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Genetic Review & Hardy-Weinberg

Sinervo©1998

1. The Neo-Darwinian Synthesis

2. Review of genetic variation

The Genotype and Phenotype

The Origins of Variation

Mutation, Segregation, and Recombination

Components of Phenotypic and Genetic Variation

Polygenic Traits

3. Hardy-Weinberg Equilibrium

A. Definitions

B. Notation

C. Algebra

D. Model

E. Numerical Example

F. Summary of H-W Predictions

4. Assumptions underlying the model


1. The Neo-Darwinian Synthesis

A list the major players and a rescue of Darwin's theory:

  1. Ronald A. Fisher -- cumulative medelian factors, this idea lead to a reconciliation of mendelian genetics and biometrical genetics
  2. See Punnet square of the most extreme case a heterozygote that has been blended away. Do a single gene with two alleles,
  3. do a Punnet square of the most extreme case a heterozygote that has been blended away and two locus case.
  4. concurrent developments by Sewall Wright and JBS Haldane solidified it:
  5. Sewall Wright believed genetic drift to be important. Fisher dismissed this and believed evolution primarily occurred in large populations.
  6. JBS Haldane worked out many aspects of selection
  7. Ernst Mayr (evolution in isolated populations leads to speciation)
  8. Stebbins (plants)
  9. Dobzhansky (experiments on selection, genetics using Drosophila)
  10. George Gaylord Simpson (more on speciation and the fossil record)
  11. Watson and Crick and the discovery of DNA
  12. Moto Kimura proposes the Neutral theory of evolution -- small changes in the DNA with no effect. The neutral theory of evolution provides the theoretical cornerstone to Molecular Evolution and the analysis of phylogenies. What is required in an analysis of phylogenies is a molecular clock.


2. Review of genetic variation

The Genotype and Phenotype

Genotype -- the sum total of all the alleles at all the loci in a organism. While precise and concise, this definition is not very useful. The genotype may also apply to genetic material at a single genetic locus and describe the alleles that an individual possesses at a particular locus. The two alleles at a genetic locus come from the male and female parents and form the basic unit of genetic variation in an individual. The process of meiosis, mendelian segregation, and recombination among all the myriad of genetic loci in an individual effectively makes each sexually-produced individual unique (see Side Box 2.1. Mutation, Segregation and Recombination). In contrast, an asexual organism is a genetic clone of its female parent. The genotype is largely static for an individual during its lifetime, except when a mutation occurs. A mutation is a lesion of DNA that changes the genetic material in one allele at a locus. If the mutation occurs in a somatic cell, there will be no consequence for transmission of material across generations. However, if the mutation occurs in a cell of the germ line, which produces sperm or eggs for sexual reproduction, then the mutation can be transmitted across generations. Mutations are the ultimate source of all genetic variation.

Phenotype -- the external expression of the genes, and the result of a gene's interaction with the environment. This includes mechanisms of development. For simplicity of analysis, we break the whole organism into phenotypic traits that are largely functional units. For example, the number of offspring that an animal produces in one bout of reproduction, termed fecundity, is a phenotypic trait related to reproduction. However, phenotypic traits are usually correlated with other phenotypic traits. For example, the number of offspring that a parent produces is related to the kind or quality of care that the parent can provide to the offspring. Usually we refer to such relationships as fitness trade-offs because an increase in one fitness trait (fecundity) has an impact on a correlated trait (quality of care). The proximate mechanisms that link these particular traits are related to energy and metabolism. Rearing large numbers of offspring requires more total energy to keep the level of care constant for each offspring. Alternatively, less energy (or lower quality care) is available for each offspring.

Environment -- the environment is anything external to the genetic material. For example, food availability in part determines body weight. In animals that lay eggs without parental care, the eggs are subjected to environmental variation in the form of temperature, hydration, or perhaps salinity. In animals which lay eggs and have extended parental care in a nest, the environment is a function of the parents and perhaps the other sib-mates in the nest. In animals with internal fertilization and some kind of gestation, the mother's physiology per se becomes an important component of the offspring's environment. Regardless of whether or not an organism requires extended parental care as a juvenile, the environment still plays a major role in that individual's development. Of course, there is a second sense in which the environment interacts with the genotype and phenotype: the environment causes natural selection. While this natural selection acts on phenotypes within a generation (e.g., the parents, its effects are transmitted to the next generation (e.g., offspring).

A simple expression describes the relationship between the variation in a particular phenotypic trait among individuals found in a single population:

P = G + E,

where P, G, and E are phenotypic, genotypic, and environmental variation in the trait. If the environmental variation is large, then little phenotypic variation arises due to genetic sources of variation. Conversely, if genetic variation is large, then the phenotypic trait is largely determined by genetic factors. As we have seen, heritable or genetic variation is central to Darwin's theory of natural selection, and evolutionary change will occur more rapidly for traits that are strongly determined by genetic factors.

The Origins of Variation

Mutation, Segregation, and Recombination

The process of natural selection is often characterized as "blind" and this is largely because the source of all genetic variation is largely a random processes. The ultimate source of genetic variation is mutations; however, segregation, and recombination provide an stochastic mechanisms for randomizing genetic variation in a population. While mutations alter DNA by changing base pairs, segregation and recombination do not alter the material content of DNA. Instead they provide powerful mechanisms for mixing up the DNA during sexual reproduction.

Mutation

The process of mutation is probabilistic. We describe the process in terms of the probability of a mutation occurring in an individual during its lifetime. Typical rates of mutation are between 1 in 10,000 (10-4) and 1 in 1,000,000 (10-6) for many organisms. Either this means that a long time must pass before a mutation occurs in a population, or the population must be very large. The population must number in excess of 106 or one million members in order to see an average of one mutation per generation for a gene with a mutation rate of 10-6. Most mutations are detrimental, and perhaps only 1 in 1,000 is beneficial. Thus, in this population of a million we might have to wait for one thousand years for a specific genetic locus to throw us a beneficial mutation (of course there are thousands of possible loci in an organism so the waiting time for any beneficial mutation is less).

Segregation

Genes are found on chromosomes. The first random event in sexual reproduction occurs during meiosis when homologous pairs of chromosomes line up on the equatorial plate in preparation for the cell division that reduces a diploid (2N) cell of the germ line to an haploid (N) gamete. During the process of segregation different homologues will randomly be distributed between the two daughter chromosomes, gametes end up with very different chromosome complements. Given that the total number of chromosomes is C, then there are 2 raised to the power of C different gametes that could be produced from segregation of chromosomes (in humans that would be 246). Segregation can produce a vast number of different gametes and is a potent mixer of variation in sexual organisms.

Recombination

Recombination between homologous pairs of chromosomes during meiosis can generate even more gamete types. The chromosome is loaded with recombinational hotspots. Any given chromosome will typically recombine with its homologue at one or two points during a given meiotic event (recombination rate on a given chromosome is a function of chromosome length). However, the recombination points for two different meiotic events can be different. Given that gametes such as sperm are generated by millions of different meiotic events, the number of potential gamete types from a single parent is vast. If one considers all possible parents in a modest-sized breeding population such as humans (5 billion), there are easily more potential recombination products than molecules in the known universe.

Components of Phenotypic and Genetic Variation

We can follow a roadmap from genotype to phenotype, but the route is rarely a one to one mapping between genes and phenotypic traits. Environment further obscures the relationship between genotype and phenotype. We have several hierarchical concepts that describe how genotype can be related to phenotypic traits. These additional terms describe different levels of genetic interaction and their effect on genotype and phenotype:

Pleiotropy -- a single gene that has an effect on the expression of two or more phenotypic traits is said to have a pleiotropic effect on the traits. For example, testosterone controls the development of what are referred to as secondary sexual characteristics (e.g., a lion's mane), but testosterone also relates to behavioral traits like aggression. Thus, a gene that controls the levels of testosterone would have a pleiotropic effect on the expression of many secondary sexual traits which are morphological, as well as behavioral. The concept of pleiotropy is intimately related to the concept of trade-off (Stearns 1976). Pleiotropy describes the proximate genetic source for many phenotypic trade-offs. If one gene controls the expression of two or more traits and those traits are related to a fitness trade-off, then we have identified the proximate source of the trade-off.

Dominance. Interactions between alleles at a single locus are termed dominance interactions. For example, if an allele is said to be recessive to another allele, then an individual that possesses a copy of the dominant allele and a copy of the recessive allele (e.g., heterozygote) will be phenotypically identical to an individual that possesses two copies of the dominant allele (e.g., homozygous). The "recessive phenotype" is only expressed if the individual is homozygous for two of the recessive alleles. If two genes are co-dominant then the heterozygote is intermediate between the two homozygous genotypes and the alleles are additive in their effect.

 

Co-dominant alleles

In the case of a co-dominant set of alleles, knowledge of genotype allows us to perfectly predict the value of the phenotype. A regression (red line) of phenotypic value on the number of A alleles or the genotypic value yields a perfect predictive line.

 

Dominant and recessive alleles

In the case of a dominant A allele, knowledge of genotypic value allows us to only imperfectly predict the value of the phenotype. Because heterozygotes and dominant homozygotes have the same phenotypic value, the fit between the regression (red line) of phenotypic value on genotypic value has error (black arrows). There is still a significant regression slope showing that some of the variation is due to additive genetic causes (red line) and some is due to dominance variation (black arrows).

 

Overdominance

In the case of overdominance, knowledge of genotype does not allow us to confidently predict the value of the phenotype. There is no slope to the regression line so none of the variation is due to "additive genetic effects" of alleles. All variation in our model is due to error in our model which arises from dominance variation (black arrows). Consider two parents that are homozygous for alternative alleles (aa and AA). If behavior were strongly determined by an overdominant allele then the heterozygous progeny would not at all resemble the homozygous parents.

 

Underdominance

Likewise, in the case of underdominance, knowledge of genotype does not allow us to predict the value of the phenotype with any confidence. There is no slope to the regression line so none of the variation is due to "additive genetic effects" of alleles. All variation in our model is due to error in our model which arises from dominance variation (black arrows).

Polygenic Traits

If two or more genes are responsible for a single trait, the phenotypic trait is said to be governed by polygenic factors. For example, growth rate is undoubtedly caused by a number of genes that act in a complex cascade. Body size, which is the result of a large number of genes, is polygenically determined. Genes that control growth hormone have a large effect on body size. Likewise, genes that control sex steroids like testosterone have some effect on body size, especially during maturation phases of growth. Undoubtedly, many genes that influence metabolism have a small, but measurable effect on body size.

Let us assume that alleles at both loci additively act in the following fashion. Whereas the allele a adds nothing to phenotype, the alternative allele A increments the phenotype by one unit. Likewise, allele b adds nothing to phenotype, the alternative allele B increments the phenotype by one unit. A Punnet square which describes the union of all possible gametic types yeilds 16 genotye combinations and five phenotype combinations (see Table 2.3).

 

 AB

 Ab

 aB

 ab

 AB

 AABB = 4

 AABb = 3

 AaBB = 3

 AaBb = 2

 Ab

 AABb = 3

 AAbb = 2

 AaBb = 2

 Aabb = 1

 aB

 AaBB = 3

 AaBb = 2

 aaBB = 2

 aaBb = 1

 ab

 AaBb = 2

 Aabb = 1

 aaBb = 1

 aabb = 0

Table 2.3. Punnett Square for the effect of two loci which have purely additive effects.

 

Effects of Environment on Phenotypic Variation

Phenotype distribution of a trait governed by two mendelian loci that act additively to produce the phenotype. If the phenotype is not affected by environmental factors then 5 distinct modes are present. As phenotype becomes more and more affected by random environmental factors, the phenotype distribution becomes more and more normally distributed.



3. Hardy-Weinberg Equilibrium

 

We are about to embark on an investigation of a field of evolutionary biology called "microevolution". Microevolution is concerned with understanding the mechanisms responsible for imparting evolutionary change within populations over relatively short time periods. Short time periods mean typically several to 10's to 1,000's of generations. It differs from "macroevolution", which deal with evolution occurring over ten to hundreds of millions of years.

According to most evolutionary biologists, macroevolution is simply the long-term consequences of microevolutionary processes. Microevolution occurs by four mechanisms: natural selection, random genetic drift, mutation, and gene flow (or migration). The foundation upon which these processes must be understood is the Hardy-Weinberg-Castle equilibrium. Under the Hardy-Weinberg equilibrium, we assume that these other forces do not operate. H-W is really a null model of how the world works, and assumes only random mating. Including the other four mechanisms, generates more complex alternative models.

A. Definitions

Consider a single locus with two alleles; A and a. Three genotypes are possible AA, Aa, aa. let:

nAA = number of AA homozygotes

nAa = number of Aa heterozygotes

naa = number of aa homozygotes

Computing Genotype frequencies:

D = nAA/N

H = nAa/N

R = naa/N,

where D + H + R = 1, and N = no. of individual

2. Allelic frequencies (what we can infer) (often mistakenly referred to as gene frequencies).

p = freq of (A) = f(A), q = freq of (a) = f(a)

B. Inferring Allele frequencies from Genotype Frequencies

C. Algebra underlying Hardy-Weinberg Equilibrium

If we know current genotype we can predict genotype in next generation, assuming random mating.


1. Genotype AA (all matings that can produce AA)
mating AAxAA AaxAA AAxAa AaxAa
frequency DxD HxD DxH RxR
fraction of offspring = AA 1 1/2 1/2 1/4


D' = DD+ 1/2 DH + 1/2 DH + 1/4 HH2
D' = (D + 1/2 H) * (D + 1/2 H)
 
D' = p * p
D' = p2

2. Genotype Aa (all matings that can produce Aa)
mating AAxaa aaxAA AAxAa AaxAA Aaxaa aaxAa AaxAa
frequency DR DR DH HD HR RH HH
fraction of offspring = Aa 1 1 1/2 1/2 1/2 1/2 1/2

H' = 2 DR + DH + HR + 1/2 HH
H' = 2 (DR + 1/2DxH +1/2HxR + 1/4HxH)
H' = 2 (D+ 1/2 H) (R + 1/2 H)
H' = 2 ( p ) (q)
3. Genotype aa = same as AA above; = q2


D. Model

E. Numerical Example

Given genotypes, what will genotype freq be @ H-W?

F. Summary of H-W Predictions



4. Assumptions underlying the model

1. infinite population size (no genetic drift)

2. no selection

3. no mutation

4. no migration (immigration or emigration)

5. random mating

In the next lecture we will see what happens when we relax the assumptions, Darwinian Medicine



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