Gauge fixing for sequence-function relationships

Quantitative models of sequence-function relationships are ubiquitous in computational biology, e.g., for modeling the DNA binding of transcription factors or the fitness landscapes of proteins. Interpreting these models, however, is complicated by the fact that the values of model parameters can often be changed without affecting model predictions. Before the values of model parameters can be meaningfully interpreted, one must remove these degrees of freedom (called “gauge freedoms” in physics) by imposing additional constraints (a process called “fixing the gauge”). However, strategies for fixing the gauge of sequence-function relationships have received little attention. Here we derive an analytically tractable family of gauges for a large class of sequence-function relationships. These gauges are derived in the context of models with all-order interactions, but an important subset of these gauges can be applied to diverse types of models, including additive models, pairwise-interaction models, and models with higher-order interactions. Many commonly used gauges are special cases of gauges within this family. We demonstrate the utility of this family of gauges by showing how different choices of gauge can be used both to explore complex activity landscapes and to reveal simplified models that are approximately correct within localized regions of sequence space. The results provide practical gauge-fixing strategies and demonstrate the utility of gauge-fixing for model exploration and interpretation.Author summary: Biophysics and other areas of quantitative biology rely heavily on mathematical models that predict biological activities from DNA, RNA, or protein sequences. Interpreting the parameters of these models, however, is not trivial. Here we address a core challenge for model interpretation–the presence of “gauge freedoms”, i.e., directions in parameter space that do not affect model predictions and therefore cannot be constrained by data. Our results provide an explicit mathematical method for removing these unconstrained degrees of freedom–a process called “fixing the gauge”–that can be applied to a wide range of commonly use models of sequence-function relationships, including models that describe interactions of arbitrarily high order. These results unify diverse gauge fixing methods that have been previously described in the literature for specific types of models. We further show how our gauge-fixing approach can be used to simplify complex models in user-specified regions of sequence space. This work thus overcomes a major obstacle in the interpretation of quantitative sequence-function relationships.