Modeling Protein Evolution - Computational Modeling of Biological Systems - page 315

where .A i ;A j / is the interaction between the type of amino acid found in positions
i and j ,andH.r 0 r ij / is a Heavyside step function equal to one if and only
if r ij , the distance between specified atoms in residues i and j , is less than some
cutoff r 0 . Early models considered only two types of amino acids, hydrophobic (H)
and polar (P), where only hydrophobic residues interact energetically [ 5 ]. Energetic
parameters for more realistic schemes have been constructed based on knowledge
of physical chemistry or, alternatively, extracted through statistical analysis of the
available protein structures [ 6 ]. The latter approach is based on two incorrect
but useful assumptions. The first is that we can integrate over all of the degrees
of freedom of the system that we consider unimportant for the simulation (such
as conformations of the side chains and solvent molecules) in order to calculate
potentials of mean force , and that the potentials of mean force can be decomposed
into a sum of uncorrelated terms that can be computed independently. The second
assumption is that the distribution of states of a single protein in its multiplicity of
possible conformations can be represented by the distribution of native states of a
multiplicity of known proteins. Under this assumption, the Boltzmann expression
relating the probability of observing a state to the free energy of that state can be
inverted to determine the free energy of an interaction based on the frequency of
that interaction in the database of known protein structures. (Finkelstein and collab-
orators have promoted an interesting evolutionary justification for this assumption
[ 7 , 8 ]). Because the potential of mean force explicitly includes the sum over all
degrees of freedom of the system, including the solvent degrees of freedom, entropic
interactions involving the solvent (and thus, the hydrophobic effect) are included in
the simulation.

2.2 Modeling Selective Constraints

Once a model of the protein and a representation of its interactions are in place,
the selective constraints acting on the protein can be determined. These can be a
mixture of structural, thermodynamic, or functional properties. Common selective
constraints include:

1. The need to fold into a well-defined structure . In general, explicit folding
simulations are too slow for all but the simplest representations of proteins or
of the evolutionary process [ 9 ]. Rather than simulating the folding process, an
alternative approach is to evaluate properties that are characteristic of folded
proteins. For instance, the presence of a nondegenerate ground state (either any
ground state or a prespecified ground state) has been considered adequate [ 10 ],
although this is difficult to justify at nonzero temperatures. Wolynes and co-
workers used concepts from spin-glass physics to suggest that an appropriate
measure of “foldability” [ 11 ] is the energy gap between the native conformation
and the distribution of the energies for random conformations [ 12 ]; this can
also be equated with a statistical Z-score. These predictions were later verified