Biology Reference
In-Depth Information
Mutations
During the takeover of a site by the winner of the competition the invading cell, that is
the copy of the winner occupying the site of the loser, can change one of its three al-
leles (chosen at random) from functional to inactive or vice versa. We call these allele
changes “mutations,” but in fact they can be due to either mutation or some other pro-
cess like transformation or even the immigration of individuals carrying the “mutant”
allele. The point in allowing allele changes both ways (losing and obtaining them) is
to maintain the presence of all six different genes (C, c, S, s, R, r) in the population so
that the system does not get stuck in any particular genetic state because of the lack of
alternative alleles. Thus, each of the six possible allele changes may have a positive
probability. Mutations are independent at the three locifor example, the quorum signal
gene S can be lost without losing the response module R at the same time; the resulting
mutant will be “mute” yet still able to respond to quorum signals.
Diffusion
Each competition step may be followed by a number (D) of diffusion steps. One dif-
fusion step consists of the random choice of a site, and the 90° rotation of the 2 ×
2 subgrid with the randomly chosen site in its upper left corner. Rotation occurs in
clockwise or anticlockwise direction with equal probability [19]. The D is the dif-
fusion parameter of the model: it is proportional to the average number of diffusion
steps taken by a cell per each competitive interaction it is engaged in. Larger D means
faster mixing in the population. Since one diffusion move involves four cells, D = 1.0
amounts to an expected number of four diffusion steps per interaction per cell. In the
simulations we use the range 0.0 ≤ D ≤ 1.0 of the diffusion parameter, and occasionally
much higher values (D = 15.0) as well.
Initial States and Output
At t = 0 the lattice is “seeded” either by the “Ignorant” (csr) genotype on all sites, or
the initial state is a random pattern of all the eight possible genotypes present at equal
proportions. We simulate pairwise competitive interactions, mutations, and diffusive
movements for N generations. One generation consists of a number of competition
steps equal to the number of sites in the lattice, so that each site is updated once per
generation on average. In the majority of simulations we have applied mutation rates
of 10
−4
both ways at each locus, which is equivalent to an average of nine mutation
events per generation within the whole habitat. The three functional alleles have a
positive cost of expression, constrained by the relation m
C
>> m
S
> m
R
(the actual
values used throughout the simulations are given in Table 1).
Simulations
With the initial conditions specified above we follow the evolution (the change in al-
lele frequencies) for both cooperation and the two components of QS. We investigate
the qualitative or quantitative effects on the evolution of cooperation and QS of the
crucial parameters of the model: the fitness reward of cooperation (r), the metabolic
cost of cooperation (m
C
), the intensity of diffusive mixing (D) and the quorum thresh-
old (n
q
). The simulations have been run until the relative frequencies of the three focal
