1 Introduction

Coined by Cannon (1926) in 1926, the idea of homeostasis has its roots in the work of Bernard (1898), and refers to a regulatory mechanism by which a feature maintains a steady state that is not perturbed by changes in the environment. Often, homeostasis involves the use of negative feedback loops (Ma et al. 2009; Golubitsky and Wang 2020) that help restore a feature to its steady state. At the scale of a whole organism, homeostasis manifests itself in many forms; some prominent examples include the maintenance of body temperature, blood sugar level, concentration of ions in body fluids with changes in the external environment. Homeostasis is also exhibited in intracellular metabolism, where certain concentration remain almost unperturbed with change in concentrations of amino acids (Reed et al. 2017).

As noted in Reed et al. (2017), homeostasis does not imply that the whole system remains invariant with change in external variables. In fact, changes in external variables can cause changes in certain internal variables, while other internal variables remain almost unchanged. In recent years there has been a lot of interest in identifying and analyzing homeostasis in mathematical models of biological interaction systems (Golubitsky and Stewart 2017; Nijhout and Reed 2014; Reed et al. 2017; Nijhout et al. 2014; Golubitsky and Stewart 2018; Tang and McMillen 2016). Some of this renewed interest started with the mathematical analysis of homeostasis in the context of the folate and methionine metabolism (Nijhout et al. 2004; Reed et al. 2017).

While there is no universally accepted mathematical definition of homeostasis, here we focus mostly on the notion of infinitesimal homeostasis for input-output systems, as introduced by Golubitsky and Stewart in Golubitsky and Stewart (2017) and further refined in Wang et al. (2020), Golubitsky et al. (2020). Our main interest is to analyze homeostasis from the point of view of reaction network theory (Feinberg 1979, 2019; Yu and Craciun 2018).

This paper is organized as follows. In Sect. 2, we define reaction networks and related notions, including the notion of injective reaction network. In Sect. 3, we introduce homeostasis as the capacity of a feature to be robust to change in the parameters of the system. We present a procedure for checking whether a reaction network may admit homeostasis by constructing a modified network and checking if it is injective (see Theorem 3.6). We also describe a sufficient condition for perfect homeostasis (see Theorem 3.8). In Sect. 4 we present several examples of reaction networks for which their capacity to exhibit homeostasis can be analyzed using the procedure described in Sect. 3.

2 Euclidean embedded graphs and reaction networks

An Euclidean embedded graph is a directed graph \({\mathcal {G}}= (V,E)\), where \(V\subset {\mathbb {R}}^n\) and E are the sets of vertices and edges respectively. Associated with every edge \(({\varvec{y}},{\varvec{y}}')\in E\) is a source vertex \({\varvec{y}}\in V\) and a target vertex \({\varvec{y}}'\in V\). An edge \(({\varvec{y}},{\varvec{y}}')\in E\) will also be denoted by \({\varvec{y}}\rightarrow {\varvec{y}}'\in E\).

A reaction network is a Euclidean embedded graph \({\mathcal {G}}= (V,E)\), where \(V\subset {\mathbb {R}}^n_{\ge 0}\) and E is the set of edges that correspond to reactions in the network (Craciun 2015, 2019). An alternative way of describing a reaction network is by specifying a set of species and a set of reactions. For example, consider the set of species \(\{X_1,X_2\}\) and the set of reactions \(\{2X_1\rightarrow 3X_2, \ X_1+X_2\rightarrow 3X_1\}\). The corresponding Euclidean embedded graph lies in \({\mathbb {R}}^2_{\ge 0}\) and has two edges: one edge from (2, 0) to (0, 3) and one edge from (1, 1) to (3, 0), where the vertex vectors are formed by the coefficients of the species \(X_1\) and \(X_2\) on the reactant side and product side, respectively.

The stoichiometric subspace of a reaction network is the linear subspace given by \(\textrm{span}\{{\varvec{y}}'-{\varvec{y}}\, |\, {\varvec{y}}\rightarrow {\varvec{y}}' \in E\}\). Given a point \({\varvec{x}}_0\in {\mathbb {R}}^n_{>0}\), the positive stoichiometric compatibility class of \({\varvec{x}}_0\) is the affine subspace \(({\varvec{x}}_0 + S)\cap {\mathbb {R}}^n_{>0}\).

There exist many choices for modelling the kinetics of reaction networks. The most common one is based on the law of mass-action (Voit et al. 2015; Guldberg and Waage 1864; Yu and Craciun 2018; Gunawardena 2003; Feinberg 1979) where, associated with each reaction \({\varvec{y}}\rightarrow {\varvec{y}}'\) there is a rate constant \(k_{{\varvec{y}}\rightarrow {\varvec{y}}'}>0\), and the dynamics of the network is given by

$$\begin{aligned} \frac{d{\varvec{x}}}{dt}=\displaystyle \sum _{{\varvec{y}}\rightarrow {\varvec{y}}'\in E} k_{{\varvec{y}}\rightarrow {\varvec{y}}'}{\varvec{x}}^{{\varvec{y}}}({\varvec{y}}'-{\varvec{y}}), \end{aligned}$$
(1)

where \({\varvec{x}}\in {\mathbb {R}}^n_{>0}\) and \({\varvec{x}}^{{\varvec{y}}}=x_1^{y_1}x_2^{y_2}\cdots x_n^{y_n}\). Let \({\varvec{k}}=(k_{{\varvec{y}}\rightarrow {\varvec{y}}'})_{{\varvec{y}}\rightarrow {\varvec{y}}'\in E}\) denote the vector of rate constants and \({\varvec{f}}({\varvec{x}},{\varvec{k}})\) denote the right-hand side of Eq. (1). The Jacobian corresponding to the dynamical system (1) is given by the matrix

$$\begin{aligned} J({\varvec{x}},{\varvec{k}})= \begin{pmatrix} \frac{\partial f_1({\varvec{x}},{\varvec{k}})}{\partial x_1} &{} \frac{\partial f_1({\varvec{x}},{\varvec{k}})}{\partial x_2} &{} \cdots &{} \frac{\partial f_1({\varvec{x}},{\varvec{k}})}{\partial x_n} \\ \frac{\partial f_2({\varvec{x}},{\varvec{k}}))}{\partial x_1} &{} \frac{\partial f_2({\varvec{x}},{\varvec{k}})}{\partial x_2} &{} \cdots &{} \frac{\partial f_2({\varvec{x}},{\varvec{k}})}{\partial x_n} \\ \vdots &{} \vdots \\ \frac{\partial f_n({\varvec{x}},{\varvec{k}})}{\partial x_1} &{} \frac{\partial f_2({\varvec{x}},{\varvec{k}})}{\partial x_2} &{} \cdots &{} \frac{\partial f_n({\varvec{x}},{\varvec{k}})}{\partial x_n} \end{pmatrix}, \end{aligned}$$

‘where \({\varvec{x}}=(x_1,x_2,...,x_n)\) and \(f_i({\varvec{x}},{\varvec{k}})\) is the right-hand side of the dynamics corresponding to species \(X_i\). A point \({\varvec{x}}_0\in {\mathbb {R}}^n_{>0}\) is said to be an equilibrium of (1) if \(\displaystyle \sum _{{\varvec{y}}\rightarrow {\varvec{y}}'\in E} k_{{\varvec{y}}\rightarrow {\varvec{y}}'}{\varvec{x}}_0^{{\varvec{y}}}({\varvec{y}}'-{\varvec{y}})=0\). An equilibrium point \({\varvec{x}}_0\in {\mathbb {R}}^n_{>0}\) of (1) is said to be complex balanced if for every vertex \({\varvec{y}}\) in the reaction network we have

$$\begin{aligned} \displaystyle \sum _{{\varvec{y}}\rightarrow {\varvec{y}}'\in E} k_{{\varvec{y}}\rightarrow {\varvec{y}}'}{\varvec{x}}_0^{{\varvec{y}}} = \displaystyle \sum _{{\varvec{y}}' \rightarrow {\varvec{y}}\in E} k_{{\varvec{y}}\rightarrow {\varvec{y}}'}{\varvec{x}}_0^{{\varvec{y}}'}. \end{aligned}$$
(2)

An equilibrium \({\varvec{x}}_0\in {\mathbb {R}}^n_{>0}\) is said to be a linearly stable equilibrium of (1) if the eigenvalues of the Jacobian of (1) evaluated at the point \({\varvec{x}}_0\) have negative real parts (Hassan 2002). It is known from Siegel and Johnston (2008, Theorem 5.2) and Feinberg (2019, 15.2.2) that complex balanced equilibria are linearly stable.

Recall the \({\varvec{f}}({\varvec{x}},{\varvec{k}})\) denotes the right hand side of Eq. (1). Now consider the function \({\varvec{x}}\rightarrow {\varvec{f}}({\varvec{x}},{\varvec{k}})\). A reaction network \({\mathcal {G}}\) is said to be injective if the function \({\varvec{x}}\rightarrow {\varvec{f}}({\varvec{x}},{\varvec{k}})\) corresponding to the dynamics given by (1) is injective for all \({\varvec{k}}\). It follows that an injective reaction network cannot have multiple equilibria.

In general, it is extremely difficult to determine whether the function \({\varvec{x}}\rightarrow {\varvec{f}}({\varvec{x}},{\varvec{k}})\) is injective or not. Necessary and sufficient conditions for a reaction network to be injective are given by Theorems 3.1, 3.2, and 3.3 in Craciun and Feinberg (2005), which we summarize here:

Theorem 2.1

Consider a reaction network \({\mathcal {G}}\) with species \(X_1,X_2,...,X_n\). Let \(J({\varvec{x}},{\varvec{k}})\) denote the Jacobian matrix corresponding to the dynamics generated by \({\mathcal {G}}\). Then the following hold:

  1. 1.

    \({\mathcal {G}}\) is injective if and only if \(det(J({\varvec{x}},{\varvec{k}}))\) is non-zero for every \({\varvec{x}}\in {\mathbb {R}}^n_{>0}\) and for all choices of rate constants \({\varvec{k}}\).

  2. 2.

    There is a one-to-one correspondence between the coefficients in the expansion of \(det(J({\varvec{x}},{\varvec{k}}))\) and products of the form \(det({\varvec{y}}_1,{\varvec{y}}_2,...,{\varvec{y}}_n)det({\varvec{y}}_1-{\varvec{y}}'_1,{\varvec{y}}_2-{\varvec{y}}'_2,...,{\varvec{y}}_n-{\varvec{y}}'_n)\) for all choices of n reactions \(\{{\varvec{y}}_1\rightarrow {\varvec{y}}'_1, {\varvec{y}}_2\rightarrow {\varvec{y}}'_2, ..., {\varvec{y}}_n\rightarrow {\varvec{y}}'_n\}\) in \({\mathcal {G}}\).

  3. 3.

    \({\mathcal {G}}\) is injective if and only if for any choice of n reactions \(\{{\varvec{y}}_1\rightarrow {\varvec{y}}'_1, {\varvec{y}}_2\rightarrow {\varvec{y}}'_2, ..., {\varvec{y}}_n\rightarrow {\varvec{y}}'_n\}\) in \({\mathcal {G}}\) all the products of the form

    $$\begin{aligned} det({\varvec{y}}_1,{\varvec{y}}_2,...,{\varvec{y}}_n)det({\varvec{y}}_1-{\varvec{y}}'_1,{\varvec{y}}_2-{\varvec{y}}'_2,...,{\varvec{y}}_n-{\varvec{y}}'_n) \end{aligned}$$

    have the same sign, and at least one such product is non-zero.

A useful tool for analyzing injectivity is the directed species reaction graph (abbreviated as DSR graph) first introduced in Banaji and Craciun (2009). In what follows, we describe some terminology in the context of DSR graphs. More details and examples can be found in Banaji and Craciun (2009), Banaji and Pantea (2016), Yu et al. (2022).

Given a reaction network \({\mathcal {G}}\), the DSR graph of \({\mathcal {G}}\) is a bipartite graph whose nodes are the species and the reactions of \({\mathcal {G}}\) (where a pair of reversible reactions shares a single reaction node; moreover, “inflow reactions" of the form \(X_i \rightarrow \emptyset \) and “outflow reactions" of the form \(\emptyset \rightarrow X_i\) are disregarded). Edges of the DSR graph always connect a species node to a reaction node (or vice-versa), and never connect two species nodes or two reaction nodes (see Fig. 1 for an example). More precisely, we have an edge between a specific species and a specific reaction if that species appears in that reaction. If a species appears in the reactant side of a reaction, then the edge is a negative edge, and if it appears in the product side of a reaction, then the edge is a positive edgeFootnote 1. We will denote a negative edge in the DSR graph with a dashed line, and a positive edge with a solid line. For irreversible reactions, edges that connect reactions to product species are directed from reactions to species. Each edge is labeled with the stoichiometric coefficient that its endpoint species has within its endpoint reactionFootnote 2. Cycles in the DSR graph play an especially important role, and we need to define several types of cycles. Let |C| denote the length of a cycle in the DSR graph. A cycle is an e-cycle (even cycle) if the number of positive edges has the same parity as the number \(\frac{|C|}{2}\). Otherwise, it is an o-cycle (odd cycle). A cycle is a s-cycle if all its edge labels are finite and

$$\begin{aligned} \displaystyle \prod _{i=1}^{|C|/2} l(e_{2i-1})= \displaystyle \prod _{i=1}^{|C|/2} l(e_{2i}), \end{aligned}$$

where l(e) denotes the stoichiometric label of the edge e. We will say that two cycles have an odd intersection if their orientation is compatible and every component of their intersection contains an odd number of edges. Figure 1 shows the DSR graph for the network consisting of the following reactions: \(\{E + S \rightleftharpoons ES, ES \rightarrow E + P, P\rightarrow S, E\rightarrow \emptyset , P\rightarrow \emptyset \}\).

Fig. 1
figure 1

DSR graph for the network given by the following reactions: \(\{E + S \rightleftharpoons ES, ES \rightarrow E + P, P\rightarrow S, E\rightarrow \emptyset , P\rightarrow \emptyset \}\)

Full size image

Theorem 2.2

A DSR criterion Banaji and Craciun (2009): Consider a reaction network \({\mathcal {G}}\). Suppose the following conditions are satisfied:

  1. 1.

    Every e-cycle is a s-cycle in the DSR graph of \({\mathcal {G}}\).

  2. 2.

    No two e-cycles have an odd intersection in the DSR graph of \({\mathcal {G}}\).

  3. 3.

    There exists a choice of n reactions \(\{{\varvec{y}}_1\rightarrow {\varvec{y}}'_1,{\varvec{y}}_2\rightarrow {\varvec{y}}'_2,{\varvec{y}}_3\rightarrow {\varvec{y}}'_3,...,{\varvec{y}}_n\rightarrow {\varvec{y}}'_n\}\) in \({\mathcal {G}}\) such that \(det({\varvec{y}}_1,{\varvec{y}}_2,{\varvec{y}}_3,...,{\varvec{y}}_n)det({\varvec{y}}'_1-{\varvec{y}}_1,{\varvec{y}}'_2-{\varvec{y}}_2,{\varvec{y}}'_3-{\varvec{y}}_3,...,{\varvec{y}}'_n-{\varvec{y}}_n)\ne 0\).

Then \({\mathcal {G}}\) is injective.

Consider the DSR graph in Fig. 1. This has two cycles given by \(E -(E+S\rightleftharpoons ES)-ES-(ES\rightarrow E+P) - E\) and \(ES-(ES\rightarrow E+P)-P-(P\rightarrow S)-S-(E+S\rightleftharpoons ES)-ES\). Both these cycles are e-cycles and s-cycles since the number of positive edges has the same parity as half the length of the respective cycle. The intersection of these two e-cycles is the path \((E+S\rightleftharpoons ES)-ES-(ES\rightarrow E+P)\) and this is not an odd intersection since it contains two edges (which is an even number). Further, if we choose four reactions from the network given by the following: \(\{E + S \rightarrow ES, ES \rightarrow E + P, E\rightarrow \emptyset , P \rightarrow \emptyset \}\), then we have \(det({\varvec{y}}_1,{\varvec{y}}_2,{\varvec{y}}_3,{\varvec{y}}_4)\cdot det({\varvec{y}}'_1-{\varvec{y}}_1,{\varvec{y}}'_2-{\varvec{y}}_2,{\varvec{y}}'_3-{\varvec{y}}_3,y'_4-{\varvec{y}}_4) = det \begin{pmatrix} 1 &{} 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ \end{pmatrix}\cdot det\begin{pmatrix} -1 &{} 1 &{} -1 &{} 0 \\ -1 &{} 0 &{} 0 &{} 0 \\ 1 &{} -1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} -1 \\ \end{pmatrix}=1\cdot 1\ne 0\). Therefore, by Theorem 2.2 this reaction network is injective.

3 Homeostasis

The ability of a feature to remain invariant when certain parameters of the system are changed is the essential idea behind homeostasis. A common example of homeostasis is exhibited when an organism maintains its body temperature despite fluctuations in the temperature of the environment. The temperature of the body varies linearly with temperature for low and high values of the environment temperature; however for moderate values of the environment temperature, the body temperature remains approximately constant. This variation of body temperature with the environment resembles the shape of a “chair" (Nijhout et al. 2004; Nijhout and Reed 2014). In Golubitsky and Stewart (2017), this “chair" form provides inspiration for a definition of homeostasis in the context of singularity theory. In particular, the idea of homeostasis corresponds to the derivative of an output (homeostasis) variable with respect to an external input being zero at a certain point. As outlined in Golubitsky and Stewart (2017), we consider the following setup: Let \({\varvec{x}}=(x_1,x_2,...,x_n)\) and consider

$$\begin{aligned} \frac{d{\varvec{x}}}{dt} = {\mathcal {F}}({\varvec{x}},\zeta ) \end{aligned}$$
(3)

given by

$$\begin{aligned} \frac{dx_1}{dt}&= f_1(x_1,x_2,...,x_n) + \zeta \nonumber \\ \frac{dx_2}{dt}&= f_2(x_1,x_2,...,x_n) \nonumber \\ \vdots&= \vdots \nonumber \\ \frac{dx_n}{dt}&= f_n(x_1,x_2,...,x_n) \end{aligned}$$
(4)

As in Golubitsky and Stewart (2017), throughout this paper we assume that the variable \(x_1\) is the input variable, and the output variable (which may or may not exhibit homeostatis) is \(x_n\). We will also assume that there exists a linearly stable equilibrium of (4) given by \(({\varvec{x}}_0,\zeta _0)\). By the implicit function theorem, there exists solutions \({\tilde{{\varvec{x}}}}(\zeta )\) in a neighbourhood of the equilibrium \(({\varvec{x}}_0,\zeta _0)\) satisfying \({\mathcal {F}}({\tilde{{\varvec{x}}}}(\zeta ),\zeta )=0\). In particular, this implies that \({\tilde{{\varvec{x}}}}(\zeta )\) is also a linearly stable equilibrium, and depends continuously on \(\zeta \) in a neighbourhood of the equilibrium \(({\varvec{x}}_0,\zeta _0)\). Following Golubitsky and Stewart (2017), Golubitsky and Wang (2020), Wang et al. (2020), we define infinitesimal homeostasis as follows.

Definition 3.1

Consider a dynamical system of the form (4). Denote by \({\tilde{{\varvec{x}}}}(\zeta )\) a linearly stable equilibrium of (4) in the neighbourhood of \(({\varvec{x}}_0,\zeta _0)\). We say that we have infinitesimal homeostasis at \(({\varvec{x}}_0,\zeta _0)\) if \(\frac{d{\tilde{x}}_n}{d\zeta }\bigg |_{({\varvec{x}}_0,\zeta _0)}=0\).

In other words, an equilibrium point exhibits infinitesimal homeostasis for the n-th variable \(x_n\) if the derivative of the value of this variable at equilibrium with respect to the external input parameter \(\zeta \) is zero. As shown in Golubitsky and Stewart (2017), this derivative is a scalar multiple of a reduced Jacobian of the system:

Theorem 3.2

Denote by J the Jacobian matrix of the right-hand side of the system (4). Under the same assumptions as in Definition 3.1, we have infinitesimal homeostasis at \(({\varvec{x}}_0,\zeta _0)\) if and only if \(det(B)\biggr |_{({\varvec{x}}_0,\zeta _0)}=0\) where B is the \((n-1)\times (n-1)\) minor of the Jacobian J obtained by deleting the first row and last column of J.

Definition 3.3

Consider a dynamical system of the form (4). Denote by \({\tilde{{\varvec{x}}}}(\zeta )\) a linearly stable equilibrium of (4) in the neighbourhood of \(({\varvec{x}}_0,\zeta _0)\). We say that we have perfect homeostasis at \(({\varvec{x}}_0,\zeta _0)\) if \(\frac{d{\tilde{x}}_n}{d\zeta }\bigg |_{({\varvec{x}}_0,\zeta _0)}=0\) for all values of \(\zeta \) on some interval containing \(\zeta _0\).

Remark 3.4

As remarked in Wang et al. (2020), there exists several forms of homeostasis. Specifically, Definition 3.1 refers to infinitesimal homeostasis, which requires the derivative of the input-output function to be zero at a point. The idea of perfect homeostasis refers to the situation when the derivative of the input-output function vanishes on an entire interval. Definition 3.3 illustrates this fact. The notion of near perfect homeostasis refers to the situation when the input-output function is approximately constant in a neighbourhood of a point.

Definition 3.5

Consider a reaction network \({\mathcal {G}}\). The homeostasis-associated reaction network of \({\mathcal {G}}\), denoted by \({\tilde{{\mathcal {G}}}}\), is obtained from \({\mathcal {G}}\) as follows

Step 1::

For each reaction in \({\mathcal {G}}\) involving the species \(X_1\), modify the reaction such that stoichiometric coefficient of \(X_1\) in the reactant is the same as the stoichiometric coefficient of \(X_1\) in the product.

Step 2::

Add the reaction \(X_n\rightarrow X_1\).

Theorem 3.6

Consider a reaction network \({\mathcal {G}}\) with species \(X_1,X_2,...,X_n\). Let \({\tilde{{\mathcal {G}}}}\) be the homeostasis-associated reaction network of \({\mathcal {G}}\). If the reaction network \({\tilde{{\mathcal {G}}}}\) is injective, then the mass-action dynamical system generated by \({\mathcal {G}}\) cannot exhibit infinitesimal homeostasis (with input \(X_1\) and output \(X_n\)) for any choices of rate constants.

Proof

Let J and \({\tilde{J}}\) denote the Jacobians coresponding to the dynamical systems generated by \({\mathcal {G}}\) and \({\tilde{{\mathcal {G}}}}\) respectively. Step 1 of the procedure in Definition 3.5 makes the first row of \({\tilde{J}}\) zero. Step 2 of Definition 3.5 generates a non-zero element in the top right corner of the \({\tilde{J}}\). Therefore, the Jacobian \({\tilde{J}}\) has the first row consisting entirely of zeros except the last element. In addition, \({\tilde{J}}\) has the same \((n-1)\times (n-1)\) minor B as obtained by deleting the first row and last column of J. Expanding along the first row of \({\tilde{J}}\), we get that \(det({\tilde{J}})=k_{n,1}det(B)\), where \(k_{n,1}\) is the rate constant corresponding to the reaction \(X_n\rightarrow X_1\). Since \({\tilde{{\mathcal {G}}}}\) is injective, by Theorem 2.1 we have \(det({\tilde{J}}({\varvec{x}}))\ne 0\) for every \({\varvec{x}}\in {\mathbb {R}}^n_{>0}\). This implies that \(det(B({\varvec{x}}))\ne 0\) for every \({\varvec{x}}\in {\mathbb {R}}^n_{>0}\) and hence \({\mathcal {G}}\) cannot exhibit infinitesimal homeostasis for any choices of rate constants. \(\square \)

Corollary 3.7

Consider a reaction network \({\mathcal {G}}\) with species \(X_1,X_2,...,X_n\). Let \({\tilde{{\mathcal {G}}}}\) be the homeostasis-associated reaction network of \({\mathcal {G}}\). If the reaction network \({\tilde{{\mathcal {G}}}}\) satisfies the conditions 1, 2 and 3 in Theorem 2.2, then the mass-action dynamical system generated by \({\mathcal {G}}\) cannot exhibit infinitesimal homeostasis (with input \(X_1\) and output \(X_n\)) for any choices of rate constants.

Proof

This follows from Theorems 3.6 and 2.2. \(\square \)

Theorem 3.8

Consider a reaction network \({\mathcal {G}}\) with species \(X_1,X_2,...,X_n\). Let \({\tilde{{\mathcal {G}}}}\) be the homeostasis-associated reaction network of \({\mathcal {G}}\). Let \({\tilde{J}}({\varvec{x}}))\) denote the Jacobian matrix corresponding to the right hand side of the dynamical system generated by \({\tilde{{\mathcal {G}}}}\). If \(det({\tilde{J}}({\varvec{x}})) = 0\) for every \({\varvec{x}}\in {\mathbb {R}}^n_{>0}\), then any mass-action dynamical system generated by \({\mathcal {G}}\) must exhibit perfect homeostasis (with input \(X_1\) and output \(X_n\)) at any linearly stable equilibrium.

Proof

Let J and \({\tilde{J}}\) denote the Jacobians coresponding to the dynamical systems generated by \({\mathcal {G}}\) and \({\tilde{{\mathcal {G}}}}\) respectively. Arguing as in the Proof of Theorem 3.6, we get that \(det({\tilde{J}})=k_{n,1}det(B)\), where \(k_{n,1}\) is the rate constant corresponding to the reaction \(X_n\rightarrow X_1\). Since \(det({\tilde{J}}({\varvec{x}})) = 0\) for every \({\varvec{x}}\in {\mathbb {R}}^n_{>0}\), we have \(det(B)= 0\) for every \({\varvec{x}}\in {\mathbb {R}}^n_{>0}\), where B is the \((n-1)\times (n-1)\) minor of the Jacobian \({\tilde{J}}\) obtained by deleting the first row and last column of \({\tilde{J}}\). This implies that any mass-action dynamical system generated by \({\mathcal {G}}\) must exhibit perfect homeostasis (with input \(X_1\) and output \(X_n\)) at any linearly stable equilibrium. \(\square \)

Corollary 3.9

Consider a reaction network \({\mathcal {G}}\) with species \(X_1,X_2,...,X_n\). Let \({\tilde{{\mathcal {G}}}}\) be the homeostasis-associated reaction network of \({\mathcal {G}}\). If the graph \({\tilde{{\mathcal {G}}}}\) does not satisfy condition 3 in Theorem 2.2, then any mass-action dynamical system generated by \({\mathcal {G}}\) must exhibit perfect homeostasis (with input \(X_1\) and output \(X_n\)) at any linearly stable equilibrium.

Proof

This follows from Theorems 3.8 and 2.2. \(\square \)

Remark 3.10

Recall that the notion of infinitesimal homeostasis (as described by Definition 3.1) assumes the existence of a linearly stable equilibrium \(({\tilde{x}}_1,{\tilde{x}}_2,...,{\tilde{x}}_n)\). For a mass-action system generated by a reaction network \({\mathcal {G}}\) this implicitly says that the dimension of the stoichiometric subspace of \({\mathcal {G}}\) must be n. In other words, the notion of infinitesimal homeostasis (as described by Definition 3.1) cannot ever apply to a mass-action system that has one or more linear conservation laws (i.e., for which the dimension of the stoichiometric subspace of \({\mathcal {G}}\) is less than n).

4 Examples

The goal of this section is to demonstrate examples of reaction networks that can or cannot exhibit infinitesimal homeostasis using the procedure outlined in Definition 3.5.

4.1 A reaction network that does not exhibit infinitesimal homeostasis for any choice of network parameters

The biological motivation for the following example comes from “sequestration networks" as defined in Joshi and Shiu (2015), Félix et al. (2016). In particular, they find instances of such networks in the transcription mechanism of E.coli. The trp operon contains genes that encode for the amino acid tryptophan. The operon is turned “off" or “on" depending upon the levels of tryptophan. When the tryptophan levels are low, it is turned “off" and when the levels of tryptophan are high, it is turned “on". In the presence of tryptophan, the trp repressor can bind to the operon sites and prevent the expression of the operon. This can be seen as a sequestration reaction \(X_1 + X_2 \rightarrow \emptyset \), where \(X_1\) is the tryptophan and \(X_2\) is the trp operon. Taking our cue from this, we consider the following sequestration network.

Example

Consider the reaction network \({\mathcal {G}}_1\) given by:

(5)

The homeostasis-associated reaction network corresponding to \({\mathcal {G}}_1\) will be denoted by \(\tilde{{\mathcal {G}}}_1\) and is given by

(6)

Note that when we apply Step 1 of the procedure listed in Definition 3.1 to the reaction \(X_4 \rightarrow X_1\), we get the reaction \(X_4 \rightarrow \emptyset \). Step 2 of the procedure then adds the reaction \(X_4 \rightarrow X_1\). As a consequence, we get the homeostasis-associated network \(\tilde{{\mathcal {G}}}_1\), where the reaction \(X_4 \rightarrow X_1\) has a larger rate constant as compared to the rate constant of the same reaction in \({\mathcal {G}}_1\). The DSR graph corresponding to the network \(\tilde{{\mathcal {G}}}_1\) is given in Fig. 2

Fig. 2
figure 2

DSR graph for network \({\tilde{G}}_1\)

Full size image

This DSR graph possesses exactly one oriented cycle given by \((X_3)-(X_3 + X_4\rightarrow \emptyset )-(X_4)-(X_4 \rightarrow X_1)-(X_1)-(X_1 + X_2 \rightarrow X_1)-(X_2)-(X_2 + X_3\rightarrow \emptyset ) - (X_3)\) which is an o-cycle. As a consequence, conditions 1 and 2 in Theorem 2.2 are satisfied. In addition, if we choose four reactions from \(\tilde{{\mathcal {G}}}_1\) given by \(\{X_1 + X_2 \rightarrow X_1,X_2 + X_3 \rightarrow \emptyset , X_3 + X_4 \rightarrow \emptyset , X_4 \rightarrow X_1\}\), then we have \(det({\varvec{y}}_1,{\varvec{y}}_2,{\varvec{y}}_3,{\varvec{y}}_4)\cdot det({\varvec{y}}'_1-{\varvec{y}}_1,{\varvec{y}}'_2-{\varvec{y}}_2,{\varvec{y}}'_3-{\varvec{y}}_3,y'_4-{\varvec{y}}_4) = det \begin{pmatrix} 1 &{} 0 &{} 0 &{} 0 \\ 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 1 \\ \end{pmatrix}\cdot det\begin{pmatrix} 0 &{} 0 &{} 0 &{} 1 \\ -1 &{} -1 &{} 0 &{} 0 \\ 0 &{} -1 &{} -1 &{} 0 \\ 0 &{} 0 &{} -1 &{} -1 \\ \end{pmatrix}=1\ne 0\). Therefore, condition 3 of Theorem 2.2 is satisfied. Using Corollary 3.7, we get that \({\mathcal {G}}_1\) cannot exhibit infinitesimal homeostasis for any choices of rate constants.

4.2 A reaction network that does exhibit infinitesimal homeostasis

Example

Consider the following network \({\mathcal {G}}_2\)

(7)

The network \({\mathcal {G}}_2\) does not have all the inflow/outflow reactions, but the stoichiometric subspace is full. Using the procedure given in Theorem 3.6, the homeostasis-associated reaction network \({\tilde{{\mathcal {G}}}}_2\) is given by the following:

(8)
Fig. 3
figure 3

DSR graph for network \({\tilde{G}}_2\)

Full size image

Let us analyze the DSR graph corresponding to the network \(\tilde{{\mathcal {G}}_2}\) as shown in Fig. 3. Since condition is not satisfied for the DSR graph of \(\tilde{{\mathcal {G}}_2}\), there is a possibility that the network \({\mathcal {G}}_2\) can exhibit infinitesimal homeostasis. In particular, \({\mathcal {G}}_2\) generates a dynamical system given by the following set of differential equations:

$$\begin{aligned} \dot{x_1}&= \zeta - x_1 -2x_1^2 + 2x_2 \nonumber \\ \dot{x_2}&= x^2_1 - x_2 \nonumber \\ \dot{x_3}&= -x_2x_3 + x_1x_3 - x^2_3 + x_3 \end{aligned}$$
(9)

This set of differential equations has the steady state given by \((x^*_1 = \zeta , x^*_2 = \zeta ^2, x^*_3 = 1 - \zeta + \zeta ^2)\). The Jacobian corresponding to 9 is given by

$$\begin{aligned} J_2= \begin{pmatrix} -1 - 4x_1 &{} 2 &{} 0 \\ 2x_1 &{} -1 &{} 0 \\ x_3 &{} -x_3 &{} -x_2 + x_1 -2x_3 +1 \end{pmatrix} \end{aligned}$$

The determinant of the \((n-1)\times (n-1)\) minor of \(J_2 \) obtained by deleting its first row and last column is given by \(x_3 - 2x_1x_3\) which is 0 at \(x_1=\frac{1}{2}\). We now check the stability of the equilibrium point given by \((x^*_1 = \frac{1}{2}, x^*_2 = \frac{1}{4}, x^*_3 = \frac{3}{4}, \zeta = \frac{1}{2})\). The Jacobian at this point is given by

$$\begin{aligned} J^*_2= \begin{pmatrix} -3 &{} 2 &{} 0 \\ 1 &{} -1 &{} 0 \\ \frac{3}{4} &{} -\frac{3}{4} &{} -\frac{1}{4} \end{pmatrix} \end{aligned}$$

The Jacobian \(J^*_2\) has eigenvalues given by \(\lambda _1 = \frac{-1}{4}, \lambda _2 = -\sqrt{3} - 2, \lambda _3 = -\sqrt{3} + 2\), which are all negative and hence the equilibrium is linearly stable. Therefore the network \({\mathcal {G}}_2\) exhibits infinitesimal homeostasis at \((x^*_1 = \frac{1}{2}, x^*_2 = \frac{1}{4}, x^*_3 = \frac{3}{4}, \zeta = \frac{1}{2})\).

4.3 A reaction network that exhibits perfect homeostasis

Example

Consider the following network \({\mathcal {G}}_3\)

(10)

The network \({\mathcal {G}}_3\) generates a dynamical system given by

$$\begin{aligned} \dot{x_1}&= \zeta - x_1 -x_1x_3 \nonumber \\ \dot{x_2}&= - x_2 + x_1x_3 \nonumber \\ \dot{x_3}&= 1 - x_3 + x_2- x_1x_3 \end{aligned}$$
(11)

The Jacobian corresponding to Eq. 11 is given by

$$\begin{aligned} J(x_1,x_2,x_3)= \begin{pmatrix} -1-x_3 &{} 0 &{} -x_1 \\ x_3 &{} -1 &{} x_1 \\ -x_3 &{} 1 &{} -1-x_1 \end{pmatrix} \end{aligned}$$

The steady-state corresponding to Eq. 11 is given by \((x^*_1,x^*_2,x^*_3)=\left( \frac{\zeta }{2}, \frac{\zeta }{2}, 1\right) \). Given this steady-state parametrization, one can show that the Jacobian with \(\zeta =1\) has all negative eigenvalues given by \(\lambda _1 = -1,\lambda _2 = \frac{-7+\sqrt{17}}{4},\lambda _3 = \frac{-7-\sqrt{17}}{4}\). Therefore the point \((\frac{1}{2},\frac{1}{2},1)\) is linearly stable.

The homostasis-associated network \({\hat{{\mathcal {G}}}}_3\) is given by the following:

$$\begin{aligned} X_3 + X_1&\rightarrow X_1 + X_2 \nonumber \\ X_2&\rightarrow X_3 \nonumber \\ X_3&\rightarrow X_1 \nonumber \\ \emptyset&\rightleftharpoons X_3 \end{aligned}$$
(12)

Every term of the form \(det(y_1,y_2,y_3)det(y'_1-y_1,y'_2-y_2,y'_3-y_3)\) from this network is zero and hence the deteminant of the Jacobian is zero. This implies that the determinant of B (which is the \((n-1)\times (n-1)\) minor of the Jacobian obtained by deleting its first row and last column) is zero. By Corollary 3.9, we get that the network \({\mathcal {G}}_3\) has perfect homeostasis at this linearly stable equilibrium.

5 Discussion

In this paper we have analyzed the notion of infinitesimal homeostasis (as introduced in Golubitsky and Stewart 2017), from the point of view of reaction network models. In particular, we have described a relationship between infinitesimal homeostasis and network injectivity, as well as a relationship between perfect homeostasis and the structure of the set of reaction vectors. Moreover, since injectivity of a network can be studied by looking at its directed species reaction graph (DSR graph) (Banaji and Craciun 2009), we have discussed how the DSR graph can be used to analyze homeostasis.

The current notion of infinitesimal homeostasis cannot apply to reaction systems with conservation laws (see Remark 3.10). An important direction for future work would be come up with an appropriate definition and analysis that removes this restriction.

Another interesting direction for future work would be the analysis of possible relationships between homeostasis (and especially perfect homeostasis) and absolute concentration robustness (ACR). The notion of ACR was first introduced in Shinar and Feinberg (2010), and refers to systems where the value of one of the variables (e.g., species concentration) is the same for all positive steady states of the system. At first, these two notions seem almost identical, but the ACR framework does not allow for any changes in parameter values. A deeper exploration of the mathematical relationships between homeostasis and absolute concentration robustness may uncover other network-level conditions for homeostasis.

Another promising direction for future work is the use of various forms of steady state parametrizations (Perez Millan and Dickenstein 2018; Feliu and Wiuf 2013) to analyze infinitesimal homeostasis. Given a certain steady state parametrization, the fact that the derivative of the output variable with respect to an input variable is zero at an equilibrium manifests itself as a property of a system of algebraic equations, whose analysis could provide useful insights into the behaviour of the system. Possible candidates for this work include toric (Pérez Millán et al. 2012) and rational (Thomson and Gunawardena 2009) steady state parametrizations.