Improving the Use of Equational Constraints in Cylindrical Algebraic Decomposition

England, Matthew; Bradford, Russell J.; Davenport, James H.

doi:10.1145/2755996.2756678

Cited by 21 publications

(55 citation statements)

References 38 publications

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“…Let M ∈ R n×s be the exponent matrix associated with the binomials in B, c.f. (4). Since no coefficient of B becomes zero under φ, the exponents of the monomials in B and Φ(B) agree.…”

Section: Binomial Network and Intermediatesmentioning

confidence: 91%

“…This approach fully characterizes whether f is surjective, but CAD is computationally expensive. In particular, the number of cells is doubly exponential in the number of variables and parameters, and depends also on the degree and number of polynomials in the system [4,Theorem 5]. Therefore the use of CAD is impractical already in relatively small examples.…”

Section: Realization Conditionsmentioning

confidence: 99%

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The Multistationarity Structure of Networks with Intermediates and a Binomial Core Network

Sadeghimanesh

Feliu

2019

Bull Math Biol

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This work addresses whether a reaction network, taken with mass-action kinetics, is multistationary, that is, admits more than one positive steady state in some stoichiometric compatibility class. We build on previous work on the effect that removing or adding intermediates has on multistationarity, and also on methods to detect multistationarity for networks with a binomial steady state ideal. In particular, we provide a new determinant criterion to decide whether a network is multistationary, which applies when the network obtained by removing intermediates has a binomial steady state ideal. We apply this method to easily characterize which subsets of complexes are responsible for multistationarity; this is what we call the multistationarity structure of the network. We use our approach to compute the multistationarity structure of the n-site sequential distributive phosphorylation cycle for arbitrary n. is multistationary or not; indeed, it suffices to find the set of inputs of our network and check whether it belongs to the multistationarity structure.The method applies to core networks that are binomial (the ideal generated by the steady state polynomials is binomial). For these networks, a method to decide on multistationarity was introduced in [13], based on the computation of sign vectors. Under some extra assumptions, another method relying on the computation of a symbolic determinant and inspection of the sign of its coefficients is presented in [12] (see Theorem 2.7).The first main result of this paper is Theorem 3.11, where we combine the results in [8,12] into a new determinant criterion for multistationarity that applies to extended networks for which the core network is binomial, even though the original network might not be binomial.The second main result is Theorem 4.2, which removes the technical realization condition in [8] for concluding that an extended network is multistationary provided the core network is. Instead, we require that both the core and the extended networks are binomial (in a compatible way).The third main contribution is Algorithm 4.8, which returns the multistationarity structure of a binomial core network based on the determinant criterion. The algorithm relies on the study of the signs of a polynomial obtained after the computation of the determinant of a symbolic matrix. Our approach is more direct than testing if the extended network is multistationary for all subsets of input complexes. We apply our method to find the multistationarity structure of the n-site distributive sequential phosphorylation cycle for arbitrary n in §4.3. This illustrates how our results allow us to study a family of networks at once.We conclude with an investigation of when the technical realization conditions in [8] are satisfied. In particular, we show that the conditions hold for typical types of intermediates like the Michaelis-Menten mechanism above. The structure of the paper is as follows. In §1 we introduce basic concepts on reaction networks and multistationarity. In §2 we introduce (co...

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“…Let M ∈ R n×s be the exponent matrix associated with the binomials in B, c.f. (4). Since no coefficient of B becomes zero under φ, the exponents of the monomials in B and Φ(B) agree.…”

Section: Binomial Network and Intermediatesmentioning

confidence: 91%

Section: Realization Conditionsmentioning

confidence: 99%

The Multistationarity Structure of Networks with Intermediates and a Binomial Core Network

Sadeghimanesh

Feliu

2019

Bull Math Biol

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“…We are grateful to all the anonymous referees of this paper, and also those of our conference papers at ISSAC 2015 (England et al, 2015) and CASC 2016 , for many helpful comments. We are particularly grateful to the referee who pointed out the mistake in the literature on operator (7) and Dr McCallum for discussing this with us.…”

Section: Acknowledgementsmentioning

confidence: 99%

Cylindrical algebraic decomposition with equational constraints

England

Bradford

Davenport

2020

Journal of Symbolic Computation

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Cylindrical Algebraic Decomposition (CAD) has long been one of the most important algorithms within Symbolic Computation, as a tool to perform quantifier elimination in first order logic over the reals. More recently it is finding prominence in the Satisfiability Checking community as a tool to identify satisfying solutions of problems in nonlinear real arithmetic.The original algorithm produces decompositions according to the signs of polynomials, when what is usually required is a decomposition according to the truth of a formula containing those polynomials. One approach to achieve that coarser (but hopefully cheaper) decomposition is to reduce the polynomials identified in the CAD to reflect a logical structure which reduces the solution space dimension: the presence of Equational Constraints (ECs). This paper may act as a tutorial for the use of CAD with ECs: we describe all necessary background and the current state of the art. In particular, we present recent work on how McCallum's theory of reduced projection may be leveraged to make further savings in the lifting phase: both to the polynomials we lift with and the cells lifted over. We give a new complexity analysis to demonstrate that the double exponent in the worst case complexity bound for CAD reduces in line with the number of ECs. We show that the reduction can apply to both the number of polynomials produced and their degree.

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“…Recent work on using machine learning to make such choice by Huang et al (2014Huang et al ( , 2016) may be applicable. Also, since MAPK problems contain many equational constraints an approach as described by England et al (2015) may be applicable for the higher dimensional CADs required to study more parameters.…”

Section: Final Thoughtsmentioning

confidence: 99%

Identifying the parametric occurrence of multiple steady states for some biological networks

Bradford

Davenport

England

et al. 2020

Journal of Symbolic Computation

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We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem using tools from Symbolic Computation. We describe how progress was made to achieve semi-algebraic descriptions of the multistationarity regions of parameter space, and compare symbolic results to numerical methods.The biological networks studied are models of the mitogen-activated protein kinases (MAPK) network which has already consumed considerable effort using special insights into its structure of corresponding models. Our main example is a model with 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment. The model also imposes positivity conditions on all variables and parameters.We apply combinations of symbolic computation methods designed for mixed equality/inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition, as well as a simplification technique adapted from Gaussian elimination and graph theory.We are able to determine multistationarity of our main example over a 2-dimensional parameter space. We also study a second MAPK model and a symbolic grid sampling technique which can locate such regions in 3-dimensional parameter space.

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Improving the Use of Equational Constraints in Cylindrical Algebraic Decomposition

Cited by 21 publications

References 38 publications

The Multistationarity Structure of Networks with Intermediates and a Binomial Core Network

The Multistationarity Structure of Networks with Intermediates and a Binomial Core Network

Cylindrical algebraic decomposition with equational constraints

Identifying the parametric occurrence of multiple steady states for some biological networks

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