  Epistasis, or the context-dependence of the effects of mutations, limits our
ability to predict the functional impact of combinations of mutations, and
ultimately our ability to predict evolutionary trajectories. Information about
the context-dependence of mutations can essentially be obtained in two ways:
First, by experimental measurement the functional effects of combinations of
mutations and calculating the epistatic contributions directly, and second, by
statistical analysis of the frequencies and co-occurrences of protein residues
in a multiple sequence alignment of protein homologs. In this manuscript, we
derive the mathematical relationship between epistasis calculated on the basis
of functional measurements, and the covariance calculated from a multiple
sequence alignment. There is no one-to-one mapping between covariance and
epistatic terms: covariance implies epistasis, but epistasis does not
necessarily lead to covariance, indicating that covariance in itself is not the
directly relevant quantity for functional prediction. Having calculated
epistatic contributions from the alignment, we can directly obtain a functional
prediction from the alignment statistics by applying a Walsh-Hadamard
transform, fully analogous to the transformation that reconstructs functional
data from measured epistatic contributions. This embedding into the Hadamard
framework is directly relevant for solidifying our theoretical understanding of
statistical methods that predict function and three-dimensional structure from
natural alignments.
