Deficiency One Theory Foundations

Abstract

This chapter has two goals. First, we want to provide not only a proof of the Deficiency One Theorem but also a guide to the thinking behind the proof. Another salt theorem, motivated by the same salt-barrel picture met in the last chapter, will play a crucial role. Second, we want to elaborate more fully on the so-called Deficiency One Algorithm described in Section 8.5. The algorithm itself will be laid out in far more detail, and its theoretical underpinnings will be explained. In this, the salt-barrel picture will find use yet again.

Appendix 17.A Proof of the Second Salt Theorem

In this appendix we provide a proof of the Second Salt Theorem, which we repeat below as Theorem 17.A.1.

Theorem 17.A.1 (Second Salt Theorem [76])

For a reaction network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ , let α be an element of ${\mathbb {R}_+^{\mathcal {R}}}$ , let $A_{\alpha }:{\mathbb {R}^{\mathcal {C}}} \to {\mathbb {R}^{\mathcal {C}}}$ be the linear transformation defined by

$$\displaystyle \begin{aligned} A_{\alpha}\, x := \sum_{{y \to y'}\ \in\ {\mathcal{R}}}\ \alpha_{{y \to y'}}x_y(\omega_{y'} - \omega_y), \end{aligned} $$

(17.A.1)

and let

$$\displaystyle \begin{aligned} g := A_{\alpha}\omega_{{\mathcal{C}}}. \end{aligned} $$

(17.A.2)

Moreover, let $\varLambda \subset {\mathcal {C}}$ be a terminal strong-linkage class, and let $b \in {\overline {\mathbb {R}}_+^{\mathcal {C}}}$ an element of $\ker \; A_{\alpha }$ with supp b = Λ. If the complexes of Λ are arranged in a sequence y(l), y(2), …, y(m) so that

$$\displaystyle \begin{aligned} b_{y(l)} \geq b_{y(2)} \geq b_{y(3)} \geq \dots \geq b_{y(m)}, \end{aligned} $$

(17.A.3)

then, for θ = 1, 2, … , m,

$$\displaystyle \begin{aligned} \sum_{i = 1}^{\theta} g_{y(i)} \geq 0. \end{aligned} $$

(17.A.4)

In fact, for each 1 ≤ θ < m, inequality holds in (17.A.4) if b _y(θ) is greater than b _y(θ+1) . When θ = m, inequality holds in (17.A.4) if and only if Λ is smaller than the linkage class containing it.

Hereafter we invoke the following notation: When, for a network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$, ${\mathcal {P}} \subset {\mathcal {C}}$ is a set of complexes, we denote, as before, the complement of ${\mathcal {P}}$ in ${\mathcal {C}}$ by ${\mathcal {P}}\,'$. When ${\mathcal {P}}$ and ${\mathcal {Q}}$ are sets of complexes, we denote by ${\mathcal {P}} \to {\mathcal {Q}}$ the set of reactions having reactant complex in ${\mathcal {P}}$ and product complex in ${\mathcal {Q}}$. That is,

$$\displaystyle \begin{aligned} {\mathcal{P}} \to {\mathcal{Q}}\ :=\ \{{y \to y'} \in {\mathcal{R}}\ : \ y \in {\mathcal{P}},\ y' \in {\mathcal{Q}}\}. \end{aligned} $$

(17.A.5)

In particular, ${\mathcal {C}} \to \{y\}$ and $\{y\} \to {\mathcal {C}}$ are what we have sometimes denoted by ${\mathcal {R}}_{\to y}$ and ${\mathcal {R}}_ {y \to }$.

Lemma 17.A.2

For a reaction network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ , let α be an element of ${\mathbb {R}^{\mathcal {R}}}$ , and let $A_{\alpha }:{\mathbb {R}^{\mathcal {C}}} \to {\mathbb {R}^{\mathcal {C}}}$ be as in (17.A.1). If x and z are elements of ${\mathbb {R}^{\mathcal {C}}}$ related by

$$\displaystyle \begin{aligned} z = A_{\alpha}\,x \end{aligned} $$

(17.A.6)

then, for each ${\mathcal {P}} \subset {\mathcal {C}}$,

$$\displaystyle \begin{aligned} \sum_{y\ \in\ {\mathcal{P}}}\,z_y\ = \ \sum_{{\mathcal{P}}\,' \to {\mathcal{P}}}\,\alpha_{y' \to y}\,x_{y'} \ - \sum_{{\mathcal{P}} \to {\mathcal{P}}\,'}\,\alpha_{y \to y'}\,x_{y}. \end{aligned} $$

(17.A.7)

Proof

From (17.A.1) it follows that

$$\displaystyle \begin{aligned} z_y = \ \sum_{{\mathcal{C}} \to \{y\}}\,\alpha_{y' \to y}\,x_{y'} \ - \sum_{\{y\} \to {\mathcal{C}}}\,\alpha_{y \to y'}\,x_{y}. \end{aligned} $$

(17.A.8)

Summing over ${\mathcal {P}} \subset {\mathcal {C}}$ we obtain

$$\displaystyle \begin{aligned} \sum_{y\ \in\ {\mathcal{P}}}\, z_y = \ \sum_{{\mathcal{C}} \to {\mathcal{P}}}\,\alpha_{y' \to y}\,x_{y'} \ - \sum_{{\mathcal{P}} \to {\mathcal{C}}}\,\alpha_{y \to y'}\,x_{y}. \end{aligned} $$

(17.A.9)

Note, however, that

$$\displaystyle \begin{aligned} {\mathcal{C}} \to {\mathcal{P}} &= ({\mathcal{P}} \to {\mathcal{P}})\ \cup \ ({\mathcal{P}} \,' \to {\mathcal{P}})\\ {\mathcal{P}} \to {\mathcal{C}} &= ({\mathcal{P}} \to {\mathcal{P}})\ \cup \ ({\mathcal{P}} \to {\mathcal{P}} \,'). \end{aligned} $$

From this and (17.A.9), equation (17.A.7) follows. □

We turn now to proof of the Second Salt Theorem.

Proof (of Theorem 17.A.1)

Let ${\mathcal {P}}$ be a subset of ${\mathcal {C}}$. Because $A_{\kappa }\,\omega _{{\mathcal {C}}} = g$ and A _κ b = 0, Lemma 17.A.2 gives the following equations:

$$\displaystyle \begin{aligned} \sum_{{\mathcal{P}}\,' \to {\mathcal{P}}}\,\kappa_{y' \to y} \ - \sum_{{\mathcal{P}} \to {\mathcal{P}}\,'}\,\kappa_{y \to y'} = \sum_{y\ \in\ {\mathcal{P}}}\,g_y \end{aligned} $$

(17.A.10)

$$\displaystyle \begin{aligned} \sum_{{\mathcal{P}}\,' \to {\mathcal{P}}}\,\kappa_{y' \to y}\,b_{y'} \ - \sum_{{\mathcal{P}} \to {\mathcal{P}}\,'}\,\kappa_{y \to y'}\,b_y = 0. \end{aligned} $$

(17.A.11)

We will begin by proving Theorem 17.A.1 for the case θ = m. In this case we identify ${\mathcal {P}}$ in equation (17.A.10) with Λ = {y(1), y(2), …, y(m)}. Because Λ is a terminal strong-linkage class, Λ → Λ′ is empty and (17.A.10) becomes

$$\displaystyle \begin{aligned} \sum_{\varLambda' \to \varLambda}\,\kappa_{y' \to y}\ =\ \sum_{i = 1}^m\,g_{y(i)}. \end{aligned} $$

(17.A.12)

If Λ is smaller than the linkage class containing it, Λ′→ Λ is not empty, and the positivity of κ ensures that both sides of (17.A.12) are positive. If Λ coincides with the linkage class containing it, Λ′→ Λ is empty, and both sides of (17.A.12) are zero. (That the right side is zero also follows from Remark 16.2.5 and the fact that g lies in im A _κ ⊂span (Δ).)

We consider next the case θ < m, for which we identify ${\mathcal {P}}$ in equations (17.A.10) and (17.A.11) with {y(1), y(2), …, y(θ)}, with θ some fixed positive integer less than m. From (17.A.3) and (17.A.11), it follows that

$$\displaystyle \begin{aligned} b_{y(\theta + 1)}\,(\sum_{{\mathcal{P}}\,' \to {\mathcal{P}}}\,\kappa_{y' \to y}) \ -\ b_{y(\theta)}\,(\sum_{{\mathcal{P}} \to {\mathcal{P}}\,'}\,\kappa_{y \to y'}) \ \geq \ 0. \end{aligned} $$

(17.A.13)

Because

$$\displaystyle \begin{aligned} b_{y(\theta)} \ \geq\ b_{y(\theta+1)}, \end{aligned} $$

(17.A.14)

we have

$$\displaystyle \begin{aligned} \sum_{{\mathcal{P}}\,' \to {\mathcal{P}}}\,\kappa_{y' \to y} \ -\ \sum_{{\mathcal{P}} \to {\mathcal{P}}\,'}\,\kappa_{y \to y'} \ \geq \ 0. \end{aligned} $$

(17.A.15)

with strict inequality holding in (17.A.15) when strict inequality holds in (17.A.14). From this and (17.A.10), we obtain the desired result (17.A.4), with strict inequality holding when strict inequality holds in (17.A.14).

This completes the proof of Theorem 17.A.1. □