Universal Motifs and the Diversity of Autocatalytic Systems

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APPLIED PHYSICAL SCIENCES definition above, these species can be considered as catalysts. We call this convention nonambiguity and assume henceforth In reaction II, the amount of species E remains unchanged, in that it is respected. contrast to the case of reaction III, where the species B experi- ences a net production. For this reason, reaction III represents Catalysis and Autocatalysis in Stoichiometric Matrices genuine autocatalysis. Although reaction II is usually referred to In this section, we will consider any possible submatrix ν¯ of ν, the as simply catalyzed in the chemistry literature, we propose to stoichiometric matrix of a reaction network, and ask whether the call it an example of allocatalysis to contrast it with the case of stoichiometry of the corresponding subnetwork, called a motif, autocatalysis, catalysis being common to both. is compatible with the definitions of allocatalysis or autocatal- We emphasize that stoichiometric considerations are neces- ysis. Note that such identification makes a priori assumptions sary but not sufficient to characterize catalysis, which, according neither on the values and signs of reaction vector coefficients nor to the definition, should also accelerate the rate of the net reac- on kinetics, nor on which species are catalytic or not. A matrix ν¯ tion. In the following, we will first generalize the stoichiometric is a restriction of ν to certain rows and columns, which respec- conditions, then examine kinetic ones. tively correspond to the species and reactions of the motif under consideration. Stoichiometric Matrix and Reaction Vectors. Reaction networks are The restriction of the rows means that the species of ν are sep- represented as a stoichiometric matrix ν (23, 26), in which arated into internal species of the motif (rows of ν¯) and external columns correspond to reactions and rows correspond to species. species (remaining rows of ν). These external species could be, The entries in a column are the stoichiometric coefficients of the in some cases, chemostatted (26), and represent feedstock com- species participating in that reaction; the coefficient is negative pounds, also called the food set (29), and waste from the point for every species consumed and positive for every species pro- of view of internal species of the motif. In Fig. 1, external species T duced. A reaction vector g = [g1, .., gr ] results in a change of have been colored in blue, while stoichiometric submatrices have species numbers ∆n = ν · g. The support of g, denoted supp(g), been boxed in yellow. Fig 1 A and D represents examples of is the set of its nonzero coordinates. A reaction cycle is a nonzero allocatalysis and autocatalysis, respectively, with their respective reaction vector c such that no net species number change occurs: submatrices ν¯0 and ν¯00, and hypergraph representations Fig. 1 C ν · c = 0, or, equivalently, c belongs to the right null space of ν. and F. Vectors bT belonging to the left null space of ν induce conser- Restriction of columns separates reactions which are part of vation laws, because, in that case, b · n represents a conserved the motif and those which occur outside of it. A motif such that quantity. The case of all coefficients bk nonnegative is referred each of its reactions has at least one reactant and at least one to as a mass-like conservation law. For example, in Fig. 1A, product is called autonomous. This means that every column conserved quantities are nE + nEA (catalysts) and nA + nEA + nB of ν¯ contains a positive and a negative coefficient. Below, we (total compounds). pose autonomy as a condition for catalysis. Indeed, it ensures Lastly, catalyzed reactions may not always be distinguished that the production of any species of the motif is conditional on from uncatalyzed ones in the stoichiometric matrix. For instance, the presence of other chemical species of the motif. Otherwise, in reactions II and III, catalysts cancel on each side, leading rate acceleration would be allowed unconditionally on an already to the same column vector as for reaction I. This is avoided present substance, in opposition to the definition of catalysis. by describing catalysis through a sequence of reactions steps Autonomy is less restrictive than former conditions for autocatal- from which it emerges, so that a participating species is either ysis (24), and is similar to the siphon concept in Petri Nets (30), a reactant or a product, but without assumption on reaction signs (SI Appendix, section 1). Note that it does not forbid that reactions outside the motif IIa IIb produce species of the motif. A + E EA E + B, [2] IIIa IIIb Criterion for Allocatalysis. By definition, allocatalysis is an ensem- A + B AB 2B. [3] ble of reactions by which a set of species remain conserved in number (the catalysts) while other external species undergo a turnover which changes their numbers. This leads to the AB1’ 2’ C conditions below. A -1 0 There exists a set of species S, a submatrix ν¯ of ν restricted to 1’ EA S, and a nonzero reaction vector c such that 1) ν¯ is autonomous; A + E EA B 0 1 c ν¯ c 1’ 2’ 2) supp( ) is included in the columns of ; 3) is a reac- 2’ + E -1 1 tion cycle of ν¯ (ν¯ · c = 0); and 4) ν · c 6= 0. The members of S EA E B which participate in c (i.e., that are consumed and produced) are EA 1 -1 E called allocatalysts: c, an allocatalytic cycle and ν¯, an allocatalytic matrix. 1’’ 2’’ Condition 1 has been discussed above. Condition 2 expresses D E A -1 0 F AB the involvement of the catalysts in the reactions c, where all 1’’ columns of ν¯ are nonzero due to condition 1, so that all reactions AB+ AB B -1 2 1’’ 2’’ of c involve catalysts. Condition 3 expresses the conservation of 2’’ AB 1-1 catalysts, and condition 4 expresses the net reaction. Since the AB 2B B reaction cycle c is a cycle of the reduced matrix but not of the original matrix, some authors have qualified it as emergent and shown that it can establish a nonequilibrium steady state driven Fig. 1. Different representations for (A–C) allocatalysis and (D–F) auto- 0 0 by the turnover of the external species (26). Note that being allocatalysis. (A) Combining reactions 1 + 2 affords an allocatalytic cycle that catalytic is not a property of the submatrix ν¯ alone but involves converts A to B. (B) Stoichiometric matrix of A; the dashed square encloses the allocatalytic submatrix ν¯0 for network B.(C) Graph representation of the larger matrix ν as imposed by condition 4. the allocatalytic subnetwork. (D) Combining 100 + 200 affords an autocatalytic cycle converting A to B. (E) Stoichiometric matrix of D; the dashed Criterion for Autocatalysis. By definition, autocatalysis is the pro- square encloses the autocatalytic submatrix ν¯00 for network E. (F) A graph cess by which a combination of reactions involves a set of species representation of the autocatalytic subnetwork. which all increase in number conditional on species in the set

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APPLIED PHYSICAL SCIENCES introduced in refs. 21 and 34 contain type I and III cores (SI fixation models have so far focused on type I networks (e.g., Appendix, Fig. S3). Fig. 2B), which have a single graph cycle containing n species. In In the GARD (Graded Autocatalysis Replication Domain) a transition step, a given species may either proceed irreversibly model for self-enhancing growth of amphiphile assemblies to the next species or disappear as a result of degradation. (4, 5), all underlying autocatalysis is described (SI Appendix, King (38) found that, if every reaction step k among n steps + + Fig. S6) by type I cycles with one fork and type II cycles built of the cycle has a success probability Πk (1 − Πk being the up from sequential nonoverlapping allocatalytic cycles (cross- degradation probability), fixation is possible for a doubling prob- n incorporation, such as N3 in Fig. 3). More generally, when such Q + ability p2 = k=1 Πk ≥ 1/2. This minimum value of p2 above catalytic cycles are compactly written as single reactions as in Eq. which fixation is possible is called the decay threshold (19, 39). 1, they can be treated in the RAF (Reflexively Autocatalytic and Bagley et al. (17) used birth–death processes to derive Pex for an Food-generated) framework (29), where they form irreducible autocatalytic loop containing one species (n = 1). Schuster (22) RAF sets (35). This formally establishes the recently suggested reported detailed time-dependent statistics for such networks in link (5, 36) between these models. various contexts. Another reported form of autocatalysis is “chemical amplifi- Here, we extend the treatment of the fixation problem so as cation” due to cavitands (37). The mechanism involves a reactive to include reversible reactions and networks beyond type I using compound in a molecular cage, whose free counterpart can react the theory of branching processes (40). In these stochastic pro- to form two species that exchange with the caged species, thus cesses, an autocatalytic species Xs is, after a sequence of reaction amplifying its release. We find that this process can be described steps in the network, replaced by k copies. Reaction sequences within our framework and corresponds to a type III core (SI yielding k copies happen with a probability pk , such that Appendix, Fig. S5). p p p Overall, previously described autocatalytic schemes comprise X −→∅0 , X −→1 X , ... X −→k kX , .... [4] types I, II, and III. We have not yet found examples of types IV s s s s s and V. The probability Pex that Xs goes extinct is then the probability that its k descendants independently go extinct, Viability of Autocatalytic Networks. Stoichiometric conditions do not guarantee that autocatalysts within motifs amplify. Whether ∞ 2 X k an initial autocatalyst amplifies or degrades depends on kinetic Pex = p0 + p1Pex + p2Pex + ... = pk Pex . [5] considerations. To address this so-called fixation problem (17, k=0 22), we examined the probability Pex of extinction (or 1 − Pex of fixation) of species within autocatalytic motifs, as a function of The main difficulty here is to derive pk from transition proba- transition probabilities of reaction steps. bilities Πk . A procedure for this is given in SI Appendix, section Considering a homogeneous system with a steady supply of 5, where branching processes are constructed from reaction net- reactants, several authors have noted that, in the highly dilute works. Below, we exemplify this method by generalizing known autocatalyst regime, appreciable rates require first-order auto- results for type I networks, solutions for other networks being catalysis (17, 22, 38); that is, each forward reaction step only detailed in SI Appendix, section 5. We then apply it to com- involves one autocatalyst. Among first-order order networks, pare the Pex of autocatalytic motifs which differ in their core structures.

Reversible Type I Cycles. Consider a type I cycle consisting of n A reaction steps, such as N1 in Fig. 3B, and let us start at the P ex ﬁrst step with species X1 (marked node). Ultimately, X1 will either be successfully converted and yield 2X1 or be degraded prematurely, which simpliﬁes Eq. 4 to

p0 p2 P ∅ ←− X1 −→ 2X1, [6] ex

with p0 + p2 = 1. The overall outcome described by Eq. 6 corresponds to the simplest type of branching process: a birth–death process. Eq. 5 then becomes a quadratic equation that yields

1 1 − 1, p2 ≥ , p2 2 Pex = 1 [7] 1, p2 < 2 . B This generalizes Bagley et al.’s (17) observation for type I networks to n > 1 and reversible reactions. For reversible reactions, p2 is found by considering all possible sequences of forward and − backward reactions along the cycle. From Xk, let Πk be the tran- + sition probability to revert to Xk−1, and Πk to convert to Xk+1. We have N N N N N n 1 2 3 4 5 Y + p2 = Πk Γk , [8] k=1 Fig. 3. (A) Pex as function of ζ (legend: Pex (ζ) for Pex < 1) for (B) ﬁve autocatalytic networks of similar size, starting at the dashed node. N1: type I ∞ X − + s 1 cycle. N : type II with one fork. N : type II, two nonoverlapping allocatalytic 2 3 Γk+1 = (Πk+1Γk Πk ) = − + , [9] cycles, a common motif in GARD with a ﬁrst-order RAF representation. N4: s=0 1 − Πk+1Γk Πk type II, allocatalytic cycles connected by intermediate steps. N5: type V. Pex after 1,000 simulated trials, detailed in SI Appendix, section 8; lines are exact where Γk recursively (Γ1 = 1) counts the statistical weight of − solution, derived in SI Appendix, section 6. all back-and-forth trajectories from Xk to itself, in terms of Πk

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Fig. and in 9 detailed section as Appendix, exchange SI selective by coupled compartments three A reaction single in derived are (P asymptotes fixation potential and extinction between i.4. Fig. ecinlmtdsurvival reaction-limited survival exchange-limited A asymptote: Slanting single rate a in degradation V from and starting U C, background: lighter a in reactions. have exchange species and Chemostatted degradation, membrane, semipermeable (C a emerges. ments, orange, in highlighted core, (B) in autocatalysis. as not perform and does network cannot network thus reaction The and Same compartment. core, single autocatalytic a any in contain species five and reactions D C A β xicinprobability Extinction (D) . β ope yteslcieecag fseisAadA and A species of exchange selective the by coupled , utcmatetatctlss ( autocatalysis. Multicompartment k B d

eaiet te eeatrato ae xdat ﬁxed rates reaction relevant other to relative , B k + eto 7 . section Appendix, SI d /k a iers otp I ois given motifs, III type to rise give can C α = safnto fecag rate exchange of function a as , u ulctdi w compartments two in duplicated but A, P √ ex 13 o utcmatetatctlssin autocatalysis multicompartment for − .Dse ht ie transition line: white Dashed 3/2. pnratrwt w compart- two with reactor Open ) ecinntokwt two with network Reaction A) ex k < ex NSLts Articles Latest PNAS ) xrsin for Expressions 1). = 2k d etclasymptote: Vertical . 2 .Atp II type A B. α k | n AB and P ex ex f7 of 5 and and k α .

APPLIED PHYSICAL SCIENCES network will guarantee the stable supply of certain products, a conditions are met for the exchange of compounds between definitive advantage when these products are key enzymes or compartments. In the example shown here (Fig. 4), this metabolites. allowed us to reuse the compounds and reactions to com- We identified these minimal motifs in known examples of plete autocatalytic cycles. The principle is more general, how- autocatalysis such as the formose reaction, central metabolic ever: Autocatalysis may also emerge from coupling phases with cycles, the GARD model, and RAF sets. Autocatalytic cores also physical–chemical conditions conducive to different reactions, provide a basis for algorithms to identify these recurring auto- as observed in liquid–solid (48) and solid–gas (49) interfaces. catalytic motifs in large chemical networks (24, 42, 43), as has Liquid–liquid interfaces in cellular organization and multiphase been done for gene regulatory networks (44). In this way, we may coacervates (41) are promising places to further explore such be able to break the complexity of large chemical networks into principles. smaller, more manageable structures (45). Additionally, auto- Overall, our framework shows that autocatalysis comes in a catalytic cores are the building block of evolution in prebiotic diversity of forms and can emerge in unexpected ways, indicating chemistries (35); thus their identification paves the way to a sys- that autocatalysis in chemistry must be more widespread than tematic exploration of the possible modes of chemical evolution previously thought. This invites a search for further extensions of (46, 47). autocatalysis, which provides new vistas for understanding how Autocatalytic motifs provide different degrees of robustness, chemistry may complexify toward life (50). which we evaluated using the notion of viability. Viability can be computed as a survival probability in an appropriately defined Materials and Methods branching process. This approach is generally applicable to auto- Theoretical methods and derivation of results are detailed in SI Appendix catalytic models upon identification of their cores, highlighting comprising the following sections: 1) terminology and definitions; 2) deriva- the interest of a unified framework. Viability results from a tion of autocatalytic cores from graph theory; 3) their chemical inter- competition between reactions that produce autocatalysts and pretation and 4) application to formose, autoinduction, metabolic cycles, side reactions such as degradation. This is intimately related to chemical amplification, RAF sets, and GARD; 5) branching process derivation and determination of P ; 6) determination of P for Fig. 3; 7) determina- the “paradox of specificity” (19, 39): Autocatalytic motifs are ex ex tion of Pex for Fig. 4D; 8) stochastic simulations; and 9) autocatalysis from more likely to be found in large networks with many different one bimolecular reaction and three compartments. chemical components engaging in many different reactions, but putting many components together favors side reactions, leading Data Availability. There are no data underlying this work. to extinction. Multicompartment autocatalysis introduced here offers a way ACKNOWLEDGMENTS. A.B. acknowledges stimulating discussions with D. van de Weem and Z. Zeravcic. A.B. and D.L. acknowledge support from around this problem: Coupled compartments effectively enlarge Agence Nationale de la Recherche (Grant ANR-10-IDEX-0001-02) and A.B., the number of species without requiring new reactions. In D.L., and P.N. acknowledge the program Paris Sciences Lettres IRIS (Initiatives multicompartment autocatalysis, cycles rely on the environ- de Recherches Interdisciplinaires et Strategiques)´ Origines et Conditions mental coupling of reaction networks, which allows access to d’Apparition de la Vie. P.N. acknowledges X. Allamigeon, S. Krishna, and C. Flamm for discussions and funding from the Centre Franco-Indien pour conditions unattainable in a single compartment. In this way, la Promotion de la Recherche Avancee.´ A.B., D.L., and P.N. acknowledge autocatalysis can emerge from reaction schemes as simple Andrew Griffiths for careful reading of the manuscript. A.B. acknowledges as a bimolecular reaction, provided certain semipermeability the donation of Jonathan Rothberg for research on artificial life.

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