Biological systems are infuriatingly complex. Across the scales of proteins, bacteria, humans, and entire ecosystems, small perturbations frequently lead to large and unpredictable systemic outcomes. While this unpredictability is commonly attributed to biological noise, it may instead arise from inadequate statistical representations of biological systems. Scientific history shows that choosing the right representation dramatically simplifies complex phenomena: the transition from geocentric to heliocentric models transformed irregular planetary paths into simple elliptical orbits. A central challenge in modern biology is therefore identifying mathematical frameworks that re-parametrize biological organization in a way that renders systemic behavior simple. Motivated by this goal, we investigated the statistical representations of related populations. We found that the evolutionary history of such biological populations is necessarily embedded within the spectral decomposition as constructed by SVD or PCA. Resolving this evolutionary history enabled identification of sub-species phylogeny in populations of human gut commensal bacteria. This sub-species phylogeny further clarified host-specific genomic adaptations resulting in loss of motility and metabolic capacities previously assumed to be stochastic variability. We additionally resolved this spectral evolutionary history across an ensemble of 60 million proteins. We found the inferred evolutionary history constrains proteins to locally small changes in phenotype and function. Further, the spectral representation enabled direct linking of evolutionary trajectories to the physical connectivity within protein structures. Together these results suggest that spectral representations provide a unifying statistical framework for linking evolutionary history, genomic variation, system behaviors, and phenotype.
