Research Project from the ICREA/Complex Systems Lab



Main Researchers: Andreea Munteanu, Ricard V. Solé 
Related projects: PACE

Introduction: The chemoton model was introduced by Gánti in 1971 (see review Gánti 2002) as a fundamental unit model of living systems. It consists in  three functionally dependent autocatalytic subsystems:   the    metabolic   chemical   network,    the template polymerization  and the  membrane subsystem  enclosing them  all. The correct functioning of  the chemoton lies in the precise stoichiometric  coupling of  the three subunits.  It ensures that both the  surface and the  inner components  evolve into doubling their  initial  value, leading to the subsequent division into two identical chemotons.  Besides the detailed introductory papers of Gànti, only a few studies of this model exist in the literature, presenting however contradictory conlusions. The present study aims toward a thorough survey of the chemoton's characteristics, such as replication period or optimal template length, in the parameters' space. Additionally, a comparative study between the deterministic approach and the stochastic one is performed.

08/02/2005   First public release and publication of this web page.
21/02/2005   Revised version following the suggestions of Prof. Tibor Gánti.
13/02/2007   Upload of the final article: J. Theor. Biol. 240, 434 (2006)



The chemoton as a basic living unit

Published on similar dates, the hypercycle model (Eigen and Schuster 1979) and the chemoton model are among the few simple theoretical models which tried to cross the border between living and non-living systems. Both models are fluid automaton models and are constructed from autocatalytic chemical systems. Tha particularities of the chemoton model are that it is separated from the environment by a membrane and is a supersystem of three coupled autocatalytic cycles. 

The self-reproducing metabolic  network transforms the  external nutrient into chemoton's  internal material necessary  for template replication and membrane growth.  The second subunit is  a self-replication network of a double-stranded template. The chain ends of the double-stranded templates can open due to inherent fluctuations, but with high probability the corresponding  monomers  of  the opposite strands  reconnect.   If  a sufficiently high  concentration of  monomers is present,  the bonding can  also take  place with  free  monomers, leading  to a  zipper-like opening of  the double-stranded complex and to  its replication (Figure 1). The third subunit  is a model of a  two-dimensional membrane forming a spherical enclosure of the system (Figure 2).

  Figure 1: Template replication

Table 1: The chemical reactions of the chemoton model

Figure 2: The chemoton model - adapted from Fernando&Di Paolo (2004). See also

Once the amount of monomers passes the required threshold value,  the template replication starts and its waste products  react with  the  membrane  precursors producing  real membrane  molecules. These molecules are spontaneously incorporated into  the membrane's molecules and the speed  of the process is  proportional to the  surface area, with  a proportionality coefficient  k10 (Csendes  1984). The  third   membrane  equation  in  Table  I illustrates "intuitively" this process of surface increase, and thus of volume increase. If the concentration  of  molecules increases rapidly,  the microsphere bursts when the  osmotic pressure reaches a critical  value.  On the other hand, if  the increase of the autocatalytic and membrane molecules is parallel and exponential, with a  similar  growth of  the  microsphere  surface,  the liquid  becomes diluted and  the sphere  decompresses.   Gánti argues  that in  order to resolve  the instability  by division  into two identical  spheres in osmotic equilibrium with their  environment, the chemoton must reach a state where  both the surface and the  inner components  have doubled their  initial  amount (Gánti ).  At  this  precise  moment  and  due  to  the concentration decrease,  an osmotic  vacuum develops and  the membrane sphere is elongated, with a neck  forming in the middle leading to the subsequent   division.   

The  model   explicitly   ensures  a   cycle stoichiometric  coupling of  the  autocatalytic cycles  such that the number of membrane molecules necessary for surface  doubling is equal to  the  number  of polymerization  iterations  needed  for complete replication of all template  molecules.  Thus  the  total number  of  templates  determines hereditarily  the geometric size  of the  chemoton  as well  as  its replication  time.   As  a  consequence of this  interdependece,  the statistical  halving of the  number of template molecules  leading to possible differences   in  the  number  of   templates molecules  in descendants might  introduce hereditary changes  and genetic diversity of chemoton populations.



Parametric study of the chemoton model

Within the  general approach to  chemical reaction  networks, simulations of  the chemoton model used the standard kinetic differential  equations providing the temporal evolution of  the metabolites' concentrations. More realistic simulations  are presented  in  Fernando&Di  Paolo  (2004)  where  a discrete  probabilistic  treatment  of  the  template  replication  is coupled with  a continuous  deterministic treatment of  the metabolism and membrane subunits.  However,  no complete stochastic simulation of the  chemoton   model  has   been  performed  so   far. 

The volume  is not  explicitly introduced in  the model  and therefore neither in  the equations system describing the temporal evolution of the concentrations.  However,  the concentrations depend intrinsically on the volume.   Thus, in order to obtain their  correct value at each step,  one must  rescale them  to the  new volume  resulting  from the increase  of surface  S.   This  implies that,  once  the template replication and  also surface growth  start, after each step  i, one must multiply   the resultant   concentrations   by   the   factor f=(Si-1/Si)3/2,  with the  resulting values  being used as input conditions for  the next integration step. In  addition, one can notice that  the division implies a factor  f=sqrt(2) and therefore the  concentrations  of the  substrates  increase  at division.   This rescaling is  not performed in  Csendes (1984), while  in Fernando&Di Paolo (2004) it  is discussed, but not included  in the simulation, as one can  see from  their Fig.  2 that the concentrations  decrease at division. This introduces a serios warning with respect to the results they obtain  and their interpretation, fact that motivated our parametric study of the chemoton model in the frameworks of step-by-step volume rescaling of the metabolites' concentration. The results are presented in the figures below and differ significantly from those of previous works. We stress the fact that we have used for simulations the system of ordinary differential equations from Fernando & Di Paolo (2004), which differs only at notation level from the system of Csendes (1984). No additional changes have been made to the system, but the present approach consists only in the updating of the concentrations after each integration step as a consequence of the volume increase.

    Strong replication condition (.C)

The replication initiates and propagates ONLY when monomers' concentration exceeds the threshold, that is V'>V*.  In other words, the rate constants associated to the initiation and propagation of the template replication are nonzero ONLY when V'>V*.  Division occurs when the surface doubles its initial value.
Figure 3: Dependence of the replication time on the monomers' concentration threshold for the case of strong replication condition with N=10 and X=100.
Figure 4: Dependence of the replication time on the monomers' concentration threshold for the case of strong replication condition with N=5 and X=10.

    Weak replication condition (.C)

The reaction rate of the replication initiatiation becomes nonzero when monomers' concentration exceeds the threshold, V'>V*, while the constant rate of the replication propagation maintains always a nonzero value.  That is, once the replication has initiated, it proceeds without obstruction characterized by the reaction rate associated to the propagation step.

Figure 5: Dependence of the replication time on the monomers' concentration threshold for the case of weak replication condition with N=10 and X=100.

Figure 6: Dependence of the replication time on the monomers' concentration threshold for the case of weak replication condition with N=5 and X=10.

Figure 7:
Temporal evolution of the replication period for N=10, X=100:
(upper panel) V*=100 and (lower panel) V*=55. See Figure 5.

    Stochastic simulation (.C)

We have also used a stochastic approach (Gillespie 1976) for simulating a stochastic version of the chemoton model. Even if the stochastic equivalent of the ordinary differential equation system of the deterministic approach is the chemical master equation, the algorithm proposed by Gillespie (1976, 1977) is not a numerical solution of the master equation, but an exact procedure to simulate the stochastic processes that master equation describes analytically. The algorithm consists in providing the answer to two questions: when will the next reaction occur and what reaction will it be. Once the next reaction is identified, the population level of the associated reactants and products are changed according to the stochiometric coefficients of the ongoing chemical reaction. The very-preliminary results are presented below, including the timeseries of the number of molecules and the temporal evolution of the replication period for two cases.  Our present work is centered on comparing the results of the two approaches, the deterministic and the stochastic, after having established an equivalence between the two approaches in terms of initial and parametric conditions.

Other particularity entailed by the stochastic approach is the fact that the stochastic rates depend on the cell volume and on the degree of the chemical reaction (Volfson et al. 2004, Wolkenhauer et al. 2004), thus they must be updated as the volume of the cell increases. Additionally, the condition for cell division must be considerd in the context of the stoichiometric coupling of the membrane and template syste, as discussed in the introduction: the template replication must provide sufficient molecules for the doubling of the membrane.

Figure 8:Time series of the number of metabolites molecules in the stochastic chemoton model. The characteristics of the simulation are: 200 Monte Carlo trajectories which have been averaged to yield these time series. The time interval was from 0 to 1, divided into 10000 intervals for the averaging of the Monte Carlo experiment.  The single strand polymer has 25 monomers  and there are 25 initial double-stranded polymers.  The other parameters are: V*=600, Y=100,X=10000. The initial conditions are: A1=3000, A2=1800, A3=1900, A4=1000, A5=450, V'=450, R=T=0, T'=1700, T*=1400, S=1000. 

Figure 9: Temporal evolution of the replication period for two stochastic simulations of the chemoton model. The two cases have been chosen such that the resultant surface, given by 2n × pV2n, is the same, S~1000. See the introduction for the stoichiometrical coupling between membrane and template.




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