Modeling Protein Evolution - Computational Modeling of Biological Systems - page 314

2.1 Modeling Protein Energetics

It is generally impractical to explicitly consider the motions and interaction energies
of all of the atoms found in a protein and its environment, and so models must be
constructed that do not consider all of the atoms and all of the motions. A common
strategy is to consider reduced representations of proteins, in which each amino acid
in the chain is represented by one or a few particles.
While the use of a reduced representation removes many conformational degrees
of freedom from the system, the number of possible conformations is still too
large to be easily modeled. To reduce the conformation space, a limited number
of possible conformations may be considered, one of which represents the native
state. These conformations can include known protein structures as well as one
of the sets of decoy structures that have been created (e.g., [ 3 ]). A more extreme
approach is to consider each amino acid as a single particle located at adjacent
vertices in a (generally cubic) lattice [ 4 ]. The covalent bonds between residues are
represented as edges in the lattice, and interactions occur only between adjacent
nonbonded residues. Possible conformations correspond to different self-avoiding
walks through these lattices; excluded volume is implemented by the requirement
that two residues cannot share the same lattice point. The lattice can either be small
(so that only the compact states are represented) or it can be big enough that entirely
unfolded proteins are possible. The number of possible compact conformations
for small proteins is reasonable (a 27-mer protein on a compact 3 3 3 lattice
has 103,346 conformations, excluding reflections and rotations), but the number of
conformations increases rapidly with the size of the protein and when noncompact
forms are considered. Small three-dimensional proteins on a regular lattice have
few residues that are internal or “buried” compared to real proteins, and so some
researchers have instead used two-dimensional lattices, which generate a more
reasonable fraction of buried residues. Such lattices have advantages for modeling
protein thermodynamics but are generally inappropriate for simulations of, for
instance, folding dynamics. The ground state can either be specified in advance or
allowed to change during the simulation.
The free energy of a protein is the sum of a number of different types of
interactions, including (a) the van der Waals contact energy between atoms, (b)
Coulomb interactions between charges, (c) hydrogen bonds, (d) the hydrophobic
effect, encompassing the entropy loss resulting from the structuring of water near
nonpolar groups, (e) bond stretches, bends, and rotations, and (f) changes in the
conformational and vibrational entropy. It is difficult to calculate an accurate value
for the free energy of a particular conformation of a protein in its full atom
representation, especially the entropic contributions. The situation becomes even
worse for coarse-grained models, as many of the atoms involved in the interactions
are not represented in the structure. Generally, modelers use highly simplified free
energy functions such as those based on contact energies, where the free energy
G.S; F / of a sequence S Df A 1 ;A 2 ; ::: A n g in fold F can then be expressed as

X

G.S; F / D .A i ;A j /H.r 0 r ij /; (1)