Cluster compartmentalization may provide resistance toparasites for catalytic networksMikael B. Cronhjort 1 and Clas Blomberg 2;3Teoretisk Fysik, Kungl Tekniska H�ogskolan, S-100 44 Stockholm, SwedenAbstractWe have performed calculations on reaction-di�usion equations with an aim tostudy two-dimensional spatial patterns. The systems explicitly studied are threedi�erent catalytic networks: A 4-component network displaying chaotic dynamics,a 5-component hypercycle network and a simple 1-component system. We have ob-tained cluster states for all these networks, and in all cases the clusters have theability to divide. This contradicts recent conclusions that only systems with chaoticdynamics may give cluster states: On the contrary, we think that any network ar-chitecture may display cluster formation and cluster division. Our conclusion is inagreement with experimental results reported for an inorganic system correspond-ing to the simple 1-component system studied in this paper. In a partial di�erentialequations model, the clusters do not provide resistance to parasites, which are as-sumed to arise by mutations: Parasites may spread from one cluster to another, andeventually kill all clusters. However, by combining the partial di�erential equationswith a suitable cut-o� rule, we demonstrate a system of partly isolated clustersthat is resistant against parasites: The parasites do not infect all clusters, and whenthe infected clusters have decayed, they are replaced by new ones, as neighbouringclusters divide.Keywords: pattern formation / compartmentalization / catalytic network /hypercycle1 IntroductionThe formation of patterns in spatial models concerning the origin of life hasrecently aroused a lot of interest [1{7]. Some models consider hypercycle net-works, in which a number of RNA-like polymers catalyse the replication of1E-mail: mic@theophys.kth.se2E-mail: cob@theophys.kth.se3To whom correspondence should be addressedPreprint submitted to Elsevier Science 25 October 1996

each other in a cyclic way [1,2,8]. The spatial ordering may provide the hyper-cycle resistance to parasites. Parasites, which are assumed to arise by muta-tions in the molecules included in the hypercycle, may be fatal to a spatiallyhomogeneous hypercycle system. Some other models consider more generalcatalytic networks, where the spatial organisation may provide a primitivecompartmentalization [3{5].In this paper, we focus on compartmentalization due to spontaneous clusterformation. It has been suggested that the cluster structure could present someof the functional advantages of cellular systems, without requiring the rela-tively complex structure of a membrane [3]. Similar clusters (\spots") haverecently been demonstrated experimentally, and modelled in numerical simu-lations, for a simple inorganic reaction-di�usion system [9{11]. These resultsprove that such patterns may arise in simple systems. We compare clusterstructures for some di�erent catalytic networks: The 4-component networkwhich displays chaotic dynamics [3,4], the 5-component hypercycle network[2], and the simplest 1-component network. The networks are de�ned in termsof partial di�erential equations, which we study numerically by approximat-ing the di�erential equations with �nite di�erence equations. We comparethe compartmentalizations that are obtained and we demonstrate that theseclusters may provide a solution to the parasite problem.2 The modelWe study a two-dimensional partial di�erential equations model of catalyticnetworks,@Xi@t =M NXj=1 kijXjXi � gXXi +DXr2Xi ; i = 1; 2; :::; N (1)@M@t = kM � gMM � LM NXi; j=1 kijXjXi +DMr2M; (2)where Xi denotes the concentration of the polymers, and M is the concen-tration of activated monomers. (M corresponds to what is called a by Nu~no,Chac�on and others in [3,4]. We do not include inactive monomers, called bin [3,4], in our model.) N is the number of di�erent polymer species. Thereplication of each polymer Xi is catalysed by each Xj at a rate constant kij.Linear (non-catalytic) growth terms are neglected. The activated monomersare produced at a constant rate, kM . gX and gM are decay rate constants, Lis the number of monomers in each polymer, and DX and DM are di�usionconstants. The parameters k, g, and D include the time unit, which de�nes2

the time scale of the system.D also includes the length unit, which de�nes thelength scale of the system. In our simulations, the partial di�erential equations(1,2) are approximated with a system of �rst order �nite di�erence equations.The basic approximations are@urs@t (t1) � urs(t1 +�t)� urs(t1)�t ; (3)r2urs � r25urs � ur+1;s + ur�1;s + ur;s+1 + ur;s�1 � 4urs(�x)2 (4)where u may be Xi or M , r and s are space coordinates, and the latticeparameter, �x, equals the length unit. Note that the time unit is not equalto the integration step, �t, which may di�er between the calculations. Whendealing with parasites, the di�erence equations are combined with a cut-o�rule, which is described in section 3.2.The simulations were performed on a 120�120 lattice with periodic boundaryconditions. In our simulations, we use di�erent kij for the di�erent catalyticnetworks. For the network displaying chaotic dynamics [3,4,12] we usekchaosij = 0BBBBBBBB@ 0:5 1:5 0:5 0:11:6 1:0 0 00 2:0 0:6 02:2 0 0:4 0 1CCCCCCCCA ; (5)and for the hypercycle network [2]khyperij = 0BBBBBBBBBBBB@ 0 0 0 0 2:62:6 0 0 0 00 2:6 0 0 00 0 2:6 0 00 0 0 2:6 0 1CCCCCCCCCCCCA : (6)For the simplest network, the 1-component system, kij has only one element,ksimpleij = k11 = 2:6: (7)The default values of the other parameters are gX = 0:1, kM = 1, gM = 0:1and L = 100. The di�usion parameters, DX and DM , are important for the3

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0.3

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(a) (b) (c)Fig. 1. Typical cluster states for the di�erent networks: a) 4-component networkwith chaotic dynamics (at t = 2000), b) the 5-component hypercycle (t = 2000),and c) the simple 1-component system (t = 5000). The time unit is de�ned througheq. 3 in the text. The grey-scale corresponds to the concentration of PN1 Xi, whiteindicating high concentrations and black low. The same grey-scale key will be usedin all pictures.cluster formation [2]: DX determines the size of the clusters, and DM theregion of monomer uptake for each cluster, i.e. the distances between clusters.Here, we have used the default values DX = 1, D chaosM = D hyperM = 200 andD simpleM = 400 (discussed below).3 Results3.1 Formation and division of clustersWe have obtained similar cluster structures for all the above mentioned cat-alytic networks (�g. 1). In the clusters, the concentration of polymers is high,and the concentration of monomers is low, as monomers are consumed by thereplication of the polymers. Between the clusters, the concentration of poly-mers is low, and monomers are present at high concentration. Monomers owin to the clusters due to the gradient in the monomer concentration. Contraryto what is concluded in [3,4], clusters can be obtained in systems withoutchaotic dynamics, e.g. the hypercycle system (eq. 6) [2]. As similar clustersare obtained also for the 1-component auto-catalytic system (eq. 7), we hereshow that the cluster formation is not dependent on a certain number of com-ponents or oscillations in the network. We have obtained similar clusters forhypercycles with 2, 3 or 4 components, although we do not show those resultshere.Clusters arise spontaneously from initial states where each polymer species4

t=2650 t=2750 t=2950Fig. 2. A sequence displaying division of clusters for a network with chaotic dy-namics. Two clusters suddenly decay spontaneously and in the subsequent divisionseveral new clusters are formed, some of which are already decaying in the lastpicture.is assigned a random concentration on each lattice point. The average of therandom values has to be above a certain threshold value (which is approxi-mately 0.005 for the 1-component system), else the polymers may decay onthe entire lattice. Once the polymers do not decay, a very small perturbation(e.g. 1%) from a spatially homogeneous state is su�cient for cluster formationto occur.The clusters have the ability to divide for all these networks. For the networkwith chaotic dynamics (eq. 5), we have not obtained a steady state, contraryto what is assumed in [3,4]. During the �rst 300 time units of our simulationsa cluster state emerges. Then, from 300 to about 1400 time units, follows aquasi-stationary state during which very little changes. After approximately1400 time units clusters begin to decay spontaneously. Where clusters havedecayed, the neighbouring clusters divide, and new clusters take the place ofthose which have decayed (�g. 2). Clusters may decay or divide one by one ormany at the same time. During a long simulation (from 1800 to 50000 timeunits) with a coarser grid (40 � 40 lattice points), clusters decay and dividecontinuously at an average rate of about once per 300 time units.For the networks without chaotic dynamics (eqs. 6, 7), spontaneous decay anddivision are not observed, but a steady state is soon established. In order toinduce a division, we have deleted one or more clusters arti�cially (�g. 3).Once the division occurs, it is similar to the division obtained for the networkwith chaotic dynamics (eq. 5). The division is rapid if many clusters havebeen deleted, but slow if only few have been deleted. The number of dividingclusters may di�er from the number of deleted clusters.Clusters are only obtained for a certain region in parameter space. For smallvariations from the default parameters speci�ed in the previous section, thedistances between clusters may increase or decrease. If the change is large,the clusters may decay altogether, merge in strings or even �ll the entire5

t=2000 t=2000

t=2340 t=3000Fig. 3. Division of hypercycle clusters. The �rst picture shows the state at t = 2000time units. In the second picture, also at t = 2000, two clusters have been deletedarti�cially, resulting in an empty (black) rectangle. The last two pictures show thesubsequent division of one neighbouring cluster.lattice uniformly. The distances between clusters also vary with the number ofcomponents, N : The fewer components, the smaller distances (�g. 1). For theauto-catalytic system with only one component (eq. 7), the clusters merge instrings when D simpleM = 200. When D simpleM = 400 the clusters remain separatedfrom each other.We have performed a systematic parameter survey for the auto-catalytic sys-tem, where we have varied DM , L and kM . Varying the parameters givessparser (�gs. 4a{c) or denser (�gs. 4d{h) systems. In dense systems, the clus-ters may merge in strings or �ll the entire lattice. The di�usion constant forthe monomers, DM , determines how large the region of monomer uptake isaround each cluster: If DM is increased, as in �g. 4a, the region of monomeruptake is larger. This gives larger distances between clusters. If instead DM isdecreased, the distances between clusters will decrease: Figs. 4d{f, where DM6

(a) (b) (c)

(d) (e) (f)

(g) (h)Fig. 4. Results obtained when the parameters DM , L and kM are varied one at atime. The default values are DM = 400, L = 100 and kM = 1:0. The pictures (att = 2000 unless other is speci�ed) show the states obtained from a certain randominitial condition with: a) DM = 1000; b) L = 200; c) kM = 0:5; d), e) and f)DM = 200 at t = 1000, t = 3000 and t = 10000, respectively; g) L = 50; and h)kM = 2:0.is half of the default value, show how clusters form so close to each other thatthey merge in strings. Indirectly, L and kM also a�ect the size of the monomeruptake region around each cluster, and thereby the distances between clusters.In �g. 4b, where L is doubled, the consumption of monomers is larger in eachcluster, resulting in a larger region depleted on monomers around each cluster,and thus larger distances between clusters. If, on the other hand, L is halvedas in �g. 4g, the distances are decreased and the clusters will merge in strings.Varying the production rate of monomers, kM , gives similar results: The lowerkM , the larger the uptake region has to be in order to provide the cluster withmonomers, and vice versa. In �g. 4c, kM is half of the default value, resulting7

in larger distances between clusters. In �g. 4h, where kM is doubled, no spatialstructure is formed: The production of monomers is su�cient to �ll the entirelattice uniformly with polymers.3.2 Clusters may provide resistance to parasitesParasites, which do not catalyse the replication of any polymers, may in gen-eral kill systems like the ones considered in this paper, if the parasites arereplicated faster than the original species, or decay slower. We have intro-duced a parasite as a species receiving stronger catalytic support than theoriginal species, but which does not give catalytic support to any species. Thecatalytic networks (eqs. 5, 6 and 7) are then altered to bekchaos+Pij = 0BBBBBBBBBBBB@ 0:5 1:5 0:5 0:1 01:6 1:0 0 0 00 2:0 0:6 0 02:2 0 0:4 0 00 0 0 3:0 01CCCCCCCCCCCCA ; khyper+Pij = 0BBBBBBBBBBBBBBB@ 0 0 0 0 2:6 02:6 0 0 0 0 00 2:6 0 0 0 00 0 2:6 0 0 00 0 0 2:6 0 00 0 0 0 3:0 01CCCCCCCCCCCCCCCA ;and ksimple+Pij = 0B@ 2:6 03:0 01CA ; (8)where the last rows and columns correspond to the catalytic support receivedand given by the parasite, respectively. As the parasite does not provide anycatalytic support, all elements of the last columns are zero. When such par-asites are introduced in the above described systems, the parasites �rst killthe cluster in which they were inserted. They do also spread to the adjacentclusters, which are subsequently killed. In general, the cluster division that isinduced by the decay of the infected clusters cannot save the systems fromthe parasites, as the clusters that divide have already been infected by theparasite before the division is completed. Finally, the parasite has infectedand killed all clusters on the lattice.However, the spread of the parasite can be restricted, if the above describedsystems are combined with a cut-o�. We have done this in the following way:The cut-o� implies that concentrations are calculated as usual, but if a con-centration value at some lattice point is found to be less than a certain cut-o�value, it is set to be equal to the cut-o� value with a probability equal to thequotient between the calculated value and the cut-o� value. Otherwise, it is8

t=2100 t=2800

t=3000 t=3500Fig. 5. At t = 2100 a parasite is inserted in the cluster at the center of the �rstpicture. The parasite soon kills that cluster, but it also spreads to a neighbouringcluster (above and left of the cluster where it was inserted) which has already beenkilled in the next picture, at t = 2800. However, the parasite does not spread furtheras the clusters are almost separated from each other by empty (black) lattice pointsdue to the cut-o�. When the infected clusters have decayed division takes place. Att = 3500 the division is completed and a new steady state is soon formed.set to be zero. This cut-o� rule is on average mass-conserving, i.e. moleculesdo not disappear or arise due to the cut-o� rule.In states where distances between clusters are large, the concentration of poly-mers is low between the clusters. It is then possible to choose a cut-o� valuewhich essentially puts the concentration of polymers between the clusters tozero, but which does not a�ect the concentrations in the clusters. We haveapplied the cut-o� rule to the state described in �g. 4c. This gives a steadystate illustrated by the �rst picture in �g. 5, where there are empty (black) re-gions between the clusters. Here the cut-o� value is 0.01. For comparison, theconcentration of polymers in the interior of the clusters is approximately 0.3.9

A parasite is introduced in one of the clusters. As the clusters have some poly-mer contact with each other, there is a small probability that the neighbouringclusters will also be infected, which is the case for one of the neighbouring clus-ters in �g. 5. Due to the cut-o�, the parasite does not spread to the rest ofthe clusters. Note that the polymer contact between clusters is not necessaryfor the cluster formation, it is merely a possible way for the parasite (or otherpolymer molecules, e.g. favourable mutants) to spread. The distances betweenclusters are determined by the monomers. When the infected clusters havedecayed, some of the surviving neighbouring clusters divide, and replace thekilled clusters with new clusters.4 Discussion and conclusionWe have obtained clusters for all the systems we have examined. Contrary toconclusions in [3,4], we have found that chaotic dynamics is not necessary forthe formation of clusters. We �nd it likely that clusters can be obtained forany network architecture. However, the networks that do not display chaoticdynamics (eqs. 6 and 7) give steady states, which we do not get for the networkwith chaotic dynamics (eq. 5).Our conclusion is also that division of clusters is not due to the chaotic dy-namics, as division can also be obtained for the other networks. Rather, wethink that the chaotic dynamics occasionally makes the clusters decay, andthen, as the monomer concentration rises, the neighbouring clusters divide ina process that does not depend on the chaotic dynamics.We also want to point out that there are two di�erent kinds of pattern forma-tion in these problems, which should be kept apart. First, systems can havealmost constant total concentration, PN1 Xi, over the entire lattice (space),and display patterns where di�erent species are present at di�erent places,e.g. spirals for the hypercycle. Second, there are patterns where the total con-centration varies signi�cantly in space, e.g. clusters: All polymer species havehigh concentrations in the clusters and low concentrations between the clus-ters. Contrary to [3], our conclusion is that the �rst kind of pattern formationhas very little to do with the second kind, i.e. with the formation of clus-ters. Some systems (e.g. the cellular automaton hypercycle in [1]) display onlythe �rst kind of pattern, others (e.g. the simple 1-component system eq. 7)only the second kind, and some systems have both kinds simultaneously (e.g.hypercycle clusters in cellular automaton models described in [13]).We do not believe that the di�erences between the model we use, and themodel used in [3,4], are signi�cant. The inactive monomers and the non-catalytic replication terms should not be important for the features of interest10

here. Rather, we believe that it is important to describe the monomers in away that gives good numerical stability to the simulation. This facilitates longsimulation times with reasonable computer resources. Our simulations wereperformed on a HP 735 work station.Resistance to parasites is not demonstrated in the partial di�erential equa-tions model described in section 2. Even though the clusters are separated byregions of low polymer concentration, the parasite di�uses from one cluster toanother, and eventually kills all clusters. As any species can grow from verylow concentrations in this model, the parasite is able to infect all clusters.To overcome this, the cut-o� rule is applied, and a suitable cut-o� value ischosen. We have chosen a value that almost isolates the clusters from eachother. The polymer contact between clusters is so low that the parasite isunlikely to di�use from one cluster to another during the life time of the in-fected cluster, which is soon killed by the parasite. When clusters remain,however, the weak contact enables favourable mutants, which do not kill orharm 'their' clusters, to spread. Actually, favourable mutants may spread evenwithout contact between clusters, as clusters containing favourable mutantsare likely to be larger than other clusters and they might be �rst to dividewhen a neighbouring cluster has been killed by a parasite.The cut-o� rule is most readily applied to the 1-component network: As theconcentration of the single polymer species is always high in the clusters, theclusters are stable when the cut-o� is applied. On the contrary, the cut-o�may be a threat to the more complicated networks (eqs. 5 and 6), as speciesmay be eliminated even in the clusters by the cut-o�, when the concentrationof a certain polymer gets too low due to the temporal oscillations.The cut-o� value corresponds to the concentration at which the medium (liq-uid) contains a single polymer molecule per unit area (volume). We think itshould be possible to �nd experimental situations where the above describedresistance to parasites can be demonstrated.For the special case with a single component, X1, the equations (1,2) can becompared with the equations of the Gray-Scott model, which describe theexperimental system studied by Pearson, Swinney and others [9{11],@V@� =UV 2 � (F + k)V +DVr2V; (9)@U@� =F (1� U) � UV 2 +DUr2U: (10)Here we choose X1 � V and M � U . In order to identify eq. (9) with eq. (1),we put � = t=k11, F + k = gX=k11 and DV = DX=k11. As we then wish toidentify eq. (10) with eq. (2), we need F = kM=k11 = gM=k11, L = 1 and DU =11

DM=k11, i.e. the necessary assumptions are kM = gM and L = 1. The �rstone of these, kM = gM , can be interpreted as a scaling in concentration whichimplies that the concentration of activated monomers will be 1 in a steadystate, if no replicating polymers are present. The second assumption, L = 1,is an important but trivial di�erence between the models: It implies that thereplicating molecules considered in the Gray-Scott model are no polymers, buteach is built from a single monomer.5 AcknowledgementsWe want to thank Paulien Hogeweg and Maarten Boerlijst for ideas about howto construct mass-conserving rules for the cut-o�, and Kristian Lindgren andan anonymous referee for valuable comments for the revision of the manuscript.References[1] M. C. Boerlijst & P. Hogeweg, Spiral wave structure in pre-biotic evolution:Hypercycles stable against parasites, Physica D 48 (1991) 17{28.[2] M. Cronhjort & C. Blomberg, Hypercycles versus parasites in a two-dimensionalpartial di�erential equations model, J. theor. Biol. 169 (1994) 31{49.[3] J. C. Nu~no, P. Chac�on, A. Moreno & F. Mor�an, Compartmentation in replicatormodels, in: F. Mor�an, A. Moreno, J. J. Merelo & P. Chac�on, eds., Advances inarti�cial life, (Springer, Berlin, 1995) 116{127.[4] P. Chac�on & J. C. Nu~no, Spatial dynamics of a model for prebiotic evolution,Physica D 81 (1995) 398{410.[5] D. Lancet et. al, A cellular automaton model for self-replication of mutuallycatalytic biopolymers, a poster presented at the Third European Conference onArti�cial Life, Granada, Spain, June 1995.[6] M. A. Nowak & R. M. May, Evolutionary games and spatial chaos, Nature 359(1993) 826{829.[7] K. Lindgren & M. G. Nordahl, Evolutionary dynamics of spatial games, PhysicaD 75 (1994) 292{309.[8] M. Eigen & P. Schuster, The hypercycle: A principle of natural self-organisation(Springer, Berlin, 1979).[9] J. E. Pearson, Complex patterns in a simple system, Science 261 (1993), 189{192. 12

[10] K. J. Lee, W. D. McCormick, Q. Ouyang and H. L. Swinney, Pattern formationby interacting chemical fronts, Science 261 (1993), 192{194.[11] K. J. Lee, W. D. McCormick, J. E. Pearson and H. L. Swinney, Experimentalobservation of self-replicating spots in a reaction-di�usion system, Nature 369(1994), 215{218.[12] M. A. Andrade, J. C. Nu~no, F. Mor�an, F. Montero & G. J. Mpitsos, Complexdynamics of a catalytic network having faulty replication into error-species,Physica D 63 (1993) 21{40.[13] C. Blomberg & M. Cronhjort, Modeling errors and parasites in the evolutionof primitive life: Possibilities of spatial self-structuring, in: J. L. Casti & A.Karlqvist, eds., Cooperation & con ict in general evolutionary processes , (JohnWiley & Sons, New York, 1995) 15{62.

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