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 subnetworks

^{3} \(\{{\mathcal {S}},{\mathcal {C}}_{\theta },{\mathcal {R}}_{\theta }\}, \theta = 1,2,\dots , p\), 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* \(\{{\mathcal {S}},{\mathcal {C}}_{\theta },{\mathcal {R}}_{\theta }\}, \theta = 1,2,\dots , p\).

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)