Cellular Automata Approaches to Enzymatic Reaction Networks - Cellular Automata - page 295

Reactive Networks: Once quantitative data is available,reaction rates for reac-
tions such as the conversion of a substrate S to a product P by an enzyme E
can be described by some rate law,usually in the form of a Michaelis-Menten
(MM) law:

d [ S ]

dt = V max [ E ][ S ]

(1)

K m +[ S ]

with the maximum conversion rate V max and the Michaelis-Menten coe 5 cient
K m .
A more detailed description would contain rates for the elementary reactions
leading to such an overall MM rate [1]:

k 1
k − 1 ES

k 2

E + S →E + P. (2)

Here,several rate constants need to be measured,which is usually not done or
not possible. Most reactions in a realistic reaction network are more complicated,
since they involve two or more substrates and two or more products,such as
energy-providing ATP. In these cases,more coe 5 cients need to be specified,and
in addition the mechanism must be specified,such as BiBiRandom (in which
case the two substrates may bind in any order) or Ordered BiBi (where the
substrates must bind in a specific order). Other effects,such as competitive
inhibition complicate the situation further.
Given a network of enzymatic reactions and the corresponding reaction rates,
one can simulate the network by solving the ODEs numerically,or using stochas-
tic simulation methods [7,8]. A static approach is to analyze steady states, pa-
rameter dependences or sensitivities,etc..

Space and Transport Phenomena. The space and diffusion or other transport
phenomena are usually only taken into account by compartmentalizing the sys-
tem where necessary,but they can also be included explicitly (as reaction-
diffusion equations,or probabilistically in the stochastic simulation approach
[6,17]). The most detailed simulation would be a full molecular dynamics simu-
lation of the cell,but this is by far not feasible yet.
In this paper I present an approach based on cellular automata for simulating
enzymatic reaction networks including diffusive transport.

2 Scales

To establish the conditions for a simulation model,let us first look at the scales
of space,time,diffusion coe 5 cients,and molecule concentrations and numbers.
First,concentrations and counts: The volume of a typical procaryotic cell is
10 − 15 l ,the typical diameter is 1 µm = 10000A. The diameter of a typical enzyme
is 100A,while the diameter of a typical metabolite is 6 A and of one atom 2A. The
concentrations of the involved molecular species differ widely. For metabolites