Hardy-Weinberg Equilibrium and Selection: Mathematical Models in Population Genetics

Hardy-Weinberg Equilibrium and Selection

Definition of Selection

Selection is a fundamental concept in population genetics, describing the process by which certain genotypes increase in frequency due to their impact on survival and reproduction. The concept has evolved from Darwin's intuitive definition to modern, formal mathematical models.

• Natural Selection: Darwin described natural selection as the struggle for life, where variations, however slight, influence survival and reproduction.

• Modern Definitions: Modern genetics uses formal proofs and statistical models to define selection. Key points include:

  • More offspring are produced than resources can support.

  • Individuals differ in their ability to survive, partly due to their genotypes.

  • Genotypes that promote survival in the current environment are more likely to reproduce, causing genotype frequencies to change over generations.

• Fitness: The proportion of individuals surviving to reproduce is quantified as viability, survivorship, or fitness.

Selection in Diploids

In diploid organisms, selection is often measured by the probability that a genotype survives from fertilization to reproductive age. Fitness can be absolute or relative.

• Viability/Survivorship: Probability that a genotype survives to reproductive age.

• Absolute Fitness: The fitness for each genotype equals its probability of survivorship.

• Relative Fitness: One genotype is set as the reference (fitness = 1), and others are scaled relative to it.

Mathematical Models of Selection

Population geneticists use mathematical variables to model selection and its effects on allele frequencies.

• Fitness Coefficients: Denoted as $\omega_{XY}$, where X and Y are allele identifiers (e.g., $\omega_{11}$ for homozygote of allele 1).

• Each genotype receives a separate coefficient of selection.

• Other variables used include $s$ (selection coefficient) and $\omega$ (average fitness).

Discrete Changes in Allele Frequency

Selection alters genotype frequencies according to their relative fitness. Hardy-Weinberg Equilibrium (HWE) provides the baseline frequencies before selection.

• Genotype Frequencies (HWE): $p^2$ (AA), $2pq$ (Aa), $q^2$ (aa), where $p$ and $q$ are allele frequencies and $p + q = 1$.

• Relative Fitness: $\omega_{11}$ (AA), $\omega_{12}$ (Aa), $\omega_{22}$ (aa).

• Post-Selection Frequencies: Multiply pre-selection frequency by relative fitness:

  • AA: $p^2 \omega_{11}$

  • Aa: $2pq \omega_{12}$

  • aa: $q^2 \omega_{22}$

Table: Genotype Frequencies and Fitness

Genotype	Pre-selection Frequency	Relative Fitness	Post-selection Frequency
AA	p2	$\omega_{11}$	$p^2 \omega_{11}$
Aa	2pq	$\omega_{12}$	$2pq \omega_{12}$
aa	q2	$\omega_{22}$	$q^2 \omega_{22}$

Average Fitness of the Population

The average fitness ($\omega$) is the sum of post-selection genotype frequencies:

• $\omega = p^2 \omega_{11} + 2pq \omega_{12} + q^2 \omega_{22}$

• After selection, the sum of genotype frequencies is less than 1, reflecting loss due to selection.

• One relative fitness value is typically 1 (reference), others are less than 1.

Normalization of Genotype Frequencies

To restore the sum of genotype frequencies to 1, post-selection frequencies are normalized by dividing by average fitness:

• AA: $\frac{p^2 \omega_{11}}{\omega}$

• Aa: $\frac{2pq \omega_{12}}{\omega}$

• aa: $\frac{q^2 \omega_{22}}{\omega}$

• Sum: $\frac{p^2 \omega_{11}}{\omega} + \frac{2pq \omega_{12}}{\omega} + \frac{q^2 \omega_{22}}{\omega} = 1$

Allele Frequency After Selection

After selection, gametes are produced according to genotype contributions:

• AA produces all A gametes.

• aa produces all a gametes.

• Aa produces half A and half a gametes.

• Frequency of A after selection ($p'$):

  • AA contributes $\frac{p^2 \omega_{11}}{\omega}$

  • Aa contributes $\frac{pq \omega_{12}}{\omega}$

  • $p' = \frac{p^2 \omega_{11} + pq \omega_{12}}{\omega}$

• Frequency of a after selection ($q'$):

  • aa contributes $\frac{q^2 \omega_{22}}{\omega}$

  • Aa contributes $\frac{pq \omega_{12}}{\omega}$

  • $q' = \frac{q^2 \omega_{22} + pq \omega_{12}}{\omega}$

Change in Allele Frequency

The change in allele frequency ($\Delta p$) quantifies the effect of selection:

• $\Delta p = p' - p$

• General formula:

  • $\Delta p = \frac{pq \left[ p(\omega_{11} - \omega_{12}) + q(\omega_{12} - \omega_{22}) \right]}{\omega}$

Selection Example

Consider a population of 'smoodges' with initial allele frequency $p = 0.500$. The relative fitness values are:

• AA: 0.900

• Aa: 0.950

• After two generations, the frequency of A is 0.474.

Random Genetic Drift

Random genetic drift is another mechanism of allele frequency change, distinct from selection.

• Definition: Random genetic drift is a change in allele frequency over generations due to random sampling.

• It is a sampling process, not directed by fitness.

• Probability of allele loss: If an allele's frequency is 0.7 in a population of 50 individuals, the probability of eventual loss is 0.3 (equal to the frequency of the alternative allele).

Table: Probability of Allele Loss

Allele Frequency	Population Size	Probability of Loss
0.7	50	0.3

Change Over Time

Allele frequencies change over generations due to selection and drift. For example, with fitness values $\omega_{11} = 0.95$, $\omega_{12} = 0.99$, $\omega_{22} = 1.0$, and initial frequency $p_0 = 0.99$, the frequency of allele A will decrease over time.

• Selection acts to reduce the frequency of less fit alleles.

• Drift can cause random fluctuations, especially in small populations.

Additional info: The notes above expand on the mathematical models and provide context for Hardy-Weinberg Equilibrium, selection, fitness, and genetic drift, as relevant to population genetics.