Information Technology Reference
In-Depth Information
4 Diffusion
Diffusion of the metabolite molecules can be simulated using a number of tech-
niques. Since we model individual molecule counts,we have to make sure that
the number of molecules is conserved by the diffusive operation. This is not
easily verified in standard cellular automata. Two techniques have been devel-
oped which make conservation of particles easy in cellular automata: Partitioned
cellular automata and block cellular automata.
The first kind is used in the lattice gas automata [2,4,5,18,19], where each
lattice site has a number of channels to the neighboring sites,which can be
occupied by at most one particle at a time. Since the transport of particles
through these channels in the synchronous cellular automaton simply represents
a permutation of the channel contents,one can easily verify that no particles get
lost (assuming correct treatment of the boundaries).
The second technique is to subdivide the lattice into blocks,and at each step
do some exchange of the contents of the sites within one block [3,12,16,20]. Since
only a few cells are involved,it is easy to verify whether conservation laws are
observed. In subsequent time steps,the block boundaries are changed to make
information exchange across the whole lattice possible. Note that such block
cellular automata are equivalent to classical cellular automata,since each can
be simulated by the other (with some duplication of cell content and extension
of the neighborhood).
Here we use this second technique to simulate diffusion. We use blocks of size
two,which are placed on the lattice in all possible orientations in subsequent
time steps (four orientations in two dimensions,six in three dimensions). For
the exchange between the two cells within a block one can use different prescrip-
tions: One possibility was already mentioned by Gillespie [7,8] as an extension
of his stochastic simulation method to spatially distributed systems: If the two
cells contain
n
1
and
n
2
particles respectively,move (
n
1
− n
2
)
/
2 particles from
cell 1 to cell 2 (or reverse,if the difference is negative). This approach mod-
els the macroscopic diffusive flux proportional to the gradient in concentration.
Better suited to the stochastic simulation is a microscopic approach: Consider
all the particles in both cells as independent,and let each particle move to the
neighboring cell with a fixed probability
p
. Then the number of particles to be
moved from cell 1 to cell 2 is obtained by sampling a binomial distribution with
parameters
p
and
n
1
,while the number of particles moving from cell 2 to cell 1 is
given by a binomial distribution with parameters
p
and
n
2
. Thus the number of
particles exchanged between the two cells is governed by the difference between
two binomial distributions. This approach leads to fluctuating particle numbers
in all cells even in the absence of reactions,which is appropriate when parti-
cles represent molecules. The particle numbers in the cells will be distributed
according to a Poisson distribution Eq. (4).
For the calculations of effective reaction rates,as in Eq. (5),the averaged
(macroscopic) diffusion operator leads to a distribution that is more compact
than the Poisson distribution,e.g. a two-valued distribution around the aver-
