Reaction Networks, Kinetics, and the Induced Differential Equations

Abstract

In the first three sections of this chapter, we’ll make precise some of the ideas that were introduced casually in Chapter 2. Section 3.1 contains our definition of a reaction network along with a small amount of auxiliary terminology. In Section 3.2 we introduce the notion of a kinetics for a network, and we discuss mass action kinetics as the archetypal example. In Section 3.3 we indicate in vectorial terms how a kinetic system—that is, a reaction network endowed with a kinetics—induces a system of differential equations.

Appendix 3.A The Kinetic Subspace

This appendix amounts to a technical digression. Although it can be skipped for now, its content sheds light on issues that will arise later on. The placement of the appendix here results from its intimate connection to ideas in Section 3.4.

For almost all that we will do, our discussion of the stoichiometric subspace, stoichiometric compatibility, and stoichiometric compatibility classes will be more than adequate. There are, however, certain instances in which we will benefit from a sharpening of observations we’ve already made. We’ll begin to lay the groundwork for that sharpening here.

For a kinetic system ${\{\mathcal {S},\mathcal {C},\mathcal {R},\mathcal {K}\}}$ our introduction of the stoichiometric subspace derived from the fact that at every composition $c \in {\overline {\mathbb {R}}_+^{\mathcal {S}}}$, the corresponding value of the species-formation-rate function, f(c), takes values in the span of the reaction vectors for the underlying reaction network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$. Thus, with im f denoting the image of f(⋅) and S denoting the network’s stoichiometric subspace, we invariably have the relation

$$\displaystyle \begin{aligned} {\mathrm{im} \,} f \subset S. \end{aligned} $$

(3.A.1)

This is to say that values of f(⋅) can’t point just anywhere. They must point along the stoichiometric subspace, and so too must values of the “velocity vector” $\dot {c}$. This is true regardless of the nature of the kinetics ${\mathcal {K}}$.

Clearly, though, we also have the relation

$$\displaystyle \begin{aligned} {\mathrm{im} \,} f \subset \mathrm{span} ({\mathrm{im} \,} f), \end{aligned} $$

(3.A.2)

where span(im f) is the smallest linear subspace of ${\mathbb {R}^{\mathcal {S}}}$ containing im f. Now it might be the case that span(im f) is a smaller linear subspace than S, in which case (3.A.2) gives a sharpening of (3.A.1). This possibility serves as motivation for the following definition:

Definition 3.A.1

The kinetic subspace K for a kinetic system ${\{\mathcal {S},\mathcal {C},\mathcal {R},\mathcal {K}\}}$ with species formation function f(⋅) is the linear subspace of ${\mathbb {R}^{\mathcal {S}}}$ defined by

$$\displaystyle \begin{aligned} K := \mathrm{span} ({\mathrm{im} \,} f). \end{aligned} $$

(3.A.3)

1.1 3.A.1 When the Kinetic Subspace Is Smaller than the Stoichiometric Subspace

To see that the kinetic subspace can indeed be smaller than the stoichiometric subspace, we begin with a concrete example that will also play a role later in the book.

Example 3.A.2

Consider the mass action system (3.A.4).

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

(3.A.4)

Note that the stoichiometric subspace for the underlying network is given by

$$\displaystyle \begin{aligned} S = \mathrm{span}\{B-A, C-A, 2A - B - C\}, \end{aligned} $$

(3.A.5)

the dimension of which is easily seen to be two. The species-formation-rate function $f:{\overline {\mathbb {R}}_+^{\mathcal {S}}} \to S$ is given by

$$\displaystyle \begin{aligned} f(c) = c_A(\alpha(B-A) + (C-A)) + c_Cc_B(2A - B - C). \end{aligned} $$

(3.A.6)

It is not difficult to see that the kinetic subspace K ⊂ S is given by

$$\displaystyle \begin{aligned} K = \mathrm{span}\{\alpha(B-A) + (C-A),\; 2A - B - C\}. \end{aligned} $$

(3.A.7)

Note that, for α ≠ 1, the vectors α(B − A) + (C − A) and 2A − B − C are not colinear. In this case the dimension of K is two, whereupon K coincides with S. On the other hand, for the very special value α = 1, the vectors α(B − A) + (C − A) and 2A − B − C are colinear. In this case the dimension of K is one, and K is actually smaller than S.

Remark 3.A.3

Example 3.A.2 teaches an important lesson: At least for certain mass action systems, the nature and, in particular, the dimension of the kinetic subspace can vary in a jarring way as values of the rate constants change. For the mass action system (3.A.4), even the smallest perturbation away from the rate constant value α = 1 causes a sudden explosion in the dimension of the kinetic subspace. The stoichiometric subspace, on the other hand, is unaffected by any kinetic parameter values, for it depends solely on the set of reactions and not at all on the rate functions associated with the reactions.

Remark 3.A.4

When the kinetic subspace is smaller than the stoichiometric subspace (as in the preceding example for the special case α = 1), there can be behavior that is troubling (or at least inconvenient). Consider, for example, a kinetic system ${\{\mathcal {S},\mathcal {C},\mathcal {R},\mathcal {K}\}}$ with species-formation-rate function f(⋅) and kinetic subspace K. Moreover, suppose that S is the stoichiometric subspace for the underlying network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$.

If K is smaller than S, then, at every composition $c^* \in {\mathbb {R}_+^{\mathcal {S}}}$, the derivative df(c ^∗) : S → S is singular. In fact, from a standard theorem of linear algebra relating the dimensions of the domain, kernel, and image of a linear transformation, we have

$$\displaystyle \begin{aligned} \dim \ S = \dim\ \ker\ df(c^*) + \dim\ {\mathrm{im} \,}\ df(c^*). \end{aligned} $$

(3.A.8)

Because f(⋅) takes values in K, it follows from (3.32) that the linear map df(c ^∗) takes values in K for each $c^* \in {\mathbb {R}_+^{\mathcal {S}}}$. Thus, for each $c^* \in {\mathbb {R}_+^{\mathcal {S}}}$ we have $\dim \ {\mathrm {im} \,}\ df(c^*) \leq \dim K < \dim \, S$, whereupon $\dim \ \ker \, df(c^*) > 0$ in which case df(c ^∗) is singular. In particular, in the sense of Section 3.6, every positive equilibrium is degenerate.

1.2 3.A.2 Should We Focus on the Kinetic Rather than the Stoichiometric Subspace?

Our discussion in Section 3.4 gave considerable geometric insight into the nature of trajectories for the differential equation $\dot {c} = f(c)$, where f(⋅) is the species-formation-rate function for the kinetic system under study. The entire discussion was predicated on the fact that f(⋅) takes values in the stoichiometric subspace for the underlying reaction network, in which case the “velocity vector” $\dot {c}$ must invariably point along the stoichiometric subspace.

As we’ve noted, however, values of f(⋅) and, consequently, of $\dot {c}$ must also point along the (perhaps smaller) kinetic subspace for the kinetic system at hand. Because $\dot {c}$ is constrained to point along the kinetic subspace K, the reasoning used in Section 3.4 can be repeated, in almost everything we said there, to replace the stoichiometric subspace S with the kinetic subspace K (and to replace stoichiometric compatibility with kinetic compatibility, defined in the obvious way). Indeed, the dilemma indicated in Remark 3.A.4 could have been mitigated if, for a particular kinetic system under study, we had regarded the derivative df(c ^∗) not as a map from S to S but instead as a map from K to K.

In view of the fact that K might be smaller than S, should we, then, change our focus from the stoichiometric subspace to the kinetic subspace? For us, the answer will be no. There are two important, intimately related, reasons for resisting the temptation:

1.2.1 3.A.2.1 Mass Action Systems for Which K ≠ S Never Have That Property Robustly

Because of their fundamental importance in the mathematical description of reacting mixtures, mass action systems play a central role in this book. Even kinetic systems that are not of mass action type often come about as approximations to more refined mass action models. In the following sense, mass action systems for which K ≠ S never have that property robustly:

If a mass action system (with specified rate constant values) has a kinetic subspace that is smaller than the stoichiometric subspace, then it is always possible to make certain reactions reversible, with arbitrarily small rate constants for the reactions added, such that the new mass action system has the same stoichiometric subspace as the old one and for which the new kinetic subspace now coincides with the stoichiometric subspace.⁸ This is discussed more fully in an appendix to Chapter 8; see also [73, 80].

Even when the reaction network itself remains unperturbed, tiny perturbations of those rate constants for which K ≠ S can make for great changes in qualitative behavior. For example, the mass action system (3.A.4) admits an infinite number of positive equilibria in each positive stoichiometric compatibility class when α = 1 (in which case the kinetic subspace is smaller than the stoichiometric subspace). On the other hand, there are no positive equilibria at all when α ≠ 1 (in which case the kinetic and stoichiometric subspaces coincide). Striking mathematical phenomena that disappear completely with the smallest perturbations of the model are of dubious relevance to the study of real chemistry.

1.2.2 3.A.2.2 The Kinetic Subspace Is Not an Attribute of a Reaction Network; It Is an Attribute of a Particular Kinetic System

It should be kept in mind that the main goal of chemical reaction network theory is to make bold statements about the behavior of reaction networks, taken perhaps with kinetics circumscribed to lie within a very broad class. This aligns well with what is typically known in chemistry: In all but the simplest networks, knowledge of kinetic parameter values (e.g., rate constants) is almost always poor. Note that the stoichiometric subspace depends only on the reaction network and not at all on kinetics. Thus, to know the stoichiometric subspace, one need only know the network, not the fine details of kinetic parameter values.⁹ By way of contrast, kinetic parameter values influence the kinetic subspace directly. Indeed, as Example 3.A.2 demonstrates, even the dimension of the kinetic subspace might, and sometimes does, hinge on certain exceptional kinetic parameter values. Focus on the kinetic subspace would simply be inappropriate to the larger goals of chemical reaction network theory. Again, the stoichiometric subspace is an attribute of a reaction network, while the kinetic subspace is an attribute of a reaction network taken with a particular kinetics.

1.3 3.A.3 Thinking (and Not Thinking) About the Kinetic Subspace

In summary, it is important to be aware that, in certain (typically non-robust) cases, the kinetic subspace might be smaller than the stoichiometric subspace and that, when this happens, certain odd behavior might appear. But we shall not let that small tail wag a very large dog. Our focus will remain on the stoichiometric subspace, which derives from the reaction network alone and not at all from peculiarities of one kinetics or another.

One of the fortuitous blessings of chemical reaction network theory is that, at least for mass action systems, there is a very large, highly robust, and easily described class of reaction networks for which the stoichiometric and kinetic subspaces coincide, no matter what values the rate constants take [73, 80]. We will discuss this class in an appendix to Chapter 8, at which point we will have more vocabulary at our disposal.¹⁰ For now it suffices to say that every network for which each reaction is reversible is a member of this class, but the class is far larger. ¹¹ For networks in this class, knowledge of the network alone suffices for knowledge of both K and S.