Cluster compartmentalization may provide resistance to parasites for catalytic networks

Cluster compartmentalization may provide resistance to parasites for catalytic networks Mikael B. Cronhjort 1 and Clas Blomberg 2 3 ;

Teoretisk Fysik, Kungl Tekniska Hogskolan, S-100 44 Stockholm, Sweden

Abstract We have performed calculations on reaction-diusion equations with an aim to study two-dimensional spatial patterns. The systems explicitly studied are three dierent catalytic networks: A 4-component network displaying chaotic dynamics, a 5-component hypercycle network and a simple 1-component system. We have obtained cluster states for all these networks, and in all cases the clusters have the ability to divide. This contradicts recent conclusions that only systems with chaotic dynamics may give cluster states: On the contrary, we think that any network architecture may display cluster formation and cluster division. Our conclusion is in agreement with experimental results reported for an inorganic system corresponding to the simple 1-component system studied in this paper. In a partial dierential equations model, the clusters do not provide resistance to parasites, which are assumed to arise by mutations: Parasites may spread from one cluster to another, and eventually kill all clusters. However, by combining the partial dierential equations with a suitable cut-o rule, we demonstrate a system of partly isolated clusters that is resistant against parasites: The parasites do not infect all clusters, and when the infected clusters have decayed, they are replaced by new ones, as neighbouring clusters divide. Keywords: pattern formation / compartmentalization / catalytic network / hypercycle

1 Introduction The formation of patterns in spatial models concerning the origin of life has recently aroused a lot of interest [1{7]. Some models consider hypercycle networks, in which a number of RNA-like polymers catalyse the replication of 1 E-mail: [email protected] 2 E-mail: [email protected] 3 To whom correspondence should

be addressed

Preprint submitted to Elsevier Science

25 October 1996

each other in a cyclic way [1,2,8]. The spatial ordering may provide the hypercycle resistance to parasites. Parasites, which are assumed to arise by mutations in the molecules included in the hypercycle, may be fatal to a spatially homogeneous hypercycle system. Some other models consider more general catalytic networks, where the spatial organisation may provide a primitive compartmentalization [3{5]. In this paper, we focus on compartmentalization due to spontaneous cluster formation. It has been suggested that the cluster structure could present some of the functional advantages of cellular systems, without requiring the relatively complex structure of a membrane [3]. Similar clusters (\spots") have recently been demonstrated experimentally, and modelled in numerical simulations, for a simple inorganic reaction-diusion system [9{11]. These results prove that such patterns may arise in simple systems. We compare cluster structures for some dierent catalytic networks: The 4-component network which displays chaotic dynamics [3,4], the 5-component hypercycle network [2], and the simplest 1-component network. The networks are de ned in terms of partial dierential equations, which we study numerically by approximating the dierential equations with nite dierence equations. We compare the compartmentalizations that are obtained and we demonstrate that these clusters may provide a solution to the parasite problem.

2 The model We study a two-dimensional partial dierential equations model of catalytic networks, @Xi

=M

@t

X N

j

=1

kij Xj Xi

=k ? g

@M

M

@t

MM

?g

X

Xi

+ D r2 X

X

? LM

X

N

i; j

=1

kij Xj Xi

i

;

i

= 1; 2; :::; N

+ D r2M;

(1) (2)

M

where X denotes the concentration of the polymers, and M is the concentration of activated monomers. (M corresponds to what is called a by Nu~no, Chacon and others in [3,4]. We do not include inactive monomers, called b in [3,4], in our model.) N is the number of dierent polymer species. The replication of each polymer X is catalysed by each X at a rate constant k . Linear (non-catalytic) growth terms are neglected. The activated monomers are produced at a constant rate, k . g and g are decay rate constants, L is the number of monomers in each polymer, and D and D are diusion constants. The parameters k, g, and D include the time unit, which de nes i

i

j

M

X

M

X

2

ij

M

the time scale of the system. D also includes the length unit, which de nes the length scale of the system. In our simulations, the partial dierential equations (1,2) are approximated with a system of rst order nite dierence equations. The basic approximations are u (t1 +
t) ? u (t1) ( t1 )  ; (3) @t
t u +1 + u ?1 ? 4u (4) r2u  r25u  u +1 + u ?1 +(
x)2 where u may be X or M , r and s are space coordinates, and the lattice parameter,
x, equals the length unit. Note that the time unit is not equal to the integration step,
t, which may dier between the calculations. When dealing with parasites, the dierence equations are combined with a cut-o rule, which is described in section 3.2. @urs

rs

rs

r

rs

;s

r

;s

r;s

r;s

rs

rs

i

The simulations were performed on a 120  120 lattice with periodic boundary conditions. In our simulations, we use dierent k for the dierent catalytic networks. For the network displaying chaotic dynamics [3,4,12] we use 1 0 0:5 1:5 0:5 0:1 C B CC B B 1 : 6 1 : 0 0 0 CC B (5) k chaos = B C; B B 0 2:0 0:6 0 C C B A @ 2:2 0 0:4 0 ij

ij

and for the hypercycle network [2] 0 1 0 0 0 0 2:6 C B B CC B B 2 :6 0 0 0 0 C B CC hyper = B B k 0 2:6 0 0 0 C B CC : B B B CC 0 0 2:6 0 0 C B @ A 0 0 0 2:6 0

(6)

ij

For the simplest network, the 1-component system, k has only one element, ij

simple = k

kij

11 = 2:6:

(7)

The default values of the other parameters are g = 0:1, k = 1, g = 0:1 and L = 100. The diusion parameters, D and D , are important for the X

X

3

M

M

M

Σ 0.5 0.4 0.3 0.2 0.1 0

(a)

(b)

(c)

Fig. 1. Typical cluster states for the dierent networks: a) 4-component network with 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 P through eq. 3 in the text. The grey-scale corresponds to the concentration of 1 X , white indicating high concentrations and black low. The same grey-scale key will be used in all pictures. N

i

cluster formation [2]: D determines the size of the clusters, and D the region of monomer uptake for each cluster, i.e. the distances between clusters. Here, we have used the default values D = 1, D chaos = D hyper = 200 and simple = 400 (discussed below). D X

M

X

M

M

M

3 Results 3.1 Formation and division of clusters

We have obtained similar cluster structures for all the above mentioned catalytic networks ( g. 1). In the clusters, the concentration of polymers is high, and the concentration of monomers is low, as monomers are consumed by the replication of the polymers. Between the clusters, the concentration of polymers is low, and monomers are present at high concentration. Monomers ow in to the clusters due to the gradient in the monomer concentration. Contrary to what is concluded in [3,4], clusters can be obtained in systems without chaotic dynamics, e.g. the hypercycle system (eq. 6) [2]. As similar clusters are obtained also for the 1-component auto-catalytic system (eq. 7), we here show that the cluster formation is not dependent on a certain number of components or oscillations in the network. We have obtained similar clusters for hypercycles with 2, 3 or 4 components, although we do not show those results here. Clusters arise spontaneously from initial states where each polymer species 4

t=2650

t=2750

t=2950

Fig. 2. A sequence displaying division of clusters for a network with chaotic dynamics. Two clusters suddenly decay spontaneously and in the subsequent division several new clusters are formed, some of which are already decaying in the last picture.

is assigned a random concentration on each lattice point. The average of the random values has to be above a certain threshold value (which is approximately 0.005 for the 1-component system), else the polymers may decay on the entire lattice. Once the polymers do not decay, a very small perturbation (e.g. 1%) from a spatially homogeneous state is sucient for cluster formation to occur. The clusters have the ability to divide for all these networks. For the network with chaotic dynamics (eq. 5), we have not obtained a steady state, contrary to what is assumed in [3,4]. During the rst 300 time units of our simulations a cluster state emerges. Then, from 300 to about 1400 time units, follows a quasi-stationary state during which very little changes. After approximately 1400 time units clusters begin to decay spontaneously. Where clusters have decayed, the neighbouring clusters divide, and new clusters take the place of those which have decayed ( g. 2). Clusters may decay or divide one by one or many at the same time. During a long simulation (from 1800 to 50000 time units) with a coarser grid (40  40 lattice points), clusters decay and divide continuously at an average rate of about once per 300 time units. For the networks without chaotic dynamics (eqs. 6, 7), spontaneous decay and division are not observed, but a steady state is soon established. In order to induce 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 network with chaotic dynamics (eq. 5). The division is rapid if many clusters have been deleted, but slow if only few have been deleted. The number of dividing clusters may dier from the number of deleted clusters. Clusters are only obtained for a certain region in parameter space. For small variations from the default parameters speci ed in the previous section, the distances between clusters may increase or decrease. If the change is large, the clusters may decay altogether, merge in strings or even ll the entire 5

t=2000

t=2000

t=2340

t=3000

Fig. 3. Division of hypercycle clusters. The rst picture shows the state at t = 2000 time units. In the second picture, also at t = 2000, two clusters have been deleted arti cially, resulting in an empty (black) rectangle. The last two pictures show the subsequent division of one neighbouring cluster.

lattice uniformly. The distances between clusters also vary with the number of components, N : The fewer components, the smaller distances ( g. 1). For the auto-catalytic system with only one component (eq. 7), the clusters merge in strings when D simple = 200. When D simple = 400 the clusters remain separated from each other. M

M

We have performed a systematic parameter survey for the auto-catalytic system, where we have varied D , L and k . Varying the parameters gives sparser ( gs. 4a{c) or denser ( gs. 4d{h) systems. In dense systems, the clusters may merge in strings or ll the entire lattice. The diusion constant for the monomers, D , determines how large the region of monomer uptake is around each cluster: If D is increased, as in g. 4a, the region of monomer uptake is larger. This gives larger distances between clusters. If instead D is decreased, the distances between clusters will decrease: Figs. 4d{f, where D M

M

M

M

M

M

6

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Fig. 4. Results obtained when the parameters D , L and k are varied one at a time. The default values are D = 400, L = 100 and k = 1:0. The pictures (at t = 2000 unless other is speci ed) show the states obtained from a certain random initial condition with: a) D = 1000; b) L = 200; c) k = 0:5; d), e) and f) D = 200 at t = 1000, t = 3000 and t = 10000, respectively; g) L = 50; and h) k = 2:0. M

M

M

M

M

M

M

M

is half of the default value, show how clusters form so close to each other that they merge in strings. Indirectly, L and k also aect the size of the monomer uptake region around each cluster, and thereby the distances between clusters. In g. 4b, where L is doubled, the consumption of monomers is larger in each cluster, resulting in a larger region depleted on monomers around each cluster, and thus larger distances between clusters. If, on the other hand, L is halved as in g. 4g, the distances are decreased and the clusters will merge in strings. Varying the production rate of monomers, k , gives similar results: The lower k , the larger the uptake region has to be in order to provide the cluster with monomers, and vice versa. In g. 4c, k is half of the default value, resulting M

M

M

M

7

in larger distances between clusters. In g. 4h, where k is doubled, no spatial structure is formed: The production of monomers is sucient to ll the entire lattice uniformly with polymers. M

3.2 Clusters may provide resistance to parasites

Parasites, which do not catalyse the replication of any polymers, may in general kill systems like the ones considered in this paper, if the parasites are replicated faster than the original species, or decay slower. We have introduced a parasite as a species receiving stronger catalytic support than the original species, but which does not give catalytic support to any species. The catalytic networks (eqs. 5, 6 and 7) are then altered to be

1 0 0 0 0 0 2 : 6 0 C BB BB 2:6 0 0 0 0 0 CCC CC BB B 0 2:6 0 0 0 0 C CC ; hyper+P = B BB k BB 0 0 2:6 0 0 0 CCC C BB B@ 0 0 0 2:6 0 0 CCA 0 0 0 0 3:0 0

1 0 0 : 5 1 : 5 0 : 5 0 : 1 0 CC B B B CC B 1:6 1:0 0 0 0 C B C B k chaos+P = B 0 2 :0 0:6 0 0 C CC ; B B B CC B 2:2 0 0:4 0 0 C B A @ 0 0 0 3:0 0 1 0 2 : 6 0 CA ; and ksimple+P = B @ 3:0 0

ij

ij

(8)

ij

where the last rows and columns correspond to the catalytic support received and given by the parasite, respectively. As the parasite does not provide any catalytic support, all elements of the last columns are zero. When such parasites are introduced in the above described systems, the parasites rst kill the cluster in which they were inserted. They do also spread to the adjacent clusters, which are subsequently killed. In general, the cluster division that is induced by the decay of the infected clusters cannot save the systems from the parasites, as the clusters that divide have already been infected by the parasite before the division is completed. Finally, the parasite has infected and killed all clusters on the lattice. However, the spread of the parasite can be restricted, if the above described systems 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 concentration 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 the quotient between the calculated value and the cut-o value. Otherwise, it is 8

t=2100

t=2800

t=3000

t=3500

Fig. 5. At t = 2100 a parasite is inserted in the cluster at the center of the rst picture. The parasite soon kills that cluster, but it also spreads to a neighbouring cluster (above and left of the cluster where it was inserted) which has already been killed in the next picture, at t = 2800. However, the parasite does not spread further as the clusters are almost separated from each other by empty (black) lattice points due to the cut-o. When the infected clusters have decayed division takes place. At t = 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. molecules do not disappear or arise due to the cut-o rule. In states where distances between clusters are large, the concentration of polymers is low between the clusters. It is then possible to choose a cut-o value which essentially puts the concentration of polymers between the clusters to zero, but which does not aect the concentrations in the clusters. We have applied the cut-o rule to the state described in g. 4c. This gives a steady state illustrated by the rst picture in g. 5, where there are empty (black) regions between the clusters. Here the cut-o value is 0.01. For comparison, the concentration 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 polymer contact with each other, there is a small probability that the neighbouring clusters will also be infected, which is the case for one of the neighbouring clusters in g. 5. Due to the cut-o, the parasite does not spread to the rest of the clusters. Note that the polymer contact between clusters is not necessary for the cluster formation, it is merely a possible way for the parasite (or other polymer molecules, e.g. favourable mutants) to spread. The distances between clusters are determined by the monomers. When the infected clusters have decayed, some of the surviving neighbouring clusters divide, and replace the killed clusters with new clusters.

4 Discussion and conclusion We have obtained clusters for all the systems we have examined. Contrary to conclusions in [3,4], we have found that chaotic dynamics is not necessary for the formation of clusters. We nd it likely that clusters can be obtained for any network architecture. However, the networks that do not display chaotic dynamics (eqs. 6 and 7) give steady states, which we do not get for the network with chaotic dynamics (eq. 5). Our conclusion is also that division of clusters is not due to the chaotic dynamics, as division can also be obtained for the other networks. Rather, we think that the chaotic dynamics occasionally makes the clusters decay, and then, as the monomer concentration rises, the neighbouring clusters divide in a process that does not depend on the chaotic dynamics. We also want to point out that there are two dierent kinds of pattern formation in these problems, which should P be kept apart. First, systems can have almost constant total concentration, 1 X , over the entire lattice (space), and display patterns where dierent species are present at dierent places, e.g. spirals for the hypercycle. Second, there are patterns where the total concentration varies signi cantly in space, e.g. clusters: All polymer species have high concentrations in the clusters and low concentrations between the clusters. Contrary to [3], our conclusion is that the rst kind of pattern formation has very little to do with the second kind, i.e. with the formation of clusters. Some systems (e.g. the cellular automaton hypercycle in [1]) display only the 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]). N

i

We do not believe that the dierences between the model we use, and the model used in [3,4], are signi cant. The inactive monomers and the noncatalytic replication terms should not be important for the features of interest 10

here. Rather, we believe that it is important to describe the monomers in a way that gives good numerical stability to the simulation. This facilitates long simulation times with reasonable computer resources. Our simulations were performed on a HP 735 work station. Resistance to parasites is not demonstrated in the partial dierential equations model described in section 2. Even though the clusters are separated by regions of low polymer concentration, the parasite diuses from one cluster to another, and eventually kills all clusters. As any species can grow from very low 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 is chosen. We have chosen a value that almost isolates the clusters from each other. The polymer contact between clusters is so low that the parasite is unlikely to diuse from one cluster to another during the life time of the infected cluster, which is soon killed by the parasite. When clusters remain, however, the weak contact enables favourable mutants, which do not kill or harm 'their' clusters, to spread. Actually, favourable mutants may spread even without contact between clusters, as clusters containing favourable mutants are likely to be larger than other clusters and they might be rst to divide when a neighbouring cluster has been killed by a parasite. The cut-o rule is most readily applied to the 1-component network: As the concentration of the single polymer species is always high in the clusters, the clusters 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 species may be eliminated even in the clusters by the cut-o, when the concentration of a certain polymer gets too low due to the temporal oscillations. The cut-o value corresponds to the concentration at which the medium (liquid) contains a single polymer molecule per unit area (volume). We think it should be possible to nd experimental situations where the above described resistance to parasites can be demonstrated. For the special case with a single component, X1, the equations (1,2) can be compared with the equations of the Gray-Scott model, which describe the experimental system studied by Pearson, Swinney and others [9{11], @V @ @U @

= U V 2 ? (F + k)V + D r2V;

(9)

= F (1 ? U ) ? U V 2 + D r2U:

(10)

V

U

Here we choose X1  V and M  U . In order to identify eq. (9) with eq. (1), we put  = t=k11, F + k = g =k11 and D = D =k11. As we then wish to identify eq. (10) with eq. (2), we need F = k =k11 = g =k11, L = 1 and D = X

V

M

11

X

M

U

D =k11 , i.e. the necessary assumptions are k = g and L = 1. The rst one of these, k = g , can be interpreted as a scaling in concentration which implies that the concentration of activated monomers will be 1 in a steady state, if no replicating polymers are present. The second assumption, L = 1, is an important but trivial dierence between the models: It implies that the replicating molecules considered in the Gray-Scott model are no polymers, but each is built from a single monomer. M

M

M

M

M

5 Acknowledgements We want to thank Paulien Hogeweg and Maarten Boerlijst for ideas about how to construct mass-conserving rules for the cut-o, and Kristian Lindgren and an anonymous referee for valuable comments for the revision of the manuscript.

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[7] K. Lindgren & M. G. Nordahl, Evolutionary dynamics of spatial games, Physica D 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.

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[10] K. J. Lee, W. D. McCormick, Q. Ouyang and H. L. Swinney, Pattern formation by interacting chemical fronts, Science 261 (1993), 192{194. [11] K. J. Lee, W. D. McCormick, J. E. Pearson and H. L. Swinney, Experimental observation of self-replicating spots in a reaction-diusion system, Nature 369 (1994), 215{218. [12] M. A. Andrade, J. C. Nu~no, F. Moran, F. Montero & G. J. Mpitsos, Complex dynamics 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 evolution of primitive life: Possibilities of spatial self-structuring, in: J. L. Casti & A. Karlqvist, eds., Cooperation & con ict in general evolutionary processes , (John Wiley & Sons, New York, 1995) 15{62.

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