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Identifying parameter regions for multistationarity

Carsten Conradi, Elisenda Feliu, Maya Mincheva, Carsten Wiuf

arXiv:1608.03993v2q-bio.MNmath.AGmath.DS

TL;DR

Identifying parameter regions with multistationarity is difficult, despite its relevance to biological decision-making and synthetic-biology design. The paper combines Brouwer degree theory and algebraic geometry to relate equilibrium multiplicity to a parameter-dependent polynomial. Under the stated assumptions, multiple equilibria exist in some compatibility classes if and only if that polynomial attains a specified sign.

  • Problem

    Identifying parameter regions associated with multistationarity is a difficult problem, and the property is linked to cellular decision-making and synthetic-biology design.

    Source: Introduction · p. 1

  • Method

    The procedure combines Brouwer degree theory and algebraic geometry, using a parameter-dependent polynomial to identify regions with unique or multiple equilibria.

    Source: Introduction · p. 2, Discussion · p. 19

  • Results

    Under Corollary 2's assumptions, some stoichiometric compatibility classes have at least two equilibria if and only if a specified multivariate polynomial can attain a particular sign.

    Source: Discussion · p. 19

  • Takeaways & Limitations

    For some reaction networks, the procedure can be formulated as an algorithm to help analyze competing models for qualitative features such as multistationarity.

    Source: Discussion · p. 20

    For the two-site phosphorylation cycle with Michaelis–Menten kinetics, a positive algebraic parameterization does not exist, so the approach cannot find its multistationarity parameter conditions.

    Source: Discussion · p. 19

Abstract

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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.

Introduction

The paper introduces a procedure for identifying parameter regions with unique or multiple equilibria in biological ODE models. It combines theoretical conditions with a parameter-dependent polynomial and applies them to reaction-network examples.

Introduction

  • Multistationarity is linked to cellular decision-making and memory-like responses, and can be a design objective in synthetic biology.The paper frames identifying the parameter values that support such qualitative properties as a useful goal for biological modeling.Source: Introduction · p. 1
  • Delimiting parameter regions is difficult: common approaches use numerical investigation and sampling, while cylindrical algebraic decomposition scales poorly for large models.Existing mathematical methods can decide whether multistationarity exists or identify specific parameter values, and some are available as software tools.Source: Introduction · p. 2
  • The theoretical results relate the number of equilibria to the signs of a parameter-dependent polynomial in species concentrations.The paper presents two results in reaction network theory, generalizing ideas in.Source: Introduction · p. 2
  • The procedure applies to dissipative reaction networks without boundary equilibria in interior stoichiometric compatibility classes, and uses positive parameterizations of positive equilibria.Its conditions can be checked manually or algorithmically; computational criteria are sufficient, and the algorithm may terminate indecisively when criteria are unmet.Source: Introduction · p. 2
  • For two protein-modification motifs, the procedure identifies mono- and multistationarity conditions involving kinase and phosphatase catalytic and Michaelis–Menten constants.Multiple equilibria also require suitable total concentrations of kinase, phosphatase, and substrate; for the second motif, the number of equilibria remains undecided when b1(κ) ≥ 0 and b2(κ) < 0.Source: Introduction · p. 2, Table 1 · p. 3

Results

The Results section uses Brouwer-degree determinant-sign tests and positive-equilibrium parameterizations to identify parameter regions with unique or multiple positive equilibria under stated network assumptions. It applies the procedure to a running example, including its multistationarity condition, while noting that computational checks may be inconclusive.

Theory

  • Under dissipativity, absence of boundary equilibria, and the stated kinetics assumption, Theorem 1 uses determinant-sign conditions to establish either one positive equilibrium or at least two in a compatibility class.In the multiple-equilibria case, at least one equilibrium is non-degenerate; if all are non-degenerate, there are at least three and always an odd number. The proof relates the determinant to Brouwer degree; the theorem does not cover the stated case where the determinant takes zero but never (−1)^(s+1).Source: Theory · p. 6
  • When the Theorem 1 assumptions hold and sign(det(M(x))) = (−1)^s throughout the positive orthant, each stoichiometric compatibility class has exactly one positive non-degenerate equilibrium.Source: Theory · p. 7
  • With a positive parameterization of equilibria, Corollary 2 applies determinant-sign tests to identify uniqueness or multiplicity in the compatibility class associated with the parameterized equilibrium.It retains the dissipativity and no-boundary-equilibria assumptions; in the multiple case, at least one equilibrium is non-degenerate, and if all are non-degenerate, there are at least three and always an odd number. Unlike Theorem 1, the compatibility class is not fixed in advance.Source: Theory · pp. 7–8
  • In the running example, Corollary 1 cannot be applied, but Corollary 2 establishes a unique positive non-degenerate equilibrium in every class with c > 0 for all reaction-rate constants.The determinant is positive for some positive concentrations, while its parameterized form has sign −1 for every positive free variable; the parameterization is essential to this conclusion.Source: Theory · p. 8
  • For κ3 > κ1, some stoichiometric compatibility class has at least two positive equilibria; for κ3 ≤ κ1, each class with a nonempty positive part has a unique positive non-degenerate equilibrium.Source: Theory · p. 9

Procedure for finding parameter regions for multistationarity

Procedure (Identification of parameter regions for multistationarity)

Applications to selected examples

The examples show how the procedure can distinguish unique from multiple equilibria and, for the hybrid histidine kinase, completely partition parameter space by a rate-constant inequality. Other cases illustrate the procedure’s scope, including a gene transcription network that requires manual checks and PTM networks whose conditions are obtained automatically.

Applications to selected examples

Two-component system

  • The two-component system has exactly one non-degenerate equilibrium in every stoichiometric compatibility class with non-empty positive part.The example’s determinant has constant sign, so the algorithm succeeds at Step 5 without Steps 6 and 7.Source: Two-component system · pp. 12, 14

Hybrid histidine kinase

  • Multistationarity in the hybrid histidine kinase is completely characterized by κ3 > κ1.When κ1 ≥ κ3, the sign test establishes one non-degenerate equilibrium per compatibility class; for κ1 < κ3, the Newton-polytope step establishes at least two positive equilibria for some class.Source: Hybrid histidine kinase · pp. 15–16

Gene transcription network

  • For the gene transcription network, the procedure terminates inconclusively, but manual verification establishes dissipativity and at least two positive equilibria for some class for all κi > 0.The network is multistationary for all parameters; the manual approach resumes after an inconclusive algorithmic step.Source: Gene transcription network · pp. 16–17

Special classes of networks

  • For two PTM network motifs, the algorithm finds conditions distinguishing mono- from multistationarity without manual intervention; an additional example is monostationary.The two motifs are conservative, admit positive parameterizations, and have strongly connected underlying substrate networks.Source: Special classes of networks · p. 18

Computational issues

Some procedure steps are computationally demanding, with complexity depending on the network and symbolic calculations sometimes exceeding software limits. The authors discuss checking strategies, tools, and ways to use known network structure to guide the analysis.

Computational issues

  • Some procedure steps are computationally demanding, and symbolic software may be unable to process some calculations.The computation may be feasible for large networks with sufficient time, but Mathematica and Maple have inherent limits.Source: Computational issues · p. 18
  • Efficient checks exist for conservativity and strong endotacticity, but the authors know of no systematic way to check Proposition 1.The former checks can use linear algebra or mixed-integer linear programming.Source: Computational issues · p. 18
  • Checking minimal siphons and non-interacting sets can require exponential time, and minimal siphons may be exponentially numerous.Removing intermediates and catalysts can often substantially reduce the complexity of finding minimal siphons.Source: Computational issues · p. 18
  • The symbolic determinant calculation often fails on common laptops for networks with more than 20 variables, with performance depending on matrix sparsity.The authors examined determinant-expansion strategies using conservation-relation rows.Source: Computational issues · p. 19
  • Existing symbolic software can find Newton-polytope vertices, and known network properties can guide the procedure to avoid unnecessary searches.Polymake and Maple are given as examples of software for the vertex calculation.Source: Computational issues · p. 19

Discussion

The procedure combines Brouwer degree theory and algebraic geometry to identify parameter regions for unique or multiple equilibria. Its scope and practical usefulness depend on the network and parameterization, and it does not answer several finer questions about equilibria.

Discussion

  • Under Corollary 2’s assumptions, a stoichiometric compatibility class has at least two equilibria if and only if a multivariate polynomial can attain a specific sign.The procedure uses this condition to identify parameter regions for unique and multiple equilibria.Source: Discussion · p. 19
  • Closed-form parameter expressions can be interpretable in biochemical terms, helping explain why multistationarity occurs.The approach addresses a difficult parameter-region problem that may otherwise require computationally expensive methods.Source: Discussion · p. 19
  • Although the procedure theoretically applies to algebraic reaction rates, it works well in practice with mass-action kinetics.For a two-site phosphorylation cycle with Michaelis–Menten kinetics, no positive algebraic parameterization exists, so the approach cannot find parameter conditions for multistationarity.Source: Discussion · p. 19
  • The procedure applies mainly to networks with conservation relations, ruling out networks where each species is produced and degraded.An algebraic parameterization of equilibria rarely exists when there are no conservation relations.Source: Discussion · p. 19
  • The procedure does not directly identify the particular compatibility classes with multiple equilibria, and exact equilibrium counts cannot be separated from those classes.In some situations, a positive parameterization using conservation relations could determine such classes up to one parameter.Source: Discussion · p. 20
  • The procedure does not determine equilibrium stability.An equilibrium satisfying Corollary 2(B) is unstable under the stated Jacobian-sign condition.Source: Discussion · p. 20
  • For some reaction networks, the procedure can be formulated as an algorithm toward “black box tools” for analysing complex dynamical systems.The supporting information contains proofs, procedure-checking details, and examples, including an extra PTM network.Source: Discussion · p. 20

Supplementary Information

The supporting information supplies proofs, procedural details, and expanded examples for the main text.

Supplementary Information

1 Preliminaries

The preliminaries establish basic language for convex sets and differentiable functions used in the subsequent analysis.

1.1 Convex sets

  • A convex set contains every segment between two of its points, and inward or outward directions describe how vectors at its boundary relate to the set.Source: §1.1 · p. 22

1.2 Functions

  • For an open bounded set, the text defines differentiable functions whose derivatives extend continuously to the boundary.It also defines the Jacobian and calls a value regular when the Jacobian is nonsingular at every preimage.Source: §1.2 · p. 22
  • Under the stated regular-value and boundary conditions, the preimage set is finite.Source: §1.2 · p. 22

2 Brouwer degree and a theorem

This section reviews Brouwer degree and develops a degree computation for a specially constructed function on a stoichiometric compatibility class.

2.1 Brouwer degree

  • The Brouwer degree is an integer that can establish the existence of solutions when it is nonzero; degree ±1 implies an odd number of solutions under the stated conditions.Source: §2.1 · p. 23
  • Homotopy invariance preserves the degree when the homotopy avoids the target value on the domain boundary.Source: §2.1 · p. 23

2.2 The Brouwer degree for a special class of functions

  • The construction ϕ_c combines f and W by using components of f and replacing selected components with Wx−c.The theorem’s setup uses a compatibility class P_c and an inward-pointing boundary condition to compute the degree of ϕ_c.Source: §2.2 · pp. 23–24
  • The proof computes the degree of ϕ_c by homotoping it to G and showing the homotopy does not vanish on the boundary.The argument uses the boundary condition on f and homotopy invariance.Source: §2.2 · pp. 24–25

3 Chemical reaction networks

The section sets up reaction-network dynamics on stoichiometric compatibility classes and establishes conditions for using Brouwer degree to analyze their equilibria. For dissipative networks without boundary equilibria, the degree calculation supports criteria for identifying mono- and multistationarity.

3.1 Setting

  • The network ODE is expressed through its stoichiometric matrix and rate functions, with trajectories confined to stoichiometric compatibility classes whose positive equilibria correspond to positive solutions of the associated class equations.The rate functions are assumed to be C1 and to vanish when a required reactant species is absent.Source: §3.1 · pp. 25–26

3.2 Conservative and dissipative networks

  • A dissipative semiflow has a compact attracting set that trajectories in each relevant compatibility class eventually enter; conservative networks are dissipative because their classes are compact.For nonconservative networks, dissipativity may still hold, but establishing it can require a suitable Lyapunov function.Source: §3.2 · pp. 26–27

3.3 Degree for dissipative semiflows

  • Under the theorem’s conditions, a constructed function and homotopy preserve the Brouwer degree, yielding degree (−1)^s for the class map.The setup assumes a dissipative semiflow and no boundary equilibria in the compatibility class.Source: §3.3 · pp. 28, 30

4 Multistationarity in dissipative networks

For dissipative reaction networks without boundary equilibria, the section relates equilibrium counts to determinant signs. A positive parameterization turns this criterion into sign tests that distinguish uniqueness from multistationarity across compatibility classes.

Theorem 4

  • If the determinant sign is (−1)^s at every positive equilibrium in a class, that class has exactly one positive, non-degenerate equilibrium.The theorem assumes dissipativity and no boundary equilibria.Source: §4 · pp. 30–31
  • If a positive equilibrium has determinant sign (−1)^(s+1), its class has at least two positive equilibria; if all are non-degenerate, there are at least three and their number is odd.The lower bound of two includes at least one non-degenerate equilibrium.Source: §4 · p. 31

Corollary 2

  • With a positive parameterization of equilibria, the determinant-sign criterion becomes a sign test on a(ˆx): sign (−1)^s gives uniqueness, while sign (−1)^(s+1) at some ˆx gives multistationarity in the corresponding class.The multistationarity case has at least two equilibria, or at least three and an odd number if all are non-degenerate.Source: §4 · p. 32

5 Details on the steps of the procedure

The section details procedure steps for checking boundary-equilibrium conditions and identifying parameter classes with multistationarity. It also describes how Newton-polytope vertices can help construct a point satisfying the polynomial sign criterion.

5 Details on the steps of the procedure

  • The detailed steps address checks needed by the procedure, including boundary-equilibrium verification and computational simplifications for larger reaction networks.Removing intermediates and catalysts can reduce the complexity of finding siphons.Source: §5 · p. 32, §5.1 · p. 33

5.1 On siphons and boundary equilibria

  • Iteratively removing intermediates or catalysts preserves whether each minimal siphon contains the support of a positive conservation relation, and a strongly connected monomolecular reduction implies no boundary equilibria in the original network’s relevant classes.For monomolecular networks, the siphon condition is equivalent to all connected components being strongly connected.Source: §5.1 · pp. 33–34

5.2 Newton polytope

  • A Newton-polytope vertex of the polynomial can provide a point where the polynomial’s sign matches that vertex’s coefficient, helping identify compatibility classes with multistationarity.The paper describes using Maple’s PolyhedralSet and VerticesAndRays commands to find vertices in its examples.Source: §5.2 · pp. 34–35

6 Details on the examples in the main text

The examples apply the algorithm to two-site phosphorylation and two-substrate catalysis to derive parameter conditions for multistationarity or uniqueness of positive equilibria. They show how the conditions are determined from polynomial coefficient signs.

6.1 Phosphorylation of two substrates

  • When α(κ) ≥ 0, all coefficients of p(bx) are positive, yielding a unique positive equilibrium in each stoichiometric compatibility class with non-empty positive relative interior.All but one coefficient are always positive; α(κ) determines the sign of the remaining coefficient.Source: §6.1 · p. 37
  • For this PTM network, multiple equilibria require κ3κ12 kM1kM4 < κ6κ9 kM2kM3 when κ3κ12 > κ6κ9, and the reverse inequality when κ3κ12 < κ6κ9.Source: §6.1 · pp. 35, 38

6.2 Two-site phosphorylation system

  • The two-site phosphorylation example recovers conditions also found in, using a direct route through ϕc that avoids changes of variables.The substrate undergoes sequential, distributive phosphorylation and dephosphorylation.Source: §6.2 · p. 38
  • For the two-site system, p(bx) has 15 terms: 9 are always positive, while the other 6 coefficients can be positive or negative; uniqueness follows if b1(κ) ≥ 0 and α(κ) ≥ 0.Five variable-sign coefficients share a factor b1(κ), while the remaining one contains α(κ).Source: §6.2 · p. 40
  • For two-site phosphorylation, b1(κ) < 0 guarantees multistationarity, while α(κ) ≥ 0 if and only if b2(κ) ≥ 0 gives the stated unique-equilibrium condition.The multistationarity conclusion follows because b1(κ) < 0 makes p(bx) negative for some bx.Source: §6.2 · p. 41
  • For two-site phosphorylation, α(κ) < 0 is not established as sufficient for multistationarity because its monomial’s exponent vector is not a Newton-polytope vertex.Source: §6.2 · p. 41

6.3 Two-substrate enzyme catalysis

  • The example establishes that the two-substrate network is conservative and dissipative, and has no boundary equilibria in positive-intersecting stoichiometric classes.The minimal siphons contain the supports of the conservation relations for c1, c2, and c3.Source: §6.3 · p. 42
  • Positive numerator and denominator coefficients in the parameterized function prove a unique positive equilibrium in each positive-intersecting stoichiometric class for two-substrate catalysis.The result follows from Corollary 2(A) with s = 4.Source: §6.3 · p. 43
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