Thermodynamic consistency of autocatalytic cycles

Significance

Describing the processes behind the origin of life requires us to better understand self-amplifying dynamics in complex chemical systems. Detecting autocatalytic cycles is a critical but challenging step in this endeavor. After characterizing the computational complexity of this problem, we investigate the impact of thermodynamic realism on autocatalysis. We demonstrate that individual cycles, regardless of thermodynamic parameters, can always be activated as long as entities may occur at any required concentration. In contrast, two cycles can become mutually incompatible due to thermodynamic constraints, and will thus never run simultaneously. These results clarify the implications of physical realism for the realization of autocatalysis.

Abstract

Autocatalysis is seen as a potential key player in the origin of life, and perhaps more generally in the emergence of Darwinian dynamics. Building on recent formalizations of this phenomenon, we tackle the computational challenge of exhaustively detecting minimal autocatalytic cycles (autocatalytic cores) in reaction networks and further evaluate the impact of thermodynamic constraints on their realization under mass action kinetics. We first characterize the complexity of the detection problem by proving its NP-completeness. This justifies the use of constraint solvers to list all cores in a given reaction network, and also to group them into compatible sets, composed of cores whose stoichiometric requirements are not contradictory. Crucially, we show that the introduction of thermodynamic realism does constrain the composition of these sets. Compatibility relationships among autocatalytic cores can indeed be disrupted when the reaction kinetics obey thermodynamic consistency throughout the network. On the contrary, these constraints have no impact on the realizability of isolated cores, unless upper or lower bounds are imposed on the concentrations of the reactants. Overall, by better characterizing the conditions of autocatalysis in complex reaction systems, this work brings us a step closer to assessing the contribution of this collective chemical behavior to the emergence of natural selection in the primordial soup.

It is increasingly recognized that producing a consistent explanation for the origination of life will require us to explain how Darwinian evolution may have gradually emerged from a nonbiological, purely physical world (1–9). Gradually rather than suddenly, that is, without assuming that natural selection only came into play once chance alone had produced the first obvious “replicators,” displaying the same heritable variance as current organisms. Under this perspective of a smooth transition from physics to biology, natural selection is hypothesized to have been active already in the “prebiotic” soup as a driver to complexity; yet in a rudimentary and currently unrecognizable fashion.

To explore this path, autocatalysis is often taken as a plausible starting point (Box 1) (6, 13–15, 19, 20). Here, more specifically, we envision autocatalytic cycles as the putative elementary components of higher-level systems that may engage in “increasingly Darwinian” dynamics. In doing so, we aim at keeping the best of the two traditionally opposed approaches to the origin of life: physicochemical realism of the metabolism-first view and evolvability of the gene-first perspective. Beyond the specifics of terrestrial life, progressing toward an articulation of Darwinian principles with physics appears to be a prerequisite to assessing their putative relevance to other physical systems (2, 3).

Box 1:

Related work on autocatalysis

The present study takes place within a flourishing body of literature taking autocatalysis as a plausible primary component of protobiotic or proto-Darwinian systems. Our model contrasts with those based on the RAF framework (10, 11) in that it follows a bottom–up approach to autocatalysis: Rather than setting catalytic relationships between components of the system and randomly picked reactions, we let the reaction network generate (or not) these relationships, as formalized by Blokhuis et al. (12). Catalysis and autocatalysis then simply emerge in the reaction network as pathways involving entities that act both as reactants and products. For example, in the reactions A+C→AC, AC+B→ABC, ABC→AB + C, entity C can be simply described as a catalyst of the reaction A+B→AB.

In taking such a bottom–up angle, our framework is much related to that of several recent studies (13–18). Some of these have considered the implications of thermodynamic constraints and mass action kinetics on specific autocatalytic motifs (14, 15, 17). Others have implemented tools for the exhaustive detection of autocatalysis (13, 16, 18). Here, we jointly consider these two components of the problem, i.e., exhaustive detection and thermodynamic realism.

On a more conceptual ground, we share with Baum et al. (6) the view that collections of autocatalytic cycles, rather than cycles alone, might constitute the scale at which incipient heritable variations may occur.

We build on recent theoretical and computational developments (12, 13, 16) to systematically search for minimal autocatalytic cycles [denoted autocatalytic cores, sensu Blokhuis et al. (12)] in reaction networks and then assess their thermodynamic consistency, i.e. the impact of thermodynamic constraints on their realization, under mass action kinetics. We first prove that finding autocatalytic cores in the network is an NP-complete problem—a question that was left open in earlier work (21, 22)—and converge with other authors in using constraint solvers as a technical solution (13, 16). We then question whether such autocatalytic cores, defined on the sole basis of the reaction network topology, can also be realized once thermodynamic constraints are introduced. To do so, we take into account the reaction kinetics that themselves depend on the Gibbs free energies and concentrations of the reactants, and the activation barriers of the reactions. We show that regardless of these physical quantities, any potential autocatalytic core may be instantiated in some region of the concentration space as long as this space is assumed unbounded. In contrast, thermodynamic constraints do restrain compatibility relationships between autocatalytic cores and will thereby impact the dynamics in complex chemical networks.

1. Framework and Definitions

We analyze networks of reversible reactions governed by mass action kinetics. This typically applies to reactions that simply consist in the association of two entities and the reciprocal dissociation (e.g. $A+B\rightleftharpoons AB$). The entities are fully defined by their composition (e.g. $A_2{B}_2$ is not distinct from $B_2{A}_2$). Given a list of entities, this rule sets the list of all possible reactions, only some of which are assumed to exist to generate a particular reaction network—this is equivalent to assuming that some reactions have an infinite activation barrier and thus have a null flow.

We can then apply the formalism of Blokhuis et al. (12) to identify autocatalytic motifs in such reaction networks. Intuitively, these can be conceived as cyclic subnetworks admitting a regime where each entity has a positive net production rate.

Definition 1:

An autocatalytic motif is defined as a set of entities E_C and reactions R_C such that

• For each reaction $R\in R_C$, at least one entity on each side is in E_C.

• There exists a vector $\overrightarrow{v}$ of flows for reactions from R_C defining a regime where the total contribution of reactions from R_C is strictly positive for each entity of E_C.

Consider for instance a reaction $A+B\rightleftharpoons AB$, with reactants A, B and product AB. Then the stoichiometry ${\sigma }_A^R$ of A in R is $-1$ (or $-2$ if A = B), and ${\sigma }_{\mathit{AB}}^R$ is 1. If an entity e does not appear in a reaction R, we set ${\sigma }_e^R=0$.

We will note v_R, the flow of the reaction at a given instant, that will be positive if the association rate is larger than the dissociation rate.

Given a motif candidate C formed of entities E_C and reactions R_C, and an entity e in E_C, we define the variation of e’s concentration due to C as

$$\begin{array}{c}\hfill {\displaystyle {\mathrm{\Delta }}_C(e)={\displaystyle \sum _{\begin{array}{c}R\in R_C\end{array}}}{\sigma }_e^R·v_R}\end{array}$$ [1]

We define a witness as a choice of v_R for each $R\in R_C$, such that for each $e\in E_C$ we have ${\mathrm{\Delta }}_C(e)>0$. This can be formalized using linear algebra, following Blokhuis et al. (12). Indeed, if M is the stoichiometric matrix restricted to E_C and R_C, then the candidate C is an autocatalytic motif if and only if there exists a witness vector $\overrightarrow{v}$ such that all coordinates of $M·\overrightarrow{v}$ are strictly positive.

Example 1.

We illustrate the definition in Fig. 1 with a simple formose-like cycle comprising three entities A₂, A₃, and A₄ and using entity A₁ as food, with detailed explanations provided in the caption.

Fig. 1.

Schematic view of a formose-like autocatalytic motif. Entities A₂, A₃, and A₄ are part of the motif while entity A₁ serves as food. The arrows indicate the net direction of each reaction, while the line width indicates their respective flows, that must be decreasing from reactions R₁ to R₃ for the cycle to run. As an example, the flow values (indicated in brackets) would produce a net increase of 1 of each entity. The Right panel shows the corresponding stoichiometric matrix M. Given the represented flow vector $\overrightarrow{v}=(5,4,3)$, we obtain $M·\overrightarrow{v}=(1,1,1)$, showing that $\overrightarrow{v}$ is indeed a witness.

Entities appearing in R_C that are not part of E_C will be called either “food” if they are consumed or “waste” if they are produced by a reaction of R_C, taking into account the sign of the witness vector that indicates the direction of reactions. Notice that an entity may simultaneously appear as food and waste in an autocatalytic motif.

Formally, a motif is said to contain another one if it includes all its entities and reactions. In contrast, a motif is minimal if it does not contain any other, in which case it corresponds to the “autocatalytic core” from Blokhuis et al. (12) and to the stoichiometric autocatalysis of Gagrani et al. (16). The present study focuses on such minimal motifs, that we denote Potential Autocatalytic Cores (PACs), to emphasize that they are defined on the sole basis of the reaction network topology, so that their realizability under thermodynamic constraints remains to be assessed.

Definition 2:

A PAC is defined as a minimal autocatalytic motif, whose identification relies solely on the reaction network topology as formalized by the stoichiometric matrix.

Example 2.

The autocatalytic motif shown in Fig. 1 is minimal, and hence constitutes a PAC.

Notice that minimality implies that a PAC witness cannot contain null reaction flows: If this were the case, the corresponding reaction could be removed without altering autocatalysis, which contradicts the minimality criterion. Furthermore, it is shown by Blokhuis et al. (12) that in a PAC, each entity is the reactant of a unique reaction, and each reaction has a unique entity of the PAC as reactant (other reactants being food). This implies that the direction of reactions is consistent across all witnesses of a given PAC: Flipping the direction of one reaction would impose flipping all the others. Therefore, each reaction of the PAC has a unique possible net direction that will be compatible with all its witness flow vectors. In SI Appendix, section A, we provide a formal proof of this property that was put forward by Blokhuis et al. (12, SI Appendix).

2. Detecting Potential Autocatalytic Cores

Our goal is to enumerate all PACs in a reaction network. To this end, we first assess the complexity of this problem, in order to determine which computational tools are required for its resolution.

2.1. NP-Completeness Proof.

Enumerating all PACs in a reaction network involves sequentially solving problems of the type: “Is there a PAC in the system besides those previously found?” We will show that a particular variant of this question is NP-complete. Namely, we will prove the NP-completeness of deciding whether a PAC exists that contains an entity A and takes food from a given subset F.

In this section, for simplicity, we will relax any compositionality constraint on reactions so that letters like $A,B,E\cdots$ will be shorthand for any kind of entity. Yet it would be straightforward (but less readable) to extend the construction to a strictly compositional framework.

Notably, the complexity of the autocatalysis detection problem has previously been considered by Andersen et al. (21) but from a different angle. These authors have specifically shown that the following problem is NP-complete: Considering a reaction network that contains a known autocatalytic core, can its resources be produced by the network? The difficulty of finding all autocatalytic cores in a reaction network, that we tackle here, has thus not yet been addressed.

In the framework of Blokhuis et al. (12), it is easy to check whether a proposed set of entities and reactions constitutes an autocatalytic core. Indeed, thanks to the linear algebra formulation summarized in Section 1, this problem is solved in polynomial time by Linear Programming. As will be shown, the difficulty rather lies in finding an autocatalytic core in a reaction network, among exponentially many possible candidates.

Formally, let PAC-DETECTION be the following algorithmic problem:

Definition 3 (PAC-DETECTION problem):

INPUT: A reaction system defined by entities $(E)$ and reactions $(R)$, a target A ∈ E, and a set of allowed foods F ⊂ E.

OUTPUT: Is there a PAC containing A and using only foods from F?

Theorem 1.

PAC-DETECTION is NP-complete.

We provide here a brief description of the proof that is fully described in SI Appendix, section B. Because a PAC candidate can be tested in polynomial time, PAC-DETECTION is in NP. It remains to be shown that it is NP-hard. To this end, we reduce from the well-known NP-complete problem SAT (23). An instance of Satisfiability (SAT) asks whether an input formula on n boolean variables $x_1,\cdots ,x_n$ is satisfiable, via a suitable assignment of variables with true/false values.

To perform the reduction, we associate each such formula φ with a reaction system S_φ, of size polynomial in φ, with a specified target entity A and a food set F. Reactions in S_φ are designed to mirror the structure of φ, ensuring that a PAC of the wanted form exists if and only if the formula is satisfiable. The only possible such PACs will actually directly encode satisfying assignments for φ.

This shows that PAC-DETECTION is NP-hard: A polynomial-time algorithm for PAC-DETECTION would yield a polynomial-time algorithm for SAT, via this reduction. We can conclude that PAC-DETECTION is NP-complete, since it is also in NP. Notice that our proof assumes a particular shape for constraints on the searched PAC (containing an entity A and using only foods from a set F). Assessing whether an unconstrained variant is NP-complete as well remains an open problem.

2.2. Implementation.

The above NP-completeness result justifies the use of an standing for Satisfiability Modulo Theories (SMT) Solver such as Z3 to enumerate all PACs. We thus implemented this approach in C++ in the EmergeNS software (24) (more generally designed to simulate the dynamics of complex physicochemical systems and down the line to trace the physical emergence of natural selection). In practice, in a reaction system defined in EmergeNS, we ask the Z3 solver to find PAC candidates and to assess, for each candidate, the existence of a PAC witness, i.e. a reaction flow vector $\overrightarrow{v}$ yielding strictly positive net production rates for all entities of the candidate. We then remove the detected PAC from the search space and repeat the process until no new PAC is found.

As remarked in Section 2.1, verifying whether a given candidate is indeed a PAC (by assessing the existence of a PAC witness) is a linear programming problem, and can thus be achieved efficiently, i.e. in polynomial time. The need for Z3 comes from the search for PAC candidates in an exponential space of possible subsets of entities and reactions.

3. PAC Consistency Under Thermodynamic Constraints

The kinetics of a reaction network must obey the second law of thermodynamics, a constraint that is not considered in the PAC definition. Indeed, this definition solely relies on the existence of a witness $\overrightarrow{v}$ of reaction flows, that may or may not be compatible with thermodynamic constraints.

More precisely, once association and dissociation constants are derived from free energies and activation barriers, they cannot be freely chosen. To clarify these constraints, recall that each entity e is associated with a chemical potential μ, which is the sum of its molar Gibbs free energy (or standard chemical potential) G and the logarithm of its activity (that we identify here with its concentration $[e]$, hence placing ourselves in the ideal solution regime):^*

$$\begin{array}{c}\hfill \mu =G +ln[e]\end{array}$$ [2]

In addition, the flow of a reaction R, denoted v_R, depends on the concentrations (i.e. activities) of its entities, under mass action kinetics:

$$\begin{array}{c}\hfill v_R=k_R^{(+)}·{\displaystyle \prod _{e_i \text{reactant}}}{[e_i]}^{-{\sigma }_{e_i}^R}-k_R^{(-)}·{\displaystyle \prod _{e_j \text{product}}}{[e_j]}^{{\sigma }_{e_j}^R},\end{array}$$ [3]

where $k_R^{(+)}$ and $k_R^{(-)}$ are respectively the forward and backward kinetic rate constants and ${\sigma }_{e_i}^R$ is the stoichiometry of e_i in reaction R (negative for reactants, hence explaining the presence of a minus sign in the above formula). The key point, to enforce the second law under mass action kinetics, is to relate the rate constants to Gibbs free energies using the local detailed balance condition (also known as Eyring’s formula):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill k_R^{(+)}& = exp(-G_R^{†}+{\displaystyle \sum _{e_i \text{reactant}}}(-{\sigma }_{e_i}^R)·G_i),\hfill \\ \hfill k_R^{(-)}& = exp(-G_R^{†}+{\displaystyle \sum _{e_j\text{product}}}{\sigma }_{e_j}^R·G_j),\hfill \end{array}\end{array}$$ [4]

where $G_R^{†}$ is the free energy of the intermediate state that induces an activation barrier. Injecting Eq. 4 in Eq. 3 leads to the following formulation that will be further used below:

$$\begin{array}{c}\hfill v_R=e^{-G_R^{†}}·\left({\displaystyle \prod _{e_i\text{reactant}}}{e}^{-{\sigma }_{e_i}^R·{\mu }_i}-{\displaystyle \prod _{e_j\text{product}}}{e}^{{\sigma }_{e_j}^R·{\mu }_j}\right).\end{array}$$ [5]

The sign of v_R is the same as that of the affinity of the reaction ${\mathcal{A}}_R=-\sum _i{\sigma }_{e_i}^R·{\mu }_i$, expressing how a positive rate can be achieved. Recall, however, that positive flows for all reactions of a PAC will not necessarily entail autocatalysis, as we aim more specifically for flows balanced in such a way that each entity has a net positive production rate (Definition 1). The question of addressing the thermodynamic consistency of PACs can thus be reformulated as follows: Can a PAC flow witness $\overrightarrow{v}$ be realized through a concentration vector $\overrightarrow{c}$ of all entities of the system? If such a vector $\overrightarrow{c}$ exists, the PAC will be considered a thermodynamically Consistent Autocatalytic Core (CAC). Here, “thermodynamic consistency” refers to mass action kinetics, with rate constants derived from Gibbs free energies and activation barriers.

Definition 4:

A CAC is a PAC for which there exists a concentration vector $\overrightarrow{c}$ yielding reaction flows forming a PAC witness.

The concentration vector $\overrightarrow{c}$ will then be called a CAC witness. Here, we reach the second key result of the present study, namely, that one can always find such a CAC witness—i.e. that a single PAC is always thermodynamically consistent—as long as the concentration space is unbounded.

Theorem 2.

Let $\overrightarrow{v}$ be a PAC witness (or any flow vector compatible with the directions of the PAC reactions). Then there exists $\lambda >0$ and a concentration vector $\overrightarrow{c}$ such that $\lambda \overrightarrow{v}$ is the flow vector induced by $\overrightarrow{c}$.

The general proof of this theorem is given in SI Appendix, section C, with an example provided in SI Appendix, section C.1. It should be noted that Theorem 2 actually has a broader scope than the problem specifically addressed in this section, since it demonstrates that any flow vector $\overrightarrow{v}$ matching the directions of the PAC reactions can be realized in the concentration space up to some proportionality factor $\lambda >0$. Notably, this holds for any values of activation barriers and Gibbs free energies, and even when food and waste concentrations are fixed to any arbitrary values.

This goes beyond the well-established result, based on affinity, that a positive flow can be achieved for any reaction by adjusting concentrations. Here, the difficulty is to aim for a fixed vector $\overrightarrow{v}$ of rates, up to some coefficient λ shared among all reactions of the PAC.

We deduce the following corollary from Theorem 2:

Corollary 1.

Any PAC is a CAC.

Proof of Corollary 1: Consider a PAC formed of entities $(e_1,\cdots ,e_n)$ and reactions $(R_1,\cdots ,R_n)$ (recall that according to Blokhuis et al. (12), the minimality of a PAC implies that it contains as many entities as reactions). Let $\overrightarrow{v}=(v_1,\cdots v_n)$ be a PAC witness. Since the inequalities to be satisfied by a PAC witness are all linear with respect to the coordinates of $\overrightarrow{v}$ (Eq. 1), for all $\lambda >0$, $\lambda \overrightarrow{v}$ is a PAC witness as well. Applying Theorem 2 gives us the existence of some $\lambda >0$ and a concentration vector $\overrightarrow{c}$ yielding flows $\lambda \overrightarrow{v}$, which is a PAC witness. Thus, $\overrightarrow{c}$ is a CAC witness, and the arbitrary PAC we started with is indeed a CAC.

4. Compatibility Among Autocatalytic Cores

In this section, we investigate whether thermodynamic constraints affect compatibility relationships among cores. To do so, we first analyze compatibility among PACs, that is, we identify sets of cores that are found compatible on the basis of the reaction network topology alone, hereafter called “multiPACs.” A set of PACs is a multiPAC if there exists a common witness $\overrightarrow{v}$ of reaction flows allowing all the PACs of the set to run simultaneously. In the framework of Gagrani et al. (16), this corresponds to a nonempty intersection of the flow-productive cones for the different autocatalytic cores considered. Here, instead of computing explicit intersections, we will use the Z3 solver to identify nonempty intersections, and directly ask for a single vector $\overrightarrow{v}$ witnessing the different PACs simultaneously. To do so, we simply concatenate the requirements already defined for each individual PAC. Let us emphasize that a multiPAC witness does not guarantee global production of each entity when reactions from all PACs are jointly considered (this can be seen for example in Fig. 4, where entity e₄ has a positive production rate within each PAC, although it is consumed overall).

Fig. 4.

Two topologically compatible but thermodynamically incompatible autocatalytic cores. The two PACS ({R₁, R₂, R₃, R₄} and {$R_1^{\prime }$, R₂, R₃, $R_4^{\prime }$}) have two reactions in common. One can check that both cores run simultaneously with flows $v_1=v_1^{\prime }=6$, $v_2=5$, $v_3=4$, and $v_4=v_4^{\prime }=7$ which allows for a production rate of 1 for all entities belonging to a PAC. However, as detailed in Box 2, it can be shown that these cores cannot be instantiated simultaneously in the concentration space, i.e. they form a multiPAC but not a multiCAC.

Interestingly, we note that incompatibilities between PACs may occur for various reasons. The simplest case, illustrated in Fig. 2, is when a reaction is shared between two PACs, but in opposite directions. This obviously prevents the existence of a common flow vector witness $\overrightarrow{v}$. Yet more subtle cases were also obtained with our software (24) using randomly generated reaction networks. One example is described in Fig. 3. Generally speaking, incompatibilities among PACs occur because of contradictory flow requirements, that is, when one PAC requires a reaction R₁ to run faster than a reaction R₂, while a second PAC requires the opposite.

Fig. 2.

A simple example of two incompatible PACs. Ignoring foods and wastes and identifying A to empty circles and B to filled circles, the PAC with solid black arrows is $AB\to A{B}_2\to A_2{B}_2\to AB+AB$ and the PAC with gray dotted arrows is $A{B}_2\to AB\to A_{2}B\to A_2{B}_3\to AB+A{B}_2$. Numbers in brackets indicate the flows of PAC reactions allowing for a (local) net production of 1 of all their entities. The two PACs share the reaction labeled $R^{\star }$, but require it to run in opposite directions.

Fig. 3.

A more subtle example of two incompatible PACs, sharing two reactions indexed with flows v₁ and v₂. Flows allowing for unitary production of PAC entities are shown in brackets for both PACs. The inner PAC depicted with solid black arrows ($A_2\to A_3\to A_4\to A_2+A_2$) imposes the flow inequalities $v_1>v_2>v_3>v_1/2$, while the gray dotted outer PAC ($A_2\to A_3\to A_4\to A_5\to A_2+A_3$) imposes inequalities $v_1+v_5>v_2>v_4>v_5>v_1$. This yields contradictory requirements: $v_1>v_2$ for the first PAC and $v_1<v_2$ for the second.

To investigate the impact of thermodynamic constraints on compatibility relationships among cores, we now assess whether pairs of compatible PACs, witnessed by a common flow vector $\overrightarrow{v}$, also constitute pairs of compatible CACs, witnessed by a common concentration vector $\overrightarrow{c}$. As before, addressing this question using the SMT solver is straightforward: It is enough to concatenate the lists of constraints required for each of the CACs under study, and to ask whether a $\overrightarrow{c}$ vector exists that simultaneously satisfies them all.

This analysis reveals that compatibility among cores inferred solely from the reaction network topology may overlook thermodynamic inconsistencies. Indeed, we find several instances of two compatible PACs making incompatible CACs. We give an example of such a behavior in Fig. 4 and provide a formal proof that these two CACs are incompatible in Box 2.

Box 2:

Thermodynamic incompatibility between CACs

For the example shown in Fig. 4, we can first show that the two PACs are compatible, since we can choose a set of reaction flows that witnesses both of them simultaneously. For instance, setting $v_1=v_1^{\prime }=6$, $v_2=5$, $v_3=4$, $v_4=v_4^{\prime }=7$ leads to a positive production rate of each entity by each PAC. More generally, this criterion is achieved for both PACs simultaneously when the following inequalities are satisfied: $v_3<v_2<v_1<v_4<2v_3$ and $v_3<v_2<v_1^{\prime }<v_4^{\prime }<2v_3$.

We will show that these inequalities are not thermodynamically achievable, i.e. that they lead to contradictory requirements in the concentration space.

Notice that the inequalities imply that all v_i are strictly positive, because $v_3<2v_3$ entails $v_3>0$, and all other v_i are larger than v₃. As proven below, this sign constraint alone is not satisfiable: Not all reactions can flow in the wanted direction.

We denote $x_i=e^{{\mu }_i}$ the exponential of the chemical potential of entity e_i (indexed as in Fig. 4), and $b_i=e^{-G_i^{†}}$, where $G_i^{†}$ is with respect to reaction R_i (similarly for $R_i^{\prime }$). Reaction flows can thus be written as follows:

$$\begin{array}{cc}{v}_1=b_1(x_1-x_B{x}_2)& v_4=b_4(x_A{x}_4-x_1)\\ v_1^{\prime }=b_1^{\prime }(x_1^{\prime }-x_A{x}_2^{\prime })& v_4^{\prime }=b_4^{\prime }(x_B{x}_4-x_1^{\prime })\\ v_3=b_3(x_3-{(x_4)}^2)& v_2=b_2(x_2{x}_2^{\prime }-x_3)\end{array}$$

From the positivity of all flows, we get

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}{x}_B{x}_2& x_1& x_A{x}_4\\ x_A{x}_2^{\prime }& x_1^{\prime }& x_B{x}_4\\ {(x_4)}^2& x_3& x_2{x}_2^{\prime }\end{array}\right.\end{array}$$

From the first line, we deduce $x_B{x}_2/x_4<x_A$, and reinjecting in the second line we obtain $x_B{x}_2{x}_2^{\prime }/x_4<x_A{x}_2^{\prime }<x_B{x}_4$. This simplifies to $x_2{x}_2^{\prime }<{(x_4)}^2$, which contradicts the condition of the third line. It should be noted that this proof is valid regardless of the activation barrier values, since the contradiction stems from the signs of the flows, while activation barriers only affect their amplitudes.

5. Discussion

Working toward the long-term goal of an explicit grounding of Darwinian dynamics into physical processes, we addressed in this study the implications of thermodynamic constraints on the existence and detection of autocatalytic cores given a reaction network. Our analysis builds on recent theoretical progress made on the formalization of autocatalysis on the sole basis of the reaction network topology (12). Under this definition, we show that the exhaustive detection of autocatalysis is an NP-complete problem. This finding fully justifies the use of constraint solvers (e.g. SMT, Integer Programming) toward which we converge with others (13, 16).

The constraints imposed by free energies and activation barriers can always be compensated by adjusting concentrations, thereby allowing any minimal autocatalytic cycle to also be thermodynamically consistent under mass action kinetics. In other words, the list of autocatalytic cores in a reaction network remains unaffected by these physical constraints, as long as concentrations are not limited by upper or lower bounds. However, as shown in SI Appendix, section D, it should be noted that heterogeneity in free energies and activation barriers do restrict the volume of the concentration space where a core can effectively run.

These conclusions on isolated cores do not readily apply to combinations of cores. Indeed, thermodynamic realism does restrict the list of mutually compatible cores, even in an unlimited concentration space, so that topologically compatible cores can turn out incompatible. Incompatibilities between two autocatalytic cores can therefore stem from two distinct sources, namely the topology of the reaction network (PAC-incompatibility) and irreconcilable demands on concentrations (CAC-incompatibility).

A stimulating next step will be to investigate the implications of autocatalysis on the system’s dynamics through time and space. Indeed, the existence of a CAC witness says little of the actual trajectories occurring in the concentration space, and on the influence of autocatalytic cores on these trajectories. Among the many autocatalytic cores that are thermodynamically achievable in a given system, which ones are actually encountered from a given starting point in the concentration space? Which ones are short-lived, long-lived, or keep running once the system has reached a steady state? Which ones run in the vicinity of this steady state, and perhaps contribute to drive the system in its direction? And finally, could autocatalysis contribute to generating more than one steady state in the concentration space? We anticipate that such multistability could enable a primordial form of heritable variation, paving the way to nascent Darwinian dynamics.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Code data have been deposited in Github (24). There are no data underlying this work.