In this section, we start from reaction networks that are specified as abstract stoichiometric relations, Eq. (1), and identify minimal constraints necessary to avoid blatantly unphysical behavior.
Notation and peliminaries
Let X be a finite set and let \(\mathscr {R}\) be a pair of formal sums of elements of X with non-negative integer coefficients according to Eq. (1). Then we call the pair \((X,\mathscr{R})\) a reaction network (RN). Equivalently, a RN is a directed, integer-weighted hypergraph with directed edges \((r^-,r^+)\) such that \(x\in r^-\) with weight \(s^-_{xr}>0\) and \(x\in r^+\) with weight \(s^+_{xr}>0\). The weights \(s^-_{xr}\) and \(s^+_{xr}\) are usually called the stoichiometric coefficients. We set \(s^-_{xr}=0\) and \(s^+_{xr}=0\) if \(x\notin r^-\) and \(x\notin r^-\), respectively. We deliberately dropped the qualifier chemical here since, as we shall see, not every RN \((X,\mathscr {R})\) makes sense as a model of a chemical system. In fact, the aim of this contribution is to characterize the set of RNs that make sense as models of chemistry.
Such directed hypergraphs are most conveniently drawn as (bipartite) König multigraphs, with distinct types of vertices representing compounds \(x\in X\) and reactions \(r\in \mathscr {R}\), respectively. Stoichiometric coefficients larger than one appear as multiple edges. See the example in Fig. 1.
For each reaction \(r\in \mathscr {R}\), we define its support as \({{\,\mathrm{supp}\,}}(r)=\{x \mid s^-_{xr}+s^+_{xr}>0\}\); that is, \(x\in {{\,\mathrm{supp}\,}}(r)\) if it appears as an educt, a product, or a catalyst in r. The stoichiometric matrix of \((X,\mathscr {R})\) is \(\mathbf {S}\in \mathbb {N}_0^{X \times \mathscr {R}}\) with entries \(\mathbf {S}_{xr}= s^+_{xr} - s^-_{xr}\).
We distinguish proper reactions r, for which there is both \(x\in X\) with \({\mathbf {S}}_{xr}<0\) and \(y\in X\) with \({\mathbf {S}}_{yr}>0\), import reactions for which \({\mathbf {S}}_{xr}\ge 0\) for all \(x\in X\), and export reactions for which \({\mathbf {S}}_{xr}\le 0\) for all \(x\in X\). We write \(\varnothing\) for the empty formula, hence \(\varnothing\) \(\longrightarrow\) A and B \(\longrightarrow\) \(\varnothing\) designate the import of A and the export of B, respectively. Note that this definition also allows catalyzed import and export reactions, e.g., C \(\longrightarrow\) C + A or B + C \(\longrightarrow\) C.
In thermodynamics, a system is closed if it does not exchange matter with its environment, but may exchange energy in the form of work or heat [60]. For a RN, this rules out import and export reactions.
Definition 1
A RN \((X,\mathscr {R})\) is closed if all reactions \(r\in \mathscr {R}\) are proper.
Given an arbitrary RN \((X,\mathscr {R})\), there is a unique inclusion-maximal closed RN contained in \((X,\mathscr {R})\), namely \((X,\mathscr {R}^\text {p})\) with
$$\begin{aligned} \mathscr {R}^p= \{r\in \mathscr {R}\mid r \text { is proper}\}. \end{aligned}$$
(2)
We will refer to \((X,\mathscr {R}^p)\) as the proper part of \((X,\mathscr {R})\).
For every reaction r, one can define a reverse reaction \(\overline{r}\) that is obtained from r by exchanging the role of products and educts. That is, \(\overline{r}\) is the reverse of r iff, for all \(x\in X\), it holds that
$$\begin{aligned} s^-_{x\overline{r}} = s^+_{xr} \quad \text {and}\quad s^+_{x\overline{r}} = s^-_{xr} . \end{aligned}$$
(3)
While thermodynamics dictates that every reaction is reversible in principle (albeit possibly with an extremely low reaction rate), it is a matter of modeling whether sufficiently slow reactions are included in the reaction set \(\mathscr {R}\).
Chemical reactions can be composed and aggregated into “overall reactions”. In the literature on metabolic networks, pathways are of this form. An overall reaction consists of multiple reactions that collectively convert a set of educts into a set of products. It can be represented as a formal sum of reactions \(\sum _{r\in \mathscr {R}} \mathbf {v}_r \, r\), where the vector of multiplicities \(\mathbf {v}\in \mathbb {N}^\mathscr {R}_0\) has non-negative integer entries. Thereby, \([\mathbf {S}\mathbf {v}]_x\) determines the net consumption or production of compound x in the overall reaction specified by \(\mathbf {v}\).
A vector \(\mathbf {v}\in \mathbb {N}_0^\mathscr {R}\) can be interpreted as an integer hyperflow in the following sense: If x is neither an educt nor a product of the overall reaction specified by \(\mathbf {v}\), then \([\mathbf {S}\mathbf {v}]_x = \sum _r (s^+_{xr}-s^-_{xr}) \mathbf {v}_r = 0\), i.e., every unit of x that is produced by some reaction r with \(\mathbf {v}_r>0\) is consumed by another reaction \(r'\) with \(\mathbf {v}_{r'}>0\).
The effect of an overall reaction can be represented via formal sums of species in two ways: as composite reactions,
$$\begin{aligned} \sum _{x\in X} \left( \sum _{r\in \mathscr {R}} s^-_{xr}\mathbf {v}_r\right) x \longrightarrow \sum _{x\in X} \left( \sum _{r\in \mathscr {R}} s^+_{xr}\mathbf {v}_r\right) x , \end{aligned}$$
(4)
or as net reactions,
$$\begin{aligned} \begin{aligned} \sum _{x\in X}&\left( \sum _{r\in \mathscr {R}} \left[ (s^-_{xr}-s^+_{xr})\mathbf {v}_r\right] _{+}\right) x \longrightarrow \sum _{x\in X} \left( \sum _{r\in \mathscr {R}} \left[ (s^+_{xr}-s^-_{xr})\mathbf {v}_r\right] _{+}\right) x . \end{aligned} \end{aligned}$$
(5)
Here we use the notation \([c]_+ = c\) if \(c>0\) and \([c]_+=0\) for \(c\le 0\). In Eq. (5), intermediates, i.e., formal catalysts are cancelled. Hence, the net consumption (or production) of a species x is \(\sum _{r\in \mathscr {R}}[(s^-_{xr}-s^+_{xr})\mathbf {v}_r]_{+}=-[\mathbf {S}\mathbf {v}]_x\) if \([\mathbf {S}\mathbf {v}]_x<0\) (or \(\sum _{r\in \mathscr {R}}[(s^+_{xr}-s^-_{xr})\mathbf {v}_r]_{+}=[\mathbf {S}\mathbf {v}]_x\) if \([\mathbf {S}\mathbf {v}]_x>0\)).
Fig. 1 shows the RN of Oro’s prebiotic adenine synthesis from HCN and the integer hyperflow \(\mathbf {v}\) corresponding to the net reaction “5 HCN \(\longrightarrow\) adenine” as an example.
While a restriction to integer hyperflows \(\mathbf {v}\in \mathbb {N}_0^\mathscr {R}\) is necessary in many applications, see e.g. [21] for a detailed discussion, it appears mathematically more convenient to use the more general setting of fluxes \(\mathbf {v}\in \mathbb {R}^\mathscr {R}_\ge\) as in the analysis of metabolic pathways. To emphasize the connection with the body of literature on network (hyper)flows we will uniformly speak of flows.
For any vector \(\mathbf {v}\in \mathbb {R}^\mathscr {R}\), we write \(\mathbf {v}\ge 0\) if \(\mathbf {v}\) is non-negative, \(\mathbf {v}>0\) if \(\mathbf {v}\) is non-negative and non-zero, that is, at least one entry is positive, and \(\mathbf {v}\gg 0\) if all entries of \(\mathbf {v}\) are positive. Analogously, we write \(\mathbf {v}\le 0\), \(\mathbf {v}<0\), and \(\mathbf {v}\ll 0\). In particular, a vector \(\mathbf {v}\in \mathbb {R}^\mathscr {R}\) is called a flow if \(\mathbf {v}\ge 0\).
A non-trivial flow satisfies \(\mathbf {v}>0\), i.e., \(\mathbf {v}\ne 0\). Two flows \(\mathbf {v_1}\) and \(\mathbf {v_2}\) are called parallel if they describe the same net reaction. In particular, we therefore have \(\mathbf {S}\mathbf {v_1} = \mathbf {S}\mathbf {v_2}\) for parallel flows.
Futile cycles in a RN are non-trivial flows for which educts and products coincide and thus the net reaction is empty.
Definition 2
A flow \(\mathbf {v}>0\) is a futile cycle if \(\mathbf {S}\mathbf {v}=0\).
We use the term futile cycle in the strict sense to describe the concurrent activity of multiple reactions (or pathways) having no net effect other than the dissipation of energy. In the literature on metabolic networks often a less restrictive concept is used that allows certain compounds (usually co-factors, ATP/ADP, redox equivalents, or solvents) to differ between products and educts, see e.g. [61,62,63,64]. In this setting, the net reaction of concurrent glycolysis and gluconeogenesis, namely the hydrolysis of ATP, is viewed as energy dissipation rather than a chemical reaction. In our setting, \({\text{ATP}} + {{\text{H}}_{2}} {\text{O}} \longrightarrow {\text{ADP}}+ {{\text{P}}_i^{-}} + {\text{H}}^{+}\), is a net reaction like any other, and hence a futile cycle would only arise if recycling of ATP, i.e., ADP + \({\text{P}}_i^{-} + {{\text{H}}^{+}} \longrightarrow {\text{ATP}} + {{\text H}_{2}}{\text{O}}\), was included as well.
If a RN has a futile cycle, it also has an integer futile cycle \(\mathbf {v}\in \mathbb {N}_0^\mathscr {R}\), since \(\mathbf {S}\) has integer entries and thus its kernel has a rational basis, which can be scaled with the least common denominator to have integer entries.
A pair \((X',\mathscr {R}')\) is a subnetwork of \((X,\mathscr {R})\) if \(X'\subseteq X\), \(\mathscr {R}'\subseteq \mathscr {R}\), and \({{\,\mathrm{supp}\,}}(r)\subseteq X'\) implies \(r\in \mathscr {R}'\). We say that a property P of a RN is hereditary if “\((X,\mathscr {R})\) has P” implies that every subnetwork “\((X',\mathscr {R}')\) has P”.
Chemical reactions are subject to thermodynamic constraints that are a direct consequence of the conservation of energy, the conservation of mass, and the reversibility of chemical reactions. In the context of chemistry, conservation of mass is of course a consequence of the conservation of atoms throughout a chemical reaction. In the following sections, we investigate how these physical principles constrain RNs. Since we have introduced RNs in terms of abstract molecules and reactions, Eq. (1), we express the necessary conditions in terms of the stoichiometric matrix \(\mathbf {S}\), which fully captures only the proper part of the RN. Throughout this work, therefore, we assume that \((X,\mathscr {R})\) is a closed RN, unless explicitly stated otherwise.
Thermodynamic constraints
Reaction energies and perpetuum mobiles
Every chemical reaction r is associated with a change in the Gibbs free energy of educts and products. We therefore introduce a vector of reaction (Gibbs free) energies \(\mathbf {g}\in \mathbb {R}^\mathscr {R}\) and write \((X,\mathscr {R},\mathbf {g})\) for a RN endowed with reaction energies. The reaction energy for an overall reaction is the total energy of the individual reactions involved. In terms of \(\mathbf {v}\in \mathbb {R}^\mathscr {R}\), it can be expressed as
$$\begin{aligned} \sum _{r\in \mathscr {R}} \mathbf {g}_r\mathbf {v}_r = \mathbf {g}^\top \mathbf {v}= \langle \mathbf {g},\mathbf {v}\rangle , \end{aligned}$$
(6)
where \(\langle \cdot ,\cdot \rangle\) denotes the scalar product on \(\mathbb {R}^\mathscr {R}\).
Futile cycles may act as a chemical version of a perpetuum mobile. This is the case whenever a flow \(\mathbf {v}> 0\) with zero formal net reaction, \(\mathbf {S}\mathbf {v}= 0\), increases or decreases energy, i.e., if \(\langle \mathbf {g},\mathbf {v}\rangle \ne 0\).
Definition 3
Let \((X,\mathscr {R},\mathbf {g})\) be a RN with reaction energies. A flow \(\mathbf {v}> 0\) is a perpetuum mobile if \(\mathbf {S}\mathbf {v}=0\) and \(\langle \mathbf {g},\mathbf {v}\rangle \ne 0\).
The classical concept of a perpetuum mobile decreases its energy, \(\langle \mathbf {g},\mathbf {v}\rangle < 0\), thereby “creating” energy for its environment. An “anti” perpetuum mobile with \(\langle \mathbf {g},\mathbf {v}\rangle > 0\) would “annihilate” energy. Either situation violates energy conservation and thus cannot be allowed in a chemical RN. Obviously, there is no perpetuum mobile if \((X,\mathscr {R})\) does not admit a futile cycle.
In fact, thermodynamics dictates that Gibbs free energy is a state function. Two parallel flows \(\mathbf {v_1}\) and \(\mathbf {v_2}\) therefore must have the same associated net reaction energies. That is, \(\mathbf {S}\mathbf {v_1}=\mathbf {S}\mathbf {v_2}\) implies \(\langle \mathbf {g}, \mathbf {v^1}\rangle = \langle \mathbf {g},\mathbf {v^2}\rangle\). Equivalently, any vector \(\mathbf {v}=\mathbf {v^1}-\mathbf {v^2} \in \mathbb {R}^\mathscr {R}\) with \(\mathbf {S}\mathbf {v}=0\) must satisfy \(\langle \mathbf {g},\mathbf {v}\rangle =0\). That is, \(\mathbf {g}\in (\ker \mathbf {S})^\perp\).
Definition 4
Let \((X,\mathscr {R},\mathbf {g})\) be a RN with reaction energies. Then \((X,\mathscr {R},\mathbf {g})\) is thermodynamic if \(\mathbf {v}\in \mathbb {R}^\mathscr {R}\) and \(\mathbf {S}\mathbf {v}=0\) imply \(\langle \mathbf {g},\mathbf {v}\rangle =0\), that is, if \(\mathbf {g}\in (\ker \mathbf {S})^\perp\).
Let \((X,\mathscr{R},\mathbf{g})\) be thermodynamic, \((X',\mathscr {R}')\) be a subnetwork of \((X,\mathscr{R})\), and \(\mathbf {g}'\) be the restriction of \(\mathbf {g}\) to \(\mathscr {R}'\). Then \(\mathbf {v}'\in \mathbb {R}^{\mathscr {R}'}\) corresponds to \(\mathbf {v}\in \mathbb {R}^{\mathscr {R}}\) with \({{\,\mathrm{supp}\,}}(\mathbf {v})\subseteq \mathscr {R}'\), and thus \(\mathbf {v}'\in \mathbb {R}^{\mathscr {R}'}\) and \(\mathbf {S}'\mathbf {v}'=0\) imply \(\mathbf {S}\mathbf {v}=0\) and further \(\langle \mathbf {g}',\mathbf {v}'\rangle =\langle \mathbf {g},\mathbf {v}\rangle =0\). Hence \((X',\mathscr {R}',\mathbf {g}')\) is again thermodynamic.
We note that the reaction energies of a reaction r and its reverse \(\overline{r}\) necessarily cancel:
Lemma 5
If r and \(\overline{r}\) are reverse reactions in a thermodynamic network \((X,\mathscr {R},\mathbf {g})\), then \(\mathbf {g}_{\overline{r}}=-\mathbf {g}_r\).
Proof
If r and \(\overline{r}\) are reverse reactions, then \(\mathbf {v}\) with \(\mathbf {v}_r=\mathbf {v}_{\overline{r}}=1\) (and \(\mathbf {v}_{r'}=0\) otherwise) satisfies \(\mathbf {S}\mathbf {v}=0\). Thus \(\langle \mathbf {g},\mathbf {v}\rangle = \mathbf {g}_r+\mathbf {g}_{\overline{r}}=0\). \(\square\)
Digression: molecular energies and Hess’ Law
Every molecular species \(x\in X\) has an associated Gibbs free energy of formation. For notational simplicity, we write \(\mathbf {G}_x\) instead of the commonly used symbol \(G_\mathrm {f}(x)\). The corresponding vector of molecular energies is denoted by \(\mathbf {G}\in \mathbb {R}^X\). Molecular energies and reactions energies \(\mathbf {g}\in \mathbb {R}^\mathscr {R}\) are related by Hess’ law: For every reaction \(r \in \mathscr {R}\), it holds that
$$\begin{aligned} \mathbf {g}_r = \sum _{x\in X} \mathbf {G}_x (s^+_{xr}-s^-_{xr}) = \sum _{x\in X} \mathbf {G}_x \, \mathbf {S}_{xr} . \end{aligned}$$
In matrix form, the relationship between reaction energies \(\mathbf {g}\) and molecular energies \(\mathbf {G}\) amounts to
$$\begin{aligned} \mathbf {g}= \mathbf {S}^\top \mathbf {G}. \end{aligned}$$
(7)
Proposition 6
Let \((X,\mathscr {R})\) be a RN and \(\mathbf {g}\in \mathbb {R}^\mathscr {R}\) be a vector of reaction energies. Then \((X,\mathscr {R},\mathbf {g})\) is thermodynamic if and only if there is a vector of molecular energies \(\mathbf {G}\in \mathbb {R}^X\) satisfying Hess’ law, Eq. (7).
Proof
By Definition 4, \((X,\mathscr {R},\mathbf {g})\) is thermodynamic if \(\mathbf {g}\in (\ker \mathbf {S})^\perp = {{\,\mathrm{im}\,}}\mathbf {S}^\top\), that is, if there is \(\mathbf {G}\) such that \(\mathbf {g}= \mathbf {S}^\top \mathbf {G}\), satisfying Hess’s law. \(\square\)
Note that the vector of molecular energies \(\mathbf {G}\) is not uniquely determined by \(\mathbf {g}\) in general.
Reversible and irreversible networks
To begin with, we consider purely reversible or irreversible RNs.
Definition 7
A RN \((X,\mathscr {R})\) is reversible if \(r\in \mathscr {R}\) implies \(\overline{r}\in \mathscr {R}\) and irreversible if \(r\in \mathscr {R}\) implies \(\overline{r}\notin \mathscr {R}\).
In reversible networks, general vectors \(\mathbf {v}\in \mathbb {R}^\mathscr {R}\) have corresponding flows \(\mathbf {{\tilde{v}}} \ge 0\) with the same net reactions and, in the case of thermodynamic networks, with the same energies.
Lemma 8
Let \((X,\mathscr {R},\mathbf {g})\) be a reversible RN (with reaction energies), and let \(\mathbf {v}\in \mathbb {R}^\mathscr {R}\) be a vector. Then there is a flow \({\mathbf{\tilde v}} \ge 0\) such that \(\mathbf {S} {\mathbf {{\tilde{v}}}} = \mathbf {S}\mathbf {v}\). If \((X,\mathscr {R},\mathbf {g})\) is thermodynamic, then further \(\langle \mathbf {g}, \mathbf {{\tilde{v}}} \rangle = \langle \mathbf {g}, \mathbf {v}\rangle\).
Proof
If \(\mathbf {v}\ge 0\), there is nothing to show. Otherwise, there are two flows \(\mathbf {v^1} \ge 0\) and \(\mathbf {v^2} > 0\) such that \(\mathbf {v}=\mathbf {v^1}-\mathbf {v^2}\). Since \((X,\mathscr {R})\) is reversible, each reaction \(r\in \mathscr {R}\) has a reverse \(\overline{r}\), and we define the reverse flow \(\mathbf {{\bar{v}}^2}>0\) such that \(\mathbf {{\bar{v}}^2}_{r} = \mathbf {v}^\mathbf {2}_{\overline{r}}\). By construction, it satisfies \(\mathbf {S}\mathbf {{\bar{v}}^2} = - \mathbf {S}\mathbf {v^2}\).
Now consider the flow \(\mathbf {{\tilde{v}}} = \mathbf {v^1}+\mathbf {{\bar{v}}^2} > 0\). It satisfies
$$\begin{aligned}\mathbf {S}\mathbf {{\tilde{v}}}= \mathbf {S}(\mathbf {v^1}+\mathbf {{\bar{v}}^2}) = \mathbf {S}(\mathbf {v^1} - \mathbf {v^2}) = \mathbf {S}\mathbf {v}.\end{aligned}$$
If the network is thermodynamic, then the reverse flow satisfies \(\langle \mathbf {g}, {{\bar{\mathbf{v}}}^2} \rangle = - \langle \mathbf {g}, \mathbf {v^2} \rangle\), by Lemma 5. Hence,
\(\displaystyle \langle \mathbf {g}, \mathbf {\tilde{v}} \rangle = \langle \mathbf {g}, \mathbf {v^1} + \mathbf {{\bar{v}}^2} \rangle = \langle \mathbf {g}, \mathbf {v^1} - \mathbf {v^2} \rangle = \langle \mathbf {g}, \mathbf {v}\rangle\). \(\square\)
By definition, a thermodynamic network cannot contain a perpetuum mobile. Conversely, by the result below, if a reversible network is not thermodynamic, then it contains a perpetuum mobile.
Proposition 9
Let
\((X,\mathscr {R},\mathbf {g})\)
be a reversible RN with reaction energies. Then, the following two statements are equivalent:
-
(i)
\((X,\mathscr {R},\mathbf {g})\) is thermodynamic.
-
(ii)
\((X,\mathscr {R},\mathbf {g})\) contains no perpetuum mobile.
Proof
Suppose \((X,\mathscr {R},\mathbf {g})\) is not thermodynamic. That is, there is \(\mathbf {v}\in \mathbb {R}^\mathscr {R}\) with \(\mathbf {S}\mathbf {v}=0\) and \(\langle \mathbf {v}, \mathbf {g}\rangle \ne 0\). By Lemma 8, there is \(\mathbf {{\tilde{v}}}\ge 0\) with \(\mathbf {S}\mathbf {{\tilde{v}}}=0\) and \(\langle \mathbf {{\tilde{v}}}, \mathbf {g}\rangle \ne 0\), that is, a perpetuum mobile. \(\square\)
The exclusion of a perpetuum mobile is not sufficient in non-reversible systems.
Example 10
Consider the following RN (with reaction energies \(\mathbf {g}\)):
It contains one futile cycle,
\({\mathrm A} {\mathop {\rightarrow }\limits ^{1}}{\mathrm B} {\mathop {\rightarrow }\limits ^{\overline{1}}}{\mathrm A}\), \(\mathbf {v}=(1,1,0,0)^\top\) with \(\langle \mathbf {g},\mathbf {v}\rangle = 0\),
but no perpetuum mobile. However, it contains two parallel flows with different energies,
\({\mathrm A} {\mathop {\rightarrow }\limits ^{1}}{\mathrm B} {\mathop {\rightarrow }\limits ^{2}}{\mathrm C}\), \(\mathbf {v}=(1,0,1,0)^\top\) with \(\langle \mathbf {g},\mathbf {v}\rangle = -2\),
\({\mathrm A} {\mathop {\rightarrow }\limits ^{3}}{\mathrm C}\), \(\mathbf {v}=(0,0,0,1)^\top\) with \(\langle \mathbf {g},\mathbf {v}\rangle = -1\).
Hence, the RN (with reaction energies \(\mathbf {g}\)) is not thermodynamic. By setting \(\mathbf {g}_3=-2\), it can be made thermodynamic.
Many RN models are non-reversible, i.e., they contain irreversible reactions whose reverse reactions are so slow that they are neglected. From a thermodynamic perspective, irreversible reactions r must be exergonic, i.e., \(\mathbf {g}_r<0\). We first consider the extreme case that all reactions \(r\in \mathscr {R}\) are irreversible.
Proposition 11
Let
\((X,\mathscr {R},\mathbf {g})\)
be an irreversible RN with reaction energies. Then, every futile cycle is a perpetuum mobile. Hence, if
\((X,\mathscr {R},\mathbf {g})\)
is thermodynamic, then there are no futile cycles.
Proof
Consider a futile cycle, that is, a flow \(\mathbf {v}> 0\) with \(\mathbf {S}\mathbf {v}=0\). Since all reactions are exergonic, \(\mathbf {v}_r>0\) implies \(\mathbf {g}_r<0\) and further \(\langle \mathbf {g}, \mathbf {v}\rangle < 0\), that is, \(\mathbf {v}\) is a perpetuum mobile. Now, if there is a futile cycle and hence a perpetuum mobile, then the network is not thermodynamic. \(\square\)
Thermodynamic soundness
We next ask whether a RN \((X,\mathscr {R})\) can always be endowed with a vector of reaction energies \(\mathbf {g}\) such that \((X,\mathscr {R},\mathbf {g})\) is thermodynamic.
Definition 12
A RN \((X,\mathscr {R})\) is thermodynamically sound if there is a vector of reaction energies \(\mathbf {g}\) such that \((X,\mathscr {R},\mathbf {g})\) is a thermodynamic network.
We note that thermodynamic soundness is a hereditary property of RNs, since we have seen above that if \((X,\mathscr {R},\mathbf {g})\) is a thermodynamic network so are all its subnetworks \((X',\mathscr {R}',\mathbf {g}')\).
Again, we first consider purely reversible or irreversible RNs.
Proposition 13
Every reversible RN is thermodynamically sound.
Proof
Since \(\mathbf {S}\ne 0\) (the zero matrix), obviously \((\ker \mathbf {S})^\perp = {{\,\mathrm{im}\,}}\mathbf {S}^\top \ne \{0\}\) (the zero vector), and hence there is a non-zero \(\mathbf {g}\in (\ker \mathbf {S})^\perp\). \(\square\)
Theorem 14
An irreversible RN is thermodynamically sound if and only if there are no futile cycles.
Proof
By Gordan’s Theorem (which is in turn a special case of Minty’s Lemma [65], see Appendix B in [66]): Either there is a negative \(\mathbf {g}\in (\ker \mathbf {S})^\perp\) or there is a non-zero, non-positive \(\mathbf {v}\in \ker \mathbf {S}\). That is, either there is \(\mathbf {g}\ll 0\) with \(\mathbf {g}\in (\ker \mathbf {S})^\perp\) (the network is thermodyn. sound) or there is \(\mathbf {v}< 0\) with \(\mathbf {v}\in \ker \mathbf {S}\); equivalently, there is a futile cycle \(\mathbf {v}>0\). \(\square\)
It is not always obvious from the specification of an artificial chemistry model whether or not it is thermodynamically sound. As an example, we consider the artificial chemistry proposed in [67]. It considers only binary reactions (two educts) that produce two products, aiming to ensure conservation of particle numbers. In one variant, the networks only contains irreversible and thus exergonic reactions. It may produce, for instance, the following set of reactions:
$$\begin{aligned} \begin{aligned} {\mathrm{A + B}}&\longrightarrow {\mathrm{C + D}} , \\ {\mathrm{A + C}}&\longrightarrow \mathrm{{E + B}} , \\ \mathrm{{B + D}}&\longrightarrow \mathrm{{F + A}} , \\ \mathrm{{E + F}}&\longrightarrow \mathrm{{A + B}} . \end{aligned} \end{aligned}$$
(9)
Their sum corresponds to the flow \(\mathbf {v}= (1,1,1,1)^\top \ge 0\) and yields the exergonic composite reaction
$$\mathrm{2A + 2B + C + D + E + F} \longrightarrow \mathrm{2A + 2B + C + D + E + F} ,$$
that is, \({\mathbf {S}}{\mathbf {v}}=0\). Thus the model admits a futile cycle composed entirely of exergonic reactions and hence a perpetuum mobile. Thus it is not thermodynamically sound.
Mixed networks
In many applications, RNs contain both reversible and irreversible reactions,
. There are two interpretations of such models:
-
(a)
In the (lax) sense used above, reversible reactions can be associated with arbitrary energies, while irreversible reactions are considered exergonic. That is, the reaction energies must satisfy \(\mathbf {g}_{r}<0\) for \(r\in \mathscr {R}_{\mathrm {irr}}\).
-
(b)
In a strict sense, the reaction energies assigned to irreversible reactions are much more negative than the reaction energies of the reversible ones. After scaling, one requires \(|\mathbf {g}_r|\le 1\) (that is, \(-1 \le \mathbf {g}_r \le 1\)) for \(r\in \mathscr {R}_{\mathrm {rev}}\) and \(|\mathbf {g}_r|\ge \gamma\) (that is, \(\mathbf {g}_r \le -\gamma\)) for \(r\in \mathscr {R}_{\mathrm {irr}}\) and (large) \(\gamma >1\). The intuition is that reactions r with \(\mathbf {g}_r \ge \gamma\) can be neglected.
The following example shows that thermodynamic soundness differs in the lax and strict senses.
Example 15
Consider the following RN (with reaction energies \(\mathbf {g}\)):
for some \(g>0\). It contains two futile cycles:
\({\mathrm A} {\mathop {\rightarrow }\limits ^{1}}{\mathrm B} {\mathop {\rightarrow }\limits ^{\overline{1}}}{\mathrm A}\), \(\mathbf {v}=(1,1,0,0)^\top\) with \(\langle \mathbf {g},\mathbf {v}\rangle = 0\),
\({\mathrm A} {\mathop {\rightarrow }\limits ^{1}}{\mathrm B} {\mathop {\rightarrow }\limits ^{2}}{\mathrm C} {\mathop {\rightarrow }\limits ^{3}}{\mathrm A}\), \(\mathbf {v}=(1,0,1,1)^\top\), \(\langle \mathbf {g},\mathbf {v}\rangle = 1-2g\).
By setting \(g=1/2\), the RN can be made thermodynamic. (Then the second futile cycle is not a perpetuum mobile.)
However, the RN in (10) cannot be seen as the limit of a thermodynamic, reversible network \(({\mathrm A}\leftrightarrow {\mathrm B}\leftrightarrow {\mathrm C}\leftrightarrow {\mathrm A})\) for large g. Thereby, one considers small \(\mathbf {g}_1,\mathbf {g}_{\overline{1}}\) and large negative \(\mathbf {g}_2,\mathbf {g}_3\) (and hence large positive \(\mathbf {g}_{\overline{2}},\mathbf {g}_{\overline{3}}\), that is, negligible reverse reactions \(\overline{2}, \overline{3}\)). Any such (limit of a) reversible RN contains a perpetuum mobile (the second futile cycle); equivalently, it is not thermodynamic.
Definition 16
A mixed network
is thermodynamically sound if there are reaction energies \(\mathbf {g}\) such that \((X,\mathscr {R},\mathbf {g})\) is thermodynamic and \(\mathbf {g}_r<0\) for \(r\in \mathscr {R}_\mathrm {irr}\).
is strictly thermodynamically sound if, for all \(\gamma >1\), there are reaction energies \(\mathbf {g}\) such that \((X,\mathscr {R},\mathbf {g})\) is thermodynamic, \(|\mathbf {g}_r| \le 1\) for \(r\in \mathscr {R}_\mathrm {rev}\), and \(\mathbf {g}_r<0\) with \(|\mathbf {g}_r| \ge \gamma\) for \(r\in \mathscr {R}_\mathrm {irr}\).
The scaling condition can also be written in the form
$$\begin{aligned} \min _{r\in \mathscr {R}_{\text {irr}}} |\mathbf {g}_r| \ge \gamma \max _{r\in \mathscr {R}_{\mathrm {rev}}} |\mathbf {g}_r| \quad \text {for all } \gamma >1. \end{aligned}$$
(11)
A more detailed justification for strict thermodynamic soundness in mixed networks will be given in the next subsection when considering open RNs. Here, we focus on the relationship of thermodynamic soundness and futile cycles.
Theorem 17
A mixed RN
is thermodynamically sound if and only if there is no irreversible futile cycle.
Proof
By a “sign vector version” of Minty’s Lemma: Either there is \(\mathbf {g}\in (\ker \mathbf {S})^\perp\) with \(\mathbf {g}_r<0\) for \(r \in \mathscr {R}_\mathrm {irr}\) (the network is thermodynamically sound) or there is a non-zero \(\mathbf {v}\in \ker \mathbf {S}\) with \(\mathbf {v}_r \le 0\) for \(r \in \mathscr {R}_\mathrm {irr}\) and \(\mathbf {v}_r = 0\) for \(r \in \mathscr {R}_\mathrm {rev}\); equivalently, there is a futile cycle \(\mathbf {v}>0\) with \({{\,\mathrm{supp}\,}}(\mathbf {v}) \subseteq \mathscr {R}_\mathrm {irr}\). \(\square\)
Theorem 18
A mixed RN
is strictly thermodynamically sound if and only if no futile cycle contains an irreversible reaction.
Proof
By Minty’s Lemma: Let \(\gamma >1\). Either there is \(\mathbf {g}\in (\ker \mathbf {S})^\perp\) with \(\mathbf {g}_r \in [-1,1]\) for \(r \in \mathscr {R}_\mathrm {rev}\) and \(\mathbf {g}_r \in (-\infty ,-\gamma ]\) for \(r \in \mathscr {R}_\mathrm {irr}\) or there is \(\mathbf {v}\in \ker \mathbf {S}\) with
$$\begin{aligned} \sum _{r \in \mathscr {R}_\mathrm {rev}} \mathbf {v}_r \, [-1,1] + \sum _{r \in \mathscr {R}_\mathrm {irr}} \mathbf {v}_r \, (-\infty ,-\gamma ] > 0. \end{aligned}$$
(12)
Thereby, a sum of intervals is defined in the obvious way, yielding an interval which is positive if each of its elements is positive. Via \(\mathbf {v}\rightarrow -\mathbf {v}\), the interval condition (12) is equivalent to: there is \(\mathbf {v}\in \ker \mathbf {S}\) with
$$\begin{aligned} \sum _{r \in \mathscr {R}_\mathrm {rev}} \mathbf {v}_r \, [-1,1] + \sum _{r \in \mathscr {R}_\mathrm {irr}} \mathbf {v}_r \, [\gamma ,\infty ) > 0 . \end{aligned}$$
(13)
As necessary conditions, we find (i) \(\mathbf {v}_{r^*} > 0\) for some \(r^* \in \mathscr {R}_\mathrm {irr}\) and (ii) \(\mathbf {v}_{r} \ge 0\) for all \(r \in \mathscr {R}_\mathrm {irr}\). By Lemma 8, (iii) there is an equivalent flow with \(\mathbf {v}_{r} \ge 0\) for \(r \in \mathscr {R}_\mathrm {rev}\). That is, there is a futile cycle \(\mathbf {v}>0\) involving an irreversible reaction. For \(\gamma\) large enough, the necessary conditions are also sufficient for the interval condition (13). \(\square\)
We may characterize strict thermodynamic soundness for mixed networks also in geometric terms:
Corollary 19
Let
, \(L_\mathrm {rev} = {{\,\mathrm{im}\,}}\mathbf {S}_\mathrm {rev}\), and \(C_\mathrm {irr} = {{\,\mathrm{cone}\,}}\mathbf {S}_\mathrm {irr}\). Then, \((X, \mathscr {R})\) is strictly thermodynamically sound if and only if it is thermodynamically sound and \(L_\mathrm {rev} \cap C_\mathrm {irr} = \{0\}\).
Figure 2 illustrates the concepts of futile cycles and (strict) thermodynamical soundness in a metabolically relevant example.
Open (mixed) networks
Opening the RN, i.e., adding transport reactions alters the representation of reaction energies. We now have to consider chemical potentials involving concentrations, i.e., we replace the (Gibbs free) energies \(\mathbf {G}_x\) by \(\mathbf {G}_x + R\,T\ln [x]\), where [x] is the activity of x, which approximately coincides with the concentrations. A reaction r then proceeds in the forward direction whenenver the chemical potential of the products is smaller than the chemical potential of the educts, i.e., if
$$\begin{aligned} \sum _x s^+_{xr} (\mathbf {G}_x + R\,T\ln [x]) < \sum _x s^-_{xr} (\mathbf {G}_x + R\,T\ln [x])\,. \end{aligned}$$
(14)
This condition can be rewritten in terms of the reaction (Gibbs free) energies and (the logarithm of) the “reaction quotient”, see e.g. [68]:
$$\begin{aligned} \mathbf {g}_r < - R\,T \sum _{x\in X} s_{xr}\ln [x] \end{aligned}$$
(15)
The activities [x] for \(x\in X\) therefore define an upper bound on the reaction energy \(\mathbf {g}_r\). In an open system, (internal) concentrations may be buffered as fixed values or are implictly determined by given influxes or external concentrations [69]. Given a specification of the environment, i.e., of the transport fluxes and/or buffered concentrations, the upper bound in Eq. (15) can have an arbitrary value. Thus, if an irreversible reaction in \(\mathscr {R}\) is meant to proceed forward for all conditions, it must be possible to choose \(\mathbf {g}_r<0\) arbitrarily small, i.e., \(|\mathbf {g}_r|\) arbitrarily large. This amounts to requiring that \((X,\mathscr {R}^p)\) is strictly thermodynamically sound. In many studies of reaction networks, one requires that a reaction proceeds forward in a given situation, but allows that it proceeds backward in other situations. In this (lax) interpretation of irreversibility it is sufficient to require that \((X,\mathscr {R}^p)\) is thermodynamically sound, but not necessarily strictly thermodynamically sound.
In Def. 16, we introduce (strict) thermodynamical soundness in terms of reaction energies, and in Thms. 17 and 18, we characterize it in terms of futile cycles. In a corresponding approach [70, 71], “extended” detailed balance is required for (closed) RNs with irreversible reactions at thermodynamic equilibrium. Thereby, activities [x], rate constants \(k_+,k_-\) and equilibrium constants K are explicitly used to formulate Wegscheider conditions for non-reversible RNs that are limits of reversible systems. The characterization of such systems in [70] is equivalent to our results.
Reversible completion
As models of chemistry, non-reversible networks are abstractions that are obtained from reversible thermodynamics networks by omitting the reverse of reactions that mostly flow into one direction.
Definition 20
Let \((X,\mathscr {R},\mathbf {g})\) be a thermodynamic RN with
. The reversible completion of \((X,\mathscr {R},\mathbf {g})\) is the RN \((X,\mathscr {R}^*,\mathbf {g}^*)\) with
and \(\mathbf {g}^*_r=\mathbf {g}_r\) for
and \(\mathbf {g}^*_{\overline{r}}= -\mathbf {g}_r\) for \(r\in \mathscr {R}_\mathrm {irr}\).
Lemma 21
If
\((X,\mathscr {R},\mathbf {g})\)
is a thermodynamic RN, then its reversible completion
\((X,\mathscr {R}^*,\mathbf {g}^*)\)
is also a thermodynamic RN.
Proof
Let \(\overline{r}\in \mathscr {R}^*\) be the reverse reaction of \(r \in \mathscr {R}_\mathrm {irr}\). By Prop. 6, for every \(r\in \mathscr {R}\) there is a vector \(\mathbf {G}\in \mathbb {R}^X\) satisfying Hess’ law. It suffices to show that \(\mathbf {G}\) still satisfies Hess’ law for \((X,\mathscr {R}^*)\). By the definition of \(\overline{r}\), its reaction energy is \(\mathbf {g}^*_{\overline{r}} = \sum _{x\in X} \mathbf {G}_x (s^+_{x{\overline{r}}} - s^-_{x{\overline{r}}}) = \sum _{x\in X} \mathbf {G}_x (s^-_{xr}-s^+_{xr}) = -\mathbf {g}_r\), as required by Def. 20. Thus \((X,\mathscr {R}^*,\mathbf {g}^*)\) is also thermodynamic. \(\square\)
The following result is an immediate consequence of Lemma 21.
Proposition 22
If the RN \((X,\mathscr {R})\) is thermodynamically sound, then its reversible completion is also thermodynamically sound, and the reaction energies \(\mathbf {g}\) can be choosen such that \(\mathbf {g}_r<0<\mathbf {g}_{\overline{r}}\) for all \(r\in \mathscr {R}_\mathrm {irr}\).
Mass conservation and cornucopias/abysses
Thermodynamic soundness is not sufficient to ensure chemical realism. As an example, consider the random kinetics model introduced in [72]. It assigns (a randomly chosen) energy G(x) to each \(x\in X\). Each reaction r is defined by randomly picking a set of educts \(e_r^-\) and products \(e_r^+\). A possible instance of this model comprises four compounds with molecular energies \(G({\mathrm A}) = -5\), \(G({\mathrm B}) = -5\), \(G({\mathrm C}) = -10\), and \(G({\mathrm X}) = -2\), and two reactions
$$\begin{aligned} \mathrm{A + B} \longrightarrow \mathrm{C + X}, \quad \mathrm{C} \longrightarrow \mathrm{A + B}. \end{aligned}$$
(16)
The first reaction is exergonic with \(\mathbf {g}_1=-2\) and the second has reaction energy \(\mathbf {g}_2=0\). The composite reaction, obtained as their sum, is \(\mathrm{A + B} \rightarrow \mathrm{A + B + X}\). Ignoring the effective catalysts A and B, the corresponding net reaction is \({\varnothing } \rightarrow \mathrm{X}\). In this universe, therefore, it is possible to spontaneosly create mass in a sequence of exergonic reactions. Reverting the signs of the energies reverts the two reactions and thus yields an exergonic reaction that makes X disappear.
We can again describe this situation in terms of flows. Recall that \([\mathbf {S}\mathbf {v}]_x\) is the net production or consumption of species x. The spontaneous creation or annihilation of mass thus corresponds to flows \(\mathbf {v}>0\) with \(\mathbf {S}\mathbf {v}>0\) or \(\mathbf {S}\mathbf {v}<0\), respectively.
Definition 23
Let \((X,\mathscr {R})\) be a RN. A flow \(\mathbf {v}>0\) is a cornucopia if \(\mathbf {S}\mathbf {v}>0\) and an abyss if \(\mathbf {S}\mathbf {v}<0\).
Systems with cornucopias or abysses cannot be considered as closed systems. The proper part of chemical reaction networks therefore must be free of cornucopias and abysses.
Since in a reversible network any vector \(\mathbf {v}\in \mathbb {R}^\mathscr {R}\) can be transformed into an equivalent flow \(\mathbf {{\tilde{v}}} \ge 0\) (with \(\mathbf {S}\mathbf {{\tilde{v}}} = \mathbf {S}\mathbf {v}\)), cf. Lemma 8, we have the following characterization.
Proposition 24
A reversible RN is free of cornucopias and abysses if and only if there is no vector \(\mathbf {v}\in \mathbb {R}^\mathscr {R}\) such that \(\mathbf {S}\mathbf {v}>0\).
In fact, mass conservation rules out cornucopias and abysses. More generally, a reaction invariant is a property that does not change over the course of a chemical reaction [8, 27, 29]. Here, we are only interested in linear reaction invariants, also called conservation laws [73], that is, quantitative properties of molecules (such as mass) whose sum is the same for educts and products.
Definition 25
A linear reaction invariant or conservation law is a non-zero vector \(\mathbf {m}\in \mathbb {R}^X\) that satisfies \(\sum _{x\in X} \mathbf {m}_x \, s^+_{xr} = \sum _{x\in X} \mathbf {m}_x \, s^-_{xr}\) for all reactions \(r\in \mathscr {R}\), that is, \(\mathbf {m}^\top \mathbf {S}=0\).
Definition 26
A RN is conservative if it has a positive conservation law, that is, if there is \(\mathbf {m}\in \mathbb {R}^X\) such that \(\mathbf {m}\gg 0\) and \(\mathbf {m}^\top \mathbf {S}=0\).
By definition, a conservative network is free of cornucopias and abysses. Conversely, by the result below, if a reversible network is not conservative, then it contains a cornucopia (and an abyss).
Theorem 27
A reversible RN
\((X,\mathscr {R})\)
is free of cornucopias and abysses if and only if it is conservative.
Proof
By Stiemke’s Theorem (which is in turn a special case of Minty’s Lemma): Either there is a non-zero, non-negative \(\mathbf {n} \in {{\,\mathrm{im}\,}}\mathbf {S}\) or there is a positive \(\mathbf {m}\in ({{\,\mathrm{im}\,}}\mathbf {S})^\perp = \ker \mathbf {S}^\top\). That is, either there is \(\mathbf {v}\in \mathbb {R}^\mathscr {R}\) with \(\mathbf {n} = \mathbf {S}\mathbf {v}> 0\) (corresponding to a cornucopia \(\mathbf {{\tilde{v}}}>0\)) or there is \(\mathbf {m}\gg 0\) with \(\mathbf {S}^\top \mathbf {m}=0\) (as claimed). \(\square\)
We therefore conclude that every closed chemical RN must have a positive reaction invariant. This is no longer true if the RN is embedded in an open system and mass exchange with the environment is allowed. By construction, each transport reaction violates at least one of the conservation laws of the closed system, since \([\mathbf {m}^\top \mathbf {S}]_{r}>0\) if r is an import reaction and \([\mathbf {m}^\top \mathbf {S}]_{r}<0\) if it is an export reaction. As discussed e.g. in [73], opening a RN by adding import or export reactions, can only reduce the number of conservation laws and cannot introduce additional constraints. Nevertheless, a RN must be chemically meaningful when the import and export reactions are turned off. That is, its proper part \((X,\mathscr {R}^p)\) must be conservative to ensure that it has a chemical realization.