ZOOL 304

Quantitative Genetics

The following notes (and quoted section headings) are adapted from Chapter 9 of Mark Ridley's textbook, EVOLUTION, 2nd ed. (1996), Blackwell Science, Inc., Cambridge MA. ISBN 0-86542-495-0.

Brief Outline

• Introductory comments
• Overview
• Checklist of important concepts and terms
• Section-by-section comments
  • Beak size in Darwin's finches.
  • Continuous traits, polygenic characters.
  • Genetic and environmental effects.
  • Genetic and environmental variance.
  • Correlation among relatives.
  • Heritability.
  • Response to selection.
  • Non-linear responses.
  • Selection reduces genetic variability.
  • Variation under stabilizing selection.
  • Selection-mutation balance .
  • Conclusion.

Introductory comments

"Given the DNA, you cannot specify the organism.  That is not because we are ignorant about it [although we are], but because all the information necessary to specify an organism is not contained in the DNA.  There is also the complete sequence of environments in which the organism develops." Richard Lewontin (from a video interview, CD accompanying the text EVOLUTION, 2nd ed., by Mark Ridley, 1996.)

Genetics is often studied as if genes, all by themselves, determined traits.  Descriptions of Mendelian genetics tend to concentrate on traits that vary sharply with genotype but are relatively unaffected by ordinary environmental variation.  Many features of biological systems are indeed designed to develop similarly under a variety of ordinary environments (i.e., they are not much disturbed by ordinary environmental perturbations or ordinary variations in genetic context).  We describe such features as being "canalized" (like water confined in a canal, or channel, which can flow only one way).

Because much of biological organization is highly canalized, we often forget how much the environment necessarily contributes to each and every phenotypic trait.  However, in our consideration of the quantitative genetics of quantitative traits, we must pay attention to the fact that phenotypic variation is influenced by both genetic variation and by environmental variation.  Selection acts on variation, but adaptive evolution follows only from that portion of variation which is heritable.

There are several issues of major significance associated with quantitative genetics.

• Quantitative genetics deals only with variance, with the quantitative differences among individuals which appear when genotypes and environments vary.  Highly canalized traits which show no variation among individuals are not the subject of quantitative genetics.

• Quantitative genetics (unlike simple versions of Mendelian genetics) recognizes that phenotypes are influenced by many factors so that particular genotypes (especially particular single-locus genotypes) do not neatly and reliably correspond with particular phenotypes.

• Most selection does not act on genotypes, but on phenotypes.
  • Therefore, the selection coefficient associated with a particular genotype is, at best, a statistical average over the various genetic contexts (haplotypes) and external environments (ecological circumstances) in which each genotype may find itself.
  • All selection coefficients are thus dependent on many factors besides the genotype itself.
     

• Heritability depends on a particular mix of genetic combinations and environments.
  • Measurement of heritability, or of genetic and environmental components for phenotypic traits, gives a description within particular circumstances, not a causal explanation.
  • Heritability is not a fixed property of a trait.
  • Heritability may change if the environment changes or if the gene pool is altered.
  • Many discussions of "heredity vs. environment" or "nature vs. nurture" are misguided.
  • Genetic determinism is a concept riddled with fallacy and paradox.
     

• Heritability can appear counter-intuitive.  "High heritability" does NOT imply "genetically determined".
  • Heritability is not defined for characters which do not vary.
  • Therefore, for characters which are the most strongly "determined" by genetics, which are so strongly canalized that they are unaltered by environmental influence, heritability is UNDEFINED.
     

• Concepts of quantitative genetics are central to artificial selection.  Most of the plant and animal breeds that dominate modern agriculture have resulted from an evolutionary process based on principles of quantitative genetics.

Although "quantitative genetics" is obviously about quantities (e.g., measurable, continuously-variable traits), we shall emphasize qualitative appreciation of the phenomena associated with quantitative genetics.

Overview

• The most essential ideas of this page are additive genetic variance and heritability.  Both of these are formal (mathematical) concepts whose definitions should be appreciated.

Variance is a mathematical statistic, equivalent to the square of the standard deviation. 

• The concept of additive genetic variance plays the same role in quantitative genetics that the concept of alleles plays in Mendelian genetics.

• Other components of measurable phenotypic variance (e.g., variance due to environmental influence, developmental noise, dominance, epistasis, and genotype-environment interaction) have the significant effect of obscuring additive genetic variance.
• But ONLY additive genetic variance contributes to heritability and to the evolutionary response to selection.
   
• By definition, "narrow sense" heritability h² is the proportion of phenotypic variance which is additive genetic variance, h² = V_A / V_P.
   
• Additive genetic variance is thus that component of total phenotypic variance which is heritable and which will respond to selection.
  • Unfortunately, there is no direct way to measure additive genetic variance.  Additive genetic variance must be calculated from its consequences.
     
• The response to selection R is equal to the heritability h² times the selection differential S, or
  R = h² S.
   
• The action of selection is expected to reduce variability (and hence reduce heritability) over time.
• However, high variance with significant heritability remains oan bservable fact for many quantitative traits that are under selection pressure.
• Explaining this mismatch between expectation and observation remains a challenging problem.

CHECK LIST of important CONCEPTS and TERMS

• Variance
• Phenotypic variance
• Additive genetic variance
• Heritability
  • h² = V_A / V_P
  • h² = R / S

• Selection differential
• Response to selection, R = h² S

Section-by-Section Comments

"Climatic changes have driven the evolution of beak size in one of Darwin's finches."

• Quantitative genetics belongs largely to the domain of artificial selection.
• In natural populations, it is difficult to obtain direct measures of either heritability or of selective differential.
• The evolutionary response of beak size in one of Darwin's finches, Geospiza fortis, provides one of the few good examples from nature of selection operating on a quantitative trait.

"Quantitative genetics is concerned with characters controlled by large numbers of genes."

• The subject of quantitative genetics is continuous variation in quantitative traits or polygenic characters.
• A particular value for a quantitative trait can be affected by both genes and environment.
• Realize that quantitative genetics presumes large numbers of genes controlling quantitative traits, but the actual number of such genes usually remains unknown.

"Variation is first divided into genetic and environmental effects."

• This distinction is made on principle. How it is done in practice will be discussed later.
• Additive genetic variance is the one source of variance that is heritable and upon which selection can operate.
• The idea of dominance effects not being heritable may at first seem counterintuitive.  Here is an example to make it perhaps seem more sensible.
  • Consider a population in Hardy-Weinburg equilibrium with A and a both equal to 0.5, and with dominant and recessive phenotypes being assigned values of 1 and 0 respectively.
  • The dominant : recessive phenotype frequencies will be 3:1, so the mean phenotype will be [(3 x 1) + (1 x 0)] / 4 = 0.75
  • Now consider a group of heterozygotes, Aa.  Since they all have the dominant phenotype of 1, they will all vary from the population mean by 0.25.  That is, the mean variance of this set of heterozygotes is 0.25
  • However, if these heterozygotes breed, they will produce offspring in the standard 3:1 ratio with mean phenotype of 0.75, the same as the original population.
  • Thus, although the mean variance of this group of parents differed from the whole population by 0.25, the mean variance of their offspring is zero.  That is, on average, the offspring do not differ from the larger population.
  • Therefore, the mean variance of the heterozygote parents is completely lost in their offspring.
  • Thus the dominance effect which caused the phenotypic variance in the heterozygotes is not heritable.  (This is not a biological fact, just a mathematical effect created by the mathematical definition of heritability.)

"The variance of a character is divided into genetic and environmental effects."

• Variance is calculated as:
   
  V_x = 1/(n-1) {summation over i from i=1 to n}(x_i - mean x)²
  (The expression in braces {} is normally symbolized by a capital Sigma.)
   
• You may be more familiar with standard deviation as a measure of variation.  By definition, standard deviation is the square-root of variance.

"Relatives have similar genotypes, producing the correlation between relatives."

• Correlation among relatives is one way to calculate V_A, additive genetic variance.
• V_A = twice the covariance between parents and offspring.
  Caution:  This relation holds only if the environmental variation is not also correlated between parents and offspring.  Otherwise, trait covariance between parent and offspring may be due to environmental covariance.
• Correlation of environments is a confounding consideration, whenever genetic correlation is being studied.

"Heritability is the proportion of phenotypic variance that is additive."

• h² is the symbol used to designate heritability.
• By definition, "narrow sense heritability" h² = V_A / V_P.

• Heritability may also be defined more intuitively, as "the quantitative extent to which offspring resemble their parents, relative to the population mean."  Again, caution.  This relation is true only if the environments of parents and offspring are not correlated.  Again, correlation of environments is a confounding consideration, whenever genetic correlation is being studied.
• It is normally very difficult to disentangle environmental correlation from additive genetic variance in natural populations, making heritability practically impossible to determine without careful experimental manipulation.
• Caution:  Heritability is UNDEFINED for characters which do not vary.  Therefore, characters which are the most strongly "determined" by genetics, which are so strongly canalized that they are unaltered by environmental influence, have NO heritability.

"A character's heritability determines its response to artificial selection."

• The response to artificial (experimental) selection offers the most reliable way to determine heritability.
• In artificial selection, the selection differential for a trait is defined as the difference between the mean of the selected individuals (those which breed the next generation) and the mean of the entire population from which those individuals were selected.
• Similarly, the response to selection is defined as the difference between the offspring mean and the mean of the entire population from which their parents were selected.
• The response to selection R is equal to the heritability h² times the selection differential S, or
  R = h² S.
• This equation tells us how the response to selection depends on h².
• But, since R and S can be directly measured in artificial selection experiments, this equation also gives us another way to determine h².
  h² = R / S

"The relation between genotype and phenotype may be non-linear, producing remarkable responses to selection."

"Selection reduces the genetic variability of a character."

• This section translates the basic ideas from the previous sections into a qualitative description of how selection on quantitative traits is expected to work in nature.
• One standard expectation is for genetic variability to be reduced by consistent directional or stabilizing selection for quantitative polygenic traits.
• Selection reduces heritability.  As less fit genotypes are eliminated by selection, the variance which defines heritability is eliminated.  Therefore, one result when selection establishes the "genetic determination" of an adaptive trait, is that the heritability of that trait disappears.  (If this sounds perverse, bear in mind that it is not a paradox but merely a consequence of the formal definition for heritability.)

"Characters in natural populations subject to stabilizing selection show genetic variation."

• More accurately, SOME (or MANY) characters in natural populations subject to stabilizing selection show genetic variation.
• Nevertheless, this observable fact appears to contradict the conclusion of the preceding section (i.e., selection acts to reduce variation).
• Resolution of this problem remains controversial.  There are several possible explanations.
  • Mutation-selection balance would requires an extremely high mutation rate.
  • Another resolution involves shifting selection pressure, frequency dependent selection, or heterozygote advantage.  (The same suspects are used to explain abundant polymorphism at Mendelian loci.)
  • A third possibility involves a role for repetitive DNA sequences as intrinsically mutable loci influencing quantitative traits.  
    • This idea remains untested and is too new for the textbooks.
    • For more information (if you're interested) see:
      Kashi, King (that's me) and Soller (1997)  Simple sequence repeats as a source of quantitative genetic variation", Trends in Genetics 13:74-78.

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