Universal Motifs and the Diversity of Autocatalytic Systems

Universal motifs and the diversity of autocatalytic systems

Alex Blokhuisa,b,c,d,1, David Lacostea, and Philippe Ngheb,1

aGulliver Laboratory, UMR CNRS 7083, Paris Sciences et Lettres University, Paris F-75231, France; bLaboratoire de Biochimie, UMR CNRS 8231, Chimie Biologie et Innovation, Ecole Superieure´ de Physique et de Chimie Industrielles de la ville de Paris, PSL University, Paris F-75231, France; cGroningen Institute for Evolutionary Life Sciences, University of Groningen, 9747 AG Groningen, The Netherlands; and dCentre for Systems Chemistry, Stratingh Institute, University of Groningen, 9747 AG Groningen, The Netherlands

Edited by Peter Schuster, University of Vienna, Vienna, Austria, and approved September 1, 2020 (received for review June 30, 2020)

Autocatalysis is essential for the origin of life and chemical evo- From this definition, we derive conditions to determine lution. However, the lack of a unified framework so far prevents whether a subnetwork embedded in a larger chemical network a systematic study of autocatalysis. Here, we derive, from basic can be catalytic or autocatalytic. These conditions provide a principles, general stoichiometric conditions for catalysis and auto- mathematical basis to identify minimal motifs, called autocat- catalysis in chemical reaction networks. This allows for a classifica- alytic cores. We found that cores have five fundamental cat- tion of minimal autocatalytic motifs called cores. While all known egories of motifs. They allow classification of all previously autocatalytic systems indeed contain minimal motifs, the classifica- described forms of autocatalysis, and also reveal hitherto uniden- tion also reveals hitherto unidentified motifs. We further examine tified autocatalytic schemes. We then study the kinetic condi- conditions for kinetic viability of such networks, which depends tions, which we call viability conditions, under which autocat- on the autocatalytic motifs they contain and is notably increased alytic networks can appear and be maintained on long timescales. by internal catalytic cycles. Finally, we show how this frame- We find that networks have different viabilities depending on work extends the range of conceivable autocatalytic systems, by their core structure, and, notably, that viability is increased applying our stoichiometric and kinetic analysis to autocatalysis by internal catalytic cycles. Finally, we expand the repertoire emerging from coupled compartments. The unified approach to of autocatalytic systems, by demonstrating a general mecha- autocatalysis presented in this work lays a foundation toward nism for their emergence in multicompartment systems (e.g., the building of a systems-level theory of chemical evolution. porous media, vesicles, multiphasic systems). This mechanism strongly relaxes chemical requirements for autocatalysis, making autocatalysis | origin of life | chemical reaction networks the phenomenon much more diverse than previously thought. Examples, Definitions, and Conventions he capacity of living systems to replicate themselves is rooted Tin a chemistry that makes more of itself, that is, an auto- Catalysis and Autocatalysis. The following reactions have the catalytic system. Autocatalysis appears to be ubiquitous in liv- same net mass balance but a different status regarding catalysis: ing systems from molecules to ecosystems (1). It is also likely (I) (II) (III) to have been continually present since the beginning of life A B, A + E B + E, A + B 2B. [1] and is invoked as a key element in prebiotic scenarios (2–5). Surprisingly, autocatalysis is considered to be a rarity in chem- Since no species is both a reactant and product in reaction I, it istry (6). Developments in systems chemistry are changing this should be regarded as uncatalyzed. Reactions II and III instead view, with an increasing number of autocatalytic systems synthe- contain species which are both a reactant and a product, species sized de novo (7–9). Chemical replicators have been endowed E in reaction II and species B in reaction III, and, following the with biomimetic properties such as protein-like folding (10) and parasitism (11). Autocatalysis has also found technologi- Significance cal applications, for example, enantiomer enrichment and acid amplification (12–14). Autocatalysis, the ability of chemical systems to make more Understanding autocatalysis represents a primary challenge of themselves, is a hallmark of living systems, as it underlies for theory. Models based on autocatalysis were first built to metabolism, reproduction, and evolution. Here, we present explain a diversity of dynamical behaviors in so-called dissipative a unified theory of autocatalysis based on stoichiometry. structures, such as bistable reactions (15), oscillating reactions, This allows us to identify essential motifs of autocatalytic and chemical waves (16). Autocatalysis then became a central networks, namely, autocatalytic cores, which come in five cat- topic in the study of self-replication dynamics in biological and egories. In these networks, internal catalytic cycles are found prebiotic systems (3, 17–19) (see refs. 20–22 for recent reviews). to favor growth. The stoichiometry approach furthermore Despite this history, a unified theory of autocatalysis is still reveals that diverse autocatalytic networks can be formed lacking. Such a theory is needed to understand the origins, diver- with multiple compartments. Overall, these findings suggest sity, and plausibility of autocatalysis. It would also provide design that autocatalysis is a richer and more abundant phenomenon principles for artificial autocatalytic systems. Here, we present a than previously thought. framework that unifies the different descriptions of autocatalysis and is based on reaction network stoichiometry (23–27). Author contributions: A.B., D.L., and P.N. designed research, performed research, and Let us start from basic definitions in chemistry as established wrote the paper.y by the International Union of Pure and Applied Chemistry (28) The authors declare no competing interest.y (see SI Appendix, section I for full definitions), where autocataly- This article is a PNAS Direct Submission.y sis is a particular form of catalysis: “A substance that increases Published under the PNAS license.y the rate of a reaction without modifying the overall standard 1 ◦ To whom correspondence may be addressed. Email: [email protected] or philippe. Gibbs energy change (∆G ) in the reaction; the process is called [email protected] catalysis. The catalyst is both a reactant and product of the reac- This article contains supporting information online at https://www.pnas.org/lookup/suppl/ tion. Catalysis brought about by one of the products of a (net) doi:10.1073/pnas.2013527117/-/DCSupplemental.y reaction is called autocatalysis.” First published September 28, 2020.

25230–25236 | PNAS | October 13, 2020 | vol. 117 | no. 41 www.pnas.org/cgi/doi/10.1073/pnas.2013527117 Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 lkuse al. et Blokhuis obnn ecin 1 reactions Combining (A) catalysis. 1. Fig. pro- species every vector for reaction positive A negative and duced. is consumed coefficient species the every reaction; the for of that coefficients in stoichiometric participating the are species column a species. in to entries correspond The rows matrix and reactions stoichiometric to correspond columns a as represented Vectors. Reaction and Matrix Stoichiometric stoichiometric the generalize ones. reac- kinetic first net examine will the then we conditions, of following, rate the the In accelerate tion. also according should which, definition, catalysis, the characterize to to sufficient of not but case sary the with to it both. contrast propose to to common we allocatalysis being literature, catalysis of autocatalysis, example chemistry an the it in call to catalyzed represents referred simply III usually experi- is reaction as II B reason, reaction species this Although the autocatalysis. in For genuine where production. unchanged, III, net remains reaction a E of catalysts. ences species case as of the to amount considered contrast the be II, can reaction species In these above, definition ersnaino h uoaayi subnetwork. autocatalytic the of submatrix representation (E autocatalytic B. the to encloses square A converting 1 cycle Combining alytic (D) subnetwork. allocatalytic the submatrix allocatalytic ( the B. to A converts either is species avoided steps participating is reactions a product, a This that of or so I. reactant sequence leading a emerges, reaction a it side, for which through each as from catalysis on vector cancel describing column catalysts by same III, the and to II instance, reactions For matrix. in stoichiometric the in ones 1A, uncatalyzed from Fig. in example, For compounds). law. (total conservation are mass-like quantities conserved a coefficients as all to of case, case that The in quantity. because, laws, vation Vectors nonzero a ν is cycle reaction A vector coordinates. reaction nonzero its of set the is numbers species AB D A AB · eepaieta tihoerccnieain r neces- are considerations stoichiometric that emphasize We aty aaye ecin a o lasb distinguished be always not may reactions catalyzed Lastly, c + + = B2B AB AE EA E r equivalently, or, 0, ifrn ersnain o (A–C for representations Different b T 1’ 2’ 2’’ 1’’ eogn otelf ulsaeof space null left the to belonging EA AB tihoercmti of matrix Stoichiometric B) c uhta ontseisnme hneoccurs: change number species net no that such ∆ + n B A A = E + + ν ν ¯ 0 n AB · B E EA E B A B A c g o network for E g h upr of support The . IIIa

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0 ctlss and (catalysts) fod nalctltcccethat cycle allocatalytic an affords b g E -1 2B. 2’’ · (C B. b 0 r loaayi n (D–F and allocatalysis ) 2’ 2 0 1 1 ν ¯ h ahdsur encloses square dashed the A; n k ] + 00 T 00 ongtv sreferred is nonnegative ersnsaconserved a represents B, o network for rp ersnainof representation Graph ) ecinntok are networks Reaction eut nacag of change a in results ν 2 + g 2,2) nwhich in 26), (23, 00 eoe supp( g denoted , ν fod nautocat- an affords F C nueconser- induce n 1’’ 1’ A h dashed the D; ( F E. + AB EA n B E graph A ) EA auto- ) + [3] [2] 2’ n 2’’ ν ), B . nkntc,nro hc pce r aayi rnt matrix A not. of or catalytic restriction are a species assumptions is nor which coefficients on priori vector nor reaction a kinetics, of on signs makes and autocatal- values identification or the on such allocatalysis neither that of definitions Note motif, the ysis. a with called compatible subnetwork, is corresponding the whether the ask of and network, stoichiometry reaction a submatrix of possible matrix any stoichiometric consider will we section, this In Matrices Stoichiometric in Autocatalysis and Catalysis respected. is it convention that this call We hc l nraei ubrcniinlo pce nteset the in species on conditional number species of in set increase a involves all reactions of which combination a which by cess Autocatalysis. for Criterion submatrix the matrix of allo- larger property being the that a Note not (26). driven is species state external catalytic steady the of nonequilibrium and turnover a emergent the establish by as can it it qualified that have shown the authors Since some matrix, reaction. original net the cycle expresses reaction 4 condition and catalysts, reactions the in of catalysts the of columns of involvement the matrix. of allocatalysts: called columns in the participate which in of included cycle tion is the supp(c) vector to 2) reaction nonzero leads a and This S, numbers. conserved their remain undergo species changes species below. of which conditions external set other turnover while a a catalysts) which (the by number in reactions of ble Allocatalysis. for motif Criterion the outside reactions that motif. forbid the of not species does (30), produce it ( Nets that Petri signs in Note reaction concept 1). on siphon assumption the without to catalysis. similar but is of and autocatal- definition (24), for conditions ysis the former than to restrictive less is opposition Autonomy in Otherwise, already substance, an motif. on present the unconditionally on allowed of be conditional species would is ensures acceleration chemical motif rate it other the of of Indeed, presence species catalysis. any the for of production condition the a that as autonomy pose and respective their submatrices with 1 respectively, Fig autocatalysis, and yellow. allocatalysis in have species boxed submatrices external point stoichiometric 1, been while the Fig. blue, In from in motif. colored waste the been of and have species (29), internal set of view food com- of the feedstock called represent also and (26), pounds, chemostatted cases, some in of of rows (rows motif (remaining the species of species internal into arated under motif the of reactions and consideration. species the to correspond tively aho t ecin a tlatoeratn n tlatone least at and reactant one that of least such called at motif is A has it. product reactions of its outside of occur which each those and motif the odto a endsusdaoe odto expresses 2 Condition above. discussed been has 1 Condition hr xssasto species of set a exists There of species the that means rows the of restriction The etito fclmssprtsratoswihaepr of part are which reactions separates columns of Restriction c ν ¯ F nov aayt.Cniin3epesstecnevto of conservation the expresses 3 Condition catalysts. involve otisapstv n eaiecefiin.Blw we Below, coefficient. negative a and positive a contains . ν ¯ ν ¯ r ozr u ocniin1 ota l reactions all that so 1, condition to due nonzero are ν ¯ 0 c and ( PNAS sacceo h eue arxbtnto the of not but matrix reduced the of cycle a is ν ν ¯ ν hsmasta vr column every that means This autonomous. · sipsdb odto 4. condition by imposed as nalctltccceand cycle allocatalytic an c, ν ¯ c c ocranrw n oun,wihrespec- which columns, and rows certain to 00 = ie,ta r osmdadpoue)are produced) and consumed are that (i.e., | n yegahrpeettosFg 1 Fig. representations hypergraph and , ;ad4) and 0); coe 3 2020 13, October ydfiiin loaayi sa ensem- an is allocatalysis definition, By nonambiguity ydfiiin uoaayi stepro- the is autocatalysis definition, By ν .Teeetra pce ol be, could species external These ). A submatrix a S, and c uhta 1) that such ν · D c | 0 6= n suehenceforth assume and ersnseape of examples represents o.117 vol. h ebr of members The . ν section Appendix, SI ¯ ν ν ¯ ¯ ln u involves but alone 3) ; ν ν ¯ ¯ of nallocatalytic an , ν | ¯ sautonomous; is ν n external and ) o 41 no. c hr all where c, etitdto restricted ν ¯ sareac- a is ν of r sep- are | ν 25231 the , C ν ¯ S

APPLIED PHYSICAL SCIENCES itself (the autocatalysts), while other species undergo a turnover. Type I Type II Type III Type IV Type V This leads to the following conditions. A There exists a set of species S, a submatrix ν¯ of ν restricted to S, and a reaction vector g such that 1) ν¯ is autonomous and 2) all coordinates of ∆n = ν¯ · g are strictly positive, or equivalently, ν¯ has no mass-like conservation laws. The members of S consumed (and produced) by g are called autocatalysts, g is an autocatalytic mode, and ν¯ is an autocatalytic matrix. C3 Condition 1 ensures the conditionality of the reactions on B C2 C S C2 C3’ autocatalysts, as it forbids cases where species of are pro- C3 duced from external reactants only, thus playing the role of C4 C4’ conditions 1 and 2 in the deﬁnition of allocatalysis. Condition 2 C5 C4’ C3’ expresses the increase in autocatalyst number. The equivalence Type I Type II C5’ between the two formulations of condition 2 is an immedi- OH ate consequence of Gordan’s theorem (31). Importantly, the O O HO O second formulation of condition 2 does not involve an autocat- HO H H H H O O O alytic mode g, so that conditions 1 and 2 can be expressed as HO H H OH HO OH O properties of a matrix itself, in contrast with allocatalysis. This O OH allows us to look for minimal autocatalytic motifs, which we do HO HO O OH O

next. Note that external species must feed the autocatalytic sys- OH O OH HO O OH OH tem in order to guarantee the net mass increase imposed by O OH HO OH OH OH condition 2. HO O OH O O H H Autocatalytic Cores. An autocatalytic core is an autocatalytic H H HO HO motif which is minimal because it does not contain any smaller OH OH autocatalytic motif. Consequently, an autocatalytic system is Fig. 2. (A) Five minimal motifs. Orange squares indicate where further either a core or it contains one or several cores. The stoichio- nodes and reactions may be added, provided this preserves the motif type (I, metric conditions show that characterizing cores is equivalent to II, III, IV, V) and minimality. (B and C) Examples of chemical networks, along finding all autonomous matrices whose images contains vectors with their autocatalytic cores. Blue, external species; yellow, autocatalysts. with only strictly positive components. This well-posed formula- (B) Type I: Breslow’s 1959 mechanism for the formose reaction (32). (C) Type tion allowed us to show that the stoichiometric matrix ν¯ of an II: Another autocatalytic cycle in the formose reaction. Species denoted as autocatalytic core must verify a number of nonobvious proper- Cx inside the nodes refer to molecules containing x carbon atoms, which are ties reported below and demonstrated in SI Appendix, sections 2 shown below in standard chemical representation. and 3. First, ν¯ must be square (the number of species equals the number of reactions) and invertible. The inverse has a chemi- a same chemical species. It is obligatory for types II and III to cal interpretation. By definition of the inverse, the kth column contain one fork with two distinct products, for type IV to con- of ν¯−1 is a reaction vector such that species k increases by tain two such forks, and for type V to contain three. In Fig. 2A, one unit, making it an elementary mode of production. Like- the orange squares on the links between the nodes indicate that wise, the reaction vector obtained by summing the columns of these links could contain further nodes and reactions in series, ν¯−1 leads to a net increase by one unit of every autocata- provided certain rules on cycles given in the next paragraph. lyst, which thus represents an elementary mode of autocatalysis. The five types differ in their number of graph cycles and the ∗ This shows how stoichiometry informs on fundamental modes of way these cycles overlap. Type I consists of a single graph cycle autocatalysis (27). that is weight asymmetric, defined as the product of the stoichio- Second, every forward reaction of a core involves only one core metric coefficients of its reaction products being different than species as a reactant. While this excludes reactions between two that of its reactants. Types II and III comprise two distinct but different core species, a single core species may react with itself. overlapping graph cycles, Type IV comprises three, and Type V As ν¯ is square, this also implies that every species of a core is more than three, where any such graph cycle involving a strict consumed (none is only produced), and thus is an autocatalyst. subset of the core species must be an allocatalytic cycle, that is, Furthermore, every species is the reactant of a single reaction. weight symmetric (it would otherwise be of type I, contradicting Overall, every species is uniquely associated with a reaction as minimality). being its reactant, so that ν¯ admits a representation with a negative diagonal and zero or positive coefficients elsewhere, at least Unification of Autocatalytic Schemes. The stoichiometric charac- one coefficient of each column being strictly positive to ensure terization of autocatalysis provides a unified approach to auto- autonomy. catalytic networks reported in the literature. The examples below These properties are constraining enough to allow an exhaus- are further detailed in SI Appendix, section 4. The formose reactive enumeration of reaction graphs that are cores. Autocatalytic tion is a classic example of autocatalysis known to contain many cores are found to belong to five categories, denoted as type I to autocatalytic cycles (33). Fig. 2 B and C shows types I and III type V. Fig. 2A represents typical members of each category as cores both found in the formose reaction. Similarly, autocat- reaction hypergraphs (see SI Appendix, Fig. S1 for general cases). alytic cores of types I and III can be found in the Calvin cycle As can be seen in these graphs, all minimal motifs contain a fork, and reverse Krebs cycle (SI Appendix, Fig. S4). Some reaction which is a reaction with a single reactant and two products. The steps in Fig. 2B may be catalyzed externally (e.g., by enzymes, presence of at least one fork is consistent with the intuition that base, ions), but external catalysis, in general, does not alter the autocatalysis requires reaction steps that amplify the amount of core. By the same token, proposed examples of autoinduction autocatalysts. In type I cores, the fork ends with two copies of the same compound, whereas, in types II to V, forks end with different compounds. Note that type I cores can be seen as a particular *Graph cycles are closed paths in the reaction hypergraph and should not be confounded case of type II if one formally allows different nodes to represent with reaction cycles which are right null vectors of the stoichiometric matrix.

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Fig. only networks, step order reaction first-order forward Among each is, autocatalyst. that one 38); auto- dilute involves 22, first-order highly (17, require the rates in catalysis that, appreciable noted regime, have autocatalyst authors several reactants, of function a steps. as reaction motifs, of autocatalytic probabilities transition within species (17, of problem fixation) probability fixation the examined so-called kinetic we this on 22), depends address degrades To or considerations. amplifies Whether autocatalyst amplify. initial motifs within an autocatalysts that guarantee not Networks. Autocatalytic IV of types Viability of examples found yet not have We V. III. and (SI and II, core I, III types type a to corresponds S5). Fig. described and thus be Appendix, species, can framework process caged our this the that within find with We exchange release. that its amplifying species react can two counterpart form free whose to cage, reactive molecular a involves a mechanism in The compound (37). cavitands to due cation” suggested models. recently these the between establishes 36) (5, formally irreducible link This form (35). they sets and where Autocatalytic RAF (Reflexively (29), RAF framework the in Eq. Food-generated) treated in be as can reactions built they single cycles as 1, written II compactly (cross- are type cycles and cycles catalytic fork allocatalytic as Appendix one such nonoverlapping (SI with incorporation, sequential cycles assemblies described from I is amphiphile type up autocatalysis by of S6) underlying Fig. all growth 5), (SI self-enhancing (4, cores for III and model I type contain 34 S3). and Fig. 21 Appendix, refs. in introduced B A N P 1 osdrn ooeeu ytmwt tayspl of supply steady a with system homogeneous a Considering comprise schemes autocatalytic described previously Overall, amplifi- “chemical is autocatalysis of form reported Another Domain) Replication Autocatalysis (Graded GARD the In ex N 2 (A) yeI ihoefork. one with II type : P ex N sfnto of function as 2 eto 6. section Appendix, SI N N 3 ζ 3 nFg ) oegnrly hnsuch when generally, More 3). Fig. in N (legend: 3 yeI,tonnvrapn allocatalytic nonoverlapping two II, type : ie r exact are lines 8 ; section Appendix, SI P P ex tihoerccniin do conditions Stoichiometric N ex 4 (ζ fetnto (or extinction of for ) P ex < v auto- five (B) for 1) N P 5 ex N 5 yeV. type : 1 − N 1 yeI type : P ex N P of ex 4 , : erdto rbblt) xto spsil o obigprob- probability degradation. doubling a success ability of for possible a is result step fixation has probability), reaction a degradation cycle every as the if disappear that, of or found species (38) irreversibly King proceed (e.g., next either networks may the species I to given type a step, on transition a focused far 2 so Fig. have models fixation tp ntentok elcdby replaced yielding network, pro- the stochastic in these steps X In species (40). autocatalytic an processes using cesses, branching I type of beyond theory networks the and reactions reversible include to in networks contexts. such various for statistics time-dependent (n detailed species reported 39). one (19, derive containing to threshold loop processes decay birth–death autocatalytic used the (17) called al. et is Bagley possible is fixation which l akadfrhtaetre rmX from trajectories back-and-forth all where X have to We revert to probability X sition From cycle. the along reactions net- backward I type for p observation (17) al.’s to works et Bagley generalizes This its that probability The with pnst h ipettp fbacigpoes birth–death a process: branching of Eq. type process. simplest the to sponds yield Eq. simplifies and which converted prematurely, successfully X be species either with step as first such steps, reaction Cycles. I Type Reversible being structures. networks other the for pare solutions in networks, known I generalizing detailed by type method for this results exemplify net- reaction we from Below, constructed works. are processes branching where 5, derive to is bilities here difficulty main The 2 ee eetn h ramn ftefiainpolms as so problem fixation the of treatment the extend we Here, sfudb osdrn l osbesqecso owr and forward of sequences possible all considering by found is p ,wihhv igegahccecontaining cycle graph single a have which B), 0 p Γ Π k X + 2 k k n k s ecnat neednl oextinct, go independently descendants Γ P P = p rcdr o hsi ie in given is this for procedure A . oishpe ihaprobability a with happen copies −→ ∅, eusvl (Γ recursively k > 2 ex ex p +1 Q ete pl tt com- to it apply then We 5. section Appendix, SI 0 1 = 5 1 = fatctltcmtf hc ifri hi core their in differ which motifs autocatalytic of k n hnbcmsaqartceuto htyields that equation quadratic a becomes then = n eesberatos o eesbereactions, reversible For reactions. reversible and =1 p h vrl ucm ecie yEq. by described outcome overall The . P PNAS X 0 s ∞ ex =0 Π P + X k + ex htX that p (Π s | ≥ 1 −→ = ∅ p P k − coe 3 2020 13, October 1 p hsmnmmvleof value minimum This 1/2. osdratp yl ossigof consisting cycle I type a Consider N ←− +1 ex 1 1 2 p 1 1 = 0 X 1, = p mre oe.Utmtl,X Ultimately, node). (marked Γ + s 1 2 s n e ssata the at start us let and 3B, Fig. in k osetnti hnteprobability the then is extinct goes X , k Y p − Π onstesaitclwih of weight statistical the counts ) =1 n 1 2 k k + 1, P −→ . . . −1 s p Π ) ex 4 2 s fe euneo reaction of sequence a after is, 2 k s k + and , to = + 2X ois ecinsequences Reaction copies. Γ p p k p X k . . . 1 | , 2 2 k 1 oisl,i em of terms in itself, to s − , Π < ≥ Π −→ o.117 vol. rmtasto proba- transition from = p Π k + k + k 2 2 1 1 p k 2X X section Appendix, SI k k − let , . , k ∞ =0 1 = (1 ocnett X to convert to +1 k 1 uhthat such , k 1 X − Γ p rb degraded be or .Shse (22) Schuster ). s Π among | k k , Π P Π k − o 41 no. k + n ex k k + etetran- the be . . . . pce.In species. . P , en the being p ex 2 6 n | corre- o an for above 1 steps 25233 k+1 will Π [4] [7] [9] [8] [6] [5] k − n .

APPLIED PHYSICAL SCIENCES and Π+. In the irreversible reaction limit Π− → 0, Γ → 1, King’s k k k A B expression for p2 is recovered (38).

Viability of Autocatalytic Cores. To investigate how autocatalytic motif structure affects survival, we calculated Pex for five different cores (N1 to N5; Fig. 3): They are of equal size (six reaction steps, six species), and all reactions proceed irreversibly with the C same success probability ζ, which plays a similar role as the tran- + sition probability Πk in the example above and is sometimes called specificity (19, 38, 39).† Fig. 3 highlights how Pex depends on ζ for each core structure. The highest ζ for extinction (Pex = 1) is observed for the type I cycle N1, and progressively lower values are found for N2 to N4, which are all of type II. Type V network N5 tolerates the lowest specificity ζ before extinction, sustaining almost 3 times higher failure rates 1 − ζ than N1. These differences can be qualitatively understood by counting the minimum number of steps needed to produce more autocatalysts. In respective order, networks N1 to D N5 in Fig. 3B do so in six, four, three, three, and two steps. In particular, given their symmetries, the Pex of N3 and N5 have the same dependence on ζ as a three- and two-membered type I cycles, respectively. It has been suggested that large networks are disfavored in general (38). The examples of Fig. 3 indicate that this can be counterbalanced by the presence of more allocatalytic cycles in the network. This is, in particular, the case for autocatalytic sets, as every net reaction must participate in an allocatalytic cycle. Extensions: Multicompartment Autocatalysis We finally show how stoichiometric criteria allow the identification of autocatalysis that emerges from compartments coupled via selective exchange, as found in systems comprising vesicles, pores, emulsions, and complex coacervates (41). The reaction Fig. 4. Multicompartment autocatalysis. (A) Reaction network with two network in Fig. 4A is incapable of autocatalysis, as it does not reactions and five species in a single compartment. The network does not contain any autocatalytic core. However, when we place this contain any autocatalytic core, and thus cannot perform autocatalysis. (B) network in two compartments α and β coupled by a mem- Same reaction network as in A, but duplicated in two compartments α brane permeable to A and A2B only, a type II core emerges and β, coupled by the selective exchange of species A and A2B. A type II (Fig. 4B). core, highlighted in orange, emerges. (C) Open reactor with two compart- The core identification indicates a possible setting for auto- ments, a semipermeable membrane, degradation, and exchange reactions. catalysis: U and V are chemostatted in α, and AB is chemostatted Chemostatted species have a lighter background: U and V in α and AB in β.(D) Extinction probability Pex for multicompartment autocatalysis in in β (Fig. 4C). The reaction involving U and V may, in principle, ex C, starting from a single Aα, as a function of exchange rate k and also take place in β, but it is not required for autocatalysis, as it is degradation rate kd, relative to other relevant reaction rates fixed at k. not part of the type II core. In Fig. 4 C and D, it is assumed that d Slanting asymptote: exchange-limited√ survival kex = 2k . Vertical asymptote: U and V are absent. reaction-limited survival kd/k = 13 − 3/2. Dashed white line: transition We now apply our viability analysis to this autocatalytic net- between extinction and potential fixation (Pex < 1). Expressions for Pex and work in the presence of degradation reactions (r6 and r7 in Fig. asymptotes are derived in SI Appendix, section 7. 4C). Let us introduce a characteristic rate k for reactions r2, r4 d and r5, a degradation rate k for reactions r6 and r7, and an ex single reaction A B + C can give rise to type III motifs, given exchange rate k for reactions r1 and r3. Fig. 4D shows the d three compartments coupled by selective exchange as detailed in extinction probability as a function of the degradation rate k ex SI Appendix, section 9 and Fig. S9. and exchange rate k , both normalized by k. To overcome the d degradation threshold (Pex < 1), the ratio k /k must lie above Discussion a certain threshold (vertical black dotted line in Fig. 4D), and We presented a theoretical framework for autocatalysis based k ex the rate of exchange should outpace the rate of degradation on stoichiometry, which allows a precise identification of the (black slanting dotted line in Fig. 4D). Stochastic simulations (SI different forms of autocatalysis. Starting with a large stoichio- Appendix, section 8 and Fig. S9) confirm autocatalytic growth in metric matrix, we provide criteria for reaction network motifs these conditions. that allow allocatalysis and autocatalysis. A detailed analysis of In this example of coupled compartments, compounds are no the graph structure contained in these reduced stoichiometric α longer restricted to one role: AB is an autocatalyst in and a matrices reveals that they contain only five possible recurrent β feedstock in . Chemical reactions are no longer restricted to motifs, which are minimal in the sense that they do not contain α one direction: The reaction used for reproduction in is reused smaller motifs. Fundamental modes of production of minimal β in to provide the missing step to close the cycle. Such multicom- autocatalytic cores are encoded in the column vectors of the partment autocatalysis is, however, more general. For instance, a inverse of the autocatalytic core submatrix. Autocatalytic cores are found to have a single reactant species for each reaction. This means that autocatalytic networks require the availability † Note that, in general, the calculation of Pex may involve reactions that are not in the of certain chemical species in their cores to operate properly, core. Here, we consider cases where this is not necessary. but also implies that the proper functioning of an autocatalytic

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