Modeling Structural and Genomic Constraints in the Evolution of Proteins - Computational Modeling of Biological Systems - page 329

free energies, yielding a correlation coefficient r D 0:92 (UB, unpublished data).
The accuracy of this method for predicting the stability effect of mutations is
comparable to state-of-the-art atomistic methods such as Fold-X [ 17 ], and its
computational simplicity allows for its use in simulating protein evolution for long
evolutionary trajectories.
Stability against unfolding is, however, not sufficient to characterize protein
stability, since one has also to assess stability against compact, incorrectly folded
conformations of low energy that can act as kinetic traps in the folding process
and, in many cases, give rise to pathological aggregation. The term positive design
indicates sequence features that favor stability against unfolding by making the
native structure more stable, obtained either through evolution or through sequence
design algorithms. On the other hand, stability against misfolding is thought to
be realized in natural proteins by increasing the energy of key contacts that are
frequently found in alternative structures, which is termed negative design [ 18 , 19 ].
Multiple sequence alignments show correlated mutations not only between positions
that are in contact in the native structure (as a consequence of positive design)
but also between positions that are distant in the native structure, which has been
interpreted as evidence for negative design [ 18 ], although functional interpretations
have also been proposed [ 20 ].
Stability against misfolded structures is difficult to estimate, and several models
of protein evolution do not consider it, despite its importance being more and
more recognized. Two simplified sets of alternative structures are most often used:
either the set of approx. 10 5 maximally compact structures on the 3 3 3
cubic lattice, which can be enumerated with affordable computational effort [ 21 ],
or the set of all compact matrices of L residues that can be obtained from non-
redundant structures in the protein database (PDB). This latter procedure, called
threading in bioinformatics jargon, guarantees that the contact matrices fulfill
physical constraints on chain connectivity, atomic repulsion, and hydrogen bonding
(secondary structure), which are not enforced in the contact energy function. We
will use the threading set in most of this chapter. Whatever the set of alternative
structures, there are several measures to assess the stability against misfolding. A
parameter that is often used is the normalized energy gap ˛. A /,definedas

E. A ; C / E. A ; C nat /
j E. A ; C nat / j

˛. A / D min

C

Œ1 q. C ; C nat / ;

(2)

where the contact overlap q. C ; C nat / measures the structural similarity between the
native (lowest energy) structure C nat and the alternative structure C . A large value of
˛. A / implies that all the low energy structures are structurally similar to the native
one and belong to its attraction basin. This is necessary for thermodynamic stability,
since the contact interaction parameters are effective free energy parameters that
depend on temperature and, if ˛ is small, a small change in these parameters
could completely change the ground state. Similarly, it is necessary for stability
against mutation and for fast folding kinetics. ˛. A / is determined by the lowest
energy structure that is structurally unrelated to the ground state and it can be