Modeling Protein Evolution - Computational Modeling of Biological Systems - page 319

Another consideration in modeling evolutionary dynamics is the timing of
reproduction and selection among individuals. It is common to follow the classic
synchronous population model that coincides with the bulk of classical mathemati-
cal theory on population genetics dynamics. In such populations, reproduction and
changes in allele frequency due to fitness differences occur at the same time for
the entire population. Each such set of steps is called a generation. Thus, for a
population of N individuals, the allele frequencies at the end of one generation
would be multiplied by their relative fitness, and then the entire population would
be randomly re-sampled with replacement from the resulting allele frequencies to
create the next generation. Mutations can occur during this re-sampling. These
can include, in addition to simple mutations, other processes such as mating and
recombination. It is traditional to hold the population size constant over the course
of a simulation.
There are a number of common variants to this scheme, including allowing
the population size to change over time, and allowing individual (or mating pair)
variation in reproductive success. Sometimes alternative schemes are utilized. For
example, in tournament selection , a subset of n Tourn individuals from the population
are chosen at random, with replacement, and only the most fit individual in this
subset is replicated in the next population. This process is then repeated N times to
fill the entire population for the next generation. Selection pressure can be controlled
by the value of n Tourn , with n Tourn D 1 corresponding to no selective pressure (the
next generation is chosen at random from the genotypes in the previous generation)
and n Tourn N corresponding to stochastic hill climbing where only the most fit
individual reproduces. A scheme that better reflects reproduction in many species
(including humans) is the Moran process , in which reproductive events occur
sequentially in the population [ 25 ]. Population size is maintained by selecting at
each time step one individual to duplicate (with probability proportional to its
fitness) and one individual to be eliminated. Again, mutations can be implemented
as part of the replication process.
In order to speed up simulations, we can also calculate, rather than simulate,
the probability of fixation. This approach relies on the assumption that mutations
that are destined to become fixed do not overlap in time, such that each mutation is
either eliminated or fixed in the population prior to the arrival of the next mutation.
In this case, the probability of fixation of the mutant P Fix has been computed by
Kimura [ 26 - 28 ]:

e 2s 1

P Fix D

e 2N s ;

(3)

1

where is equal to 1 for haploid organisms and 2 for diploid organisms, and the
selection coefficient s is given by

! mut ! wt
! mut

s D

; (4)

where ! mut and ! wt are the relative fitnesses of the mutant and wild-type alleles,
respectively. This equation holds when s<<1, in which case we can choose a