Branching process models for mutant genes in nonstationary populations

Area of Science:

• Population genetics
• Mathematical biology
• Genetics

Background:

• Deleterious genes reach a population balance through selection and mutation.
• Understanding this balance is crucial, especially when the normal population is not in equilibrium.

Purpose of the Study:

• To explore the stochastic balance of deleterious genes when the normal population is not at equilibrium.
• To derive explicit formulas and recurrence relations for the probability distribution of mutant individuals.
• To compute expectations for genetic models involving various inheritance patterns and linked markers.

Main Methods:

• Modeling new mutations as a Poisson process.
• Utilizing continuous-time branching processes for gene evolution.
• Deriving explicit formulas and recurrence relations for probability distributions.
• Applying Laplace transforms for exponentially growing populations.

Main Results:

• Explicit formulas and recurrence relations for the probability distribution of mutant individuals were derived.
• Expectations for random variables in autosomal dominant and X-linked disease models were computed.
• The framework accommodates haplotype information for linked markers, aiding linkage disequilibrium studies.

Conclusions:

• The study provides a robust mathematical framework for understanding deleterious gene dynamics under non-equilibrium conditions.
• The derived formulas are applicable to various genetic models and population structures.
• This research offers insights into strategies like positional cloning in population isolates.