1 Autocatalysis in chemical networks: unifications and extensions 1, 2 1 2 2 Alex Blokhuis, David Lacoste, and Philippe Nghe 1) 3 Gulliver Laboratory, UMR CNRS 7083, PSL University, 10 rue Vauquelin, Paris F-75231, 4 France 2) 5 Laboratoire de Biochimie, Chimie Biologie et Innovation, ESPCI Paris, PSL University,10 rue Vauquelin, 6 Paris F-75231, France 7 (Dated: May 16, 2020) 8 Autocatalysis is an essential property for theories of abiogenesis and chemical evolution. However, the 9 different formalisms proposed so far seemingly address different forms of autocatalysis and it remains unclear 10 whether all of them have been captured. Furthermore, the lack of unified framework thus far prevents 11 a systematic study of autocatalysis. Here, we derive general stoichiometric conditions for catalysis and 12 autocatalysis in chemical reaction networks from basic principles in chemistry. This allows for a classification 13 of minimal autocatalytic motifs, which includes all known autocatalytic systems and motifs that had not 14 been reported previously. We further examine conditions for kinetic viability of such networks, which 15 depends on the autocatalytic motifs they contain. Finally, we show how this framework extends the range 16 of conceivable autocatalytic systems, by applying our stoichiometric and kinetic analysis to autocatalysis 17 emerging from coupled compartments. The unified approach to autocatalysis presented in this work lays a 18 foundation towards the building of a systems-level theory of chemical evolution. 19 PACS numbers: 05.40.-a 82.65.+r 82.20.-w 20 INTRODUCTION 53 different descriptions of autocatalysis and is based on 27–30 54 reaction network stoichiometry . 21 Life’s capacity to make more of itself is rooted in a 22 chemistry that makes more of itself. Autocatalysis ap- 23 pears to be ubiquitous in living systems from molecules 55 Let us start from basic definitions in chemistry as 1 31 24 to ecosystems . It is also likely to have been continu- 56 established by IUPAC (see Supplementary Informa- 25 ally present since the beginning of life and is invoked 57 tion Section I for full definitions), where autocatalysis 2–5 26 as a key element in prebiotic scenarios . However, 58 is a particular form of catalysis: A substance that in- 6 27 autocatalysis is considered to be a rarity in chemistry . 59 creases the rate of a reaction without modifying the ◦ 28 This would suggest that prebiotic molecules may not 60 overall standard Gibbs energy change (∆G ) in the re- 4,5 29 be that different from biomolecules (lipids , nucleic 61 action; the process is called catalysis. The catalyst is 7–10 2,3 30 acids , and peptides ). 62 both a reactant and product of the reaction. Catalysis 63 brought about by one of the products of a (net) reaction 31 Developments in systems chemistry are changing 64 is called autocatalysis. 32 this view, with an increasing number of autocatalytic 11–13 33 systems synthesized de novo . Chemical replicators 34 have been endowed with biomimetic properties such as 14 15 35 protein-like folding and parasitism . Autocatalysis 65 From this definition, we derive conditions to de- 36 also finds technological applications, e.g. eniantomere 66 termine whether a subnetwork embedded in a larger 16–18 37 and acid amplification . 67 chemical network, can be catalytic or autocatalytic. 68 These conditions provide a mathematical basis to iden- 38 The elucidation of autocatalysis represents a pri- 69 tify minimal motifs, called autocatalytic cores. Cores 39 mary challenge for theory. Models based on autocatal- 70 come in five structural motifs. They allow classification 40 ysis were first built to explain a diversity of dynami- 71 of all previously described forms of autocatalysis, and 41 cal behaviors in so called dissipative structures, such 19 72 also reveal not yet identified autocatalytic schemes. We 42 as bistable reactions , oscillating reactions, chemical 20 73 then study the kinetic conditions, which we call viabil- 43 waves and chemical chaos. Autocatalysis then be- 74 ity conditions, under which autocatalytic networks can 44 came a central topic for studying the self-replication 3,21–23 75 appear and be maintained on long times. We find that 45 dynamics of biological and prebiotic systems 24–26 76 networks have a different viability depending on their 46 (see for recent reviews). 77 core structure, notably that internal catalytic cycles 47 Despite this long history, a unified theory of au- 78 increase robustness. Finally, we expand the repertoire 48 tocatalysis is still lacking. Such a theory is needed to 79 of autocatalytic systems, by demonstrating a general 49 understand the origins, diversity and plausibility of 80 mechanism for its emergence on a multicompartment 50 autocatalysis. It would also provide design principles 81 level. This mechanism strongly relaxes chemical re- 51 for artificial autocatalytic systems. 82 quirements for autocatalysis, making the phenomenon 52 Here, we present a framework that unifies the 83 more abundant and diverse than previously thought. 2 1 2 84 EXAMPLES, DEFINITIONS AND CONVENTIONS a) b) c) 1 A -1 0 EA A + E EA B 0 1 85 Catalysis and autocatalysis 2 1 2 EA E +B E -1 1 EA 1 -1 E 86 The following reactions have the same net mass 87 balance but a different status regarding catalysis: d) e) 1 2 f) (I) (II) (III) A -1 0 AB A −−* B, A+E −−* B+E, A+B −−−* 2B. 1 )−− )−− )−−− + B -1 2 (1) A B AB 1 2 2 AB 1 -1 88 Since no species is both a reactant and product AB 2B B 89 in reaction (I), it should be regarded as uncatalyzed. 90 Reactions (II) and (III) instead contain species which Figure 1: Different representation for allocatalysis (a,b,c) 91 are both a reactant and a product, species E in reaction and autocatalysis (d,e,f). a) Combining reactions 92 (I) and species B in reaction (III) and following the (1’)+(2’) affords an allocatalytic cycle that converts 93 definition above, these species can be considered as A to B. b) stoichiometric matrix of a), the dashed 94 catalysts. In reaction (II), the amount of species E square encloses the allocatalytic submatrix ν¯0 for 95 remains unchanged, in contrast to the case of reaction network b). c) Graph representation of the allocat- 96 (III), where the species B experiences a net production. alytic subnetwork. d) Combining (1”)+(2”) affords 97 For this reason, reaction (III) represents genuine au- an autocatalytic cycle converting A to B. e) stoichio- 98 tocatalysis. Although reaction (II) is usually referred metric matrix of d), the dashed square encloses the 00 99 to as simply catalyzed in the chemistry literature, we autocatalytic submatrix ν¯ for network e). f) a graph representation of the autocatalytic subnetwork. 100 propose to call it an exemple of allocatalysis to contrast 101 it with the case of autocatalysis. Then, we can reserve 102 the word catalysis for any instance and combination of 103 allocatalysis, autocatalysis and reverse autocatalysis. 132 matrix. For instance, in reactions (II-III), catalysts 133 cancel on each side leading to the same column vector 104 We emphasize that such stoichiometric consider- 134 as for (I). One way to avoid this is to describe enough 105 ations are necessary but not sufficient to characterize 135 intermediate steps so that a participating species is 106 catalysis, which according to the definition should also 136 either a reactant or a product: 107 accelerate the rate of the net reaction. In the following, 108 we will first generalize the stoichiometric conditions, −−IIa* −−IIb* 109 then examine kinetic ones. A + E )−− EA )−− E + B, (2) IIIa IIIb A + B )−−−−* AB )−−−−* 2B . (3) 110 Stoichiometric matrix and reaction vectors 137 We call this convention non-ambiguity and assume 138 henceforth that it is respected. 111 Reaction networks are represented as a stoichio- 27,30 112 metric matrix ν , in which columns correspond to 139 CATALYSIS AND AUTOCATALYSIS IN 113 reactions and rows to species. The entries in a column 140 STOICHIOMETRIC MATRICES 114 are the stoichiometric coefficients of the species partic- 115 ipating in that reaction, the coefficient is negative for 116 every species consumed and positive for every species 141 We seek to identify candidate motifs (subnetworks) T 117 produced. A reaction vector g = [g1, .., gr] results 142 for allocatalysis and autocatalysis within reaction net- 118 in a change of species numbers ∆n = ν · g. The sup- 143 works based the only knowledge of the stoichiometric 119 port of g (denoted supp(g)) is the set of its non-zero 144 matrix ν. Such identification does not make a priori 120 coordinates. A reaction cycle is a non-zero reaction 145 assumptions on the values or signs of reaction vector 121 vector c such that no net species number change occurs 146 coefficients, or on kinetics, which we will examine in the 122 : ν · c = 0, or equivalently, c belongs to the right null 147 next section. A motif has a stoichiometric submatrix ν¯ T 123 space of ν. Vectors b belonging to the left null space 148 obtained by restricting ν to certain rows and columns. 124 of ν induce conservation laws, because in that case 149 Restriction to rows delimitates species within the 125 30 b · n represents a conserved quantity. The case of all 150 system from external species , such as feedstock com- 126 32 coefficients bk nonnegative is refered to as a mass-like 151 pounds (also so-called food set ) and waste. In Fig. 1, 127 conservation law. For example in Fig.1a, conserved 152 external species have been colored in blue, while stoi- 128 quantities are nE + nEA (catalysts) and nA + nEA + nB 153 chiometric submatrices have been boxed in yellow.