Impact of composition on the dynamics of autocatalytic sets
IImpact of composition on the dynamics of autocatalytic sets
Alessandro Ravoni
Department of Mathematics and Physics,University of Roma Tre,Via della Vasca Navale 84, 00146 Rome,Italy
Autocatalytic sets are sets of entities that mutually catalyse each other’s productionthrough chemical reactions from a basic food source. Recently, the reflexively autocat-alytic and food generated theory has introduced a formal definition of autocatalytic setswhich has provided promising results in the context of the origin of life. However, thelink between the structure of autocatalytic sets and the possibility of different long-termbehaviours is still unclear. In this work, we study how different interactions amongautocatalytic sets affect the emergent dynamics. To this aim, we develop a model inwhich interactions are presented through composition operations among networks, andthe dynamics of the networks is reproduced via stochastic simulations. We find thatthe dynamical emergence of the autocatalytic sets depends on the adopted compositionoperations. In particular, operations involving entities that are sources for autocatalyticsets can promote the formation of different autocatalytic subsets, opening the door tovarious long-term behaviours.
Keywords: Autocatalytic sets; Origin of life; Network Composition; Stochastic Petri nets; Binary polymer model
I. INTRODUCTION
The exact sequence of events that led to the formationof the first living organisms from non-living matter isstill a topic under debate (Benner et al., 2012; Bernhardt,2012; Luisi, 2016; Rasmussen et al., 2004; Szostak, 2017).On the other hand, some particular qualities of earlyorganisms are commonly accepted and are well known.Among these, we can point out the self-replication abilityof the early life forms (Higgs and Lehman, 2015; Kauff-man, 1993; Luisi, 2016; Nghe et al., 2015; Rasmussen etal., 2016). In this scenario, the autocatalytic sets (ASs)are of great interest. Introduced by Kauffman (Kauff-man, 1971, 1986, 1993), ASs are sets of entities capableof spontaneous emergence and self-reproduction throughcatalytic reactions, starting from a finite set of entitiesassumed to be available in the environment. There areseveral definitions of ASs in the literature (see, for in-stance, (Jain and Krishna, 1998; Sharov, 1991)). Re-cently, Hordijk et al. (2011) have introduced the notion ofreflexively autocatalytic and food generated (RAF) sets,a formal definition of ASs in the framework of chemi-cal reaction systems (CRSs) (see Section II for defini-tions). Properties of RAF sets have been studied by var-ious authors, and among the most important results (see(Hordijk and Steel, 2017, 2018) for more details) note theimplementation of a polynomial time algorithm able toidentify the presence of a RAF set in a general networkof interacting entities (Horidjk and Steel, 2004; Hordijket al., 2011; Hordijk and Steel, 2012) and the detectionof an autocatalytic structure in the metabolic network ofEscherichia Coli (Sousa et al., 2015) and ancient anaero-bic autotrophs (Xavier et al., 2020). Moreover, RAF theory has successfully proved thatRAF sets have a hierarchical structure, where the largestset of reactions within a CRS satisfying RAF prop-erty (the so-called maxRAF) has many subsets thatare smaller RAF sets themselves (Hordijk et al., 2012;Hordijk and Steel, 2014, 2018). The latter is a pecu-liarity that makes RAF sets potentially suitable for ex-periencing adaptive evolution (a feature that is gener-ally referred to as evolvability); i.e., to collect evolution-ary changes beneficial for survival and reproduction in agiven environment. Indeed, it has been argued that theautocatalytic subsets present within the structure of themaxRAF could be the elementary units on which natu-ral selection can act (Hordijk and Steel, 2014; Hordijk etal., 2018b; Vasas et al., 2012): the availability of sponta-neous reactions would allow the occurrence of mutationsand, consequently, the appearance of novel autocatalyticsubsets able to replicate themselves with different ratesand competing with each other.However, first results show that the asymptotic dy-namics of simple RAF sets eventually reaches the statein which all the reactions of the maxRAF occur catalyt-ically (Hordijk et al., 2018b; Vasas et al., 2012). In thisstate, all the elementary autocatalytic units coexist with-out effectively competing with each other, thus leavingno room for adaptive evolution (Hordijk et al., 2018b;Vasas et al., 2010, 2012). The evolvability of RAF setscan be restored by embedding them into compartmentsand allowing the sharing of resources and the exchangeof chemical molecules (Hordijk et al., 2018b; Kauffman,2011; Serra and Villani, 2019; Vasas et al., 2012). In fact,through numerical simulations, it has been observed thatRAF sets enclosed in semipermeable protocells can reach a r X i v : . [ n li n . AO ] S e p different asymptotic states (Serra and Villani, 2019), andthat spatially separated RAF sets consuming the samefood source can give rise to different combinations ofcompeting autocatalytic subsets (Hordijk et al., 2018b),suggesting that the evolvability of RAF sets is related tothe interactions among RAF sets themselves.In this work we investigate this latter point. In par-ticular, we study the role of various interactions amongRAF sets in order to understand how these interactionsaffect the emergent dynamics. To this aim, we use thestochastic Petri nets (SPNs) formalism (Haas, 2006; Mol-loy, 1982) to represent and evolve RAF sets. Further-more, we introduce some composition operations actingon nets, which correspond to different interactions amongRAF sets. In this framework, assuming that the entiremaxRAF set always emerges in an isolated RAF set,our goal is to find some composition operations underwhich the dynamical appearance of the maxRAF is notinvariant. This means that the corresponding interac-tion causes only some of the maxRAF subsets to emerge,allowing the existence of multiple long-term behavioursrequired for the evolvability of RAF sets.The paper is organised as follows. In Section II we in-troduce the definitions of RAF sets and SPNs. In SectionIII we describe the model we use to evolve nets and weintroduce the composition operations. In Section IV wepresent and analyse the results obtained by simulatingthe dynamics of various composed RAF sets. Finally, inSection V we discuss and demonstrate the conclusions. II. BACKGROUNDA. Reflexively autocatalytic and food generated sets
In RAF theory (Horidjk and Steel, 2004; Hordijk etal., 2011), a network of interacting entities is representedby a CRS. Following previous definitions (Horidjk andSteel, 2004; Hordijk and Steel, 2017; Lohn et al., 1998),we introduce a CRS as a tuple (
S, R, C ) such that:– S is a set of entities;– R is a set of reactions, ρ → π , where ρ, π ∈ S arethe reactants and products of a reaction, respec-tively;– C is a catalysis set, that is, a set of pairs { ( s, r ) , s ∈ S , r ∈ R } indicating the entity s as the catalyst ofreaction r .We also define a food set F ⊂ S such that entities s ∈ F are assumed to be available from the environment. Wedescribe a CRS as a bipartite graph such that:– nodes are of two kinds V = S ∪ R ;– edges are of two kinds E = E r ∪ C ; – ∅ is a pseudo-entity representing the environmentand s ∈ S is a food entity if there exists a reaction r ∈ R such that ( ∅ , r ) ∈ E r and ( r, s ) ∈ E r .Note that E r is called the spontaneous reactions set.The edges ( s, r ) ∈ E r can be interpreted as source entity s is consumed by reaction r , while in edges ( r, s ) ∈ E r entity s is produced by reaction r . i s denotes a reactionsuch that ( ∅ , i s ) , ( i s , s ) ∈ E r , implying an input reactionproducing a food entity. Moreover, we introduce outflowreactions o s such that ( s, o s ) , ( o s , ∅ ) ∈ E r . Thus, theCRS is a flow reactor that allows inflow and outflow ofentities from and towards the environment.Let R (cid:48) represent a subset of R . The closure cl R (cid:48) ( F )is defined to be the (unique) minimal subset of S thatcontains F together with all entities that can be producedfrom F by repeated applications of reactions in R . Notethat cl R (cid:48) ( F ) is well defined and finite (Horidjk and Steel,2004). Given a CRS ( S, R, C, F ), a RAF set is a set ofreactions R (cid:48) ⊆ R (and associated entities) that satisfiesthe following properties:– Reflexively autocatalytic: for each reaction r ∈ R (cid:48) there exists at least one entity s ∈ cl R (cid:48) ( F ) suchthat ( s, r ) ∈ C ;– F -generated: for each reaction r ∈ R (cid:48) and for eachentity s ∈ S such that ( s, r ) ∈ E r , it is s ∈ cl R (cid:48) ( F ).Thus, a RAF set is a set of reactions able to catalyticallyproduce all its source entities starting from a suitablefood set. It is also possible to define the closure of a setof reactions, introducing the notion of closed RAF sets(Smith et al., 2014). Given a CRS ( S, R, C, F ), a subset R (cid:48) of R is said to be a closed RAF set if:– R (cid:48) is a RAF set;– ∀ r such that all its source entities and at least onecatalyst are either part of the set F or are producedby a reaction from R (cid:48) , it is r ∈ R (cid:48) .Authors of (Hordijk et al., 2018a) describe a procedureto detect closed RAF sets in a generic CRS. It hasbeen argued that closed RAF sets are associated withthe attractors of the dynamics of a CRS and there-fore represent the relevant units capable of experienc-ing adaptive evolution (Hordijk et al., 2018a,b; Smith etal., 2014; Vasas et al., 2012). FIG. 1 shows an exam-ple of a RAF set and its constituent closed RAF sets.We exemplify our point by introducing a set of entities S = { A, B, C, D, E, F, G, H, I, L, M, N, O, P } and a foodset F = { A, B, D, F, I, M } . In this example, the RAF setis composed of the following reactions (an entity abovethe arrow of a reaction indicates the catalyst associated FIG. 1 RAF set example. Entities are represented by let-ters (dark-green for food-entities, black for non-food entities).The food set is surrounded by a dark-green dashed rectangle.Reactions are displayed by coloured squares. A black arrowemerging from a letter towards a square (from a square point-ing at a letter) indicates the corresponding entity is a source(a product) for that reaction. Red dashed arrows indicatecatalysis. The maxRAF set consists of three closed RAF sets: R (1) = { r , r } (green squares), R (2) = { r , r , r , r } (ma-genta and green squares), R (2) = { r , r , r , r , r , r } (blueand green squares). with that reaction): r : A + B M −−→ C r : A + D O −−→ E r : C + F A −−→ G r : B + E N −−→ H r : C + I P −−→ L r : C + M E −−→ N r : D + M H −−→ O r : F + L D −−→ PThe maxRAF set consists of three closed RAF sets: R (1) = { r , r } , R (2) = { r , r , r , r } , R (2) = { r , r , r , r , r , r } . Note that the closed RAF set R (1) = { r , r } is also a constructively autocatalytic andfood generated (CAF) set (Mossel and Steel, 2005); i.e.a RAF set for which reactions can take place on the con-dition that their catalysts are already made available bythe occurrence of catalysed reactions (starting from F ).More restrictive conditions of this kind result in higherrates of catalysis that form a CAF set (Mossel and Steel,2005). All the RAF sets studied in this work contain oneRAF subset which in turn is a CAF set. B. Stochastic Petri nets
A Petri net consists of (see Petri and Reisig (Petri andReisig, 2008) for further details):– a finite set of place P ;– a finite set of transitions W ;– functions b, e : P × W → N .Here b ( p, w ) and e ( p, w ) are the number of edges fromplace p to transition w and from transition w to place p ,respectively. The sets b ( w ) ⊂ P and e ( w ) ⊂ P are thesets of places connected to transition w by at least oneedge. A marking X of a Petri net is a map X : P → N that assigns a number of tokens to each place. In fact, amarking X identifies a state of the system in the space ofpossible configurations of tokens available in each place.With x p , we indicate the number of tokens of place p available in marking X . Firing a transition w consumes b ( p, w ) tokens from each of its input places p ∈ b ( w ),and produces e ( p (cid:48) , w ) tokens in each of its output places p (cid:48) ∈ e ( w ). For each marking X , a transition w is enabled(it may fire) if there are enough tokens in its input placesmaking the consumption possible. This shall occur, ifand only if X ( p ) ≥ b ( p, w ), ∀ p ∈ P . A stochastic Petrinet (Haas, 2006; Molloy, 1982) is a Petri net for whicheach transition is equipped with a (possibly marking de-pendent) rate for the exponentially distributed transitionfiring times. λ denotes the set of firing rates of a SPN.Note that the evolution of a SPN with exponentially dis-tributed transition rates is isomorphic to continuous-timeMarkov chain (Molloy, 1982).Petri net formalism provides a suitable environmentfor studying the composition of networks, with both acomputational and theoretical approach; the latter, inparticular, in the context of category theory (Baez andPollard, 2017). Moreover, this formalism can describenets with different dynamics (Vazquez and Silva, 2011).The authors will investigate these aspects in a forthcom-ing work. III. THE MODELA. Building the net
Given a CRS (
S, R, C, F ) (or a set of reactions andassociated entities that is a RAF set), we build a SPN byadding a place p for each species s ∈ S and a transition w for each reaction r ∈ R such that b ( w ) = ρ ( r ) and e ( w ) = π ( r ), where ρ ( r ) and π ( r ) are the set of all sources andtargets entities of edges ( s, r ) , ( r, s ) ∈ E r , respectively.Note that with this notation we consider both inflowingand outflowing transitions. Moreover, for each catalysis( s, r ) ∈ C we add a transition w such that b ( w ) = ρ ( r ) ∪ s and e ( w ) = π ( r ) ∪ s . Hereafter, we use S and R toindicate both species and reactions of a CRS and placesand transitions of a SPN. The rates λ ( r ) associated witheach transition r are marking dependent rates: λ ( r ) = h r ( X ) λ r . (1)Here, λ r is a fixed constant depending on the type of itscorresponding reaction in the CRS ( λ r = { λ s , λ c , λ i , λ o } in which λ r specifies spontaneous, catalysed, inflowingand outflowing reactions, respectively) and h r ( M ) is avalue proportional to the number of combinations of to-kens available in the input places of transition r in thestate X . Thus, explicitly, we shall have: h r ( X ) = (cid:81) j b ( j, r ) V | b ( r ) |− (cid:89) j (cid:18) x j b ( j, r ) (cid:19) , (2)where the product is among all the input places of tran-sition r . We set functions b and e such that the inflowingtransitions do not consume tokens of the pseudo-entity ∅ and produce a fixed value of tokens of the food enti-ties, while the outflowing transitions consume a token ofthe outflowing entities and do not produce tokens of thepseudo-entity ∅ . Thus, the rate of outflowing transitionsis proportional to the amount of tokens of the outflowingentity, while the rate of inflowing transitions is indepen-dent of the state of the system. It is noteworthy thatthe inflowing of food elements still remains a stochasticevent. We add inflowing transitions for entities not be-longing to the originary food set F (setting the rate ofsuch transition equal to zero), if required for the purposeof composition between nets (see Section III.B). The dy-namics of the obtained SPN is described by the stochasticmass action kinetics, that is the classical dynamics usedto represent chemical reactions, assuming a well-stirredsystem (Anderson and Kurtz, 2011).The CRSs used in this work are generated accordingto the binary polymer model (BPM) (Kauffman, 1986).The BPM produces a CRS where the entities set S con-sists of all bit strings up to (and including) a maximumlength N , and the reaction set R consists of condensationand cleavage reactions. Condensation reaction is a con-catenation of two bit strings resulting in a longer string,and cleavage reaction cats a bit string into two smallerones. The food set is represented by all entities witha length less than or equal to a fixed length l f (we set l f = 2), and each entity can be a catalyst of each reactionwith a certain probability fixed a priori.We chose to only allow the condensation reactions tooccur in the system. Note that the technique we use isalso applicable in the case where cleavage reactions areallowed. In fact, none of the operations we introducein Section III.B is affected by the reversibility of thenetwork reactions. Including cleavage reactions wouldproduce networks, in principle, with different dynamics,since a network with irreversible reactions can have dif-ferent topology with respect the equivalent reversible net- work (Feinberg, 1995). The authors are currently inves-tigating this aspect in an upcoming work.With this limitation, all the spontaneous reactions ofour model are binary reactions (i.e., reactions with tworeactants, possibly of the same entity type), while allthe catalysed reactions are ternary reactions. Even ifternary reactions are rare, they can represent a first ap-proximation of two or more elemental reactions, such asthe sequence of reactions of an enzyme catalysis (Gille-spie, 2007). B. Composition
We model interactions between CRSs as compositionoperations between SPNs. First, note that RAF sets sat-isfy the following conditions (hereafter, we refer to theseas the inclusion conditions) (Horidjk and Steel, 2004): • if R (cid:48) is RAF in ( S , R , C , F ), it is RAF alsoin ( S , R , C , F ), if conditions S ⊆ S , R ⊆ R , C ⊆ C , F ⊆ F are satisfied; • if R (cid:48) is RAF ( S , R , C , F ) and R (cid:48) is RAF in( S , R , C , F ), then R (cid:48) ∪ R (cid:48) is RAF in ( S ∪ S , R ∪ R , C ∪ C , F ∪ F ).Thus, if composition does not remove entities from thefood set or reactions belonging to a RAF set, it will nothave any impact on its RAF property. However, the dy-namical behaviour of the composed system can be, gen-erally, different from that of the starting one.Let ( S , R , b , e , λ ) and ( S , R , b , e , λ ) be twoSPNs and let I, O be subsets of their inflowing and out-flowing transitions sets. Let ∼ be the equivalence relationsuch that s ∼ s (cid:48) if i s ∈ I and o s (cid:48) ∈ O for some choice of I, O . S ∼ denotes the set of places identified by relation ∼ . We define the following composition operations:CO I : ( S , S ) → S ∗ = S ∪ S ;( R , R ) → R ∗ = ( R ∪ R ∪ R I ) \ ( O ∪ I );( b , b ) → b ∗ = b ∪ b ∪ b I ;( e , e ) → e ∗ = e ∪ e ∪ e I ;( λ , λ ) → λ ∗ = λ ∪ λ ∪ λ I ; R I := { r | b ( r ) = s, e ( r ) = s (cid:48) , ∀ s ∈ S , s (cid:48) ∈ S such that s ∼ s (cid:48) } ; λ I := { λ ( r ) | λ ( r ) = h r ( X ) λ f , ∀ r ∈ R I } .CO II : ( S , S ) → S ∗ = S ∪ S ;( R , R ) → R ∗ = ( R ∪ R ∪ R II ) \ ( O ∪ I );( b , b ) → b ∗ = b ∪ b ∪ b II ;( e , e ) → e ∗ = e ∪ e ∪ e II ;( λ , λ ) → λ ∗ = λ ∪ λ ∪ λ II ; R II := { r | b ( r ) = b ( r (cid:48) ) , e ( r ) = b ( r (cid:48)(cid:48) ) , ∀ r (cid:48)(cid:48) ∈ R such that b ( r (cid:48)(cid:48) ) ⊂ S ∼ and b ( r (cid:48) ) ∼ b ( r (cid:48)(cid:48) ) } ; λ II := { λ ( r ) | λ ( r ) = h r ( X ) λ f , ∀ r ∈ R II } .CO III : ( S , S ) → S ∗ = { S (cid:116) S } / ∼ ;( R , R ) → R ∗ = R ∪ R ;( b , b ) → b ∗ = b ∪ b ;( e , e ) → e ∗ = e ∪ e ;( λ , λ ) → λ ∗ = λ ∪ λ .Here λ f is a constant value and ( S ∗ , R ∗ , b ∗ , e ∗ , λ ∗ ) is thecomposed SPN.To summarise, all the composition operations we de-fine relate a set of places S that are input for outflowingtransitions of a SPN, together with a set of places S (cid:48) thatare input for inflowing transitions of another SPN. Theformal addition of inflowing transitions for places not be-longing to the food set enlarges the possible compositionoperations between SPNs.Given a set S ∼ , operation CO I adds a transition from S to S (cid:48) for each pair of places in S ∼ , while operationCO II adds a transition from S to S (cid:48) for each combina-tion of places that appears as input of a transition inthe inflowing net. Each combination corresponds to thedefinition of complex in the framework of chemical re-action networks (Feinberg, 1995). In fact, given a set ofchemical species, a complex is defined as a member ofthe vector space generated by the species that providesthe inputs (or the outputs) of a reaction (Anderson andKurtz, 2011; Feinberg, 1995).Both operation CO I and CO II introduce a flux of en-tities from one net to another. The composite networkcan therefore be seen as the union of two separate net-works that evolve in parallel, communicating only with(asymmetrical) exchange of chemical species. This couldbe, for instance, the case of two spatially separated pro-tocells, one of which can release molecules towards theother. According to this interpretation, the flowing rate λ f is a parameter that encompasses the characteristicsof the flow process (for example, cell permeability). Op-eration CO III , instead, merges each pair of places in S ∼ ,allowing transitions of the two original SPNs to operateon the glued set of places. Composing nets via opera-tion CO III actually produces a new single network. Inthis case, one can think of composite net as the resultof mutations that enlarge a network (for instance, net1), introducing new possible reactions and, consequently,new chemical species (corresponding to net 2). Note that complexes play a major role in the framework of chem-ical reaction networks theory. For instance, the deficiency theo-rems (Feinberg, 1995) are able to predict whether the dynamicsof a large class of networks will have a stationary distribution,starting from the topology of the reaction graph having com-plexes as nodes.
It is worth to underline that, if there exists a s ∈ S ∼ such that s ∈ F , operations CO I and CO II can actu-ally modify the RAF property of net 2. In particu-lar, if transitions r ∈ R ( I,II ) are assumed to be spon-taneous transitions ( λ f < λ c ), net 2 could not be cat-alytically produced starting from the food set F . Inthis case, the composed net contains a RAF set R (cid:48) suchthat R ⊆ R (cid:48) ⊂ ( R ∪ R ), with R = R (cid:48) if F ⊆ S ∼ . In-stead, if transitions r ∈ R ( I,II ) are assumed to be (auto)catalysed transitions ( λ f ≥ λ c ), the whole composed netshall be a RAF set. IV. RESULTS AND DISCUSSION
In this Section, we present the results regarding theimpact of composition on the dynamics of RAF sets asfollows: we first introduce the characteristics of simula-tions and the quantities taken into account. In SectionIV.A we present results for the non interacting nets, inorder to have a reference model for the interacting cases,presented in Section IV.B.Starting from different instances of the BPM with
N <
8, we use the RAF algorithm introduced in (Hordijket al., 2011) in order to detect and select three differentRAF sets, each of which contains more than one closedRAF set. Note that, even if different RAF sets have thesame species set S , the set of reactions E will be different;i.e., various RAF sets have different chemistry. The RAFsets identified through this procedure constitute the col-lection on which we will carry out the study. Even if sucha small collection cannot be taken as a solid statisticalbasis, it is still sufficient for providing us with interest-ing information. We duplicate each RAF set and let itinteracts with its copy by switching to representation ofRAF set as an SPN and using one of the compositionoperations introduced above. We simulated the dynam-ics of the system using the standard Gillespie algorithm(Gillespie, 1976, 1977), setting the volume of the systemat V = 1 (arbitrary units). For each simulation, we per-form 100 independent runs of 10 time steps. One ofthe necessary conditions for a RAF set is the ability toproduce itself starting from the elements of the food set.Indeed, the initial state of the SPN is set such that: (cid:40) x s ( t = 0) = x if s ∈ F ; x s ( t = 0) = 0 otherwise . (3)Here x is an arbitrary constant. The values of x , λ c , λ i and λ o are set such that the number of tokens of foodplaces at t → ∞ is equal to 10 , for an SPN with inflow,outflow and all (and only) binary transitions having, asinput, food places only (and firing rate λ c ). The valueof λ s is fixed at λ s = λ c /
10, while λ f varies such that λ f ∈ [ λ c − , λ c ]. It is noteworthy that the valuesof these parameters are not taken from “in vivo” data,but they have phenomenological motivations. Thus, al-though we can reasonably generalise the characteristicsof the dynamics, quantities such as the species’ produc-tion rate or the time evolution of the concentrations maydiffer from those in other similar stochastic simulations(Hordijk and Steel, 2012; Hordijk et al., 2018b).We focus our attention on the effective appearance ofa maxRAF set during the evolution of the system. Inparticular, we introduce the following quantities:1) M s ( t ) = (cid:80) s x s ( t ) , ∀ s / ∈ F ;2) τ i = min { t | n ( r ) ≥ i , ∀ r ∈ R } .Here n ( r ) is the number of executions of transition r ,and R is the maxRAF set. Note that the natural con-dition x s ( t ) ≥ M s ( t ) can possess, at most, the valueone. Both M s ( t ) and τ i are calculated for each singlenet that forms the composed net. Let m M = M s ( t ) bethe mean value of M s ( t ) for t → ∞ . If all the non-foodentities of a maxRAF set are efficiently produced, then M s ( t ) → t → ∞ and m M = 0. However, the defi-nition of a RAF set does not ensure that all the entitiesassociated with such a set are present in large amountduring the evolution of the system. For instance, a non-food entity that is a source for a transition of a RAFset could be continuously consumed by that transition assoon as it is produced, resulting in a fluctuating evolutionof its number of tokens. For this reason, we consider aless restrictive condition than m M = 0 for the emergenceof a maxRAF set. In particular, for large t , we requirea strictly positive concentration for all non-food entitiesand a high concentration for most of them. This impliesthat the relation x s < M s , and that the relation x s ≈ m M < . (4)We assume that, if condition (4) holds, the entiremaxRAF set emerges.Note, moreover, that different growing rates among en-tities of a RAF set correspond to different effective firingrates of the transitions. Therefore, even if a maxRAF R set appears, the time τ necessary to perform all thetransition of R can exhibit different slopes during theevolution of the net, based on the various effective ratesof subsets of R . We use the slope m τ of the straight line y τ ( i ) = m τ i + q τ that approximates τ i for i → ∞ in orderto compare the efficiency of the (total) self-production ofdifferent RAF sets. We summarise the results obtainedfrom the simulated composition operations and the dif-ferent RAF sets of our collection through a scatter plotshowing the values ( m M , m τ ) averaged over independentruns (FIG. 7). FIG. 2 Isolated RAF set. Number of tokens over time ofnon-food entities obtained in a simulation run of the isolatedRAF set shown in FIG. 1. After a transient time of t ≈ .
006 (arbitrary units) all the entities are present in a largeamount, with the exception of the entity C which is repeatedlyconsumed by more than one reaction of the RAF set. A. Non interacting nets
We start investigating the non interacting nets by sim-ulating the evolution of three isolated RAF sets that con-stitute the collection, in order to obtain the dynamicsthat will be the benchmark for the evolution of the com-posite nets. FIG. 2 shows the time evolution of the num-ber of tokens of non-food entities for the isolated RAF setrepresented in FIG. 1. Hereafter, we refer to this simula-tion as “reference”, since it will be used as a benchmarkfor the other simulations.It is evident that, after a transient time of approxi-mately 0 .
06 time units, all the entities associated withthe RAF set grow in number, with the exception of en-tity C : C is the only entity in the set to be a sourcefor more than one transition (FIG. 1). The green linein FIG. 3 shows the corresponding trend of M s ( t ). Asexpected, after the same transient time of ≈ .
06 timeunits, M s ( t ) decreases down to values close to zero, theentire maxRAF set appears and condition (4) is satisfied(FIG. 7).We find that the asymptotic dynamics of simple iso-lated RAF sets is not affected by changing the initialstate. This is obvious from FIG. 3 and FIG. 4 by com-paring the trends of M s ( t ) and τ i obtained for the sameRAF set for various initial conditions. In particular, weperform different simulations by setting x s ( t = 0) equalto a random number less than 10 for all non-food enti-ties of the maxRAF set, and by setting x s ( t = 0) = 10 for only those entities associated with a particular closedRAF set. For all the RAF sets in the collection, the re-sulting values of m M and m τ are in agreement with thosecorresponding to the initial conditions described by Eq. FIG. 3 Total production of non-food entities by isolated nets. M s ( t ) obtained in a simulation run of the isolated RAF setshown in FIG. 1 for different initial conditions: solid greenline (reference simulation): x s ( t = 0) = 100 if s ∈ F , x s ( t =0) = 0 otherwise; dotted magenta line: x s ( t = 0) = 100if s ∈ F , x s ( t = 0) = rand (0 , ) otherwise; dashed blueline: x s ( t = 0) = 100 if s ∈ F , x s ( t = 0) = 10 if s ∈{ E, H, N, O } , x s ( t = 0) = 0 otherwise. After a transient timeof t ≈ .
006 (arbitrary units) M s ( t ) always takes values closeto zero, indicating that the maxRAF set has emerged for allthe different tested configurations. Dash-dotted red line: netthat failed to be a RAF set, obtained by switching off thecatalysis ( D,
8) in the RAF set shown in FIG. 1. In this case, M s ( t ) is always greater than 1, indicating that at least onenon-food entity is not produced by transitions of the system. (3) (FIG. 7). Conversely, the dynamics emerging in anet that failed to be a RAF set is significantly different(FIG. 3 and FIG. 4, red lines).These results suggest once again that simple RAF setshave an effective advantage in self-reproduction over non-RAF set. Also, the structure of RAF sets alone is notsufficient to guarantee the presence and the dynamicalselectability of different long-term behaviours. Using thedynamics of isolated RAF sets as reference, we can nowmove on to the dynamics of composed RAF sets. B. Composite nets
In order to compose nets, we choose five different sets S ∼ : the set of places belonging to the food set, the setof places non belonging to the food set, the set of placescorresponding to the molecules of length l = l f + 1 and l ≤ l f + 1 of the BPM and the set of places that are notinput places for spontaneous transitions (not selected foroperation CO II ). We compose copies of the RAF setsaccording to the composition operations CO I , CO II and CO III . The initial states of the nets are set accordingto Eq. (3). Moreover, for operation CO III , simulationsare performed in which the transitions belonging to net
FIG. 4 Cycles of transitions completed by isolated nets. τ i obtained for the isolated RAF set shown in FIG. 1 with differ-ent initial conditions: solid green line (reference simulation): x s ( t = 0) = 100 if s ∈ F , x s ( t = 0) = 0 otherwise; dotted ma-genta line: x s ( t = 0) = 100 if s ∈ F , x s ( t = 0) = rand (0 , )otherwise; dashed blue line: x s ( t = 0) = 100 if s ∈ F , x s ( t = 0) = 10 if s ∈ { E, H, N, O } , x s ( t = 0) = 0 otherwise.The similar slope of τ i for t → ∞ associated with differenttested configurations indicates that, after a transient time,the efficiency of performing all the transitions of the RAF setdoes not depend on the initial conditions. Red circle: the netthat failed to be a RAF set, obtained by switching off thecatalysis ( D,
8) in the RAF set shown in FIG. 1. In this case,the net is not able to perform all its transitions. timesteps. Hereafter, we refer to this particular configurationas the “delayed configuration”.We find that composition operations do not have anyimpact on the emergence of the maxRAF sets for anychoice of S ∼ that does not include food entities (FIG. 7).However, a primordial form of biological interactions canbe established; namely, facilitation and cheating. In par-ticular, in nets composed through operations CO I and CO II , inflowing of external entities can facilitate the ap-pearance and the sustenance of a RAF set, improvingits production efficiency. At the same time, withdrawingentities produced by a RAF set can counteract its pro-duction. These aspects are well highlighted by varioustrends of the generated m τ due to different conditions.On the other hand, as expected, composition opera-tions involving the food set have a major role on theemergence of the maxRAF set. FIG. 5 and FIG. 6 showthe behaviour of M s ( t ) and τ i associated with two RAFsets that are composed by operations CO I , CO II and CO III for S ∼ = F . Similar trends are obtained fromthe composition of the other RAF sets in our collection.Results show that operations CO I and CO II prevent theentire maxRAF set in at least one of the two copies fromemerging. In particular, we find that, if the rate λ f oftransitions allowing the flux of food elements from net 1 FIG. 5 Total production of non-food entities by compositenets. M s ( t ) is obtained for two copies of the RAF set shownin FIG. 1 with different composition operations. For eachcomposite net, two M s ( t ) are calculated, each from the enti-ties associated with the original copies. Dash-dotted magentaline: operation CO I , net 1, λ f = 10 λ c . Dashed cyan line:operation CO II , net 1, λ f = 10 λ c . Solid yellow line: oper-ation CO III , net 1. Dash-dotted violet line: operation CO I ,net 2, λ f = 10 λ c . Solid blue line: operation CO II , net 2, λ f = 10 λ c . Dotted dark-yellow line: operation CO II , net2, delayed. For all composite operations, M s ( t ) of net 2 doesnot satisfy the condition stated by Eq. (4) that results in theemergence of the maxRAF set. For operation CO II , both themaxRAF sets of net 1 and net 2 do not emerge. to net 2 is low enough ( λ f < λ c ), operation CO I pre-vents the emergence of the maxRAF set in net 2, whilenet 1 exhibits the same dynamics of the isolated net, ascan be concluded from FIG. 7. On the other hand, forthe same composition operation, if the rate λ f is suffi-ciently high ( λ f ≥ λ c ), the flow of food elements issuch that the maxRAF set of net 1 does not contain avail-able resources to perform all its transitions, while net 2evolves as if it is isolated and able to drawn food directlyfrom the environment (FIG. 7). For operation CO II , wehave achieved significantly different results. In this case,we find different threshold values λ f = ( λ f , λ f ) (de-pending on the specific RAF set of the collection) suchthat the emergence of the maxRAF set in net 2 is pre-vent for λ f ≤ λ f , while the opposite situation is ob-tained for λ f ≥ λ f . Moreover, for intermediate values λ f < λ f < λ f , no maxRAF set emerges but only someof the closed RAF sets (FIG. 5 and FIG. 6). There-fore, for this range of values, the flux of entities is suchthat both nets 1 and 2 have enough food elements tofire transitions and perform (complementary) subsets ofthe maxRAF set, namely, the closed RAF sets. Further-more, the stochastic nature of the flow process allows theemergence of different closed RAF sets in each run, henceshowing the possible selectability of asymptotic dynamicsfor composite nets. FIG. 6 Cycles of transitions completed by composite nets. τ i is obtained for two copies of the RAF set shown in FIG. 1with different composition operations. For each compositenet, two τ i are calculated, each from the transitions associatedwith the textcolorredoriginal copies. Dash-dotted magentaline: operation CO I , net 1, λ f = 10 λ c . Solid yellow line:operation CO III , net 1. Large cyan circle: operation CO II ,net 1, λ f = 10 λ c . Intermediate violet circle: operation CO I ,net 2, λ f = 10 λ c . Intermediate blue circle: operation CO II ,net 2, λ f = 10 λ c . Small dark-yellow circle: operation CO II ,net 2, delayed. For all composite operations, net 2 is not ableto execute all its transitions. For operation CO II , both net1 and net 2 do not perform a complete cycle of transition ofthe RAF set. The observed dynamics lead to some important con-siderations: first, it is clear that the actual availability ofresources is a crucial element in the dynamical realisationof a RAF set, and the rate at which food elements enterthe net is as relevant as the definition of the food set it-self, as expected. Moreover, the differences emerging dueto the impact of the CO I and CO II operations suggestthat the complexes play an important role in the dy-namics of the RAF sets, even if they are defined startingfrom the single entities. We will investigate these pointsin a following work. Finally, as previously observed in(Hordijk et al., 2018b), biological interactions differentfrom competition among RAF sets are plausible.It is also intriguing that an effective competition canemerge if two nets share the same food, as in case ofcomposition operation CO III and S ∼ . In particular, weobserved that in the delayed configuration, the presenceof the maxRAF of net 1 prevents the emergence of themaxRAF of net 2. In fact, once the maxRAF of net 1has had enough time to appear, the number of tokens ofits associated entities increases. Since the effective rateof a transition is proportional to the number of tokensof its sources, entities of net 1 have an higher chance ofreacting with respect to their counterparts belonging tothe delayed net 2. Most of the food elements are there-fore consumed by transitions of net 1. Once activated,only some of the transitions of net 2 are able to be per-formed efficiently, leading to the emergence of only someof the closed RAF sets which constitute the maxRAF setof net 2. Different runs show that the emerging closedRAF sets can vary due to the stochastic nature of thesystem evolution, thus guaranteeing the selectability ofthe different long-term behaviours.This result is in contrast with the previous ones whereit has been observed that isolated RAF sets are not ableto experiment different asymptotic dynamics. In fact,composing two RAF sets using operation CO III producesa composite net in which all transitions form a (larger)RAF set (see the inclusion conditions, Section III.B), andthe effect of the delay can be seen as a selection of aparticular initial state. However, the same compositenet is not able to produce competition if the delay isnot introduced. Also, an effective competition betweenclosed RAF sets has not been observed in an isolatednet with initial conditions containing RAF sets alreadyemerged at time t = 0 (FIG. 7, FIG. 3 and FIG. 4).We suggest that key elements for this form of compe-tition are both the structure of the composite RAF setand the particular choice of initial conditions. In fact, innets composed by operation CO III and S ∼ = F , eachtransition that consumes at least one food entity as asource or a catalyst, always has at least one competi-tor represented by its copy. By contrast, the hierarchicalstructure of RAF sets does not guarantee such level ofcompetition between different close RAF sets. Moreover,the delay brings the system into a state that promotes theformation of some subsets of the RAF set. The systemcan hardly reach this state only through random fluctua-tions. The results presented in this paper show that thesetwo conditions allow RAF sets to have different accessibleasymptotic dynamics. V. CONCLUSIONS
In this work we study the impact of composition opera-tions on the dynamics of simple RAF sets. This allows usto test whether the interactions among RAF sets permitdifferent long-term dynamical behaviour of the RAF setsthemselves, that is a necessary condition for evolvability.To this aim, we generate various RAF sets starting fromdifferent instances of the BPM and represent RAF setsas SPNs. Moreover, we introduce composition operationsthat, acting on SPNs, correspond to interactions amongRAF sets. We find that, if the composition operationsdo not involve the food entities of a RAF set, the dy-namics of the system always reaches a state in which allthe composed RAF sets appear. However, how fast theRAF sets emerge and their efficiency in self-reproductiondepends on the exchange of entities, showing that thecomposition operations can give rise to interactions withecological features.
FIG. 7 Impact of composition operations on the emergenceof the maxRAF set. On the x -axis: slope m τ of the straightline fitting τ i for t → ∞ , averaged over 100 independent runs.On the y -axis: m M = M s ( t ), averaged over 100 independentruns. Here · · · is the average over time for t → ∞ . For eachrun, both m τ and m M are calculated for t ≥ t f , where t f signifies the end of the simulation run. Green triangles: RAFset 1 (Fig. 1). Red circles: RAF set 2. Blue squares: RAF set3. Empty points: net 1. Filled points: net 2. Yellow shadedregion indicates values corresponding to m τ → ∞ , obtainedfor composition operations CO I , CO II , CO III (delayed) and S ∼ = F . On the other hand, composition operations involvingfood entities hinders the appearance of, at least, one ofthe two original RAF sets, giving rise to possible differentlong-term behaviours. In particular, if the food entitiescan be exchanged between one RAF set and another, asthe case of composition operations CO I and CO II , theemergence of the maxRAF set of the starting nets de-pends on the rate of the exchange transitions (the flow).For sufficiently low rates, only the RAF set capable ofacquiring food directly from the environment is able toform. Conversely, for sufficiently high rates the elementsof the food are exchanged in such portions to allow onlythe receiving RAF set to appear. Furthermore, if com-plexes of a reaction are involved in the exchange and notindividual entities (as in the case of the composition op-eration CO II ), we find an intermediate interval of flowrate values within which the exchange of food elementsbetween the nets allows the emergence of just some ofthe closed RAF sets that constitute the starting maxRAFsets. For these intermediate flow rates, therefore, the nec-essary evolvability conditions are met. Finally, we findthat only some closed RAF sets within a maxRAF setcan appear if the maxRAF set shares the food entitieswith its copy that has already fully emerged. The latteris a relevant result, since sharing the same food set bytwo RAF sets produces a maxRAF set that is the unionof the two starting RAF sets. The dynamics observed in0this case shows that different long-term behaviours arepossible for a single RAF set, at least as long as the sys-tem is in a particular initial state and the subsets of theRAF set compete with each other for each reaction thatneeds food elements.In previous works it was theorised that separate RAFnetworks could compete for shared resources (Hordijkand Steel, 2014; Kauffman, 2011; Vasas et al., 2012), anda competitive dynamics was observed in spatially sepa-rate RAF sets (Hordijk et al., 2018b). Results presentedin this paper confirm the possible evolvability of a sys-tem of RAF sets separated into compartments. We alsonoticed that isolated (composite) RAF sets can experi-ence different asymptotic dynamics. Furthermore, recentresults show the presence of RAF sets in real biologicalsystems (Sousa et al., 2015; Xavier et al., 2020), confirm-ing their biological interest. However, since all the simple(tested) isolated RAF sets experience the emergence ofthe entire maxRAF set, the definition of the RAF setsdoes not seem to be sufficient for implying dynamics withmultiple selectable long-term behaviours.In order to further clarify this last point, we will ex-plore a larger ensemble of RAF sets in a forthcomingpaper. Moreover, in the future it might be interestingto study the composition of RAF sets enclosed in pro-tocells, investigating how the coupling between internalnetworks and boundaries affects the global dynamics ofthe system. ACKNOWLEDGMENTS
We would like to thank Marco Pedicini for his ideas,insightful help and valuable discussions. We want to ac-knowledge helpful suggestions from Marco Villani andFatemeh Zahara Majidi, the latter also for improving themanuscript with a precise and accurate proof-reading.
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