Concordant Reaction Networks: Architectures That Promote Dull, Reliable Behavior Across Broad Kinetic Classes

Abstract

In this chapter we’ll resume a narrative that ran through Chapters 7 and 8. There our interest was largely in the relationship between the structure of a reaction network and its capacity (or lack of it) to give rise to instabilities or multiple equilibria within a positive stoichiometric compatibility class. The principal theorems centered mostly on a reaction network’s deficiency or, in some cases, on the deficiencies of the network’s linkage classes. In this chapter, however, deficiencies will play almost no role at all.

Appendices

Appendix 10.A Deducing Behavior of a Reaction Network from What Is Known About Its Behavior in a Fully Open Context

The body of work alluded to in Section 10.2 was aimed at understanding the surprisingly dull behavior exhibited, over a very wide range of different chemistries, by (isothermal) classical continuous-flow stirred-tank reactors (CFSTRs). In their underlying arguments, these results relied heavily on the mathematical features of the fully open setting—features that are apt for classical CFSTRs but not broadly. In biological applications, for example, we should not expect that, for every species , there will be a degradation reaction of the form .

When results derived in the fully open context are available, it is natural to wonder about the extent to which the fully open requirement can be relaxed. Stated differently, we would like to know when a statement that holds true for a network’s fully open extension, modified to account for stoichiometric compatibility, extends to the network itself. In this chapter there was considerable discussion of that very issue, but always in consideration of what can be said about a network’s behavior when it is known that the network’s fully open extension is concordant.

It is the purpose of this appendix to discuss, briefly and informally, certain results along these lines, results that transcend concordance considerations or even restrictions to particular classes of kinetics. For the sake of concretion, we will restrict our attention to information that derives from knowledge of the impossibility of multiple positive equilibria in a fully open setting.

It is reasonable to conjecture that, for a given chemistry, the impossibility of multiple positive equilibria in a fully open CFSTR context (including a feed stream) implies the impossibility of multiple positive stoichiometrically compatible equilibria for the chemistry itself, removed from the fully open setting. That conjecture, unmodified, would be false. Consider, for example, the pathological mass action system (10.A.1) considered earlier in Section 8.A.3. Recall that in each positive stoichiometric compatibility class there are multiple equilibria, in fact an infinite number of them. (See Figure 8.A.2.) Yet, it can be confirmed that, in a fully open CFSTR context, there is precisely one positive equilibrium, so long as there is some A in the feed stream.

$$\displaystyle \begin{aligned} B \overset{1}{\leftarrow} &A \overset{1}{\to} C\\ C + B &\overset{1}\to 2A \end{aligned} $$

(10.A.1)

The conjecture is, however, almost true. Implicit-function-theorem arguments in [47] indicate that if a reaction network, taken with a smooth kinetics, admits two distinct stoichiometrically compatible nondegenerate ²¹ positive equilibria, then there are parameter values for a classical CFSTR, within which the same kinetic system is operative, such that there are again two distinct positive equilibria. Therefore, a theorem that serves to deny the capacity for multiple positive equilibria in the fully open CFSTR context also gives information about the capacity for multiple stoichiometrically compatible positive equilibria for the original chemistry: There can be multiple stoichiometrically compatible positive equilibria for the original chemistry only if all but perhaps one of those equilibria are degenerate. Note that all positive equilibria of the mass action system (10.A.1) are degenerate.

Appendix 10.B Mass Action Injectivity

The principal theorems of this chapter circumscribe behavior for the very large class of concordant reaction networks when the kinetics is subject to very weak constraints (e.g., when the kinetics is differentiably monotonic). To a great extent, these theorems have significant roots in prior work restricted to mass action kinetics. In fact, some of the earlier theory tailored specifically to mass action systems permit similar theorems for a still broader class of networks, including some discordant ones. It is the purpose of this appendix to illuminate, briefly, some relationships between results in this chapter and results that derive from earlier mass-action-specific theory.

1.1 10.B.1 Two Similar Theorems, One Broad and One Mass-Action-Specific

By way of example, we will state two theorems—Theorems 10.B.2 and 10.B.3—one very general in the kinetics it invokes and one specific to mass action kinetics. Once these are in place, we will be in a position to consider an example that indicates how mass-action-restricted theorems can give information when broader theorems are silent.

As a precursor to the statement of Theorem 10.B.2, we recall Theorem 10.5.15, repeated below as Theorem 10.B.1. (See Section 10.B.3.1 for remarks about the proof.)

Theorem 10.B.1

For reaction network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ the following are equivalent:

• (i) ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ is concordant

• (ii) For every choice of a differentiably monotonic kinetics ${\mathcal {K}}$ the derivative of the species-formation-rate function for the kinetic system ${\{\mathcal {S},\mathcal {C},\mathcal {R},\mathcal {K}\}}$ is nonsingular at every positive composition.

Items (i) and (ii) in Theorem 10.B.1 are then equivalent to item (ii) in Theorem 10.5.5 and item (iii) in Proposition 10.5.8. For the purposes of this appendix, the important equivalence is the one expressed in the following theorem, which is merely a corollary of Theorems 10.B.1 and 10.5.5:

Theorem 10.B.2

For reaction network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ the following are equivalent:

• (i) For every weakly monotonic kinetics ${\mathcal {K}}$ the kinetic system ${\{\mathcal {S},\mathcal {C},\mathcal {R},\mathcal {K}\}}$ is injective.

• (ii) For every choice of a differentiably monotonic kinetics ${\mathcal {K}}$ the derivative of the species-formation-rate function for the kinetic system ${\{\mathcal {S},\mathcal {C},\mathcal {R},\mathcal {K}\}}$ is nonsingular at every positive composition.

In preparation for the statement of a mass action version of Theorem 10.B.2, we will want to qualify the meaning of kinetic-system injectivity just slightly. Recall that a kinetic system is injective if its species-formation-rate function f(⋅) has the property that f(c ^∗) ≠ f(c ^∗∗) whenever c ^∗ and c ^∗∗ are distinct stoichiometrically compatible compositions, at least one of which is positive. We shall say that a kinetic system is positive-composition injective if f(c ^∗) ≠ f(c ^∗∗) whenever c ^∗ and c ^∗∗ are distinct stoichiometrically compatible compositions, both positive.

Proof of the following theorem is given in Section 10.B.3.2. See also Remark 10.B.4.

Theorem 10.B.3

For reaction network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ the following are equivalent:

• (i) For every choice of rate constants $k \in {\mathbb {R}_+^{\mathcal {R}}}$ the mass action system ${\{\mathcal {S},\mathcal {C},\mathcal {R},k\}}$ is positive-composition injective.

• (ii) For every choice of rate constants $k \in {\mathbb {R}_+^{\mathcal {R}}}$ the derivative of the species-formation-rate function for the mass action system ${\{\mathcal {S},\mathcal {C},\mathcal {R},k\}}$ is nonsingular at every positive composition.

If a network satisfies either of the two equivalent conditions in Theorem 10.B.3, we will say that the network is mass action injective. For a network of interest, condition (ii) provides an attractive route to a computational test for mass action injectivity. Such a test is available in [62]. Computational aspects are discussed later in Section 10.B.4 and more fully in [115].

In fact, the two equivalences given by Theorems 10.B.2 and 10.B.3 have computational virtues: in both cases they tie difficult-to-study global properties in (i) of the nonlinear species-formation-rate function $f: {\overline {\mathbb {R}}_+^{\mathcal {S}}} \to S$ directly to properties in (ii) of a more local object, the linear transformation df(c) : S → S.

Remark 10.B.4

Theorem 10.B.3 has origins in the Ph.D. thesis of Gheorghe Craciun [44], which focused on fully open reaction networks. For the fully open case, proofs appeared there and in [46], along with determinant-based theorems for assessing injectivity of fully open mass action systems. These in turn provided foundations for Species-Reaction Graph mass action theorems given in [47] and [52].

For networks that are not fully open, stoichiometric considerations exert themselves, and the proof becomes slightly more complicated. For this more general case, Theorem 10.B.3 was stated in Remark 6.4 of [49], where determinant conditions for mass action injectivity were again provided, this time for networks that are not presumed to be fully open.²² For the purpose of documenting the mass-action-injectivity computational module of [62], a proof of Theorem 10.B.3—essentially the one given in Section 10.B.3.2—was included in the Ph.D. thesis of Haixia Ji [115]. A proof was also given [84], where a determinant-based Mathematica-like script for assessment of mass action injectivity was provided.

1.2 10.B.2 Two Instructive Examples

In this section we examine two networks that teach very different lessons.

Example 10.B.5

Consider first network (10.B.2).

(10.B.2)

The network is discordant, as can be determined by means of [62]. Theorem 10.5.5 then tells us that there is a weakly monotonic kinetics for the network such that the resulting kinetic system is not injective. In fact, Theorem 10.5.10 tells us more: there is for the network a weakly monotonic kinetics for which the resulting kinetic system admits a pair of distinct positive stoichiometrically compatible equilibria. Moreover, Theorem 10.B.1 ensures for the network the existence of a differentiably monotonic kinetics such that at some positive composition the derivative of the species-formation-rate function is singular. In particular, Theorem 10.7.7 tells us that the kinetics can be chosen in such a way as to engender a degenerate positive equilibrium.

On the other hand, the network is mass action injective; this too can be determined by means of [62]. Theorem 10.B.3 then tells us that, while there is indeed some kinetics for the network that give rise to the various phenomena just described, there is no mass action kinetics for which these phenomena will be admitted, no matter what rate constants are assigned to the various reactions.²³

The example demonstrates concretely that, for at least certain networks, the equivalent behavior-restricting conditions in the mass action Theorem 10.B.3 might be satisfied, while those in the more kinetically expansive Theorem 10.B.2 are not. Indeed, theory focused exclusively on the archetypal mass action kinetics has a special richness all its own.

Example 10.B.6

There is a possible source of confusion that we would do well to mitigate by means of an example. Recall network (10.B.3), which we met earlier in Section 10.4.

$$\displaystyle \begin{aligned} A+B &\rightleftarrows P \\ B+C &\rightleftarrows Q\\ C &\rightleftarrows 2A \end{aligned} $$

(10.B.3)

The network is not concordant, nor is it mass action injective [62]. In particular, neither of the two equivalent conditions in Theorem 10.B.3 is satisfied. This tells us that there is a rate constant specification such that the resulting mass action system admits a pair of stoichiometrically compatible positive compositions, c ^∗ and c ^∗∗, at which the species-formation-rate function takes identical values: f(c ^∗) = f(c ^∗∗). Moreover, there is a (perhaps different) rate constant specification such that at some positive composition, the derivative of the species-formation-rate function has a singular derivative.

Note, however, that network (10.B.3) has a deficiency of zero (n = 6, ℓ = 3, s = 3).

The Deficiency Zero Theorem (Theorem 7.1.1) tells us that for no assignment of rate constants can the resulting mass action system admit a pair of stoichiometrically compatible positive equilibria. This is to say that, although for some rate constant assignment the species-formation-rate function can indeed take the same value at two different stoichiometrically compatible positive compositions, that value cannot be zero.

Moreover, the local version of the Deficiency Zero Theorem (Theorem 7.3.1 ) tells us that for no assignment of rate constants can there be a positive equilibrium that is degenerate. This is to say that, although for some rate constant assignment there can be a positive composition at which the derivative of species-formation-rate function is singular, that composition cannot be an equilibrium.

The lesson here is this: Although mass action injectivity is a network attribute that does indeed preclude multiple stoichiometrically compatible positive equilibria and degenerate positive equilibria, regardless of rate constant values, the same preclusions might result even for networks that are not mass action injective.

1.3 10.B.3 Some Proofs

In this section we provide proofs of Theorems 10.B.1 and 10.B.3.

1.3.1 10.B.3.1 Proof of Theorem 10.B.1

The implication (i) ⇒ (ii) is the subject of Theorem 10.5.14, which was proved in [157]. Proof that (ii) ⇒ (i) is essentially contained in the proof of Theorem 10.7.7, also given in [157]; we provide the relevant part of that argument below. (Theorem 10.7.7 more or less gives what we want here, but its emphasis is on the stability of positive equilibria.)

Proof ((ii)⇒ (i))

Suppose that (ii) holds but that the network is discordant. In this case, Proposition 10.6.25 tells us that we can choose $\{p_{{y \to y'}}\}_{{y \to y'} \in {\mathcal {R}}} \subset {\overline {\mathbb {R}}_+^{\mathcal {S}}}$, with ${\mathrm {supp} \,} p_{{y \to y'}} = {\mathrm {supp} \,} y$, $\forall \mbox{{$y \to y'$}} \in {\mathcal {R}}$, such that the linear transformation T : S → S given by

$$\displaystyle \begin{aligned} T\sigma \; := \sum_{{y \to y'}\, \in \, {\mathcal{R}}}p_{y \to y'} \cdot \sigma(y' - y) \end{aligned} $$

(10.B.4)

is singular. Now let $c^* \in {\overline {\mathbb {R}}_+^{\mathcal {S}}}$ be some fixed positive composition, and for the network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ let ${\mathcal {K}}$ be the kinetics defined, for each ${y \to y'} \in {\mathcal {R}}$, by

$$\displaystyle \begin{aligned} {\mathcal{K}}_{{y \to y'}}(c) := \eta_{{y \to y'}}c^{q_{{y \to y'}}}, \end{aligned} $$

(10.B.5)

where

$$\displaystyle \begin{aligned} q_{{y \to y'}} := c^*p_{{y \to y'}} \quad \mathrm{and} \quad \eta_{{y \to y'}} := [(c^*)^{q_{{y \to y'}}}]^{-1}. \end{aligned} $$

(10.B.6)

For this choice, the derivative at c ^∗ of the species-formation-rate function for the differentiably monotonic kinetic system ${\{\mathcal {S},\mathcal {C},\mathcal {R},\mathcal {K}\}}$ coincides with the singular linear transformation T : S → S given by (10.B.4). This contradicts (ii). □

1.3.2 10.B.3.2 Proof of Theorem 10.B.3

Recall from Remark 3.6.2 that, for a mass action system ${\{\mathcal {S},\mathcal {C},\mathcal {R},k\}}$ with stoichiometric subspace S, the derivative of the species-formation-rate function f(⋅) has a special form: At composition $a \in {\mathbb {R}_+^{\mathcal {S}}}$ the derivative df(a) : S → S is given by the requirement that, for each σ ∈ S,

$$\displaystyle \begin{aligned} d f(a)\sigma = \sum_{{y \to y'} \in {\mathcal{R}}}\eta_{{y \to y'}}(y*_{a}\sigma)(y'-y), \end{aligned} $$

(10.B.7)

where, for each ${y \to y'} \in {\mathcal {R}}$,

$$\displaystyle \begin{aligned} \eta_{{y \to y'}} := k_{{y \to y'}}(a)^y \end{aligned} $$

(10.B.8)

and “∗_a” indicates the scalar product in ${\mathbb {R}^{\mathcal {S}}}$ defined by

(10.B.9)

With this in mind, we can restate Theorem 10.B.3 in the following way, essentially asserting, in more detail, the equivalence of the negations of (i) and (ii) in the original statement.

Theorem 10.B.7 ( [49, 115])

For reaction network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ with stoichiometric subspace S the following are equivalent:

• (i) There is a choice of rate constants $k \in {\mathbb {R}_+^{\mathcal {R}}}$ and two distinct positive compositions c ^∗ and c ^∗∗ , with c ^∗− c ^∗∗∈ S, such that

  $$\displaystyle \begin{aligned} \sum_{y \to y' \in \mathcal{R}}k_{y \to y'}(c^*)^y(y'-y) = \sum_{y \to y' \in \mathcal{R}}k_{y \to y'}(c^{**})^y(y'-y). \end{aligned} $$

  (10.B.10)

• (ii) There is a choice of $\eta \in {\mathbb {R}_+^{\mathcal {R}}}$ and $a \in {\mathbb {R}_+^{\mathcal {S}}}$ such that the linear transformation T _η,a : S → S given by

  $$\displaystyle \begin{aligned} T_{\eta,\,a}\sigma\; := \sum_{{y \to y'}\ \in\ {\mathcal{R}}}\eta_{{y \to y'}}(y*_{a}\sigma)(y' - y) \end{aligned} $$

  (10.B.11)

  is singular.

Proof

(i) ⇒ (ii): With k, c ^∗, and c ^∗∗ as in (i), we can rewrite (10.B.10) in the following form:

$$\displaystyle \begin{aligned} 0\;\; &= \sum_{y \to y' \in \mathcal{R}}k_{y \to y'}(c^*)^y\,[\frac{(c^{**})^y}{(c^{*})^y} - 1 ](y'-y)\\ &= \sum_{y \to y' \in \mathcal{R}}k_{y \to y'}(c^*)^y\,[e^{y\cdot\mu} - 1 ](y'-y), \end{aligned} $$

(10.B.12)

where “⋅” is the standard scalar product in ${\mathbb {R}^{\mathcal {S}}}$ and

$$\displaystyle \begin{aligned} \mu := \ln c^{**} - \,\ln c^{*}. \end{aligned} $$

(10.B.13)

Now let

$$\displaystyle \begin{aligned} \sigma := c^{**} - c^* \in S. \end{aligned} $$

(10.B.14)

Note that, componentwise, sgn μ = sgn σ. Therefore, there is an $a \in {\mathbb {R}_+^{\mathcal {S}}}$ such that $\mu = \frac {1}{a}\sigma$. Then, in (10.B.12), we can replace y ⋅ μ by y ∗_a σ, where “∗_a” is the scalar product in ${\mathbb {R}^{\mathcal {S}}}$ given by (10.B.9).

Note also that ${\mathrm {sgn} \,}(e^{y*_a\sigma }- 1) = {\mathrm {sgn} \,}(y*_a\sigma )$. Thus, for each $y \in {\mathcal {C}}$, there is a positive number p _y such that $e^{y*_a\sigma }- 1 = p_y(y*_a\sigma )$. If, for each ${y \to y'} \in {\mathcal {R}}$, we set $\eta _{y\to y'} := k_{y \to y'}(c^*)^yp_y$, it follows from (10.B.12) that

$$\displaystyle \begin{aligned} \sum_{y \to y' \in \mathcal{R}} \eta_{y \to y'}(y*_a\sigma)(y' - y) = 0. \end{aligned} $$

(10.B.15)

Because σ is not zero, it follows from (10.B.15) that the linear transformation T _η,a : S → S defined in (ii) is singular.

(ii) ⇒ (i): Suppose that $\eta \in {\mathbb {R}_+^{\mathcal {R}}}$, $a \in {\mathbb {R}_+^{\mathcal {S}}}$, and nonzero σ ∈ S are such as to make (10.B.15) true. Let

$$\displaystyle \begin{aligned} \mu := \frac{1}{a}\sigma, \end{aligned} $$

(10.B.16)

in which case (10.B.15) becomes

$$\displaystyle \begin{aligned} \sum_{y \to y' \in \mathcal{R}} \eta_{y \to y'}(y\cdot\mu)(y' - y) = 0. \end{aligned} $$

(10.B.17)

Because sgn (y ⋅ μ) = sgn (e ^y⋅μ − 1), it is not difficult to see that we can choose $\bar {\eta } \in {\mathbb {R}_+^{\mathcal {R}}}$ such that

$$\displaystyle \begin{aligned} \sum_{y \to y' \in \mathcal{R}} \bar{\eta}_{y \to y'}(e^{y\cdot\mu} - 1)(y' - y) = 0. \end{aligned} $$

(10.B.18)

Because, componentwise, sgn (μ) = sgn (σ), we also have sgn (e ^μ − 1) = sgn σ. Thus, there is c ^∗∈ ${\mathbb {R}_+^{\mathcal {S}}}$ such that σ = c ^∗(e ^μ − 1). Now let c ^∗∗ := c ^∗ e ^μ. Note that c ^∗∗≠ c ^∗, c ^∗∗− c ^∗ = σ ∈ S, and $\mu = \ln c^{**} - \ln c^*$. After choosing k ∈ ${\mathbb {R}_+^{\mathcal {R}}}$ to satisfy $\bar {\eta }_{y\to y'} = k_{y \to y'}(c^*)^y, \forall \, y \to y' \in \mathcal {R}$, we can begin with (10.B.18) and then reverse the steps in (10.B.12) to obtain (10.B.15). □

1.4 10.B.4 A Route to the Determination of Mass Action Injectivity

Condition (ii) of Theorem 10.B.7 suggests several routes to establishing that a given network is or is not mass action injective. In particular, it is easy to see that (ii) is equivalent to the following:

(iii) There is a choice of $\eta \in {\mathbb {R}_+^{\mathcal {R}}}$ such that the linear transformation $\bar {T}_{\eta }:{\mathbb {R}^{\mathcal {S}}} \to S$ given by

$$\displaystyle \begin{aligned} \bar{T}_{\eta}\gamma\; := \sum_{{y \to y'}\ \in\ {\mathcal{R}}}\eta_{{y \to y'}}(y \cdot \gamma)(y' - y) \end{aligned} $$

(10.B.19)

has in its kernel a vector that is sign-compatible with S.

Here “⋅” denotes the standard scalar product in ${\mathbb {R}^{\mathcal {S}}}$. Recall that a vector $\gamma ^{*} \in {\mathbb {R}^{\mathcal {S}}}$ is sign-compatible with S (Definition 8.5.1) if there exists σ ∈ S such that, componentwise, sgn γ ^∗ = sgn σ. This formulation permits a connection with the linear transformation $L:{\mathbb {R}^{\mathcal {R}}} \to S$, given by

$$\displaystyle \begin{aligned} L\alpha = \sum_{{y \to y'} \in {\mathcal{R}}}\alpha_{{y \to y'}}(y' - y), \end{aligned} $$

(10.B.20)

the linear transformation that appears in the definition of concordance. Clearly, (iii) is equivalent to the following:

(iv) There exist $\alpha \in \ker L$ and a nonzero $\gamma \in {\mathbb {R}^{\mathcal {S}}}$, sign-compatible with S, such that ${\mathrm {sgn} \,}(y \cdot \gamma ) = {\mathrm {sgn} \,} \alpha _{{y \to y'}}$ for all ${y \to y'} \in {\mathcal {R}}$.

Mass action injectivity (or lack of it) can be determined—as it is in [62]—by means of sign-checking along lines described in (iv). For algorithmic aspects see [115].