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Multistationarity and Bistability for Fewnomial Chemical Reaction Networks

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Abstract

Bistability and multistationarity are properties of reaction networks linked to switch-like responses and connected to cell memory and cell decision making. Determining whether and when a network exhibits bistability is a hard and open mathematical problem. One successful strategy consists of analyzing small networks and deducing that some of the properties are preserved upon passage to the full network. Motivated by this, we study chemical reaction networks with few chemical complexes. Under mass action kinetics, the steady states of these networks are described by fewnomial systems, that is polynomial systems having few distinct monomials. Such systems of polynomials are often studied in real algebraic geometry by the use of Gale dual systems. Using this Gale duality, we give precise conditions in terms of the reaction rate constants for the number and stability of the steady states of families of reaction networks with one non-flow reaction.

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Acknowledgements

This work was partially funded by the Independent Research Fund of Denmark.

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Correspondence to Martin Helmer.

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Appendix A: More on Gale Dual Systems

Appendix A: More on Gale Dual Systems

Let \({\mathcal {V}}\) be the zero dimensional subscheme of \(({\mathbb {R}}_{>0})^{n}\) defined by the system of n Laurent polynomials

$$\begin{aligned} Cx^W= \begin{bmatrix} f_1(x_1,\dots ,x_{n}) \\ \vdots \\ f_n(x_1,\dots ,x_{n}) \end{bmatrix} =0, \end{aligned}$$
(46)

with W and C having Gale dual matrices Q and D, respectively, chosen as in §2.2. Recall that W and C are \(n\times (n+l+1)\) matrices, while Q and D are \((n+l+1)\times (l+1)\) matrices. We can now define a homomorphism of algebraic groups specified by the monomial map determined by the exponents of (46)

$$\begin{aligned}&\varphi _{W} :({\mathbb {R}}_{>0})^{n} \rightarrow ({\mathbb {R}}_{>0})^{n+l} \times \{1\} \subset {\mathbb {P}}_{{\mathbb {R}}}^{n+l}\\&\varphi _{W}:x\mapsto x^W= \left[ x^{w_1}:\dots :x^{w_{n+l}}:1\right] ^T. \end{aligned}$$

The homomorphism \(\varphi _{W}\) is dual to the homomorphism of free abelian groups \(\iota _{W}:{\mathbb {Z}}^{l+n}\rightarrow {\mathbb {Z}}^n\) which maps the standard \(i^{th}\) basis vector of \({\mathbb {Z}}^{l+n}\) to the column vector \(w_i\). Let \([z_1:\cdots :z_{n+l+1}]\) be coordinates for \({\mathbb {P}}_{{\mathbb {R}}}^{n+l}\), then the polynomials \(f_i\) are the pullbacks of linear forms \(\Lambda _i\) under the monomial map \(\varphi _{W} \), that is

$$\begin{aligned} f_i=\varphi _{W}^*(\Lambda _i),\qquad \text {where}\quad \Lambda _i=\sum _{j=1}^{n+l+1} c_{i,j}z_j. \end{aligned}$$
(47)

The scheme \({\mathcal {V}}\) defined by (46) is the pullback of the linear space \(L=V(\Lambda _1,\dots ,\Lambda _n) \subset {\mathbb {P}}_{{\mathbb {R}}}^{n+l}\). Let \({\mathbb {Z}}W\) denote the integer lattice spanned by the columns of W. Since \({\mathbb {Z}}W={\mathbb {Z}}^{n}\), we have that the intersection \(Y=L\cap \varphi _{W}(({\mathbb {R}}_{>0})^{n} )\) is proper (that is the intersection has the expected dimension) and the map \(\varphi _{W}\) defines a scheme theoretic isomorphism between \({\mathcal {V}}\) and Y; see, for example, Proposition 1.1 of Bihan and Sottile 2008.

Define the map \(\psi _{{\mathcal {V}}}:{\mathbb {R}}^l\rightarrow {\mathbb {P}}_{{\mathbb {R}}}^{l+n}\) given by \( \psi _{{\mathcal {V}}}(y) = [d_1(y): \cdots : d_{n+l}(y):1] \) where the \(d_i(y)\) are the linear forms in \({\mathbb {R}}[y_1,\dots ,y_{l}]\) defined by the rows of \(D\cdot [y_1,\dots ,y_{l},1]^T\) as in (8). Note that by construction \(\psi _{{\mathcal {V}}}\) is an isomorphism from \({\mathbb {R}}^l\) to the linear subspace L of \({\mathbb {P}}^{n+l}\). Hence, we have an isomorphism of schemes given by \(\psi _{{\mathcal {V}}}^{-1}\circ \varphi _W \) so that \( {\mathcal {V}}\cong Z=\psi _{{\mathcal {V}}}^{-1}(\varphi _W({\mathcal {V}})) \subset {\mathbb {R}}^l.\) The resulting isomorphic scheme Z is referred to as the Gale dual scheme of \({\mathcal {V}}\).

We now give the equations which describe Z. Every integer linear relation, \( \sum _i \beta ^{(i)} w_i=0\) with \(\beta ^{(i)}\in {\mathbb {Z}}\), among column vectors \(w_i\) in W, corresponds to the Laurent monomial equality \( \prod _{i=1}^{n+l} z_i^{\beta ^{(i)}}=1 \) on \(\varphi _{W}(({\mathbb {R}}_{>0})^{n} ) \subset {\mathbb {P}}_{{\mathbb {R}}}^{l+n}\); here, we have that \(z_i>0\). Pulling this relation back under the map \(\psi _{{\mathcal {V}}}\) gives the relation

$$\begin{aligned} \prod _{i=1}^{n+l} d_i(y)^{\beta ^{(i)}}=1 \;\;\;\mathrm{in} \;\; {\mathbb {R}}[y_1,\dots , y_l]\;\;\; \mathrm{with}\; \; d_i(y)>0. \end{aligned}$$
(48)

There is one such relation (48) for each row of Q giving the following system of l equations in \({\mathbb {R}}[y_1,\dots ,y_l]\)

$$\begin{aligned} \prod _{i=1}^{n+l} d_i(y)^{q_{i,1}} = 1 , \dots , \prod _{i=1}^{n+l} d_i(y)^{q_{i,l}} = 1 ,\;\;\; \mathrm{such\; that}\; \; d_i(y)>0. \end{aligned}$$
(49)

By construction \(Z\subset ({\mathbb {R}}_{>0})^l\) is the set of solutions to the system of equations (49) and \(Z\cong {\mathcal {V}}\) as schemes. The system (49) is referred to as the Gale dual system of the original system (46). Hence, the one-to-one correspondence in Proposition 2.3 is an isomorphism of schemes.

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Feliu, E., Helmer, M. Multistationarity and Bistability for Fewnomial Chemical Reaction Networks. Bull Math Biol 81, 1089–1121 (2019). https://doi.org/10.1007/s11538-018-00555-z

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