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Polynomial Superlevel Set Representation of the Multistationarity Region of Chemical Reaction Networks
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Abstract
In this work a new representation of the multistationarity region of reaction networks is introduced using the polynomial superlevel sets. The advantages of using the polynomial superlevel set representation to the former existing representations such as CAD, the finite and the grid representations are discussed. And finally the algorithms to compute this new representation are provided. The results are given in a general mathematical formalism of parametric system of equations and therefore can be used in other applied areas.
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Motivated by problems of uncertainty propagation and robust estimation we are interested in computing a polynomial sublevel set of fixed degree and minimum volume that contains a given semialgebraic set K. At this level of generality this problem is not tractable, even though it becomes convex e.g. when restricted to nonnegative homogeneous polynomials. Our contribution is to describe and justify a tractable L1-norm or trace heuristic for this problem, relying upon hierarchies of linear matrix inequality (LMI) relaxations when K is semialgebraic, and simplifying to linear constraints when K is a collection of samples, a discrete union of points.
We introduce CoNtRol, a web based framework for analysis of chemical reaction networks (CRNs). It is designed to be both extensible and simple to use, complementing existing CRN-related tools. CoNtRol currently implements a number of necessary and/or sufficient structural tests for multiple equilibria, stable periodic orbits, convergence to equilibria and persistence, with the potential for incorporation of further tests.
Reference implementation: reaction-networks.net/control/. Source code and binaries, released under the GPLv3: reaction-networks.net/control/download/. Documentation: reaction-networks.net/wiki/CoNtRol.
pete.donnell@port.ac.uk.
Multisite phosphorylation networks are encountered in many intracellular
processes like signal transduction, cell-cycle control or nuclear signal
integration. In {\em Wang and Sontag, 2008}, the authors study the number of
steady states in general $n$-site sequential distributive phosphorylation and
show that there are at most $2n-1$ steady states. They furthermore conjecture
that, for odd $n$, there are at most $n$ and that, for even $n$, there are at
most $n+1$ steady states. Building on earlier work in {\em Holstein et.al.,
2013}, we present a scalar determining equation for multistationarity which
will lead to $5$ steady states for a $3$-site and to $7$ steady states for a
$4$-site phosphorylation system and hence to counterexamples to the conjecture
of Wang and Sontag. We conclude with a brief biological interpretation of the
inherent geometric properties of multistationarity.
We give necessary and sufficient conditions in terms of sign vectors for the
injectivity of families of polynomials maps with arbitrary real exponents
defined on the positive orthant. Our work relates and extends existing
injectivity conditions expressed in terms of Jacobian matrices and
determinants. In the context of chemical reaction networks with power-law
kinetics, our results can be used to preclude as well as to guarantee multiple
positive steady states. In the context of real algebraic geometry, our results
reveal the first partial multivariate generalization of the classical
Descartes' rule, which bounds the number of positive real roots of a univariate
real polynomial in terms of the number of sign variations of its coefficients.
A new Maple package for solving parametric systems of polynomial equations and inequal-ities is described. The main idea for solving such a system is as follows. The parameter space R d is divided into two parts: the discriminant variety W and its complement R d \W . The dis-criminant variety is a generalization of the well-known discriminant of a univariate polynomial and contains all those parameter values leading to non-generic solutions of the system. The complement R d \W can be expressed as a finite union of open cells such that the number of real solutions of the input system is constant on each cell. In this way, all parameter values leading to generic solutions of the system can be described systematically. The underlying techniques used are Gröbner bases, polynomial real root finding, and cylindrical algebraic decomposi-tion. This package offers a friendly interface for scientists and engineers to solve parametric problems, as illustrated by an example from control theory.
Checking non-negativity of polynomials using sum-of-squares has recently been popularized and found many applications in control. Although the method is based on convex programming, the optimization problems rapidly grow and result in huge semidefinite programs. Additionally, they often become increasingly ill-conditioned. To alleviate these problems, it is important to exploit properties of the analyzed polynomial, and post-process the obtained solution. This technical note describes how the sum-of-squares module in the MATLAB toolbox YALMIP handles these issues.
Recent years have seen a dramatic increase in the use of mathematical modeling to gain insight into gene regulatory network behavior across many different organisms. In particular, there has been considerable interest in using mathematical tools to understand how multistable regulatory networks may contribute to developmental processes such as cell fate determination. Indeed, such a network may subserve the formation of unicellular leaf hairs (trichomes) in the model plant Arabidopsis thaliana.
In order to investigate the capacity of small gene regulatory networks to generate multiple equilibria, we present a chemical reaction network (CRN)-based modeling formalism and describe a number of methods for CRN analysis in a parameter-free context. These methods are compared and applied to a full set of one-component subnetworks, as well as a large random sample from 40,680 similarly constructed two-component subnetworks. We find that positive feedback and cooperativity mediated by transcription factor (TF) dimerization is a requirement for one-component subnetwork bistability. For subnetworks with two components, the presence of these processes increases the probability that a randomly sampled subnetwork will exhibit multiple equilibria, although we find several examples of bistable two-component subnetworks that do not involve cooperative TF-promoter binding. In the specific case of epidermal differentiation in Arabidopsis, dimerization of the GL3-GL1 complex and cooperative sequential binding of GL3-GL1 to the CPC promoter are each independently sufficient for bistability.
Computational methods utilizing CRN-specific theorems to rule out bistability in small gene regulatory networks are far superior to techniques generally applicable to deterministic ODE systems. Using these methods to conduct an unbiased survey of parameter-free deterministic models of small networks, and the Arabidopsis epidermal cell differentiation subnetwork in particular, we illustrate how future experimental research may be guided by network structure analysis.
Gene regulatory networks have an important role in every process of life, including cell differentiation, metabolism, the cell cycle and signal transduction. By understanding the dynamics of these networks we can shed light on the mechanisms of diseases that occur when these cellular processes are dysregulated. Accurate prediction of the behaviour of regulatory networks will also speed up biotechnological projects, as such predictions are quicker and cheaper than lab experiments. Computational methods, both for supporting the development of network models and for the analysis of their functionality, have already proved to be a valuable research tool.
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.
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.
We study the existence of linear and nonlinear conservation laws in biochemical reaction networks with mass-action kinetics. It is straightforward to compute the linear conservation laws as they are related to the left null-space of the stoichiometry matrix. The nonlinear conservation laws are difficult to identify and have rarely been considered in the context of mass-action reaction networks. Here, using the Darboux theory of integrability, we provide necessary structural (i.e., parameter-independent) conditions on a reaction network to guarantee the existence of nonlinear conservation laws of a certain type. We give necessary and sufficient structural conditions for the existence of exponential factors with linear exponents and univariate linear Darboux polynomials. This allows us to conclude that nonlinear first integrals only exist under the same structural condition (as in the case of the Lotka-Volterra system). We finally show that the existence of such a first integral generally implies that the system is persistent and has stable steady states. We illustrate our results by examples.
The following problem is considered: Given a deficiency one network, determine whether there exist rate constants for it such that the corresponding isothermal mass action differential equations admit multiple positive steady states. A procedure is given to make this determination for any deficiency one network, no matter how intricate, so long as it satisfies certain weak structural conditions. When there do exist rate constants that give rise to multiple steady states, such rate constants can in fact be exhibited. If multiple steady states are observed in a laboratory reactor with poorly understood chemistry, the theory provides sensitive means to screen candidates for the operative chemical mechanism. Even when measurements of the steady state compositions are fragmentary, the theory will sometimes indicate that a candidate network which does admit multiple steady states will nevertheless be unable to account for the particular measurements made. A catalytic CFSTR is considered as an example.
The dynamics of complex isothermal reactors are studied in general terms with special focus on connections between reaction network structure and the capacity of the corresponding differential equations to admit unstable behavior. As in some earlier work, the principal results rely on a classification of reaction networks by means of an easily computed non-negative integer index called the deficiency. This index often provides nontrivial information about the kind of dynamics that can be expected. Part of the previously reported Deficiency Zero Theorem is substantially generalized by the Deficiency One Theorem. The foundation is laid for a companion article containing a theory of multiple steady states generated by reaction networks of deficiency one.
Typescript (photocopy). Thesis (Ph. D.)--University of Rochester. Dept. of Chemical Engineering, 1998. Includes vita and abstract. Includes bibliographical references (leaves 293-295).
SeDuMi is an add-on for MATLAB, that lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large scale optimization problems are solved efficiently, by exploiting sparsity. This paper describes how to work with this toolbox. KEY WORDS: Symmetric cone, semidefinite programming, second order cone programming, self--duality, MATLAB, SeDuMi. SeDuMi stands for Self-Dual-Minimization: it implements the self-dual embedding technique for optimization over self-dual homogeneous cones. The self-dual embedding technique as proposed Ye, Todd and Mizuno [23], essentially makes it possible to solve certain optimization problems in a single phase, leading either to an optimal solution, or a certificate of infeasibility. Optimization over self--dual homogeneous cones, or more concisely, optimization over symmetric cones, was first studied by Nesterov and Todd [16], and is currently an...
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- Amirhosein Sadeghimanesh
AmirHosein Sadeghimanesh. Algebraic tools in the study of Multistationarity of Chemical Reaction Networks. PhD thesis, University of Copenhaegn, 10 2018.