Aspects of Reaction Network Structure

Abstract

Our aim in this chapter is to provide a vocabulary with which reaction network structure might be discussed. Although some of the terms will find use well before others, they are all intimately related. For this reason we will find it advantageous to locate their definitions in one place. At the end of the chapter, there is an appendix, in which we discuss a sense in which a reaction network can sometimes be decomposed into independent subnetworks. This is an important aspect of reaction network structure, one that belongs naturally in this chapter. Nevertheless, the discussion of the decomposition has been placed in an appendix because it is slightly more technical than the rest and is not essential to an understanding of the chapters immediately following.

Appendix 6.A Independent Subnetworks

In Section 6.4 we saw that it is not necessarily true that the rank of a reaction network is the same as the sum of the ranks of its linkage classes, nor is it invariably true that the deficiency of a network is the same as the sum of the deficiencies of its linkage classes. As we shall see, networks for which equality does hold often have nice properties, for then certain aspects of network behavior derive from the behavior of subnetworks that, in a sense, act independently of one another. Because this idea will arise later on more than once, it is worthwhile to elaborate on the decomposition of a reaction network into “independent subnetworks” [73] early and in a single place. That is the purpose of this appendix.

We begin with some elementary considerations in linear algebra [98]. Let V  be a vector space and let U be a linear subspace of V . When U ₁, U ₂, …, U _p are linear subspaces of U, we say that U is the sum of U ₁, U ₂, …, U _p if each vector x ∈ U has at least one representation of the form

$$\displaystyle \begin{aligned} x = x_1 + x_2 + \dots + x_p, \quad x_{\theta} \in U_{\theta},\; \theta = 1, 2, \dots, p. \end{aligned}$$

In this case we write

$$\displaystyle \begin{aligned} U = U_1 + U_2 + \dots + U_p. \end{aligned}$$

We say that U is the direct sum of U ₁, U ₂, …, U _p if, in addition, the equation

$$\displaystyle \begin{aligned} 0 = x_1 + x_2 + \dots + x_p, \quad x_{\theta} \in U_{\theta},\; \theta = 1, 2, \dots, p \end{aligned}$$

is satisfied only if x _θ = 0, θ = 1, 2, …, p. In this case we write

$$\displaystyle \begin{aligned} U = U_1 \oplus U_2 \oplus \dots \oplus U_p. \end{aligned}$$

Here is a necessary and sufficient condition for direct summing: When U is the sum of U ₁, U ₂, …, U _p, the sum is direct if and only if the dimensions of these various subspaces satisfy the equation

$$\displaystyle \begin{aligned} \dim U = \sum_{\theta =1}^p \dim U_{\theta}. \end{aligned}$$

Now let ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ be a reaction network with stoichiometric subspace S and rank s (= $\dim S$). Suppose that, for one reason or another, we choose to partition the reaction set ${\mathcal {R}}$ into subsets ${\mathcal {R}}_1, {\mathcal {R}}_2, \dots {\mathcal {R}}_p$. In this case, we can think of the partition as giving rise to a partition of the parent reaction network into subnetworks³ $\{{\mathcal {S}},{\mathcal {C}}_{\theta },{\mathcal {R}}_{\theta }\}, \theta = 1,2,\dots , p$, where

$$\displaystyle \begin{aligned} {\mathcal{C}}_{\theta} := \{y \in {\mathcal{C}}: \mbox{there exists}\ y \to y' \in {\mathcal{R}}_{\theta}\ \mathrm{or}\ y' \to y \in {\mathcal{R}}_{\theta}\}. \end{aligned} $$

(6.A.1)

Each subnetwork $\{{\mathcal {S}},{\mathcal {C}}_{\theta },{\mathcal {R}}_{\theta }\}$ will have its own stoichiometric subspace S _θ ⊂ S and its own rank s _θ (= $\dim S_{\theta }$). Because

$$\displaystyle \begin{aligned} S = \mathrm{span}\, \{y' - y \in {\mathbb{R}^{\mathcal{S}}}: y \to y' \in {\mathcal{R}}\}, \end{aligned}$$

and

$$\displaystyle \begin{aligned} S_{\theta} = \mathrm{span}\, \{y' - y \in {\mathbb{R}^{\mathcal{S}}}: y \to y' \in {\mathcal{R}}_{\theta}\}, \quad \theta = 1,2,\dots,p \end{aligned}$$

it is easy to see that

$$\displaystyle \begin{aligned} S = S_1 + S_2 + \dots + S_p. \end{aligned}$$

In fact, we have direct summing,

$$\displaystyle \begin{aligned} S = S_1 \oplus S_2 \oplus \dots \oplus S_p, \end{aligned} $$

(6.A.2)

precisely when the rank of the network and the ranks of the subnetworks satisfy the equation

$$\displaystyle \begin{aligned} s = s_1 + s_2 + \dots + s_p. \end{aligned} $$

(6.A.3)

In this case we say that the parent network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ has been partitioned into independent subnetworks $\{{\mathcal {S}},{\mathcal {C}}_{\theta },{\mathcal {R}}_{\theta }\}, \theta = 1,2,\dots , p$.

Remark 6.A.6

Even when there is no a priori specification of a reaction set partition, we shall sometimes find it convenient to say that a subnetwork $\{{\mathcal {S}},{\mathcal {C}}_{\ast },{\mathcal {R}}_{\ast }\}$ of a parent network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ is an independent subnetwork of the parent network. In this case, we mean that the subnetwork and its complement, derived from the reaction subset ${\mathcal {R}} \setminus {\mathcal {R}}_{\ast }$, constitute a partition of the parent network into two independent subnetworks. This is equivalent to saying that the ranks of the subnetwork and of its complement sum to the rank of the parent network.

Partition of a reaction network into independent subnetworks can sometimes be of considerable help in the study of a kinetic system derived from the network. In fact, suppose that ${\{\mathcal {S},\mathcal {C},\mathcal {R},\mathcal {K}\}}$ is a kinetic system and that a partition of the reaction set ${\mathcal {R}}$ induces a partition of the network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ into independent subnetworks $\{{\mathcal {S}},{\mathcal {C}}_{\theta },{\mathcal {R}}_{\theta }\}, \theta = 1,2,\dots , p$. In this case we can think of the kinetic system ${\{\mathcal {S},\mathcal {C},\mathcal {R},\mathcal {K}\}}$ as having been partitioned into kinetic subsystems $\{{\mathcal {S}},{\mathcal {C}}_{\theta },{\mathcal {R}}_{\theta },{\mathcal {K}}_{\theta }\}, \theta = 1,2,\dots , p$, where ${\mathcal {K}}_{\theta }$ is just the restriction of ${\mathcal {K}}$ to reactions in ${\mathcal {R}}_{\theta }$.

Recall that the species-formation-rate function $f:{\overline {\mathbb {R}}_+^{\mathcal {S}}} \to S$ for the kinetic system ${\{\mathcal {S},\mathcal {C},\mathcal {R},\mathcal {K}\}}$ is given by

$$\displaystyle \begin{aligned} f(c) := \sum_{ {\mathcal{R}}}{\mathcal{K}}_{{y \to y'}}(c)(y' - y) = \sum_{\theta = 1}^p\; \sum_{{\mathcal{R}}_{\theta}}{\mathcal{K}}_{{y \to y'}}(c)(y' - y). \end{aligned} $$

(6.A.4)

We can associate with each kinetic subsystem $\{{\mathcal {S}},{\mathcal {C}}_{\theta },{\mathcal {R}}_{\theta },{\mathcal {K}}_{\theta }\}$ its own species-formation-rate function $f_{\theta }:{\overline {\mathbb {R}}_+^{\mathcal {S}}} \to S_{\theta }$ in a natural way:

$$\displaystyle \begin{aligned} f_{\theta}(c) := \sum_{{\mathcal{R}}_{\theta}}{\mathcal{K}}_{{y \to y'}}(c)(y' - y). \end{aligned} $$

(6.A.5)

Then we clearly have, for each $c\in {\overline {\mathbb {R}}_+^{\mathcal {S}}}$,

$$\displaystyle \begin{aligned} f(c) = f_1(c) + f_2(c) + \dots + f_p(c) \end{aligned} $$

(6.A.6)

with f _θ(c) ∈ S _θ, θ = 1, 2, …p.

Now suppose that $c^{*} \in {\overline {\mathbb {R}}_+^{\mathcal {S}}}$ is an equilibrium of the kinetic system under discussion, in which case f(c ^∗) = 0 and, therefore,

$$\displaystyle \begin{aligned} 0 = f_1(c^*) + f_2(c^*) + \dots + f_p(c^*). \end{aligned} $$

(6.A.7)

By supposition the subnetworks $\{{\mathcal {S}},{\mathcal {C}}_{\theta },{\mathcal {R}}_{\theta }\}, \theta = 1,2,\dots , p$ are independent, so we have the direct sum (6.A.2), whereupon we must have

$$\displaystyle \begin{aligned} f_{\theta}(c^*) = 0,\quad \theta = 1, 2,\dots, p. \end{aligned} $$

(6.A.8)

This is to say that if c ^∗ is an equilibrium of the parent kinetic system ${\{\mathcal {S},\mathcal {C},\mathcal {R},\mathcal {K}\}}$ then it must be an equilibrium of each of the kinetic subsystems $\{{\mathcal {S}},{\mathcal {C}}_{\theta },{\mathcal {R}}_{\theta },{\mathcal {K}}_{\theta }\}$, θ = 1, 2, …, p separately. This simple idea will usually find use in the following way: Suppose that $E \subset {\overline {\mathbb {R}}_+^{\mathcal {S}}}$ is the set of equilibria for the kinetic system ${\{\mathcal {S},\mathcal {C},\mathcal {R},\mathcal {K}\}}$ and that $E_\theta \subset {\overline {\mathbb {R}}_+^{\mathcal {S}}}$, θ = 1, 2, …, p are the equilibrium sets of the kinetic subsystems $\{{\mathcal {S}},{\mathcal {C}}_{\theta },{\mathcal {R}}_{\theta },{\mathcal {K}}_{\theta }\}$, θ = 1, 2, …, p. Then

$$\displaystyle \begin{aligned} E = \bigcap_{\theta=1}^{p}E_{\theta}. \end{aligned} $$

(6.A.9)

Example 6.A.7 (A variant of the Edelstein network)

We show in (6.A.10) an augmented version of the Edelstein mass action network displayed earlier as (E.2) and as (6.A.11) below. The difference is in the addition of the reactions $A \rightleftarrows C$, taken with rate constants k and k′ of unspecified value. (The example, as structured, will prove useful—and, in fact, crucial—when we discuss the Deficiency One Theorem.)

(6.A.10)

It is easily determined that the rank of the augmented Edelstein network shown in (6.A.10) is three. Note that the network underlying the original Edelstein mass action system, shown below as (6.A.11), has a rank of two, while network underlying the small added mass action system (6.A.12) has a rank of one.

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} &\displaystyle A&\displaystyle \overset{8.5}{\underset{1}\rightleftarrows} 2A\\ A + B \overset{1}{\underset{1}\rightleftarrows} &\displaystyle C&\displaystyle \overset{1}{\underset{0.2}\rightleftarrows} B \end{array} \end{aligned} $$

(6.A.11)

$$\displaystyle \begin{aligned} A \overset{k}{\underset{k'}\rightleftarrows} C \end{aligned} $$

(6.A.12)

Because the two ranks sum to the rank of the parent network shown in (6.A.10), the two smaller networks constitute a partition of the parent network into independent subnetworks. Thus any equilibrium of the parent mass action system (6.A.10) must, in fact, be an equilibrium of each of the smaller mass action systems (6.A.11) and (6.A.12) separately.

Recall that the locus of equilibria for the mass action system (6.A.11) was drawn schematically in Figure E.3. On the other hand, the equilibria for the small mass action system (6.A.12) are those compositions that satisfy the equation

$$\displaystyle \begin{aligned} c_C = \frac{k}{k'}c_A. \end{aligned} $$

(6.A.13)

Represented in the context of Figure E.3, such compositions would constitute that part of the positive orthant consisting of a plane containing the B-axis and passing through the line in the C-A plane given by (6.A.13). Such a plane intersects the curve of equilibria for (6.A.11) in precisely one point. That point is the sole equilibrium of the mass action system (6.A.10).

As we shall see, the fact that there is precisely one positive equilibrium is an immediate consequence of the Deficiency One Theorem, discussed in Chapter H.

Remark 6.A.8 (When the deficiencies of the linkage classes sum to the deficiency of the network)

We shall see that an important condition in the hypothesis of the Deficiency One Theorem is that, for the network under consideration, the deficiencies of its linkage classes sum to the deficiency of the network as a whole. As we pointed out in Section 6.4, this is equivalent to the requirement that the ranks of the individual linkage classes sum to the rank of the network as a whole. If we associate a subnetwork with each of the ℓ linkage classes in the obvious way, then the ranks of the subnetworks and the rank of the entire network will satisfy

$$\displaystyle \begin{aligned} s = s_1 + s_2 + \dots + s_{\ell}, \end{aligned}$$

and the stoichiometric subspace of the entire network will be the direct sum of the stoichiometric subspaces associated with the individual linkage classes:

$$\displaystyle \begin{aligned} S = S_1 \oplus S_2 \oplus \dots \oplus S_{\ell}. \end{aligned} $$

(6.A.14)

In this case, the collection of subnetworks associated with the ℓ linkage classes constitute a partition of the parent network into independent subnetworks.

If the network is given a kinetics and c ^∗ is an equilibrium, then we must have

$$\displaystyle \begin{aligned} f_{\theta}(c^*) = 0,\quad \theta = 1, 2,\dots, \ell, \end{aligned} $$

(6.A.15)

where $f_{\theta }(\cdot ): {\overline {\mathbb {R}}_+^{\mathcal {S}}} \to S_{\theta }$ is the species-formation-rate function associated with linkage class ${\mathcal {L}}^{\theta }$. That is, equilibration must take place separately in the kinetic system associated with each linkage class.