Geometric framework for biological evolution
This work provides a foundational geometric framework for understanding biological evolution, potentially impacting evolutionary biology and theoretical modeling, though it is incremental in extending existing concepts like maximum entropy and covariance matrices.
The authors developed a generally covariant framework for evolutionary dynamics that connects genotype and phenotype spaces, showing that evolution can be modeled as a learning process on fitness landscapes, with the Lande equation emerging as a covariant gradient ascent equation.
We develop a generally covariant description of evolutionary dynamics that operates consistently in both genotype and phenotype spaces. We show that the maximum entropy principle yields a fundamental identification between the inverse metric tensor and the covariance matrix, revealing the Lande equation as a covariant gradient ascent equation. This demonstrates that evolution can be modeled as a learning process on the fitness landscape, with the specific learning algorithm determined by the functional relation between the metric tensor and the noise covariance arising from microscopic dynamics. While the metric (or the inverse genotypic covariance matrix) has been extensively characterized empirically, the noise covariance and its associated observable (the covariance of evolutionary changes) have never been directly measured. This poses the experimental challenge of determining the functional form relating metric to noise covariance.