Main Researchers: Andreea Munteanu, Ricard V.
Solé


The
chemoton as a basic living unit
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 k_{10}
(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.

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=(S_{i1}/S_{i})^{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 stepbystep 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.
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.
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 verypreliminary 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.

References T. Csendes A simulation study on the chemoton Kybernetes, 13 (2): 79, 1984. M. Eigen and P. Schuster, The Hypercycle: A Principle of Natural SelfOrganization Berlin: Springer 1979. C. Fernando and Di E.A., Paolo The Chemoton: A model for the origin of long RNA templates. Proceedings of the Ninth International Conference on the Simulation and Synthesis of Living Systems, ALIFE'9 Boston, September 12th15th, MIT Press 2004. T. Gánti. On the early evolution of biological periodicity. Cell. Biol. Int. , 26 (8): 729, 2002. T. Gánti. The principles of life Oxford University Press 2003. D. Gillespie A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions J.Comp. Phys., 22: 403, 1976 D. Gillespie, Exact Stochastic Simulation of Coupled Chemical Reactions J. Phys. Chem., 81:2340, 1977. Lu, T. Volfson, D. Tsimring, L. Hasty Cellular growth and division in the Gillespie algorithm Systems Biology, IEE, 1(1):121, 2004 O. Wolkenhauer, M. Ullah, W. Kolch, K.H. Cho Modeling and simulation of intracellular dynamics: choosing an appropriate framework IEEE Trans Nanobioscience. 3(3):200, 2004. 
