Phenotypic Diversity and Chaos in a Minimal Cell Model

ARTICLE IN PRESS

Journal of Theoretical Biology 240 (2006) 434–442 www.elsevier.com/locate/yjtbi

Phenotypic diversity and chaos in a minimal cell model

Andreea Munteanua,Ã, Ricard V. Sole´a,b

aICREA-Complex Systems Lab, Universitat Pompeu Fabra (GRIB), Dr. Aiguader 80, 08003 Barcelona, Spain bSanta Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA

Received 4 April 2005; received in revised form 10 October 2005; accepted 12 October 2005 Available online 5 December 2005

Abstract

Gańti’s chemoton model (Gańti, T., 2002. On the early evolution of biological periodicity. Cell. Biol. Int. 26, 729) is considered as an iconic example of a minimal protocell including three key subsystems: membrane, metabolism and information. The three subsystems are connected through stoichiometrical coupling which ensures the existence of a replication cycle for the chemoton. Our detailed exploration of a version of this model indicates that it displays a wide range of complex dynamics, from regularity to chaos. Here, we report the presence of a very rich set of dynamical patterns potentially displayed by a protocell as described by this implementation of a chemoton-like model. The implications for early cellular evolution and synthesis of artificial cells are discussed. r 2005 Elsevier Ltd. All rights reserved.

Keywords: Protocell; Origins of life; Chemoton; Cellular networks; Chaos

1. Introduction pursued in the experiments. For the latter, no intentional implications concerning the birth of primordial cells are Cells are the basic building blocks of all life on our sought and its proper functioning is based on present planet. They are the minimal systems able to self-replicate biogenic and/or abiogenic chemistry. in a regular manner and evolve through changes in genetic What are the minimal building blocks of a self- information. Understanding how cellular life emerged is replicating protocell? This is a fundamental question of one of the most challenging problems in life sciences. The both theoretical and practical relevance. At its fundamental early origin of cellular life certainly could not have level, it becomes linked with early conjectures about what is consisted of a complex membrane with a long genome required in order to have self-replication. Pioneering work and intricate metabolic pathways. Instead, it is believed to by von Neumann revealed that a machine able to replicate have been the result of coupled, simple chemical reactions itself needed a few basic subsystems which can be easily that allowed evolution to act on replicating nanostructures. identified with the key components found in real cells in Theoretically, this hypothesis is the basis of the most our current biosphere (von Neumann, 1966; Sipper and relevant modelling approaches available in the literature Reggia, 2001; Freitas and Merkle, 2004). In this context, (see Segreánd Lancet, 2004, for a review). Experimentally, reproduction is a system-property of the total ensemble of the moment of the synthesis of an artificial nanocell based components (Emmeche, 1994). Cells are living systems with on this theory seems to be closer than ever (Bartel and a kinematic structure close to von Neumann’s view. More Unrau, 1999; Szostak et al., 2001; Luisi, 2002; Pohorille precisely, all cells consist of three basic components of and Deamer, 2002; Rasmussen et al., 2004). However, cellular networks: metabolism, information and membrane there is a conceptual difference between an origin-of-life (Gańti, 1975; Alberts et al., 2002). These components allow protocell and an artificial protocell whose synthesis is building of new constituents and eventually self-replication. Modern cells use template polymerization to replicate ÃCorresponding author. Tel.: +34 935422834; fax: +34 932213237. their genetic information, which is stored in the same linear E-mail address: [email protected] (A. Munteanu). chemical code (the DNA), thus being a sequence-based

0022-5193/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2005.10.013 ARTICLE IN PRESS A. Munteanu, R.V. Sole´ / Journal of Theoretical Biology 240 (2006) 434–442 435 information system, with the base-pairing mechanism allowing for unlimited heredity (Szathma´ry and Maynard Smith, 1997). In order to sustain life, cells have to be in a none- qulibrium state, taking free energy from the environment and using raw materials to drive the chemical reactions inside the cell body. In the same spirit, simple minimal protocells should be able to sustain a reliable replication cycle, properly coupling the components in a way that they support each other’s proper functioning, with the driving forces being chemical energy or light. Clearly, once a threshold of chemical complexity is reached, nonlinear phenomena are likely to develop. In this context, it is relevant to ask what types of dynamical patterns are expected to occur when a given set of components is used to build a protocell: are stable replication cycles a generic feature of a coupled set of cellular components? Can nonlinearities jeopardize the presence of a stable cell cycle and potential heredity? Is the complexity of templates a key ingredient in sustaining reliable replication? In this paper, we want to address these questions by means of a numerical study of a minimal cell model in the framework of artificial protocell design. Thus, our inves- Fig. 1. The chemoton: the metabolic unit, the template (information) tigation is directed neither toward extracting explicit subunit and the membrane subsystem. The dots in the template subsystem indicate the iterative development of the template replication from pV V 1 implications related to the origins-of-life studies nor n to pV nV n1. Adapted from Gańti (2002). toward modeling present-day cell functioning. However, it is natural that related implications be the by-products of such an investigation. consumption in the copying process, depending on the Our simulations concern a version of the minimal cell number and the length of the template molecules. As the model introduced by Gańti (1975) and referred to as the templates are doubled, the number of monomers should chemoton model. A few distinct implementations of the also double in order to ensure the functioning of the chemoton’s chemical reactions network exist in the automata, a condition fulfilled by the autocatalytic nature literature (Be´ke´s, 1975; Csendes, 1984; Fernando and Di of the metabolic subunit. Paolo, 2004) and in the same spirit of these studies, ours is The third subunit is a model of a two-dimensional not aimed at characterizing the chemoton model itself, but membrane enclosing the metabolic and the template a chemoton-like one. We report here on very complex polymerization subsystems. Once the concentration of dynamical patterns, thus suggesting a wealth of potential monomers passes the required threshold value, the dynamical behaviors exhibited by protocells. We also template replication starts and its by-products react with comment on the differences between the studies existent the actual membrane precursors producing real membrane in the literature. molecules which are spontaneously incorporated in the membrane. Thus the membrane’s surface increases and 2. Chemoton model implicitly the chemoton’s volume (for a recent review on replicating vesicles, see Hanczyc and Szostak, 2004). The chemoton model consists of three stoichiometrically The correct functioning of the chemoton lies in the coupled autocatalytic subsystems: the metabolic chemical precise stoichiometric coupling of the three subunits, more network, the template polymerization and the membrane precisely the coordination between the accumulation of subsystem enclosing them all (Fig. 1). The self-reproducing molecules and the surface increase in order to achieve an metabolic network transforms the external nutrients into equilibrium of the osmotic pressure relative to the the chemoton’s internal material necessary for template environment. If the concentration of molecules increases replication and membrane growth (see Gańti, 2002, 2003). rapidly, the microsphere bursts when the osmotic pressure The second subunit consists in the self-replication cycle of a reaches a critical value. On the other hand, if the increase homopolymer whose by-product is a specific precursor of the cytosolic and membrane molecules (and thus molecule necessary for membrane growth. The template microsphere surface) is parallel and exponential, the liquid replication has a program-controlling role consisting in the becomes diluted and the sphere decompresses. Gańti (2002) fact that the monomers would start to polymerize only argues that the latter instability is solved by the division when they have reached a certain threshold concentration. into two identical spheres in osmotic equilibrium with their The control implies the regulation of the monomers’ environment. At this precise moment, it is implicitly ARTICLE IN PRESS 436 A. Munteanu, R.V. Sole´ / Journal of Theoretical Biology 240 (2006) 434–442 assumed that, due to the concentration decay, an osmotic treatment of the template replication is coupled with a vacuum develops and the membrane sphere is elongated, continuous deterministic treatment of the metabolism and with a neck forming in the middle leading to the membrane subunits. However, no complete stochastic subsequent division. simulation of the chemoton model has been performed so As defined in chemoton theory, the model is based on far. A stochastic implementation of the chemoton’s stoichiometric coupling of the autocatalytic cycles such chemical reactions providing the temporal evolution of that the number of membrane molecules necessary for the number of molecules would constitute a more faithful surface doubling is equal to the number of polymerization simulation of the chemoton as introduced by Gańti (1975), iterations needed for complete replication of all template at least as far as the template replication is concerned. This molecules. For example, assuming a sphere of 10 million remark is related to the fact that, in Ga` nti’s original molecules membrane and 105 simple strand templates, in theoretical model of the chemoton, all template macro- order for the chemoton to double its surface and template molecules are replicated only once per replication cycle, as molecules, the strands must be composed of 102 nucleo- occurs with the DNA macromolecule in modern cells. This tides (Gańti, 2003). As defined here, the template can be feature is essential to the stoichiometric coupling of the regarded as an example of non-enzymatic informational subsystems. Thus, if all the templates have been replicated, replicating system, where the information resides not in the but their replication has produced less membrane pre- sequence (as in RNA or DNA macromolecules), but in the cursors as necessary for membrane’s surface duplication, length of the polymers. A more appropriate informational then the chemoton would ultimately burst as a conse- template would consist of two (or more) monomers, quence of accumulation of cytosolic molecules. allowing for infinite information capacity (Gańti, 1975, As the subsystems of the chemoton model must be 2003), but this extension implies challenging and demand- stoichiometrically coupled, the initial conditions must ing simulations. ensure that the doubling of one subsystem’s components implies the doubling of the other subsystems’ components. 3. The kinetics This condition prevents the breaking of the stoichiometric coupling and possibly the replication cycle. However, in Studying the chemoton theory, one can remark that view of simulating a model of a protocell, we saw no reason neither of the works on simulated models mentioned in why the replication of a template molecule should be Section 1 is equivalent to the original chemoton model as limited to one iteration per replication cycle and thus introduced by Gańti (1975). Instead, they all include allowed for our simulations multiple template replications, slightly modified assumptions mainly required by the if needed. We are aware that this produces a significant simulation process itself. Therefore, they can be regarded departure from the original chemoton model, but we have as chemoton-based models and not simulations of the focused on the implications for a simpler cell as is the case chemoton theory, as they are not faithful implementations of the bacterial plasmids (small rings of DNA) with high of it. From this point of view, our study joins the former number of copies per cell which carry non-essential genetic approach. material, but provide certain fitness properties (e.g. Within the general approach to chemical reaction resistance to antibiotics). Additionally, we remark that networks, simulations of the chemoton-based models used the studies from the literature on the chemoton model the standard kinetic differential equations providing the discussed before appear to follow the latter approach temporal evolution of the metabolites’ concentrations. rather than the former. More realistic simulations are presented by Fernando and Table 1 includes the chemical reactions of the chemoton Di Paolo (2004, Model II), where a discrete probabilistic model, where ki is the rate constant. The members of the

Table 1 Chemical reactions of the chemoton model. Initial membrane T½m contains a number of m membrane molecules

Metabolism Membrane Template

k k k 21 0 8 26 A1 þ X A2 T ! T Replication initiation: pV n þ V pV nV 1 þ R k0 k0 1 k 6 T þ R 29 T 2k2 k0 k7 A2 A2 þ Y 9 Replication propagation: pV nV i þ V ! pV nV iþ1 þ R; k0 2 k T þ T½mþi ! T ½mþiþ1 2k3 A3 A4 þ V 0 i m 1 n 1 equations with 1pipn 1 k0 o o 3 2k4 0 A4 A5 þ T pV nV n ! pV n þ pV n k0 4 k5 A5 2 A1 þ A1 k0 5 ARTICLE IN PRESS A. Munteanu, R.V. Sole´ / Journal of Theoretical Biology 240 (2006) 434–442 437

Table 2 Kinetic differential equations of the chemoton model

_ 0 0 ½A1¼2ðk5½A5k5½A1½A1Þ k1½A1½Xþk1½A2 _ 0 0 ½A2¼k1½A1½Xk1½A2k2½A2þk2½A3½Y _ 0 0 ½A3¼k2½A2k2½A3½Yk3½A3þk3½A4½V _ 0 0 0 ½A4¼k3½A3k3½A4½Vk4½A4þk4½A5½T _ 0 0 0 ½A5¼k4½A4k4½A5½T k5½A5þk5½A1½A1 2Pn1 _ 0 0 ½V¼k3½A3k3½A4½Vþk6½pV 2nV 1½Rk6½pV 2n½V k7½pV 2nV i½V i¼1 2Pn1 _ 0 0 ½R¼k6½pV 2n½Vk6½pV 2nV 1½Rþk9½Tk9½T ½Rþ k7½pV 2nV i½V i¼1 _0 0 0 0 ½T ¼k4½A4k4½A5½T k8½T _ 0 0 ½T ¼k8½T k9½T ½Rþk9½T _ 0 ½T¼k9½T ½Rk9½Tk½TS S_ ¼ k½TS _ 0 ½pV n¼2k7½pV nV n1½Vþk6½pV nV 1½Rk6½pV n½V _ 0 ½pV nV 1¼k6½pV n½Vk6½pV nV 1½Rk7½pV nV 1½V _ ½pV nV iþ1¼k7½pV nV i½Vk7½pV nV iþ1½V with 1pion 1

metabolic autocatalytic cycle are denoted as A1; ...; A5. In a very general framework, several types of linear These elements are produced from A1 through a closed polymeric macromolecules can be envisioned as templates. cycle involving five reactions: the consumption of the The theory (Gańti, 2003) considers a double-stranded external nutrient X, the formation of waste Y, of template template formed of two simple strands of length n whose and membrane precursors, V and T 0, respectively, and the replication occurs in a zipper-like manner. In this case, the closing step of the autocatalytic cycle producing two A1 template’s ends can open due to inherent fluctuations and if (see Table 1). It is assumed that a large quantity of nutrient a sufficiently high concentration of monomers is present X is available and that the waste Y is eliminated through (½VX½V), the bonding with free monomers can take the membrane in such manner that the concentration of X place, initiating the replication. However, as implemented and Y can be considered as constant, denoted as ½X and by Csendes (1984) and Be´ke´s (1975), the template appears ½Y. One can note that at least one of the metabolism to be a simple strand of length n, while it remains unclear in molecules must have nonnull concentration at an initial Fernando and Di Paolo (2004) due to distinct notations. moment in order to ensure the metabolism functioning and We have considered this approach in our simulations, but thus chemoton’s survival. In accordance with these one can notice that under the present implementation, the chemical reactions, the kinetic differential equations behavior resultant from an n-single-strand template is describing the temporal evolution of the chemical system equivalent to the behavior of a n=2-double-strand template, are included in Table 2, where the upper dot implies a for even values of n. As additional discussion and as derivative with respect to time and the quantities in mentioned by Fernando and Di Paolo (2004), we found no brackets denote concentrations. meaningful physical interpretation for Csendes’ variables As a particularity, the k8-equation was considered of template replication. Furthermore, we found no necessary by Csendes (1984) ‘‘since otherwise in the case justification for the lack of summing over intermediate 0 0 of certain ki and ki the accumulated T might make the states of replication in Be´ke´s (1975). autocatalytic cycle slow down still before R, originated The reaction rate constant of the initiation for the from the pV n synthesis, might extract it.’’ However, we template replication is k6 ¼ 0 for ½Vo½V . The template have not verified this statement. replication initiates once the condition ½VX½V is The membrane subsystem obtains the precursor mole- fulfilled. At this moment, k6 acquires a nonzero value cules from the other two subunits and transforms them into and the surface, and implicitly the volume, starts to grow. a proper membrane molecule T (second column of Table When the surface doubles its initial value, the system is 1). These molecules are spontaneously incorporated into assumed to divide, a process which translates into the the membrane’s molecules and the speed of the process is halving of its surface and all its metabolites, and the system proportional to the membrane surface area S (itself of differential equations resumes integration with the new proportional with the number of constituent T molecules), values of the metabolites’ concentrations. Thus, we assume with a proportionality coefficient k, as considered also by that the descendants share the inner material equally and Csendes (1984); Fernando and Di Paolo (2004) and in a subsequently we follow the evolution of one descendant slightly different view, by Be´ke´s (1975). One can see thus only. that k is not a reaction rate constant as defined in the The volume is not explicitly introduced in the model and standard chemical kinetics. therefore neither in the equations describing the temporal ARTICLE IN PRESS 438 A. Munteanu, R.V. Sole´ / Journal of Theoretical Biology 240 (2006) 434–442 evolution of the concentrations. However, the concentra- 0.30 tions depend intrinsically on the volume. Thus, in order to obtain their correct value at each step, one must rescale 0.25 them to the new volume denoted by Q resulting from the increase of surface S. This implies that, once the template 0.20 replication and also surface growth start, after each step i, one must multiply the resultant concentrations by the 0.15 factor Cycle duration 0.10 3=2 Qi1 Si1 f ¼ ¼ , (1) 0.05 Qi Si with the resulting values being used as input conditions for 0.00 the next integration step. Since the concentrations are given 0204060 80 100 120 in moles per volume, the division, as defined here, implies [V]* the halving of both moles and surface. This entails the Fig. 2. Dependence of the replication time on the monomers threshold for multiplication of thep concentrationsffiffiffi at division with a the case of weak replication condition with n ¼ 10 and ½X¼100. factor div ¼ 0:5 f ¼ 2, with f ¼ 23=2. It is a consequence, on one hand, of the halving in moles and, on the other hand, of the volume rescaling factor as Sb ¼ 2Sa, with b and a referring to the values before and and template length, n. The simulations revealed a highly after division, respectively. One can note that in this nonlinear dependence on ½V , with multiple self-replica- approach the concentrations of the substrates increase at tion periods for a given set of parameters. An example of splitting. the irregular dependence of the replication period on the Even if volume rescaling is discussed by both Csendes template threshold can be seen in Fig. 2. Additional, we (1984) and Fernando and Di Paolo (2004), it is apparent show in Fig. 3 an example of the temporal evolution from their that the concentrations decrease at division, of the metabolites’ concentrations illustrating several being thus inconsistent with their line of argument. This replication cycles. We also considered relevant to include introduces a warning with respect to the results they obtain in Fig. 4 an illustration of the temporal evolution of the and their interpretation. Their main results concern the replication time for a regular and an irregular case of dependence of the replication time on the main parameters replication. of the model, such as monomers threshold ½V, template As for the dependence of the replication period on the length n or nutrient concentration ½X. Interesting enough, template length, we present our results in Fig. 5, where we Csendes (1984) obtains that longer templates have lower have used the same parameters which lead Fernando and replication times at low levels of nutrient concentration, Di Paolo (2004) to their results. The differences between implying an increased fitness in harsher environments. our results and the ones existent in the literature suggest Fernando and Di Paolo (2004) disprove this result, but that they are a consequence of the way the chemoton discuss the kinetic conditions that could lead longer- responds to the increase of volume. templates into faster replication. From Fig. 5a, b one can note that the scarcity of nutrients (low values of ½X) has a significant impact on the dynamics, leading to a more irregular replication at low 4. Results template lengths. In other words, nutrients’ scarcity pushes the existence of uniperiodic cycles towards longer template Using the set of from Table 2 (as Fernando and Di lengths. An even stronger effect in this direction is obtained Paolo, 2004) and the rescaling of concentrations from Eq. by increasing the rate constant of the template replication 1, we can analyze the expected dynamical patterns and reaction ðk7Þ (Fig. 5c)—as performed in Fernando and Di transitions to be observed in a protocell as defined above. Paolo (2004)—and based on a more realistic approach in The simulations were based on a embedded Runge–Kutta which the bottleneck step for the replication is the Prince–Dormand (8,9) method with adaptive step-size initiation, with the propagation occurring on a much control. The initial conditions of the dynamic system as shorter time scale. well as the parameters’ values are identical to the ones used Fernando and Di Paolo (2004) obtained an optimum in Fernando and Di Paolo (2004). value for the template length showing longer replication time for either higher or lower template length compared to 4.1. Replication cycle that optimum value. However, we have to remark from their Fig. 5b that the difference in the shortest and longest We centered our study on the dependence of the replication time is of the order of the integration time step, replication time on several ‘‘genotypic’’ characteristics of extremely diminishing the significance of an ‘‘optimum’’ the model, such as monomers’ concentration threshold ½V value. No optimum value of the template length is ARTICLE IN PRESS A. Munteanu, R.V. Sole´ / Journal of Theoretical Biology 240 (2006) 434–442 439

2.5 60 2 50 V A1 1.5 40 4 0.4 3.5

3 R 0.2 A2 2.5 4 0.0 3.5 21 18 3 T’ A3 2.5 15 4 12 3.5 30 3 25 A4 2.5 T* 20 2.0 20

S 1.5 A5 16 12 1.0 10.0 10.2 10.4 10.6 10.8 11.0 11.2 11.4 10.0 10.5 11.0 11.5 time time

Fig. 3. Time series of metabolites concentration for: ½X¼100, n ¼ 10, ½V ¼ 55. See Fig. 7.

0.30 0.20 (a) 0.20 0.16

0.10 0.12

Cycle duration 0.08 0.20 Cycle duration 0.15 0.04

0.10 (b) 1.20 0.05 Cycle duration 0.80 0 50 100 150 200 250 300 350 400

Divisions 0.40 Cycle duration Fig. 4. Temporal evolution of the replication period for n ¼ 10, ½X¼100: (upper panel) ½V ¼ 100 and (lower panel) ½V ¼ 55. See (c) Fig. 7. 0.25 0.20 0.15 apparent from our results, not even considering an average 0.10 period for every template length n. Cycle duration 0.05 0.00 4.2. Dependence on the initial conditions 10 20 30 40 50 60 70 80 90 100 110 120 130 n

From the results mentioned so far, one can notice that a Fig. 5. Dependence of the replication time on the template length for the small change in the parameters’ values can lead to a weak replication condition for the cases of (a) ½X¼100, ½V ¼ 35 and signiﬁcantly different behavior. Such an example is the k7 ¼ 10; (b) ½X¼1, ½V ¼ 35 and k7 ¼ 10; (c) ½X¼100, ½V ¼ 35 and k ¼ 100. dynamics around ½V 50 in Fig. 2. This remark 7 motivated us to investigate also the dependence of the resultant dynamics on the initial conditions, that is on the divisions. In order to identify the type of replication cycle, initial concentrations of the metabolites. Our simulations we have introduced a measure of the difference between revealed a strong dependence, with the type of replication two consecutive descendants deﬁned as their Euclidean cycle (uniperiodic, biperiodic or multi-periodic) being a distance in the concentration space consequence not only of the parameters’ values, but also of "# 1=2 the initial conditions. 10Xþn d ðc c Þ2 , (2) We have studied several sets of parameters’ values by mi di i¼1 considering initial conditions with random values lower birth than 21.0, with the exception of the templates’ variables where the cm and cd refer to the concentrations of the where the upper random value was 0.01. For each set, 100 ‘‘mother’’ and the ‘‘daughter’’, respectively, calculated at models of random initial conditions were integrated for 200 birth. It is a simple indicator of the replication type, for ARTICLE IN PRESS 440 A. Munteanu, R.V. Sole´ / Journal of Theoretical Biology 240 (2006) 434–442

50 Lancet (1999). One can note from Fig. 5 that increasing the 40 template length (and thus ‘‘genome’’ complexity) leads to a 30

δ reduction in the variance associated to the cycle periodicity. 20 Thus, an increase in the complexity of template molecules 10 40 might be able to produce a uniperiodic replication cycle. 30

δ 20 10 5. Division considerations 40 30 It has been assumed so far that the protocell system

δ 20 divides once the inner compounds, including the mem- 10 brane, are doubled. As discussed above, this condition 0 implies that instead of a factor 2 in the increase of volume, 0 1 2345 6 7 8910 a factor of 23=2 is obtained, leading to an osmotic time disequilibrium which is supposed to be the trigger of Fig. 6. Evolution of the mother-daughter distance in the concentration division. In this context, we imagined two plausible space for different random initial conditions. The cases selected division schemes illustrated in Fig. 7. For the results correspond to ½X¼100 and: (upper panel) ½V ¼ 60, n ¼ 10, k7 ¼ 10; discussed until now, we have used the scheme from panel (middle panel) ½V ¼ 35, n ¼ 46, k ¼ 100; (lower panel) ½V ¼ 35, 7 (a): a protocell of initial surface S and volume Q grows n ¼ 60, k7 ¼ 100. The filled-circles timeseries correspond to the initial 0 0 condition used throughout this study. spherically until its surface is doubled with the volume being greater than 2Q0, followed by a rapid deformation and subsequent division into two spheres equal to the example: a value of d ¼ 0 implies a uniperiodic replication primordial one. This situation corresponds, for example, to cycle, while a constant nonzero value indicates a biperiodic experimental division of vesicles when subjected to strong cycle. For clarity, a small representative selection is shown shear stresses, either by extrusion (being forced through in Fig. 6. It illustrates the temporal evolution of the d- small pores under pressure) or sonication (Hanczyc and difference. With filled-circles it is represented the case Szostak, 2004). The deformation phase can be taken into employed in the simulations from the previous section, account continuously if the protocell’s growth takes place while with triangles, we draw the cases which have reached through a gradual departure from a spherical shape, as a uniperiodic replication cycle ðd ¼ 0Þ. For the upper-panel illustrated in Fig. 7b, such that the surface doubling set of parameters (see also Fig. 2), the replication cycle of coincides with the volume doubling. In order to simulate the 100 chemotons with random initial conditions followed such a case, a nonspherical volume-surfacepffiffiffi dependence was 3=2 one of the two types illustrated here: an almost biperiodic employed, i.e. V / S ð1 þðS S0Þð1= 2 1Þ=S0Þ. For cycle or an irregular one. For the middle-panel set of completeness, we represent in Fig. 8 the bifurcation parameters, the 100 chemotons tended either to the d ¼ 0 diagram as in Fig. 5, but following the division scheme case or to the d ¼ 23 case, with the trajectory represented from Fig. 7b. One can remark that for the latter approach here with crosses acting as a separatrix. With stars and no rescaling at division is necessary and thus the division squares we illustrated examples of intermediate cases. implies no discontinuous transition in concentration as it Interesting enough, for the lower-panels set of parameters, does in the former case. all the models presented an irregular replication cycle (filled-circles) but one case, which tended to d ¼ 0 (triangles). From the nonlinear dynamics theory and these results concerning the dependence on the initial conditions, we conclude that there exist basins of attraction associated to the attractor states identified, i.e. uniperiodic, biperiodic, (a) multiperiodic replication cycles. In this context, it is tempting to address the question whether there exists at least one initial condition for every set of parameters leading to a uniperiodic replication cycle. Our future investigations will be directed also toward answering this (b) question. This issue is relevant imagining that a unique replication cycle might be considered advantageous in an Fig. 7. Plausible division scenarios for the present model: (a) the protocell is supposed to grow spherically until its surface is doubled, followed by the evolutionary context in which the compositional informa- division into two spheres, (b) the protocell’s growth takes place through a tion—the concentration of metabolites—must be reliably gradual departure from a spherical shape, doubling both the surface and transmitted to the descendants, as suggested by Segreánd the volume prior to division. ARTICLE IN PRESS A. Munteanu, R.V. Sole´ / Journal of Theoretical Biology 240 (2006) 434–442 441

0.14 In other words, the uni-periodic cases imply that the 0.13 accumulation and consumption of monomers are ba- 0.12 lanced—satisfying the stoichiometrical relation. Thus, if 0.11 one assumes a scenario in which the heredity of composi- 0.10 tional information is important, than the cases which lead

Cycle duration 0.09 to this balance must be identified. On the other hand, we 0.08 10 20 30 40 50 60 70 80 90 100 110 120 130 expected that the replication and division affect the (a) n replication dynamics, leading to a smoother or a rougher 0.16 replication dynamics, and our results support such a 0.14 hypothesis. Contrary to the existent studies of simulated 0.12 models based on the chemoton theory and performed using 0.10 the deterministic reaction rate equations (Be´ke´s, 1975; Csendes, 1984; Fernando and Di Paolo, 2004), our study of 0.08 Cycle duration the parameters’ space (template length and replication 0.06 0204060 80 100 120 threshold) reveals parameters’ ranges yielding not only a (b) [V]* unique replication cycle, but also ranges for which various replication periods exist. 50 Our investigations performed so far did not reveal 40 explicit cause of the variability in the replication cycle 30 observed from the results. However, we can express an δ 20 educated guess based on similar behaviour resultant from a 10 population dynamics model. These cases are a conse- 0 quences of an unbalanced consumption-production of 0 5 10 15 20 25 30 35 40 (c) time monomers, but also may be attributed to the intrinsic reset of the division process. Gamarra and Sole´(2002) Fig. 8. The case of gradual deformation: (a) the case from Fig. 7c, (b) the analytically prove that their population dynamics system case from Fig. 7, using also the same initial conditions, (c) The evolution described by continuous plus discrete equations population of the mother-daughter distance for the case ½V ¼ 100. generation behaves as a discrete map producing the well- known period doubling route to chaos. In the case of the 6. Discussion present model, the reset produced by the division interrupts the intrinsic exponential growth of the system and yields As discussed in the introduction, the chemoton model the discrete temporal evolution of the replication cycle. The (Gańti, 1975) is an important step toward an integrated chemoton-like dynamics is considerably more complex simple model of a protocell capable of performing the than the model of Gamarra & Soleánd the former case is essential functions of a living system as we know it today. analytically intractable in the spirit of the latter approach. It is the simplest existent model including the three However, we conjecture that the variability observed in the stoichiometrically-coupled processing units: the metabo- replication cycle is also related to this reset procedure. lism, the membrane compartment and the ‘‘genetic The presence of these rich dynamical patterns might information’’. The genetic macromolecule is a linear have nontrivial implications. Just looking at the two template consisting of a certain number of monomers, with parameters explored here, selection towards regular cell the number of monomers being intrinsically linked with the cycles seems to be possible provided that initiation of growth of the chemoton system as the by-products of template replication is efficient enough: low thresholds ½V template replication constitute the membrane precursor- would be searched for. On the other hand, increasing molecules. In modern cells, growth and division—the template complexity might have more benefits than costs: if replication cycle—are coordinated by internal feedback selection towards fast-replicating cells is at work, it might processes ensuring the doubling of all cell’s constituents also require a reduction of fluctuations in the replication prior to division. A similar coordinated growth-and-division cycle, which our results suggest that occurs for larger process occurs in the chemoton model as a consequence of information carriers. The importance of reduced fluctua- the existence of a replication threshold for the genetic tions can be envisioned in the context of the ‘‘composi- subsystem. More precisely, the template replication initiates tional information’’. In this framework, it is crucial that once the monomers’ concentration has reached a critical any useful configuration of the metabolites concentrations value, followed by an unhindered replication propagation. be transmitted accurately to the descendants. In other Our search regarded the relation between the genotype words, the replication cycle should be uniperiodic. On the (parameters values) and a phenotypic feature (replication other hand, it has to be mentioned that the uni-periodic time), with the phenotypic feature being affected primarily case is indeed the behavior expected from an exact by the tight interdependence between the accumulation and implementation of the original chemoton model, as a the consumption of monomers in the replication process. consequence of the stoichiometrical coupling. ARTICLE IN PRESS 442 A. Munteanu, R.V. Sole´ / Journal of Theoretical Biology 240 (2006) 434–442

The existence of irregular replication cycles is relevant be drawn from our study. This limitation motivates us to especially when considering their absence in the previous direct out future investigations toward constructing and studies. This behavior may suggest an intrinsic variability subsequently investigating models inspired from feasible of the replication cycle, if the variability is related to the experimental scenario of protocell formation. heredity of compositional information. In this context, we remark that small changes in the concentration levels may Acknowledgements lead to drastic changes in the phenotypic characteristics of The authors thank Harold Fellermann, Javier Macia, the chemoton, the most important being the replication Josep Sardanyes and Sergi Valverde for useful Venetian period. In the advent of simulating a population of such discussions around protocells and life. We also thank models in order to identify selection effects and population Chrisantha Fernando, Steen Rasmussen, John McCaskill, dynamics, the diversity of replication cycles, and implicitly Norman Packard and Mark Bedau for useful comments. of ‘‘phenotypes’’ might prove to have unpredictable This work has been supported by EU PACE grant within consequences on the population dynamics. This uncer- the 6th Framework Program under contract FP6-002035 tainty can be resolved only by the simulation itself, and we (Programmable Artificial Cell Evolution), by MCyT grant consider this issue as our future investigation. FIS2004-05422 and by the Santa Fe Institute. It has to be remarked that the ‘‘genome’’ as considered in the simulated models based on the chemoton by Be´ke´s References (1975),Csendes (1984) and Fernando and Di Paolo (2004, Model I) is not a real template as only a single type of Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., Walter, P., 2002. Molecular Biology of the Cell. Garland, New York. monomers is contemplated and thus no information can be Bartel, D., Unrau, P., 1999. Constructing an RNA world. Trends Cell. stored in such a macromolecule. In other words, the Biol. 9, M9. template polymers in this approach do not and cannot Be´ke´s, F., 1975. Simulation of kinetics of proliferating chemical systems. contain a sequence-based ‘‘genetic’’ information, instead Biosystems 7, 189. only the number of their constituent monomers can be Csendes, T., 1984. A simulation study on the chemoton. Kybernetes 13, 79. Emmeche, C., 1994. The Garden in the Machine. Princeton University regarded as information carrier as long as this feature affects Press, Princeton, NJ. directly the replication time of the chemical system. Gańti Fernando, C., Di Paolo, E., 2004. The chemoton: a model for the origin of (1975) dwelled upon a concept of a genetic subsystem based long rna templates. 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