Cellular Automata Approaches to Enzymatic Reaction Networks - Cellular Automata - page 302

in the first model. Therefore only the reaction A → B takes place,with rate four
times higher),and diffusion of all species between neighboring compartments is
introduced. The lines in the time evolution plot correspond to the solution of
this ODE.
The large differences in the two solutions show that spatial dependences
should not be ignored when whole-cell models are constructed.
6 Conclusion
We have described a cellular automaton model for enzymatic reaction networks.
This model is based on block-cellular automata to ensure conservation of par-
ticles and assumes that at most one enzyme can be present at any lattice site.
The enzymes can change state by binding to metabolite molecules,and the state
changes are described by probabilistic rules derived from the enzyme kinetics to
be simulated. The quantitative correctness of the rules has been demonstrated
by analytic arguments and by comparison of simulations with the ODE solution.
This model can incorporate complex processes,where an enzyme binds several
ligands,possibly in specific order. The explicit inclusion of space makes possible
the detailed investigation of phenomena that depend on diffusion of different
species. As an example we have demonstrated that simply placing the enzymes
at different regions leads to a significantly different average behavior. In this case
the behavior can be well approximated by a compartmentalized ODE model,but
in more complex geometries of biological relevance,this is not necessarily the
case.
The model presented here is a microscopic model,since individual enzyme
molecules are explicitly represented,and the corresponding fluctuations take
place,which sets the model apart from numerical methods for solving the av-
eraged PDE. We showed that enzymatic reactions cannot be simulated by the
same techniques used e.g.,in reactive lattice gas automata [2],since the rate law
is not polynomial.
As further steps,we will apply this cellular automaton approach to more
complex enzymatic reaction networks and try to obtain biologically meaningful
spatial distributions of the individual enzymes.
References
[1] Hans Bisswanger. Enzymkinetik . Wiley-VCH, Weinheim, 2000.
[2] Jean Pierre Boon, David Dab, Raymond Kapral, and Anna Lawniczak. Lattice
gas automata for reactive systems. Physics Reports , 273(2):55-147, 1996.
[3] B. Chopard and M. Droz. Cellular automata approach to diffusion problems.
In P. Manneville, N. Boccara, G. Y. Vichniac, and R. Bidaux, editors, Cellu-
lar automata and modelling of complex physical systems , pages 130-143, Berlin,
Heidelberg, 1989. Springer-Verlag.