Unexpected evolutionary dynamics in a string based artificial chemistry

This work investigates closure in Cell Signaling Networks, which is one research area within the ESIGNET project. We employ a string-based Artificial Chemistry based on Holland's broadcast language (Molecular Classifier System, Broadcast
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  Unexpected Evolutionary Dynamics in a String Based Artificial Chemistry James Decraene, George Mitchell, Barry McMullin Artificial Life Laboratory - Research Institute for Networks and Communications EngineeringSchool of Electronic Engineering - Dublin City University, IrelandEmail: Abstract This work investigates closure in Cell Signaling Networks,which is one research area within the ESIGNET project 1 .We employ a string-based Artificial Chemistry based onHolland’s broadcast language (  Molecular Classifier System, Broadcast Language , or MCS.b). We present a series of experiments focusing on the emergence and evolution of self-maintaining molecular organizations. Such experimentsnaturally relate to similar studies conducted in artificialchemistries such as Tierra, Alchemy and Alpha-Universes.However, our results demonstrate some counter-intuitive out-comes, not indicated in previous literature. Each of these “un-expected” evolutionary dynamics (including an elongationcatastrophe phenomenon) are examined and explained bothinformally and formally. We also demonstrate how the elon-gation catastrophe can be prevented using a multi-level se-lectional model of the MCS.b (which acts both at the molec-ular and cellular level). This work provides complementaryinsights into the understanding of evolutionary dynamics inminimal artificial chemistries. Introduction Cell Signaling Networks (CSNs) are complex biochemicalnetworks of interacting molecules (proteins, ions, secondarymessengers, etc.) occurring in living cells. Through com-plex molecular interactions (e.g., signal transduction), CSNsareabletocoordinatecriticalcellularactivities(e.g., celldif-ferentiation, apoptosis) in response to internal and externalstimuli.As CSNs occur in cells, these networks have to replicatethemselves prior to the cellular division. This allows thereplicated CSNs to be “distributed” to the offspring cells.Errors may occur during this replication process, e.g., anoffspring cell may inherit only a partial CSN. Thus resultingin potentially defective cells which would lead to a varietyof undesired effects (e.g., premature cell death). As a result,the“fitness”ofacellisimplicitlyrepresentedbythe survival and performance of a cell in achieving self-maintenance and  cell-level replication. 1 ESIGNET: Evolving Cell Signaling Networks in silico , an EUFP6 project, contract no. 12789, Based on the above assumption, we hypothesize thatCSNs may be regarded as subsets of closed (and thus self-maintaining) systems. The latter would have the additionalability to replicate themselves as a whole (cellular division).The signal processing ability of CSNs would emerge fromthe closure properties of these systems.Examining such phenomena relates closely to otherstudies which have been conducted on Holland’s Alpha-Universes (Holland, 1976), Tierra(Ray,1991) and Alchemy (Fontana and Buss, 1994). Although these ArtificialChemistries (ACs) were developed for different purposesand were implemented differently, these systems exhibitedcommon evolutionary phenomena such as the emergence of (collectively) autocatalytic reaction networks (Dittrich et al.,2001;McMullin,2000). Inthisinvestigation, suchclassesof network are of interest as they would allow CSNs to self-maintain and replicate themselves. Moreover, as demon-strated in several ACs, it is commonly accepted that theemergence and maintenance of such collectively autocat-alytic reaction networks is relatively trivial.We introduce the Molecular Classifier System, Broadcast  Language System , or MCS.b(J.Decraene et al.,2007). This addresses the reflexive nature of molecular species and au-tomatically gives rise to an implicit molecular fitness func-tion represented by the “replication” ability of the individualmolecular species. We present a series of experiments fo-cusing on the emergence of self-maintaining organizationsand finally we examine the outcomes of these experimentstogether with possible modifications for further work. Molecular Classifier Systems Molecular Classifier Systems are a class of string-rewritingbased AC inspired by the broadcast language (BL; seeHol-land, 1992). As opposed to more traditional string-rewritingsystems, operations are stochastic and reflexive (no distinc-tion made between operands and operators). The behav-ior of the condition (binding) properties and action (enzy-matic functions) is defined by a language specified withinthe MCS. This “chemical” language defines and constrainsthe complexity of the chemical reactions that may be mod-  eled and simulated. In this AC, all reactants are catalyticin the sense that they are not consumed during reactions.These reactions result from successful molecular interac-tions which occur at random. When a reaction occurs, aproduct molecule is inserted in the reactor whereas anothermolecule, selected at random, may be removed from the re-actor space (designating the system outflow).A molecule may contain several condition/action ruleswhich define the binding and enzymatic properties. A reac-tionbetweenmoleculesoccursifatleastoneconditionalpartfrom any rules in a molecule A matches a target molecule B . A is regarded as an enzyme whereas B is regarded asa substrate molecule. When a reaction occurs, the actionpart from the satisfied rule in A is utilized to perform theenzymatic operations upon the bound substrate molecule B .This operation results in the production of another offspring(product). If several rules in A are satisfied by B , then oneof these rules is picked at random and employed to carry outthe enzymatic function.We proposed a simplification of the BL(J.Decraene et al.,2007)whichisusedastheMCSchemicallanguageresultingin the MCS.b system. MCS.b has some similarity with theLearning Classifier Systems, also pioneered by John Hol-land(Holland and Reitman,1978); however there are also a number of differences. For example, the LCS strings arefixed length on an alphabet of  λ = { 1 , 0 , # } ; whereas theBL strings are of variable length using a significantly largeralphabet of  Λ = { 1 , 0 , ∗ , : , ♦ ,  ,  ,  } . BL stringsare referred to as broadcast devices . A broadcast device isparsed into zero, one or more broadcast units , where eachunit represents a single condition/action rule. The symbol ∗ separates broadcast units within a broadcast device. Thesymbol : separates a condition from an action within a sin-gle broadcast unit. { ♦ ,  , } are single/multiple characterwildcards that may also copy matched (sub-)strings into out-put strings. A detailed description is omitted in this paper,see (J.Decraene, 2006) for full specification of our BL im-plementation. Autocatalytic organizations A series of experiments using the MCS.b is now outlined.These experiments first examine both the self-maintenanceand the spontaneous emergence of autocatalytic molecules(i.e., molecules that can self-replicate). Both spontaneousemergence and self-maintenance were reported as easily ob-tained in Alchemy. Spontaneous emergence was not ex-pected or reported for the srcinal Tierra system; however, itdid arise in the related Amoeba system, specifically devisedfor this purpose (Pargellis,2001). No selective advantages for universal replicases An artifact of the BL’s syntax is that it is moderately difficultto observe the spontaneous emergence of an individually au-tocatalytic molecule. Specifically, there are 4 8 ( 65 , 536 ) dis-tinct molecules of length 4 symbols (the minimal length toconstruct a functional/enzymatic molecule), of which only asingle one ( R 0 = ∗  :  ) is autocatalytic. Although theprobability of spontaneously obtaining such autocatalyticmolecules is therefore quite low in MCS.b, the intuition wasthat, once such a molecule does appear, it should be able torapidly fill the reaction space. This phenomenon was indeedobserved in Alchemy and was expected to occur in MCS.b.We present here a series of experiments which explore andtest this conjecture.The behavior of the minimal self-replicase, R 0 , is as fol-lows. The matching condition is defined by a single symbol,  , which designates a multiple character wildcard. This in-dicates that R 0 may bind to any molecule. In addition whena reaction occurs between R 0 and a substrate molecule I  0 ,  is assigned a value, being the matched substring of  I  0 .In this case, this will be the complete string I  0 . A uniquesymbol  also constitutes the action part of  R 0 . This spec-ifies that the output string of  R 0 is exactly the string boundby the  in the condition part, i.e., a copy of  I  0 . Thereforethe broadcast device R 0 is actually a “universal” replicase;which, by definition, means that it is also a self- replicase (inthe special case that it binds to another instance of itself, i.e., I  0 = R 0 ). The “specificity” of  R 0 is said to be null .Fig.1presents a first experiment examining the behav-ior of  R 0 averaged over 30 simulation runs. The broadcast“universe” (reaction space) is configured as follows: • The system is seeded with 900 randomly generatedmolecules, each of length 10 symbols. • In addition, 100 instances of  R 0 are inserted. • n max designates the fixed maximum number of molecules that may be contained in the universe, n max =1000 . • Molecular interactions occur as follows: two molecules A and B arepickedatrandom, A isconsideredasanenzymeand B as a substrate. If  A can bind and react with B thena molecule C  is produced. If the current size of the popu-lation, n , is less than n max then C  is simply added to thepopulation (and n increases by 1); otherwise a moleculeis picked at random and is replaced with C  (and the pop-ulation size remains unchanged at n ). • No mutation may occur in these experiments. • A single “timestep” is arbitrarily defined as 50 molecularinteractions.A high concentration (0.1) of  R 0 was chosen to minimiseearly extinction due simply to stochastic fluctuation.From Fig.1it is clear that the species R 0 never growsto take over the population; on the contrary, it consistentlydiminishes, contrary to the srcinal, informal, prediction. A  -0.0200. ⋅ 10 0 1 ⋅ 10 3 2 ⋅ 10 3 3 ⋅ 10 3 4 ⋅ 10 3 5 ⋅ 10 3 6 ⋅ 10 3 7 ⋅ 10 3 8 ⋅ 10 3 9 ⋅ 10 3 1 ⋅ 10    P  o  p  u   l  a   t   i  o  n  p  r  o  p  o  r   t   i  o  n Timestep Figure 1: Relative population growth of replicators R 0 aver-aged over 30 simulation runs. Solid line is average concen-tration; error bars denote standard deviation.formalexplanationofthisoutcomeisgivenbymodellingthesystemwiththe(approximate, continuous)catalyticnetwork equation(Stadler et al., 1993). The state of the system is de-scribed by the concentration vector x = ( x 1 ,...,x n ) with x 1 + ... + x n = 1 and x i > 0 , where x i refers to the concen-tration of a molecular species (or collection of “chemicallyequivalent” species) s i . The general dynamic behaviour isthen given by: ˙ x k = n  i =1 n  j =1 α kij x i x j − x kn  i,j,l =1 α lij x i x j (1)with k = 1 ,...,nα kij are the rate constants for each reaction s i + s j → s i + s j + s k . In this experiment, these simplify to: α kij =  1 if  s i + s j → s i + s j + s k 0 otherwise(2)For simplicity, consider the simple case where only uni-versal replicases ( R 0 ) and non-enzymatic molecules ( NE  )(that may only act as substrates) are present. This is clearlythe most  favourable case for the growth of  R 0 . Denote themolecular concentrations of  R 0 and NE  by x 1 and x 2 re-spectively. Then α 1 ij = 1 if  i = 1 ,j = 1 ; otherwise α 1 ij = 0 .Similarly, α 2 ij = 1 if  i = 1 ,j = 2 ; otherwise α 2 ij = 0 . In-serting into Eq.1,we obtain: ˙ x 1 = x 21 − x 1 ( x 21 + x 1 x 2 ) (3)But given that x 2 = 1 − x 1 : ˙ x 1 = x 21 − x 31 − x 21 + x 31 ˙ x 1 = 0 (4)whereas the growth rate of molecules NE  is: ˙ x 2 = x 1 (1 − x 1 ) − (1 − x 1 )[ x 21 + x 1 (1 − x 1 )] (5) ˙ x 2 = x 1 − x 21 − (1 − x 1 )( x 21 + x 1 − x 21 )˙ x 2 = x 1 − x 21 − x 1 + x 21 ˙ x 2 = 0 (6)Thus, both molecular species R 0 and NE  share a com-mon zero “expected” growth. Under the stochastic condi-tions of the reactor this would yield a random drift in relativeconcentrations—as opposed to a quasi-deterministic growthof the R 0 species. Qualitatively this is due to the fact thatany (self-)replicase having low or zero specificity, such as R 0 , will not only replicate itself but also replicate any othermolecules; and therefore cannot selectively displace thesemolecules. But recall that this was the best case situationfor growth of  R 0 , where none of the other molecules had anyenzymatic activity. In the practical case of Fig.1the collec-tionofsuchadditionalsidereactionswillgiveanettnegativegrowth rate for R 0 , which therefore, quasi-deterministically,decays. Specificity and domination of the replicases To confirm the importance of specificity, we proceeded to aseries of experiments in which we incrementally increasedthe specificity of the (self-)replicases. Table1shows the dif-ferent replicases employed in these experiments. R 1 des-ignates a molecule that would only react with moleculeswhose strings end with the symbol “1”. As the latter oc-curs at the rightmost position of  R 1 , it may react with itself,producing another instance of  R 1 . Similarly, R 2 only bindsto molecular strings containing the suffix 01 . This “signa-ture” forms a constraint on the replicases, allowing them toreact only with a progressively more restricted set of sub-strate molecules. This impacts directly on these molecules’binding specificity.Replicase Informational string R 0 ∗  :  R 1 ∗  1 :  1 R 2 ∗  01 :  01 R 3 ∗  101 :  101 R 4 ∗  0101 :  0101 Table 1: (self-)replicases with increasing specificityThe results depicted in Fig.2confirm the importanceof specificity upon the system dynamics. The ability of a(self-)replicase to dominate and sustain itself, against a ran-dom initial population of molecules, increases progressivelywith its binding specificity. As in the previous section, wecan explain and demonstrate this behavior through the useof a simple ODE model. 0 ⋅ 10 0 2 ⋅ 10 2 4 ⋅ 10 2 6 ⋅ 10 2 8 ⋅ 10 2 1 ⋅ 10 3 1 ⋅ 10 3 1 ⋅ 10 3 2 ⋅ 10 3 2 ⋅ 10 3 2 ⋅ 10    P  o  p  u   l  a   t   i  o  n  p  r  o  p  o  r   t   i  o  n Timestep R 0 R 1 R 2 R 3 R 4 Figure 2: Population growth of replicators R 0 ,R 1 ,R 2 ,R 3 and R 4 . Each line represents the average concentration of corresponding replicase over 30 simulation runs.In this case, we consider a reactor containing only the fol-lowing molecular species: • Replicases R 1 which only replicate molecules terminat-ing with the symbol “1” (which includes R 1 moleculesthemselves). • A variety of non-enzymatic molecules NE  which arerandomly generated. NE  1 ⊆ NE  is the subset of molecules whose strings terminate with the designatedsymbol. These molecules contained in NE  1 can be repli-cated by molecules R 1 .The concentration vector is given by x =( x 1 ,x 2 ,...,x n ) with x 1 + x 2 + ... + x n = 1 where x 1 is the concentration of  R 1 and x 2 is the sum of con-centrations of molecules in NE  1 . The growth rate of thedifferent molecular species in this reactor are as follows: ˙ x 1 = x 21 − x 1 ( x 21 + x 1 x 2 ) (7) ˙ x 1 = x 21 − x 31 − x 21 x 2 ˙ x 1 = x 21 (1 − x 1 − x 2 ) (8)The growth rate of molecules NE  1 is: ˙ x 2 = x 1 x 2 − x 2 ( x 21 + x 1 x 2 ) (9) ˙ x 2 = x 1 x 2 − x 21 x 2 − x 1 x 22 ˙ x 2 = x 1 x 2 (1 − x 1 − x 2 ) (10)Since x 1 + x 2 + ... + x n = 1 , we have x 1 + x 2 < 1 andtherefore ˙ x 1 > 0 and ˙ x 2 > 0 . Whereas the growth rate of any other molecules (that may be not replicated by R 1 ) inthe reactor space is given by: ˙ x = 0 − x i ( x 21 + x 1 x 2 ) (11)with 2 < i ≤ n In Eq.12,we note that any given molecules s =( s 3 ,...,s n ) possess a negative growth rate which indicatethat these molecules would be displaced by molecules R 1 and NE  1 .In this model, only NE  1 molecules are able to parasitethe replicases R 1 . By increasing the specificity of repli-cases, we decrease the range of molecule that may parasitethe replicases. This explains the behavior observed in Fig.2,in which replicases with higher specificity are more likely totake over the reactor space.Therefore in this system, for replicase molecules to suc-cessfully sustain themselves and/or to dominate the molec-ular population, a significant binding specificity is required.We conjecture that this underlying phenomenon may havebeen implicated in the dynamics of a variety of previouslyreported artificial chemistries; but, to our knowledge, it hasnot previously been explicitly isolated in the manner pre-sented here. Spontaneous emergence of replicases In the previous set of experiments, mutation was turned off in order to facilitate our investigation on replicases, whichwere hand-designed and inserted into the initial population.This led to a limited diversity in the population. To examinethe spontaneous emergence of autocatalytic molecules, weperformed a second series of experiments in which no repli-cases are specified and molecular mutation could occur. Thelatter is implemented as follows: • When a new molecule is produced, a mutation with prob-ability p sym = 0 . 001 may be applied to each of its sym-bols. Therefore, the longer the molecule, the higher theprobability of mutation occurring. • Three types of mutation are distinguished and are appliedwith equal probabilities: – Symbol flipping: The current symbol is replaced with asymbol picked uniformly at random from Λ . – Symbol insertion: A symbol is picked uniformly at ran-dom from Λ and inserted after the current symbol. – Symbol deletion: The current symbol is removed. • To maintain diversity in the event of low ongoing reactionactivity, a global mutation technique occurring every 100timesteps is also available. A subset ( r mut = 0 . 01 ) of thepopulationisselectedatrandomandoneofthethreetypesof mutation mutation (chosen as above) is then appliedto a single symbol picked uniformly at random in eachmolecule of this subset.As mutation now occurs, diversity is maintained duringlong term evolution. The spontaneous appearance of repli-cators was expected. Results indicated that (self-)replicasesdo emerge, however they never manage to self-sustain.  This is explained as follows: • As already noted, the BL syntax does not strongly facili-tate the spontaneous emergence of replicators. This syn-tactical constraint may discourage the spontaneous emer-gence of self-replicators. The BL syntax may also have animpact on the robustness of these self-replicators againstmutation effects. • Secondly if self-replicators do emerge, they would be re-quired to possess a specificity higher than null to sustainthemselves. • Finally, replicators are likely to possess a low molecularconcentration when emerging. This low concentration di-minishes the capacity of these molecular species to persistagainst side reactions and mutation events.These three factors, when combined, significantly lowerthe probability of having a replicator spontaneously emergeand self-sustain in the MCS.b.We examined the nature of the (self-)replicases that mayemerge during evolution. An additional set of experimentswas specified as follows: • Each simulation run was initialised with 100 randomlygenerated, 10-symbol long, molecules. • n max = 1000 (i.e., the population initially grew withoutany displacement; but once the total number of moleculesreached 1000 it was limited to this value, by displacingone random molecule for each new molecule generated,as previously described). • 30 simulation runs were performed, each for 100000timesteps.To identify spontaneously emerging self-replicases, ev-ery molecule was tested at each timestep for self-replicationfunctionality. The spontaneously emerging self-replicasesidentified in these experiments are listed in Table2. Thisshows that 15 distinct self-replicases appeared. However,note that it is a property of the BL syntax that some symbolsare ignored when functionally interpreted (they are, in a cer-tain sense, “junk” symbols). Thus, although 15 distinct self-replicases were identified, it turns out that the core broadcastunits (the “active sites”, after discarding “junk” symbols)are, in fact, identical for 14 of these; and are all equiva-lent to the srcinal universal self-replicase, R 0 = ∗  :  ,discussed earlier. Only the broadcast device ∗  0  :  0 possesses a core broadcast unit of a different form, namely ∗  0 :  0 . This is an alternate form of  R 1 , having just theminimal specificity of one symbol.In the 30 experimental runs, the highest concentrationachieved by any of these spontaneously occurring self-replicases was 0 . 001 —i.e., just a single isolated molecule.Self-replicases 00   ∗  :   ∗ 0 1  0 ∗  :  00   ∗  :  ♦ ∗ 0 1   0 ∗  :  : 1 ∗  : ♦ : 1 ∗  : ♦ : 0  ∗  :   : ∗  :  ∗ 01 ∗  ∗  :     1 ♦ : ∗  :  : ∗  : ∗  :    ∗  :  ∗  :  ∗  0  :  0 ♦ ∗  ∗  : ♦  1 ∗  : ♦ Table 2: Spontaneously emergent self-replicases in MCS.bThis is consistent with the comments earlier in this sec-tion, and the results of the previous section. It is progres-sively more difficult for self-replicases of higher specificityto spontaneously arise by chance (due to their greater length,and relatively rare frequency as defined by the BL syntax);but self-replicases of very low specificity (which do sponta-neously occur) cannot grow to significant concentrations.The spontaneous emergence of a “sustainable” self-replicase (i.e., of sufficient specificity to establish itself)remains theoretically possible in MCS.b. However, boththe experimental results and the informal analysis presentedhere suggest that the expected emergence time would be ex-tremely (perhaps infeasibly) long. While we have not for-mally quantified this, it appears that MCS.b therefore sharesthis property with the Tierra system. Rise and fall of the fittest In the Tierra system, a hand-designed molecule called the“ancestor” is manually introduced into the space. This ini-tially grows to saturate the available core memory. The pop-ulation subsequently evolves into a variety of collectivelyautocatalytic reaction networks (where Tierra “creatures” orprograms are here considered analogous to “molecules”).Accordingly, ournextstepistomirrorthismethodology, andintroduce a hand-designed self-replicase of relatively highspecificity into the MCS.b system.However, the results indicate that MCS.b does not  exhibitan evolutionary dynamic at all comparable to Tierra in thiscase. Fig.3presents an example of such an experiment.The “ancestor” self-replicators do, at first, quickly fill thereaction space ( n max = 1000 ), just as expected. However,this population immediately collapses again. The averagemolecularlengththenincreasesdramatically, whiletheover-all reaction rate (indicating the average rate of binding be-tween random molecules in the population) also collapses.In this particular run, molecules were arbitrarily limited toa maximum length of  BD lmax = 500 . Other experiments,without such a limit, indicated that the growth in molecu-lar length appeared to continue indefinitely, subject only toavailable physical (computer) resources.As with the experiments discussed earlier, these resultswerenotexpected. Infact, certainmutantsoftheoriginalau-
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