Determinant Expansions of Signed Matrices and of Certain Jacobians

Helton, J. William; Klep, Igor; Gomez, Raul

doi:10.1137/080718838

Cited by 18 publications

(21 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To prove the converse, first note that we have that v(ρ cv ) = (S ⊥ b ) Tṽ (ρ cv ) verifies v(ρ cv ) T S b = 0 and v(ρ cv ) > 0 for all ρ cv ∈ P cv . This proves the equality and the first inequality in (25). Observe now that for any ρ u ∈ P u , there exists a nonnegative matrix ∆…”

Section: Bimolecular Networksupporting

confidence: 54%

See 1 more Smart Citation

Robust and Structural Ergodicity Analysis and Antithetic Integral Control of a Class of Stochastic Reaction Networks

Briat

Khammash

2018

Preprint

View full text Add to dashboard Cite

Controlling stochastic reactions networks is a challenging problem with important implications in various fields such as systems and synthetic biology. Various regulation motifs have been discovered or posited over the recent years, a very recent one being the so-called Antithetic Integral Control (AIC) motif [3]. Several appealing properties for the AIC motif have been demonstrated for classes of reaction networks that satisfy certain irreducibility, ergodicity and output controllability conditions. Here we address the problem of verifying these conditions for large sets of reaction networks with time-invariant topologies, either from a robust or a structural viewpoint, using three different approaches. The first one adopts a robust viewpoint and relies on the notion of interval matrices. The second one adopts a structural viewpoint and is based on sign properties of matrices. The last one is a direct approach where the parameter dependence is exactly taken into account and can be used to obtain both robust and structural conditions. The obtained results lie in the same spirit as those obtained in [3] where properties of reaction networks are independently characterized in terms of control theoretic concepts, linear programs, and graph-theoretic/algebraic conditions. Alternatively, those conditions can be cast as convex optimization problems that can be checked efficiently using modern optimization methods. Several examples are given for illustration.

show abstract

Section: Bimolecular Networksupporting

confidence: 54%

“…Sign-matrices have also been considered in the context of reaction networks, albeit much more sporadically; see e.g. [1,23,[25][26][27][28]. In this case, again, the conditions obtained in [1,27] can be stated as a very simple linear program that can be shown to be equivalent to some graph theoretical conditions.…”

Section: Introductionmentioning

confidence: 99%

Robust and Structural Ergodicity Analysis and Antithetic Integral Control of a Class of Stochastic Reaction Networks

Briat

Khammash

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…Note that the only situation that is not covered by Theorem 1 is when sign(det(M (x))) takes the value 0 for some x, but never the value (−1) s+1 . The determinant of M (x) is the same as the core determinant in [30,Lemma 3.7]. See also [10,Remark 9.27].…”

Section: Equilibriamentioning

confidence: 99%

Identifying parameter regions for multistationarity

Conradi¹,

Feliu²,

Mincheva³

et al. 2017

PLoS Comput Biol

184

View full text Add to dashboard Cite

Mathematical modelling has become an established tool for studying the dynamics of biological systems. Current applications range from building models that reproduce quantitative data to identifying systems with predefined qualitative features, such as switching behaviour, bistability or oscillations. Mathematically, the latter question amounts to identifying parameter values associated with a given qualitative feature. We introduce a procedure to partition the parameter space of a parameterized system of ordinary differential equations into regions for which the system has a unique or multiple equilibria. The procedure is based on the computation of the Brouwer degree, and it creates a multivariate polynomial with parameter depending coefficients. The signs of the coefficients determine parameter regions with and without multistationarity. A particular strength of the procedure is the avoidance of numerical analysis and parameter sampling. The procedure consists of a number of steps. Each of these steps might be addressed algorithmically using various computer programs and available software, or manually. We demonstrate our procedure on several models of gene transcription and cell signalling, and show that in many cases we obtain a complete partitioning of the parameter space with respect to multistationarity.

show abstract

“…For mass-action systems, conditions that exclude the possibility of degenerate equilibria are described in [14]. For non-mass-action systems such conditions are described in [16,25].…”

Section: The Main Resultsmentioning

confidence: 99%

Graph-Theoretic Analysis of Multistability and Monotonicity for Biochemical Reaction Networks

Crăciun

Pantea²,

Sontag³

2011

Design and Analysis of Biomolecular Circuits

View full text Add to dashboard Cite

Mathematical models of biochemical reaction networks are usually high dimensional, nonlinear, and have many unknown parameters, such as reaction rate constants, or unspecified types of chemical kinetics (such as mass-action, Michaelis-Menten, or Hill kinetics). On the other hand, important properties of these dynamical systems are often determined by the network structure, and do not depend on the unknown parameter values or kinetics. For example, some reaction networks may give rise to multiple equilibria (i.e., they may function as a biochemical switch) while other networks have unique equilibria for any parameter values. Or, some reaction networks may give rise to monotone systems, which renders their dynamics especially stable. We describe how the speciesreaction graph (SR graph) can be used to analyze both multistability and monotonicity of networks.

show abstract

Determinant Expansions of Signed Matrices and of Certain Jacobians

Cited by 18 publications

References 18 publications

Robust and Structural Ergodicity Analysis and Antithetic Integral Control of a Class of Stochastic Reaction Networks

Robust and Structural Ergodicity Analysis and Antithetic Integral Control of a Class of Stochastic Reaction Networks

Identifying parameter regions for multistationarity

Graph-Theoretic Analysis of Multistability and Monotonicity for Biochemical Reaction Networks

Contact Info

Product

Resources

About