A path integral formulation of the Wright-Fisher process with genic selection.

Area of Science:

• Population Genetics
• Theoretical Biology
• Mathematical Biology

Background:

• The Wright-Fisher model is a cornerstone of population genetics theory.
• Traditional analyses often employ diffusion approximations, typically within a partial differential equations framework.
• Limitations exist in the traditional diffusion approximation for certain selection scenarios.

Purpose of the Study:

• To introduce a novel path integral formalism for the Wright-Fisher process with selection.
• To develop a new perturbation series for approximating the transition density.
• To provide a probabilistic interpretation of selective events using Feynman diagrams.

Main Methods:

• Development of a path integral formalism for the Wright-Fisher process.
• Derivation of a perturbation series based on the path integral.
• Probabilistic interpretation of the perturbation series using Feynman diagrams.

Main Results:

• A simple perturbation series approximation for the transition density was obtained.
• The perturbation series demonstrates a clear probabilistic interpretation linked to selective events.
• The approximation is accurate for weak selection and arbitrarily accurate for any selection coefficient.

Conclusions:

• The path integral formalism offers a powerful alternative to traditional diffusion approximations.
• The perturbation series provides an accurate and interpretable method for analyzing the Wright-Fisher process with selection.
• This approach enhances the theoretical toolkit for studying evolutionary dynamics.