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Fig. 1.
An example of the 25
×
25 grid with a possible A3 topology. The full squares
(MS demes) indicate the places where mutation and selection occur. Arrows indicate
the eight possible directions for a migration event.
in primary immune responses. In order to obtain a variance due to stochastic
events each simulation was repeated 20 times.
3R su s
3.1
Some Results from Genetics Population Theory
We first summarize some analytical results from population genetics that are
relevant to understand the results shown for this model of GC evolution. Let
us consider a large population of individuals (
e.g.,
B cells) undergoing mutation
at rate
U
d
per individual per generation. Lets assume that every mutation has
a negative effect, decreasing the fitness (
anity) by an amount
s
d
. Then,
after approximately 1
/s
d
generations (each constituting a cycle of mutation and
selection), the distribution of bad mutations in the population is Poisson with
mean
U
d
/s
d
. This means two things: first, if
s
d
is small it takes a lot of time to
achieve this distribution; second, when it is achieved it can have a very large mean
and variance. In the simulations
s
d
was around 10% the initial fitness so that
the equilibrium distribution was reached in a period shorter than the time of a
typical GC reaction of a primary immune response. Let
a
(
t
)bethemeannumber
of negative mutations at time
t
after the start of the SHM process, then the
∝
distribution at time
t
is Poisson with mean given by:
a
(
t
)=
1
s
d
)
t
U
d
/s
d
[6]. Population genetics theory also shows that, if the population is not very
large and/or
s
d
is small, the equilibrium above is not stable and a continuous
accumulation of deleterious mutations can occur [7]. This is likely to happen if
the condition
N
−
(1
−
U
d
/s
d
) is satisfied, where
N
is population size.
If positive (a
nity increasing) mutations are allowed to occur at rate
U
a
per
cell per generation then for
U
a
U
d
the distribution of negative mutations
(decreasing anity or deleterious) stays close to a Poisson [8].
×
Exp
(
−
