Quasi-Thermodynamic Kinetic Systems

Abstract

In Chapter 7 we discussed, briefly, origins of the Deficiency Zero Theorem. There we introduced the idea of complex balancing, a major generalization by Horn and Jackson [109] of an earlier related idea, detailed balancing. We also hinted at connections of both ideas to classical thermodynamics. In this chapter we will elaborate on thermodynamic roots underlying arguments to come, largely to provide motivation for purely mathematical proof techniques that might otherwise seem improvisatory.

Appendices

Appendix 13.A Proof of Theorem 13.3.3

Our aim in this appendix is to complete the proof of Theorem 13.3.3. We will begin with Proposition 13.A.1, which appeared in 1979 [71] and which has some interest in its own right, divorced from its use in chemical reaction network theory. Although by ${\mathcal {S}}$ and S in the proposition statement we obviously have in mind the species set and the stoichiometric subspace, the proposition is stated more abstractly. Theorem 13.3.3 will amount to a corollary of Proposition 13.A.1. The same proposition will find a different use in Appendix 13.B.

Proposition 13.A.1 ( [71, 76])

Let ${\mathcal {S}}$ be any finite set, let ${\mathbb {R}^{\mathcal {S}}}$ be the vector space generated by ${\mathcal {S}}$ , let S be a linear subspace of ${\mathbb {R}^{\mathcal {S}}}$ , and let a and b be members of ${\mathbb {R}_+^{\mathcal {S}}}$ . There is a unique vector μ ∈ S ^⊥ such that

$$\displaystyle \begin{aligned} a e^{\;\mu} -\; b \end{aligned}$$

is an element of S.

Proof

Let $\phi : {\mathbb {R}^{\mathcal {S}}} \to \mathbb {R}$ be defined by

(13.A.1)

Straightforward computation shows that the gradient of ϕ at x is given by

$$\displaystyle \begin{aligned} \nabla \phi (x) \equiv ae^{x} - b \end{aligned} $$

(13.A.2)

and that the Hessian of ϕ at x, $H(x): {\mathbb {R}^{\mathcal {S}}} \to {\mathbb {R}^{\mathcal {S}}}$, is given by

$$\displaystyle \begin{aligned} H(x)\gamma \equiv (ae^x)\gamma. \end{aligned} $$

(13.A.3)

Moreover, for each $x \in {\mathbb {R}^{\mathcal {S}}}$, H(x) is positive-definite: For all nonzero $\gamma \in {\mathbb {R}^{\mathcal {S}}}$

(13.A.4)

Thus, the function ϕ is strictly convex [143].

Next we want to show that, for any nonzero $x \in {\mathbb {R}^{\mathcal {S}}}$,

$$\displaystyle \begin{aligned} \lim\limits_{\lambda \to \infty} \phi(\lambda x) = \infty. \end{aligned} $$

(13.A.5)

Note that

(13.A.6)

Note also that, for , the positivity of and gives

(13.A.7)

while for we have

(13.A.8)

Thus, for x ≠ 0, (13.A.6)–(13.A.8) imply (13.A.5).

Now let $\bar {\phi }:S^{\perp } \to \mathbb {R}$ be the restriction of ϕ to S ^⊥. Since ϕ is continuous and convex, so is $\bar {\phi }$. The continuity of $\bar {\phi }$ and a standard result for convex functions ensure that the set

$$\displaystyle \begin{aligned} C:=\{ x \in S^{\perp} : \bar{\phi }(x)\le \bar{\phi}(0)\} \end{aligned} $$

(13.A.9)

is closed, convex (and obviously contains the zero vector). Moreover, it follows from (13.A.5) that C contains no half line with endpoint 0. Since, in a finite-dimensional vector space, every unbounded closed convex set containing 0 must contain a half line with endpoint 0 (see [163], p. 105), it follows that C is bounded and therefore compact.

Thus, there exists μ ∈ C such that

$$\displaystyle \begin{aligned} \bar{ \phi}(\mu ) \le \bar{ \phi}(x), \quad \forall x \in C. \end{aligned} $$

(13.A.10)

In fact, from the definition of C, we have

$$\displaystyle \begin{aligned} \bar{ \phi}(\mu ) \le \bar{ \phi}(x), \quad \forall x \in S^{\perp}. \end{aligned} $$

(13.A.11)

Therefore, for all γ ∈ S ^⊥,

$$\displaystyle \begin{aligned} \begin{array}{rcl} 0 &\displaystyle =&\displaystyle \frac{d}{{d\theta }}\bar{\phi}(\mu + \theta \gamma )\left| {{}_{\theta = 0}}\right. \end{array} \end{aligned} $$

(13.A.12)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle =&\displaystyle \frac{d}{{d\theta }}\phi(\mu + \theta \gamma )\left| {{}_{\theta = 0}} \right. \end{array} \end{aligned} $$

(13.A.13)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle =&\displaystyle \nabla \phi(\mu)\cdot\gamma. \end{array} \end{aligned} $$

(13.A.14)

It follows that ∇ϕ(μ) must lie in S so that, from (13.A.2), we have the inclusion

$$\displaystyle \begin{aligned} a\;e^{\;\mu} -\; b \; \in\; S. \end{aligned} $$

(13.A.15)

Thus, μ ∈ S ^⊥ satisfies the requirements of the proposition.

To prove uniqueness we presume that μ′∈ S ^⊥ also satisfies the inclusion

$$\displaystyle \begin{aligned} a\;e^{\;\mu'} -\; b \; \in\; S. \end{aligned} $$

(13.A.16)

From (13.A.15) and (13.A.16), we have

$$\displaystyle \begin{aligned} a\;(e^{\;\mu'} - e^{\;\mu}) \in S, \end{aligned} $$

(13.A.17)

and, since μ′− μ lies in S ^⊥, we also have

(13.A.18)

Since each is positive and the exponential function is strictly monotonically increasing, (13.A.18) can hold only if for all —that is, only if μ′ = μ. □

Remark 13.A.2

There is an interesting observation that can be made here. With ${\mathbb {R}^{\mathcal {S}}}$ and S as in Proposition 13.A.1, it is a standard result in linear algebra that any $b \in {\mathbb {R}^{\mathcal {S}}}$ has a unique representation

$$\displaystyle \begin{aligned} b = x_1 + x_2, \quad x_1 \in S^{\perp},\ x_2 \in S. \end{aligned}$$

Taking for all in Proposition 13.A.1, we obtain a similar (but deeper) result: Any positive $b \in {\mathbb {R}^{\mathcal {S}}}$—that is, any $b \in {\mathbb {R}_+^{\mathcal {S}}}$—admits a unique representation

$$\displaystyle \begin{aligned} b = e^{\mu_1} + \mu_2, \quad \mu_1 \in S^{\perp},\ \mu_2 \in S. \end{aligned}$$

The following corollary of Proposition 13.A.1 is essentially a restatement of Theorem 13.3.3, proved in 1972 by Horn and Jackson [109] but in a different way.

Corollary 13.A.3

Let ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ be a reaction network with stoichiometric subspace S. For any $c^* \in {\mathbb {R}_+^{\mathcal {S}}}$ the set

$$\displaystyle \begin{aligned} E: = \{ c \in {\mathbb{R}_+^{\mathcal{S}}}:\; \ln\, c\; -\; \ln\, c^* \in S^{\perp}\} \end{aligned} $$

(13.A.19)

meets each positive stoichiometric compatibility class in precisely one point.

Proof

Let p be an arbitrary element of ${\mathbb {R}_+^{\mathcal {S}}}$. We will show that E meets the positive stoichiometric compatibility class containing p in precisely one point. That there can be at most one such point was proved just after the statement of Theorem 13.3.3. To prove the existence of such a point, we note that Proposition 13.A.1 ensures the existence of μ ∈ S ^⊥ such that

$$\displaystyle \begin{aligned} c^*\,e^{\mu} - p \in S. \end{aligned} $$

(13.A.20)

Now let c ^† be defined by

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

(13.A.21)

From (13.A.20) and (13.A.21), it follows that c ^† is in the positive stoichiometric compatibility class containing p. Taking logarithms in (13.A.21), we obtain

$$\displaystyle \begin{aligned} \ln\, c^{\dagger}\; -\; \ln\, c^* = \mu \in S^{\perp} \end{aligned} $$

(13.A.22)

Thus, c ^† is a member of E as well. □

Appendix 13.B Existence of a Positive Equilibrium for a Reversible Star-Like Mass Action System

Our objective here is to prove that, when ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ is a reversible star-like network and $k \in {\mathbb {R}_+^{\mathcal {R}}}$ is an arbitrary rate constant assignment, the mass action system ${\{\mathcal {S},\mathcal {C},\mathcal {R},k\}}$ admits a positive equilibrium. As in Section 13.5, we denote by y ₀ the complex central to the star network and by ${\mathcal {C}}^*$ the remaining complexes.

What we want to show is that there exists $c \in {\mathbb {R}_+^{\mathcal {S}}}$ such that

$$\displaystyle \begin{aligned} \sum_{y\;\in\; {\mathcal{C}}^* }k_{y\; \to\; y_0}\;c^y(y_0 - y) + \sum_{y \;\in\; {\mathcal{C}}^*}k_{y_0 \;\to\; y}\;c^{y_0}\;(y - y_0) = 0. \end{aligned} $$

(13.B.1)

This can be written in the form

$$\displaystyle \begin{aligned} e^{\;y_0\;\cdot\;\ln c}\sum_{y\;\in\; {\mathcal{C}}^* }(k_{y\; \to\; y_0}\;e^{\;(y - y_0)\;\cdot\;\ln c}- k_{y_0\; \to\; y})(y - y_0) = 0. \end{aligned} $$

(13.B.2)

Clearly, it will be enough to show the existence of $x\in {\mathbb {R}^{\mathcal {S}}}$ such that

$$\displaystyle \begin{aligned} \sum_{y\;\in\; {\mathcal{C}}^* }(k_{y\; \to\; y_0}\;e^{\;(y - y_0)\;\cdot\;x}- k_{y_0\; \to\; y})(y - y_0) = 0, \end{aligned} $$

(13.B.3)

for then we can take c = e ^x.

With these things in mind, we let $T:{{\mathbb {R}^{\mathcal {C}^*}}} \to {\mathbb {R}^{\mathcal {S}}}$ be the linear transformation defined by

$$\displaystyle \begin{aligned} T\alpha = \sum_{y \in {\mathcal{C}}^*}\alpha_y(y - y_0), \quad \forall \alpha \in {{\mathbb{R}^{\mathcal{C}^*}}}. \end{aligned} $$

(13.B.4)

Before proceeding further we record the following remark:

Remark 13.B.1

Suppose that V and W are finite-dimensional vector spaces, each with a scalar product, and that L : V → W is a linear transformation. By the transpose (sometimes called the adjoint) of L, we mean the linear transformation L ^T : W → V such that

$$\displaystyle \begin{aligned} (Lv) \cdot w = v\cdot L^T w \quad \forall v \in V, w \in W. \end{aligned} $$

(13.B.5)

It is a standard fact of linear algebra that ${{\mathrm {im} \,}} L^T = (\ker L)^{\perp }$ [98].

Note that the transpose of T, $T^T:{\mathbb {R}^{\mathcal {S}}} \to {{\mathbb {R}^{\mathcal {C}^*}}}$, is given by

$$\displaystyle \begin{aligned} T^Tz = \sum_{y \in {\mathcal{C}}^*}[(y - y_0)\cdot z]\;\omega_{\;y}, \quad \forall z \in {\mathbb{R}^{\mathcal{S}}}, \end{aligned} $$

(13.B.6)

where $\{\omega _{\;y}\}_{y \in {\mathcal {C}}^*}$ is the standard basis for ${\mathbb {R}^{\mathcal {C}^*}}$. (Recall Section 2.2.2.) Let a and b in ${\mathbb {R}^{\mathcal {C}^*}}$ be defined by

$$\displaystyle \begin{aligned} a_y = k_{y \to y_0} \quad \mathrm{and} \quad b_y = k_{y_0 \to y}, \quad \forall y \in {\mathcal{C}}^*. \end{aligned} $$

(13.B.7)

In view of (13.B.3), (13.B.6), and (13.B.7), our proof will be complete if we can show that there is a $\mu \in {\mathrm {im} \,} T = (\ker T)^\perp$ such that

$$\displaystyle \begin{aligned} a e^{\mu} - b \end{aligned}$$

is a member of $\ker T$. In fact, the existence of such a μ is ensured by Proposition 13.A.1. □