Deficiency One Theory

Abstract

The Deficiency Zero Theorem tells us, among other things, that for all weakly reversible deficiency zero networks taken with mass action kinetics, the induced differential equations admit precisely one equilibrium in each positive stoichiometric compatibility class. This holds true regardless of values the rate constants take and regardless of how intricate the differential equations might be. It turns out that there is an easily described but even broader class of networks for which the same statement can be made. This is the subject of the Deficiency One Theorem.

Appendix 8.A Why Mass Action Models with an Excess of Terminal Strong-Linkage Classes Are Problematic

The Deficiency One Theorem and the Deficiency One Algorithm preclude from consideration t > ℓ mass action systems—that is, mass action systems in which the underlying reaction network has more terminal strong-linkage classes than linkage classes. In fact, this is no great loss, for such systems often exhibit phenomena (e.g., the existence of multiple equilibria within a positive stoichiometric compatibility class) that disappear when the system is perturbed in the slightest way. With respect to their precise reflection in real chemistry, ephemeral mathematical phenomena of this kind should be viewed with skepticism.

That t > ℓ mass action systems should behave in this way has roots in Appendix 3.A, where we considered distinctions between the kinetic and stoichiometric subspaces. There we indicated how, when the kinetic subspace for a particular kinetic system is smaller than the stoichiometric subspace for the underlying reaction network, strange non-robust behavior can result. Here we extend that discussion. In particular, for mass action systems, we relate the lack of coincidence of the kinetic and stoichiometric subspaces directly to an excess of terminal strong-linkage classes.

1.1 8.A.1 The Kinetic Subspace Revisited

We will begin by reviewing and elaborating somewhat on ideas from Appendix 3.A. For a kinetic system ${\{\mathcal {S},\mathcal {C},\mathcal {R},\mathcal {K}\}}$ , the kinetic subspace, denoted K, is the smallest linear subspace of ${\mathbb {R}^{\mathcal {S}}}$ that contains the image of the species-formation-rate function. Because the species-formation-rate function takes values in the stoichiometric subspace S for the underlying network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ , we always have the inclusion K ⊂ S. For a particular kinetics ${\mathcal {K}}$, the kinetic subspace might be smaller than the stoichiometric subspace.

Recall that the differential equation for a kinetic system ${\{\mathcal {S},\mathcal {C},\mathcal {R},\mathcal {K}\}}$ is

$$\displaystyle \begin{aligned} \dot{c} = f(c), \end{aligned} $$

(8.A.1)

where f(⋅) is the system’s species-formation-rate function. Because the “velocity vector” $\dot {c}$ takes values in the image of the species-formation-rate function, it invariably points not only along S but, indeed, along K. As we pointed out in Appendix 3.A, the fact that K might be smaller than S gives rise to a sharpening of ideas in Chapter 3.

In particular, using ideas very much like those in Chapter 3, we can see without difficulty that a composition c′ might follow a composition c along a solution of (8.A.1) only if c′− c is a member not only of the stoichiometric subspace S but also of the possibly smaller kinetic subspace K ⊂ S. This provides motivation for the following definitions, all related to kinetic compatibility, which parallel our earlier definitions related to stoichiometric compatibility.

Definition 8.A.3

Let ${\{\mathcal {S},\mathcal {C},\mathcal {R},\mathcal {K}\}}$ be a kinetic system, and let $K \; \subset \; {\mathbb {R}^{\mathcal {S}}}$ be its kinetic subspace. Two vectors c′ and $c \in {\overline {\mathbb {R}}_+^{\mathcal {S}}}$ are kinetically compatible if c′− c lies in K. Kinetic compatibility is an equivalence relation that induces a partition of ${\overline {\mathbb {R}}_+^{\mathcal {S}}}$ into equivalence classes called the kinetic compatibility classes for the kinetic system. In particular, the kinetic compatibility class containing $c \in {\overline {\mathbb {R}}_+^{\mathcal {S}}}$ is the set $(c + K) \cap {\overline {\mathbb {R}}_+^{\mathcal {S}}}$.

When, for a particular kinetic system, K and S coincide, there is no distinction between kinetic and stoichiometric compatibility classes. However, when K is smaller than S, then each stoichiometric compatibility class is stratified into a collection of finer kinetic compatibility classes. This will be illustrated later on in Figure 8.A.2 when we consider a concrete example. A trajectory of (6.1.1) that begins in a particular kinetic compatibility class resides entirely within that same kinetic compatibility class.

Hereafter in this appendix, our focus will mostly be on mass action systems. The discussion is close to one given in [73].

1.2 8.A.2 The Kinetic Subspace for a Mass Action System

Consider a mass action system ${\{\mathcal {S},\mathcal {C},\mathcal {R},k\}}$. For each $y \in {\mathcal {C}}$ we denote by ${\mathcal {R}}_{y\to }$ the set of all reactions having y as their reactant complex. That is,

$$\displaystyle \begin{aligned} {\mathcal{R}}_{y\to} := \{\bar{y} \to y' \in {\mathcal{R}}: \bar{y} = y\}. \end{aligned} $$

(8.A.2)

The species-formation-rate function (3.8) for the system can be rearranged:

$$\displaystyle \begin{aligned} f(c) := \sum_{y \in {\mathcal{C}}}c^y\sum_{{\mathcal{R}}_{y\to}}k_{{y \to y'}}(y' - y). \end{aligned} $$

(8.A.3)

Thus, we can write

$$\displaystyle \begin{aligned} f(c) := \sum_{y \in {\mathcal{C}}}c^yd_y, \end{aligned} $$

(8.A.4)

where

$$\displaystyle \begin{aligned} d_y := \sum_{{\mathcal{R}}_{y\to}}k_{{y \to y'}}(y' - y). \end{aligned} $$

(8.A.5)

When ${\mathcal {R}}_{y\to }$ is empty—that is, when y is not a reactant complex for any reaction—it is understood that d _y = 0.

From (8.A.4) it follows that, for the mass action system under consideration, the kinetic subspace K—again, the smallest linear subspace of ${\mathbb {R}^{\mathcal {S}}}$ containing the image of f(⋅)—is contained in the span of the set $\{d_y: y \in {\mathcal {C}}\}$. In fact, we actually have equality⁶:

$$\displaystyle \begin{aligned} K = \mathrm{span}\{d_y: y \in {\mathcal{C}}\}. \end{aligned} $$

(8.A.6)

Because each d _y is a linear combination of reaction vectors, it becomes evident in another way that K resides in S, the stoichiometric subspace for the reaction network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$. Whether K coincides with S or is smaller than S might depend on the particular values that the rate constants take, for these values determine the vectors $\{d_y: y \in {\mathcal {C}}\}$.

We will eventually see that there is a surprisingly large class of reaction networks having the property that, for every assignment of rate constants, the kinetic and stoichiometric subspaces are identical. First, we will examine the kinds of anomalous behavior that can result for networks outside this class when, for a particular rate constant assignment, the kinetic subspace is smaller than the stoichiometric subspace.

1.3 8.A.3 An Instructive Mass Action Example Revisited

It will be helpful to revisit the mass action system (8.A.7) introduced earlier in Appendix 3.A, this time with a more detailed discussion of the behavior the system admits:

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

(8.A.7)

The differential equations for the system in component form are

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \dot{c}_A &\displaystyle =&\displaystyle -(\alpha + 1)c_A + 2c_Bc_C \\ \dot{c}_B &\displaystyle =&\displaystyle \alpha c_A - c_Bc_C\\ \dot{c}_C &\displaystyle =&\displaystyle c_A - c_Bc_C. \end{array} \end{aligned} $$

(8.A.8)

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

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

(8.A.9)

its dimension is two. Stoichiometric compatibility classes take the form of triangles, as shown in Figure 8.A.1.

Fig. 8.A.1

Stoichiometric compatibility class and equilibria for the mass action system (8.A.7)

Fig. 8.A.2

Phase portraits for the mass action system (8.A.7), α = 2 and α = 1

From (8.A.8) it is easy to see that for α ≠ 1 there are no positive equilibria at all: as indicated in Figure 8.A.1, the equilibria are those composition states in which only B or C is present. On the other hand, when α = 1 there is an explosion of equilibria. In fact, within each positive stoichiometric compatibility class, there is an infinite number of equilibria, depicted schematically in the right panel of Figure 8.A.1.

To a great extent, this singular behavior is connected to the fact that the stoichiometric and kinetic subspaces coincide except in the exceptional case α = 1. It is helpful to examine the nature of the kinetic subspace in light shed by the relation (8.A.6). For the mass action system under consideration, we have

$$\displaystyle \begin{aligned} d_A = \alpha(B-A) + (C-A) \quad d_{A+B} = 2A - C - B \quad d_B = 0 \quad d_C = 0. \end{aligned} $$

(8.A.10)

Clearly, then

$$\displaystyle \begin{aligned} K = \mathrm{span}\{d_A,d_{A+B}\}. \end{aligned} $$

(8.A.11)

Note that d _A and d _A+B constitute an independent set whenever α ≠ 1, in which case K ⊂ S is two-dimensional and therefore coincides with S. However, when α = 1 d _A and d _A+B are colinear, in which case K is one-dimensional and is smaller than S.

Composition trajectories for both the typical α = 2 and the exceptional α = 1 cases are sketched in Figure 8.A.2. For α = 2 the kinetic and stoichiometric compatibility classes are identical, and trajectories are free to meander freely within such a class. The situation for α = 1 is very different. As depicted in the right panel of Figure 8.A.2, the kinetic subspace K is a line residing in the two-dimensional stoichiometric subspace S. The triangular stoichiometric compatibility class depicted previously is now shown striated into kinetic compatibility classes that appear as line segments. For α = 1 composition trajectories are no longer free to wander: instead, they must reside entirely within the straight line segments corresponding to the kinetic compatibility classes.

For a kinetic system in which the kinetic subspace is smaller than the stoichiometric subspace, Figure 8.A.2 provides an indication of why it is that a stoichiometric compatibility class that contains a positive equilibrium will usually contain an infinite number of them: Consider a positive equilibrium c ^∗ that resides within a particular stoichiometric compatibility class for the underlying network. When K is smaller than S, that stoichiometric compatibility class is stratified into an infinite number of finer kinetic compatibility classes, one of which contains c ^∗. As the right panel of Figure 8.A.2 suggests, the dynamics in the kinetic compatibility class containing c ^∗ can be expected to be replicated very closely in “nearby” kinetic compatibility classes. In particular, it is natural to expect that every nearby kinetic compatibility class will contain an equilibrium close to c ^∗. (This can be made precise using an implicit function theorem argument that applies to smooth kinetics, not just mass action kinetics.)

These considerations tell us something important about theorems or algorithms (such as the Deficiency One Theorem or the Deficiency One Algorithm) that purport to ensure that, for a particular reaction network taken with kinetics constrained only to lie within a large family (e.g., mass action), no positive stoichiometric compatibility class can ever contain more than one equilibrium: Their hypotheses must, perhaps implicitly, preclude the possibility that the kinetic subspace can be smaller than the stoichiometric subspace for any member of the indicated kinetics family.

As we shall see in the remainder of this appendix, when the kinetics is mass action, that preclusion is tied in a surprisingly simple and powerful way to the number of terminal strong-linkage classes possessed by the particular network under study. We shall also see why mass action models in which there is an excess of terminal strong-linkage classes make for problematic descriptions of real chemistry.

1.4 8.A.4 The Importance of the Number of Terminal Strong-Linkage Classes

Each linkage class in a reaction network contains at least one terminal strong-linkage class (Section 6.6). Thus, we always have ${\mathcal {t}} \geq \ell$, and we might or might not have ${\mathcal {t}} = \ell$. The following theorem [73, 80] tells us that if ${\mathcal {t}} = \ell$, then, so long as the kinetics is mass action, the kinetic subspace must coincide with the stoichiometric subspace, no matter what values the rate constants take. It also tells us circumstances in which we can be sure that the kinetic subspace is smaller than the stoichiometric subspace. The roots of the theorem are discussed in Chapter P of Part III. See, in particular, the proof provided in Appendix P.B.

Theorem 8.A.4

Consider a reaction network ${\{\mathcal {S},\mathcal {C},\mathcal {R}\}}$ having ℓ linkage classes, ${\mathcal {t}}$ terminal strong-linkage classes, deficiency δ, and stoichiometric subspace $S \subset {\mathbb {R}^{\mathcal {S}}}$ . Suppose that for a particular $k \in {\mathbb {R}_+^{\mathcal {R}}}$ , the mass action system ${\{\mathcal {S},\mathcal {C},\mathcal {R},k\}}$ gives rise to the kinetic subspace $K \subset {\mathbb {R}^{\mathcal {S}}}$ . Then the following statements hold true:

• (i) If ${\mathcal {t}} = \ell$ , then, regardless of rate constant values, K = S. That is, if each linkage class in the underlying network contains no more than one terminal strong-linkage class, there is no distinction between the kinetic and stoichiometric subspaces.

• (ii) If ${\mathcal {t}} - \ell \ >\ \delta$ , then, regardless of rate constant values, K is smaller than S. In fact,

  $$\displaystyle \begin{aligned} \dim S - \dim K\ \geq \ {\mathcal{t}} - \ell - \delta. \end{aligned} $$

  (8.A.12)

• (iii) If ${\mathcal {t}}\ > \ \ell$ , if ${\mathcal {t}} - \ell \ \geq \ \delta$ , and if the rate constants are such that the mass action system ${\{\mathcal {S},\mathcal {C},\mathcal {R},k\}}$ admits a positive equilibrium, then K is smaller than S. In fact,

  $$\displaystyle \begin{aligned} \dim S - \dim K \ > \ {\mathcal{t}} - \ell - \delta. \end{aligned} $$

  (8.A.13)

Remark 8.A.5

For a proof see Appendix P.B. Parts (i) and (ii) were proved in [80]. Part (iii) was discussed in [73] and is an easy extension of arguments in [80]. As we shall see in Appendix P.B, the requirement in (iii) that there be a positive equilibrium is stronger than it need be. It is enough that there be an equilibrium $c^* \in {\overline {\mathbb {R}}_+^{\mathcal {S}}}$ such that for some nonterminal complex y, it is true that supp y ⊂supp c ^∗. In Appendix P.B and in [80], it is shown that for mass action systems in which the underlying network has a deficiency of zero, we always have the equality $\dim S - \dim K = {\mathcal {t}} - \ell$.

Because for weakly reversible networks the linkage classes are identical to the terminal strong-linkage classes, we have the following important corollary of Theorem 8.A.4:

Corollary 8.A.6

For any mass action system in which the underlying reaction network is weakly reversible, the kinetic and stoichiometric subspaces are identical. In particular the kinetic and stoichiometric subspaces are identical if the underlying reaction network is reversible.

We should keep in mind that this chapter is about deficiency one theory. In this case, Theorem 8.A.4 and, in particular, the inequality (8.A.13) provide considerable insight about what our expectations should be. The essential fact is summarized in the following corollary.

Corollary 8.A.7

Consider a deficiency one reaction network having more terminal strong-linkage classes than linkage classes. For any assignment of rate constants such that the resulting mass action system admits a positive equilibrium, the kinetic subspace is smaller than the stoichiometric subspace.

The corollary tells us why it was inevitable that ${\mathcal {t}} > \ell$ networks be excluded in the formulation of the deficiency one results described earlier, the aim of which was to describe mass action systems that could not admit multiple positive equilibria within a stoichiometric compatibility class. For reasons discussed earlier, when the kinetic subspace is smaller than the stoichiometric subspace, a positive equilibrium in a stoichiometric compatibility class will usually be accompanied by an infinite number of them.

Example 8.A.8

We return to the network shown in (8.A.7), this time in light shed by Theorem 8.A.4. For the network δ = 1, ℓ = 2, and ${\mathcal {t}} = 3$. Thus, we have ${\mathcal {t}} > \ell$ and ${\mathcal {t}} - \ell = \delta$. When the rate constants for the network are such that the corresponding mass action system admits no positive equilibrium, Theorem 8.A.4 is silent; indeed, despite the fact that ${\mathcal {t}} > \ell$, the kinetic and stoichiometric subspaces nevertheless coincide. On the other hand, when there is a positive equilibrium—in particular, when α = 1 in (8.A.7)—Theorem 8.A.4 tells us that the kinetic subspace is smaller than stoichiometric subspace, with $\dim S - \dim K = 1$. In this case there is an infinite number of equilibria in each positive stoichiometric compatibility class.

1.5 8.A.5 The Fragility of Phenomena Emerging from Mass Action Models Having an Excess of Terminal Strong-Linkage Classes

We have seen that there are certain dramatic phenomena exhibited by mass action systems—for example, the α = 1 equilibria explosion exhibited in Figure 8.A.1—that ultimately derive from the fact that the kinetic subspace is smaller than the stoichiometric subspace. Even if we leave aside the fact that these phenomena might be highly singular, occurring for only very singular rate constant values, there is still something else to consider: their lack of persistence against seemingly inconsequential perturbations of the reaction network itself.

To make this apparent, we repeat below the special mass action system we’ve been studying,

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

(8.A.14)

in order that we might contrast its underlying network with a slightly different one:

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

(8.A.15)

Note that this second network differs from the first only to the extent that the reaction A → C has been made reversible. But now there are only two terminal strong-linkage classes, {B} and {2A}, whereas before there were three, {B}, {C}, and {2A}. Before, we had ${\mathcal {t}} > \ell$, whereupon Corollary 8.A.7 ensured that the kinetic and stoichiometric subspaces do not coincide, so long as the rate constants are such that there is a positive equilibrium. Now, for network (8.A.15), the same theorem ensures that the kinetic and stoichiometric subspaces must coincide, no matter what rate constants are assigned to the various reactions. Thus, dramatic phenomena that, for (8.A.14), rely on K ≠ S will be absent for any mass action system in which (8.A.15) is the underlying reaction network.

Clearly, any reaction network for which ${\mathcal {t}} > \ell$ can, as in the example, be perturbed to a ${\mathcal {t}} = \ell$ network by the addition of sufficiently many reverse reactions. It should be understood that chemists often insist that all reactions are reversible, if only to a very small extent (i.e., perhaps with some reverse reactions having, in the mass action context, almost negligible rate constants). For this reason, it would not seem wise to place untempered credence in dramatic mass-action-model phenomena that derive from a lack of coincidence of the kinetic and stoichiometric subspaces (and, therefore, from the presence of an excess of terminal strong-linkage classes).

It is one thing to employ, for practical modeling purposes, networks in which there is a lack of reversibility in some or all reactions. It is quite a different thing to expect in nature precise replication of certain dramatic phenomena that result only from networks that offend chemists’ sensibilities. Indeed, it is fitting that otherwise broad theorems should, in their hypotheses, exclude such networks from consideration at the outset. The exclusion of ${\mathcal {t}} > \ell$ networks in the Deficiency One Theorem not only makes its conclusion possible, the exclusion also aligns with what chemists believe.

1.6 8.A.6 Summary: The Good News That Theorem 8.A.4 Contains

We first introduced the seemingly troublesome distinction between the kinetic and stoichiometric subspaces in an appendix to Chapter 3. There we emphasized that the stoichiometric subspace is an attribute of a reaction network alone, while the more fundamental kinetic subspace is an attribute of a kinetic system that might change from one kinetics to another, perhaps as kinetic parameters vary ever so slightly.

However, that appendix ended on a happy note (Section 3.A.3): we asserted that, at least for mass action systems, there is a very large and highly robust family of reaction networks for which there is no distinction between the kinetic and stoichiometric subspaces. To know the stoichiometric subspace for a network in that family is to know the kinetic subspace; the two subspaces coincide no matter what values the mass action rate constants take.

At the end of Chapter 3, we did not yet have the language available to describe that congenial reaction network family meaningfully, but now we do. The kinetic and stoichiometric subspaces coincide for any mass action system such that, in the underlying reaction network, no linkage class contains two or more terminal strong-linkage classes. The family contains all weakly reversible networks and, in particular, all reversible networks, but it is far broader.

To the extent that a mass action model might exhibit a lack of coincidence of the kinetic and stoichiometric subspaces, it cannot do so robustly: the lack of coincidence will vanish in a “nearby” mass action system in which one or more reactions are made reversible, perhaps with tiny rate constants for the reactions added. That modification is one that chemists would bless.