Gauge freedoms, symmetries, and the interpretability of sequence-function relationships

Quantitative models of sequence-function relationships are ubiquitous in post-genomic biology, e.g., for describing the activities of gene regulatory sequences or to model the fitness landscapes of proteins. These models usually exhibit many gauge freedoms—directions in parameter space along which transformations do not affect model output. But in contrast to the central role that gauge freedoms play in theoretical physics, the origins, properties, and consequences of gauge freedoms in sequence-function relationships have received little attention. Here we study gauge freedoms in both linear and linear-nonlinear sequence-function relationships. In particular, we connect the gauge freedoms that arise in linear models with one-hot sequence features to character permutation symmetry, and discuss the importance of such symmetry for biological interpretability. We also propose specific gauge fixing strategies that are especially useful in the context of these linear models. Finally, we identify two distinct classes of gauge freedoms that arise in linear-nonlinear models which are not present in linear models. This work thus establishes mathematical and conceptual tools for better understanding how sequence encodes biological function.