Geometric framework for biological evolution

TL;DR

This paper introduces a covariant geometric framework for biological evolution, linking genotype and phenotype spaces, and models evolution as a learning process on the fitness landscape with a focus on the metric tensor and noise covariance.

Contribution

It develops a unified geometric description of evolutionary dynamics, connecting the metric tensor to the covariance matrix and framing evolution as a gradient ascent learning process.

Findings

Identifies the inverse metric tensor with the covariance matrix.

Reveals evolution as a covariant gradient ascent process.

Highlights the challenge of experimentally measuring noise covariance.

Abstract

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.