Kac-Rice formulas and the number of solutions of parametrized systems of polynomial equations

Feliu, Elisenda; Sadeghimanesh, AmirHosein

doi:10.1090/mcom/3760

Cited by 3 publications

(16 citation statements)

References 41 publications

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“…But in some cases it is possible to compute the average number of the solutions without solving the system. One such case is introduced in [ 16 ]. Instead of solving the system for many points, it is enough to compute one integral called the Kac-Rice integral.…”

Section: Bisection Algorithms For Rectangular Representationmentioning

confidence: 99%

“…A similar algorithm was presented in [ 16 , Section 2]. The first difference is that the input to the bisection algorithm in [ 16 ] is not necessarily a system with two general number of solutions. The second difference is that the output, there, is not only the two lists and (compare Fig.…”

Section: Bisection Algorithms For Rectangular Representationmentioning

confidence: 99%

“…The second difference is that the output, there, is not only the two lists and (compare Fig. 9 d–f with [ 16 , Figure 1c]).…”

Section: Bisection Algorithms For Rectangular Representationmentioning

confidence: 99%

“…To illustrate this method we use Example 2.1 of [ 16 ], shown here in Fig. 9 a, for which the Kac-Rice integral is already derived.…”

Section: Bisection Algorithms For Rectangular Representationmentioning

confidence: 99%

“…Recently in [ 16 ] a new approach to get a description of the multistationarity region is proposed. In this method one does not need to solve the system of equations to count the number of equilibriums.…”

Section: Introductionmentioning

confidence: 99%

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Polynomial superlevel set representation of the multistationarity region of chemical reaction networks

Sadeghimanesh

England

2022

BMC Bioinformatics

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In this paper we introduce a new representation for the multistationarity region of a reaction network, using polynomial superlevel sets. The advantages of using this polynomial superlevel set representation over the already existing representations (cylindrical algebraic decompositions, numeric sampling, rectangular divisions) is discussed, and algorithms to compute this new representation are provided. The results are given for the general mathematical formalism of a parametric system of equations and so may be applied to other application domains.

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Section: Bisection Algorithms For Rectangular Representationmentioning

confidence: 99%

Section: Bisection Algorithms For Rectangular Representationmentioning

confidence: 99%

“…The second difference is that the output, there, is not only the two lists and (compare Fig. 9 d–f with [ 16 , Figure 1c]).…”

Section: Bisection Algorithms For Rectangular Representationmentioning

confidence: 99%

“…To illustrate this method we use Example 2.1 of [ 16 ], shown here in Fig. 9 a, for which the Kac-Rice integral is already derived.…”

Section: Bisection Algorithms For Rectangular Representationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Polynomial superlevel set representation of the multistationarity region of chemical reaction networks

Sadeghimanesh

England

2022

BMC Bioinformatics

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Absolute concentration robustness and multistationarity in reaction networks: Conditions for coexistence

Kaihnsa,

Nguyen,

Shiu

2024

Eur. J. Appl. Math

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Many reaction networks arising in applications are multistationary, that is, they have the capacity for more than one steady state, while some networks exhibit absolute concentration robustness (ACR), which means that some species concentration is the same at all steady states. Both multistationarity and ACR are significant in biological settings, but only recently has attention focused on the possibility for these properties to coexist. Our main result states that such coexistence in at-most-bimolecular networks (which encompass most networks arising in biology) requires at least three species, five complexes and three reactions. We prove additional bounds on the number of reactions for general networks based on the number of linear conservation laws. Finally, we prove that, outside of a few exceptional cases, ACR is equivalent to non-multistationarity for bimolecular networks that are small (more precisely, one-dimensional or up to two species). Our proofs involve analyses of systems of sparse polynomials, and we also use classical results from chemical reaction network theory.

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Topological descriptors of the parameter region of multistationarity: Deciding upon connectivity

Telek

Feliu²

2023

PLoS Comput Biol

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Switch-like responses arising from bistability have been linked to cell signaling processes and memory. Revealing the shape and properties of the set of parameters that lead to bistability is necessary to understand the underlying biological mechanisms, but is a complex mathematical problem. We present an efficient approach to address a basic topological property of the parameter region of multistationary, namely whether it is connected. The connectivity of this region can be interpreted in terms of the biological mechanisms underlying bistability and the switch-like patterns that the system can create. We provide an algorithm to assert that the parameter region of multistationarity is connected, targeting reaction networks with mass-action kinetics. We show that this is the case for numerous relevant cell signaling motifs, previously described to exhibit bistability. The method relies on linear programming and bypasses the expensive computational cost of direct and generic approaches to study parametric polynomial systems. This characteristic makes it suitable for mass-screening of reaction networks. Although the algorithm can only be used to certify connectivity, we illustrate that the ideas behind the algorithm can be adapted on a case-by-case basis to also decide that the region is not connected. In particular, we show that for a motif displaying a phosphorylation cycle with allosteric enzyme regulation, the region of multistationarity has two distinct connected components, corresponding to two different, but symmetric, biological mechanisms.

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Kac-Rice formulas and the number of solutions of parametrized systems of polynomial equations

Cited by 3 publications

References 41 publications

Polynomial superlevel set representation of the multistationarity region of chemical reaction networks

Polynomial superlevel set representation of the multistationarity region of chemical reaction networks

Absolute concentration robustness and multistationarity in reaction networks: Conditions for coexistence

Topological descriptors of the parameter region of multistationarity: Deciding upon connectivity

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