Modeling Structural and Genomic Constraints in the Evolution of Proteins - Computational Modeling of Biological Systems - page 332

where N is the effective population size and f. A / is the exponential growth rate
of the phenotype associated to sequence A , which will be called fitness in the
following. This formula enormously simplifies both the analytic and the numeric
study of evolution. It has been noted that the above formula, multiplied by the
mutation probability from A to A 0 , can be interpreted as the transition probability
of a Markov process in sequence space. Such a Markov process admits a stationary
distribution in which fitness fluctuates around an equilibrium value, with events of
fitness increase and decrease being on the average equally likely. The stationary
distribution can be computed analytically [ 39 , 40 ], and it is given by

P evol . A / P mut . A / expŒN log f. A / ; (6)

where P mut . A / is the probability to obtain sequence A under mutation alone. The
factor expŒN log f. A / is equivalent to a Boltzmann distribution, where the effective
population size N plays the role of inverse temperature and the logarithm of fitness
plays the role of minus energy. Thus the model predicts that smaller populations
reach lower fitness, which means that their macromolecules are less stable and the
mutational entropy in sequence space is larger.
Wright [ 28 ] generalized the stationary distribution ( 6 ) to the case where the
product N is not small. This stationary distribution has also a deep analogy with
statistical physics [ 42 ]. However, it has a simple expression only in the case of two
alleles, in which case the probability to find the first allele with frequency x 1

and

the second one with frequency x 2 D 1 x 1 is

P.x 1 ;x 2 / / x V 1 1
1

e x 1 log f 1 x V 2 1
2

e x 2 log f 2 ; (7)

where f i is the fitness of allele i and V 1 D N u 21 =. u 12 C u 21 /. The last factor

represents the mutation bias from allele 2 to allele 1.
It would be interesting to further develop the analogy of statistical mechanics
also in the case of potentially infinite alleles. This has only been done in the infinite
population limit, where population dynamics can also be studied analytically, as
a mutation can be fixed only if it is advantageous or neutral. In this limit, a
large number of mutants arise at each generation, and the population can be
represented as a distribution in the space of all possible genotypes, which is
called a quasispecies [ 43 ]. Also, in this limit there is a formal analogy between
population dynamics and statistical physics, where mutation rate plays the role of
temperature [ 44 , 45 ]. In the single peak landscape, where a unique master sequence
has higher fitness than the sea of mutant sequences, this model undergoes a phase
transition in which at low mutation rate a sizeable fraction of the population adopts
the master sequence, whereas at high mutation rate the population is dispersed in a
sea of less stable mutant sequences.