Paper List

期刊: ArXiv Preprint
发布日期: 2026-03-16
Theoretical BiologyEvolutionary Dynamics

Geometric framework for biological evolution

Unknown

Vitaly Vanchurin

30秒速读

IN SHORT: This paper addresses the fundamental challenge of developing a coordinate-independent, geometric description of evolutionary dynamics that bridges genotype and phenotype spaces, revealing evolution as a learning process.

核心创新

  • Methodology Establishes a generally covariant framework for evolutionary dynamics that operates consistently across genotype and phenotype spaces, enabling coordinate-independent analysis.
  • Theory Demonstrates through maximum entropy principle that the inverse metric tensor equals the covariance matrix, transforming the Lande equation into a covariant gradient ascent equation.
  • Methodology Models evolution as a learning process where the specific optimization algorithm is determined by the functional relationship g(κ) between metric tensor and noise covariance.

主要结论

  • The maximum entropy principle yields fundamental identification: g^{αr,βs} = c^{αr,βs} (inverse metric equals genotypic covariance matrix).
  • The Lande equation transforms to covariant gradient ascent: dx̄^i/dt = G^{ij}(x̄) ∂ℱ(x̄)/∂x̄^j, where G^{ij} = C^{ij} (inverse phenotype metric equals phenotypic covariance).
  • Evolution implements specific learning algorithms determined by functional relation g(κ) between metric and noise covariance, with three regimes identified: quantum (α=1), efficient learning (α=1/2), and equilibration (α=0).
研究空白： Current evolutionary frameworks lack coordinate-independent descriptions and fail to capture the geometric structures underlying evolutionary processes, particularly the unmeasured relationship between metric tensor and noise covariance.

摘要: 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.