[Paper Review] Coexisting Stable Equilibria in a Multiple-allele Population Genetics Model

Linlin Su, Colbert L Sesanker+1 more|arXiv (Cornell University)|Aug 25, 2011

Evolution and Genetic Dynamics13 references3 citations

TL;DR

This paper classifies all possible coexisting stable equilibria in continuous-time, single-locus population genetics models with three to six alleles under selection. Using analytical methods for three alleles and extensive computer simulations for higher allele counts, the authors identify 14 patterns for the three-allele model, 117 for four alleles, at least 2351 for five alleles, and over 60,000 for six alleles, revealing an exponential increase in pattern complexity with allele number.

ABSTRACT

In this paper we find and classify all patterns for a single locus three- and four-allele population genetics models in continuous time. A pattern for a $k$-allele model means all coexisting locally stable equilibria with respect to the flow defined by the equations $\dot{p}_i = p_i(r_i-r), i=1,...,k,$ where $p_i, r_i$ are the frequency and marginal fitness of allele $A_i$, respectively, and $r$ is the mean fitness of the population. It is well known that for the two-allele model there are only three patterns depending on the relative fitness between the homozygotes and the heterozygote. It turns out that for the three-allele model there are 14 patterns and for the four-allele model there are 117 patterns. With the help of computer simulations, we find 2351 patterns for the five-allele model. For the six-allele model, there are more than 60,000 patterns. In addition, for each pattern of the three-allele model, we also determine the asymptotic behavior of solutions of the above system of equations as $t o \infty$. The problem of finding patterns has been studied in the past and it is an important problem because the results can be used to predict the long-term genetic makeup of a population.

Motivation & Objective

• To classify all possible patterns of coexisting stable equilibria in multiple-allele population genetics models under continuous-time selection.
• To determine the number and configuration of stable equilibria for three-, four-, five-, and six-allele models.
• To understand the long-term genetic dynamics by analyzing the asymptotic behavior of solutions as t → ∞.
• To investigate whether the number of patterns grows exponentially with the number of alleles.
• To provide a comprehensive catalog of stable equilibrium configurations for use in predicting long-term population genetic outcomes.

Proposed method

• Uses the continuous-time replicator system: dp_i/dt = p_i(r_i - r), where r_i is marginal fitness and r is mean fitness.
• Applies linear stability analysis by computing the Jacobian matrix at each equilibrium and evaluating the signs of its eigenvalues to determine stability.
• Employs analytical proofs for the three-allele model, rigorously classifying all 14 possible patterns and their separatrices.
• Uses computer simulations with random fitness matrices satisfying r_11 > r_22 > ... > r_kk to sample equilibrium configurations and count stable equilibria.
• Develops a sorting algorithm to identify and catalog distinct patterns based on the number and location of stable equilibria across all possible equilibria.
• Runs large-scale simulations (up to 400 million iterations on 20 processors) for five- and six-allele models to estimate the total number of unique patterns.

Experimental results

Research questions

• RQ1 How many distinct patterns of coexisting stable equilibria exist in a three-allele population genetics model under selection?
• RQ2 What is the number of stable equilibrium configurations for four-allele and five-allele models, and how do they scale with allele count?
• RQ3 Can computer simulations reliably identify the full set of possible patterns for higher-allele models, and what is the computational complexity involved?
• RQ4 Does the number of patterns grow exponentially with the number of alleles, as suggested by the observed data?
• RQ5 What is the asymptotic behavior of solutions in each pattern, and how do separatrices divide the state space in the three-allele model?

Key findings

• There are exactly 14 distinct patterns of coexisting stable equilibria in the three-allele model, with a complete classification of their asymptotic dynamics and separatrices.
• For the four-allele model, 117 distinct patterns of stable equilibria exist, as determined through computer simulations.
• At least 2351 distinct patterns were identified for the five-allele model, though a few patterns may remain undetected due to simulation limitations.
• Over 60,000 distinct patterns were found for the six-allele model using large-scale parallel simulations.
• The number of patterns appears to grow exponentially with the number of alleles, suggesting a rapid increase in genetic system complexity under selection.
• The study confirms that stable equilibria can coexist in complex configurations, and that the full set of patterns cannot be predicted by simple dominance rules or prior conjectures like Cannings and Vickers’ subset conjecture.