Modeling Protein Evolution - Computational Modeling of Biological Systems - page 316

using folding simulations [ 9 ]. Evolutionary simulations can be performed so that
proteins with a foldability larger than some critical value are considered viable,
with fitness equal to one, while proteins with lesser foldability are considered
nonviable, with fitness equal to zero [ 13 , 14 ]. (Evolutionary models where
fitnesses are either a constant high value or a constant low value depending upon
some criterion are called truncation selection .)
2. The need to be sufficiently stable in the native state conformation .Thisis
quantified as either a free energy of folding .G Folding / or as the fraction of
time spent in the folded state at equilibrium. The added difficulty here is the need
to consider the enormous ensemble of unfolded conformations. One approach
is to consider a large number of alternative conformations, recognizing that the
number of conformations considered represents a vast underestimation of the
total possible number of such states. Alternatively, one can extrapolate from a
small ensemble of proteins to the properties of a larger ensemble. For instance,
imagine that the distribution of free energies of alternati ve structures U .G/
is approximated by a Gaussian distribution with average G and variance 2 .
We can consider sampling a large number of structures from this distribution,
representing the entire ensemble of unfolded states, to calculate G U , the free
energy of the ensemble of alternative unfolded states

¢ 2 2kT G

2kT

G U D C kT ln N U ; (2)

where N U is the total number of alternative states, k is the Boltzmann constant,
and T is the temperature [ 15 ]. Calculation of the free energies of a limited number
of altern ati ve structures may be sufficient to characterize U .G/ and allow us to
estimate G and 2 .
Another alternative is to consider that we often do not have to calculate stabilities,
but only the change in stability .G Folding / due to a mutation. Such changes in
stability can be calculated using, for instance, FoldX [ 16 ]. While this can be more
accurate than the simple representation of protein energetics, the complexity of these
calculations limits evolutionary simulations.

3. The need to be functional . In addition to being foldable and stable, proteins
generally fulfill one or more functions. In contrast to foldability and stability,
the functional requirements for proteins are highly specific, making modeling of
functional constraints difficult. One class of models that have been investigated
involves the consideration of binding properties. For instance, Hirst and co-
workers considered the ability of a protein to construct an appropriate binding
pocket [ 17 , 18 ]. Alternatively, a protein's fitness can be modeled as a function of
how well it binds a specified peptide or other protein [ 19 , 20 ]. The intermolecular
binding interaction can be calculated using the same parameters used for the
intramolecular interactions, and we can consider fundamental properties of
protein function such as specificity by modeling binding to competing peptides
as well. We can also consider what happens in the evolutionary simulation when