Fitness Landscape Ruggedness Arises from Biophysical Complexity

Epistasis, in which the effect of a mutation depends on the genetic background it is introduced into, drives protein evolution and design by shaping the fitness landscape. Quantifying how constrained by epistasis (and thus rugged ) a fitness landscape is when data is combinatorially incomplete, however, remains challenging. Here, we propose a spectral graph theory approach to measure fitness landscape ruggedness as the characteristic time step of a heat diffusion process. Our approach is conceptually analogous to modelling a fitness signal as heat and measuring the ruggedness of the fitness landscape as the length of time that signal has diffused over a network graph for. Through theory and simulation, we show that our diffusion-based approach describes the global ruggedness of a fitness landscape, even when only a fraction of the fitness map is experimentally characterized. Applying this approach to empirical fitness landscapes that encompass deep mutational scans, combinatorially complete datasets, and sparse sampling of homologous sequences, we find large variation in ruggedness both among and within different protein folds and functions. Finally, we use spectral graph theory to demonstrate that ruggedness is intimately related to the fraction of sequence space that maintains sufficient fitness to overcome purifying selection, which we term the solution set of the fitness function. These findings offer a theoretically grounded explanation for why complex biophysical traits often exhibit strong epistasis, while simpler traits, such as fold stability, evolve over smooth fitness landscapes. Our results underscore how spectral methods can help unify and quantify the interplay between fitness constraints and evolutionary outcomes in protein sequence space.