Symmetry, gauge freedoms, and the interpretability of sequence-function relationships

Abstract

Quantitative models that describe how biological sequences encode functional activities are ubiquitous in modern biology. One important aspect of these models is that they commonly exhibit gauge freedoms, i.e., directions in parameter space that do not affect model predictions. In physics, gauge freedoms arise when physical theories are formulated in ways that respect fundamental symmetries. However, the connections that gauge freedoms in models of sequence-function relationships have to the symmetries of sequence space have yet to be systematically studied. Here we study the gauge freedoms of models that respect a specific symmetry of sequence space: the group of position-specific character permutations. We find that gauge freedoms arise when model parameters transform under redundant irreducible matrix representations of this group. Based on this finding, we describe an “embedding distillation” procedure that enables analytic calculation of the number of independent gauge freedoms, as well as efficient computation of a sparse basis for the space of gauge freedoms. We also study how parameter transformation behavior affects parameter interpretability. We find that in many (and possibly all) nontrivial models, the ability to interpret individual model parameters as quantifying intrinsic allelic effects requires that gauge freedoms be present. This finding establishes an incompatibility between two distinct notions of parameter interpretability. Our work thus advances the understanding of symmetries, gauge freedoms, and parameter interpretability in sequence-function relationships.Significance StatementGauge freedoms—diections in parameter space that do not affect model predictions—are ubiquitous in mathematical models of biological sequence-function relationships. But in contrast to theoretical physics, where gauge freedoms play a central role, little is understood about the mathematical properties of gauge freedoms in models of sequence-function relationships. Here we identify a connection between specific symmetries of sequence space and the gauge freedoms present in a large class of commonly used models for sequence-function relationships. We show that this connection can be used to perform useful mathematical computations, and we discuss the impact of model transformation properties on parameter interpretability. The results fill a major gap in the understanding of quantitative sequence-function relationships.