The secant condition for instability in biochemical feedback control—I. The role of cooperativity and saturability

Thron, C. D.

doi:10.1007/bf02460724

Cited by 59 publications

(88 citation statements)

References 24 publications

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“…This means the continuous negative feedback model (17) cannot the generate oscillations. This result corresponds to the secant condition when n ¼2 (Tyson and Othmer, 1978;Thron, 1991;Sontag, 2006).…”

Section: Generalizationsupporting

confidence: 50%

“…As Tyson and Othmer (1978), Thron (1991) and Kurosawa et al (2002) proved, the ODE model with two variables never oscillates, regardless of its function form. We found that the discretized and ultradiscretized models can oscillate depending on the values of the parameters.…”

Section: Discussionmentioning

confidence: 91%

“…This condition is known as the secant condition (Tyson and Othmer, 1978;Thron, 1991;Sontag, 2006). In particular, if n ¼2, any values of α i and β i cannot satisfy the condition (3) because the right-hand side of Eq.…”

Section: Introductionmentioning

confidence: 97%

See 2 more Smart Citations

Discrete and ultradiscrete models for biological rhythms comprising a simple negative feedback loop

Gibo

Ito

2015

Journal of Theoretical Biology

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Section: Generalizationsupporting

confidence: 50%

Section: Discussionmentioning

confidence: 91%

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Discrete and ultradiscrete models for biological rhythms comprising a simple negative feedback loop

Gibo

Ito

2015

Journal of Theoretical Biology

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“…An alternative is that the transitions are very sharp although not of first order and represent a strong negative feedback which, when coupled with a cascade of rate-limiting chemical or mechanical processes, can also undergo an oscillatory feedback instability [4, 21,72,73,121,122]. As noted earlier, such a combination may be operative in the Teorell oscillator for thick membranes.…”

Section: Discussionmentioning

confidence: 99%

Oscillatory Systems Created with Polymer Membranes

Siegel¹

2010

Nonlinear Dynamics With Polymers

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“…A  C R N We now apply the results of the previous sections to cyclic reaction networks, where the end product of a sequence of reactions activates or inhibits the first reaction upstream. To evaluate the local stability properties of cyclic reaction networks with inhibitory feedback, Tyson and Othmer [19] and Thron [20] studied the system matrix:…”

Section: Lemma 5 ( [2]) Consider the Reaction-diffusion Modelmentioning

confidence: 99%

A New Sufficient Condition for Additive D-Stability and Application to Cyclic Reaction-Diffusion Models

Arcak

2009

2009 American Control Conference

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Matrix A is said to be additively D-stable if A − D remains Hurwitz for all nonnegative diagonal matrices D. In reaction-diffusion models, additive D-stability of the matrix describing the reaction dynamics guarantees stability of the homogeneous steady-state, thus ruling out the possibility of diffusion-driven instabilities. We present a new criterion for additive D-stability using the concept of compound matrices. We first give conditions under which the second additive compound matrix has nonnegative offdiagonal entries. We then use this Metzler property of the compound matrix to prove additive D-stability with the help of an additional determinant condition. This result is then applied to investigate stability of cyclic reaction networks in the presence of diffusion. I. IThe concept of diagonal stability and its variants are commonly used in the study of dynamic models in economics, ecology, and control theory, as surveyed by Kaszkurewicz and Bhaya [1]. A square matrix A is said to be diagonally stable if there exists a diagonal matrix S > 0 such that A T S + S A < 0. Two related properties that are less restrictive than diagonal stability but possibly more restrictive than the Hurwitz property of A are multiplicative D-stability, which means that DA is Hurwitz for all diagonal matrices D > 0, and additive D-stability, which means that A − D is Hurwitz for all diagonal D ≥ 0.Additive D-stability is particularly useful for the study of reaction-diffusion systems where the matrix A represents the linearization of the reaction dynamics at a steady-state. Denoting by D the diagonal matrix of diffusion coefficients for each species, Casten and Holland [2] showed that the stability of the reactiondiffusion PDE is determined by the simultaneous stability of the family of matrices A − λ k D, where λ k ≥ 0, k = 1, 2, 3, · · · are the eigenvalues for Laplace's equation with Neumann boundary condition on the given spatial domain. Additive D-stability thus guarantees stability of the spatially homogeneous steady-state and rules out the possibility of diffusion-driven instabilities which constitute the basis of Turing's mechanism for pattern formation [3], [4]. Wang and Li [5] further studied the connection between additive D-stability and reactiondiffusion models, and gave several algebraic sufficient conditions that guarantee either stability or instability in the presence of diffusion.In this paper, we present a new sufficient condition for additive D-stability using the concept of additive compound matrices [6]-[8] defined below. The key property employed in this condition is a special sign structure of matrix A which ensures that its second additive compound matrix A [2] is Metzler; that is, the off-diagonal entries of A [2] are nonnegative. Among the systems that exhibit this sign structure are cyclic reaction networks with negative feedback, where the end product of a sequence of reactions inhibits the first reaction upstream. For this class of networks, [9] established stability of the homogeneous stead...

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The secant condition for instability in biochemical feedback control—I. The role of cooperativity and saturability

Cited by 59 publications

References 24 publications

Discrete and ultradiscrete models for biological rhythms comprising a simple negative feedback loop

Discrete and ultradiscrete models for biological rhythms comprising a simple negative feedback loop

Oscillatory Systems Created with Polymer Membranes

A New Sufficient Condition for Additive D-Stability and Application to Cyclic Reaction-Diffusion Models

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