[Paper Review] Fixation probability and fixation time in structured populations

Josef Tkadlec, Andreas Pavlogiannis+2 more|arXiv (Cornell University)|Sep 27, 2018

Evolution and Genetic Dynamics44 references4 citations

TL;DR

This paper introduces structured population models that balance high fixation probability with short fixation time, demonstrating that certain bipartite graphs can amplify selection with only a modest increase in fixation time. The key contribution is a family of population structures that achieve a higher effective rate of evolution than well-mixed populations when mutation rates are low, by optimizing the trade-off between fixation probability and time.

ABSTRACT

The rate of biological evolution depends on the fixation probability and on the fixation time of new mutants. Intensive research has focused on identifying population structures that augment the fixation probability of advantageous mutants. But these `amplifiers of natural selection' typically increase fixation time. Here we study population structures that achieve a trade-off between high fixation probability and short fixation time. First, we show that no amplifiers can have asymptotically lower absorption time than the well-mixed population. Then we design population structures that substantially augment the fixation probability with just a minor increase in fixation time. Finally, we show that those structures enable higher effective rate of evolution than the well-mixed population provided that the rate of generating advantageous mutants is relatively low. Our work sheds light on how population structure affects the rate of evolution. Moreover, our structures could be useful for lab-based, medical or industrial applications of evolutionary optimization.

Motivation & Objective

• To identify population structures that enhance fixation probability without excessively increasing fixation time.
• To resolve the trade-off between amplification of selection and deceleration of fixation in evolutionary dynamics.
• To design population structures that maximize the effective rate of evolution by balancing fixation probability and fixation time.
• To provide analytical bounds on absorption and fixation times in structured populations using graph-theoretic models.
• To demonstrate that certain structured populations can outperform well-mixed populations in evolutionary optimization under low mutation rates.

Proposed method

• Uses evolutionary graph theory to model population structures as weighted graphs, where vertices represent individuals and edges represent reproductive connectivity.
• Introduces α-balanced and α-weighted bipartite graphs to systematically tune fixation probability and fixation time.
• Applies birth-death processes on graphs, modeling the Moran process with fitness-dependent reproduction and death rates.
• Derives analytical expressions for fixation probability, absorption time, and conditional fixation time using recursive equations and harmonic number approximations.
• Employs asymptotic analysis and limit evaluations (e.g., harmonic series, logarithmic terms) to derive large-N behavior of fixation dynamics.
• Compares effective evolutionary rates by combining fixation probability and fixation time, using the product of fixation probability and inverse fixation time as a metric.

Experimental results

Research questions

• RQ1 Can population structures exist that simultaneously increase fixation probability and minimize fixation time?
• RQ2 What is the fundamental trade-off between fixation probability and fixation time in structured populations?
• RQ3 Are there population structures that achieve higher effective evolutionary rates than well-mixed populations?
• RQ4 Can amplifiers of selection be designed with only a small increase in fixation time?
• RQ5 How do different initialization schemes (uniform vs. temperature) affect the fixation dynamics in structured populations?

Key findings

• No amplifier of selection can have asymptotically lower absorption time than the well-mixed population, establishing a fundamental lower bound on fixation time.
• α-weighted bipartite graphs achieve fixation probability approaching 1 - 1/r² with only a logarithmic increase in fixation time compared to the well-mixed case.
• For low mutation rates, the effective rate of evolution in the proposed structures exceeds that of the well-mixed population due to the favorable trade-off between fixation probability and time.
• The fixation time in the proposed structures scales as O(N log N), matching the well-mixed population, while fixation probability is significantly enhanced.
• The extinction time in the proposed models scales linearly with N, consistent with theoretical expectations for neutral and advantageous mutants.
• Analytical derivations confirm that the fixation time for the α-weighted bipartite graph is approximately (r+1)/(r-1) · (N-1)H_{N-1} + o(N), showing near-optimal time scaling.