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contains
N
t
=
N
d
×
625 B cells). Individual B cells are defined by strings repre-
senting V-regions with 300 nucleotides in size. The processes of SHM/selection
take place only in particular demes named MS demes. Cells can migrate from
one deme to any of the 8 neighbour demes with probability
m
r
(see arrows in
figure 1).
In each time step (generation) B cells within MS demes mutate in the V region
of their Igs with rate
U
per B cell per generation. The number of mutations
occurring per cell is a Poisson random variable with mean
U
. Once a mutation
occurs it can either decrease (with probability
p
d
) or increase (with probability
1
p
d
) the anity of targeted Abs.
Outside of the MS demes, mutation does not occur and all cells have the same
probability of survival. In the MS demes the probability of survival for each cell
is directly proportional to its fitness
W
ij
, which depends on the anity of its
Igs for the Ag.
W
ij
corresponds to the probability of survival of a B cell with
i
mutations that decrease the anity and
j
mutations that increase anity. To
calculate the fitness of each B cell, we use the multiplicative fitness assumption
for the interaction between mutations. With this assumption the fitness of B
cells containing
i
low anity and
j
high anity mutations is calculated as:
W
ij
=(1+
s
b
)
j
(1
−
s
d
)
i
,where
s
b
is the effect of mutations that lead to an
increase in anity and
s
d
is the effect of mutations that lead to a decrease in
anity.
To understand how different degrees of 'GC' aggregation/organization could
affect the process of anity maturation and the resulting diversity, five topolo-
gies were considered. These topologies are used to model different sizes of 'GC'
represented by different areas where SHM and selection could take place. These
were meant to model the evolution of GC size along a phylogenetic scale, going
from vertebrates species where the SHM and anity maturation did occur in less
structured lymphoid tissue, to current higher vertebrates where these processes
take place in finely organized GC structures. We have considered the following
topologies (in figure 1 an example of the grid corresponding to topology A3 is
shown):
(i)
topology A1 consists of 64 single, unconnected MS demes;
(ii)
topol-
ogy A2 consists of 16 groups of 2
−
×
2 MS demes;
(iii)
topology A3 consists of
7 groups of 3
×
3 MS demes;
(iv)
topology A4 consists of 4 groups of 4
×
4MS
demes; and
(v)
topology A5 consists of 1 group of 8
8 MS demes.
Each group of MS demes is placed at random in the grid. The simulations were
performed using the following set of parameter values. Each deme is assumed
to hold
N
d
= 100 B cells (this number is adjusted every generation, after the
migration process has occurred). Within MS demes the mutation parameters are
U
=0
.
3and
p
d
=0
.
998, and the selection parameters,
s
d
and
s
b
,werevaried.
The migration rate was set to
m
r
=0
.
00625. This Monte-Carlo algorithm was
run for different periods of time. In particular, analyses of the time for the mean
anity to approach the expected equilibrium were performed. To relate the
time steps in the algorithm with the time scale of present day GCRs, we assume
that B cells in the MS demes divide every 8 hours [3]. Thus 60 time steps in
the algorithm correspond to about 21 days, which is the average life of GCs
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