, if the initial condition is close enough to the nontrivial critical point, and this growth may be characterized by oscillations in tumor and vascular mass. We suggest that angiogenesis per se does not suffice for explaining the observed stabilization of vascular tumor size.

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Biological Delay Systems and the Mikhailov Criterion of Stability

Cited by 42 publications

References 16 publications

Stochastic Models of Gene Expression with Delayed Degradation

Stochastic Models of Gene Expression with Delayed Degradation

General model of a cascade of reactions with time delays: Global stability analysis

Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models

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