Cellular Automata Approaches to Enzymatic Reaction Networks - Cellular Automata - page 296

they can be between 1 µM and 5 mM ,while for enzymes they can be in the
range of 10 nM... 1 µM . This means that a complete cell with volume 10 − 18 m 3 ,
we find between 1 and 60000 molecules of a given enzyme species,and between
10 5 and 10 9 molecules of a given metabolite species. Simply from the size of the
enzyme,we can calculate that in a space volume of size (100 A) 3 ,there can be
O(1) enzyme molecules,but many metabolite molecules. In this paper I consider
a simulation with lattice sites which have approximately this size and can contain
at most one enzyme molecule of any species (exclusion principle) and an arbitrary
number of metabolite molecules. Since we use integers to count the molecules
instead of calculating with concentrations,our results will be more similar to
the stochastic models than to the PDE models.
The diffusion coe 5 cient for a typical metabolite is around 10 − 9 m 2

s ,for an
enzyme that is not bound to a membrane it varies,but is at least two orders
of magnitude smaller. The time scales of events inside a cell vary greatly,from
10 − 12 s (dissociation events) to 100 s (fastest cell division). In the cellular au-
tomaton model,our diffusion method dictates a connection between time and
space scales and diffusion coe 5 cients. In our case,a lattice spacing of 100 A and
typical metabolite diffusion coe 5 cients imply a time step of around 2 . 5 ∗ 10 − 8 s ,
which means that millions of time steps are necessary to simulate dynamical
changes of metabolite concentrations.

3 Modeling Enzymes

In our model,each lattice site (we do not use the term cell as usually used in
cellular automata to avoid confusion with the biological cell) can contain at most
one enzyme molecule but many metabolites. We first consider the options for
modeling the enzymatic reaction in one such site.
We show in section 3.1 that directly using the Michealis-Menten rate law is
not possible in the discretized setting of cellular automata,then demonstrate in
section 3.2 how to obtain correct results by directly simulating the mechanism
that was approximated by the Michealis-Menten rate law.

3.1 Michaelis-Menten Rate Law

The first possibility is to directly use a Michaelis-Menten rate law. The rate law
in Eq. (1) contains the concentration of the enzyme [ E ],which in one site is
always zero or one. If we assume that the concentrations of metabolites can be
simply calculated from the metabolite numbers present in the cell,we have the
following cellular automata rule: In time ∆t ,the probability of converting one
molecule S to a molecule P is zero if there is no enzyme E present,otherwise

Q S→P ( S )= ∆t

αS
K m + αS ,

α V max

(3)

where α is a scaling factor to convert molecule counts per lattice site into con-
centrations. Here we will show that in our setting this method does not give