5.1: Complex Traits

Introduction

Many traits are quantitatively inherited. A quantitatively inherited trait is controlled by many genes at different loci, with each gene — known as a polygene — contributing a small effect to the expression of the character. Polygenes are also known quantitative trait loci (QTL). QTLs involved in expression of a quantitative character act cumulatively to determine the phenotype of the trait. Their mode of inheritance is called quantitative genetics. Quantitative genetics describes the connection between phenotype and genotype and provides tools to show how phenotypic selection of complex characters changes allele frequencies.

A quantitative phenotype is usually modeled as a function of its genotype as modified by the environment.

Phenotype = Genotype + Environment + (Genotype x Environment)

\[P = G + E + (G \times E)\]

Some characters are more responsive or sensitive to environmental conditions than others. Qualitative traits such as flower color are not strongly influenced by the environment. On the other hand, quantitative traits such as grain yield or abiotic stress tolerance are influenced markedly by the environment. The degree of sensitivity or the range of potential responses to the environment is determined by the genetic composition of the individual plant or population of plants. Sources of variation include:

• Environmental variation
• Genetic variation and
• Interaction of genetic and environmental variation

 

Inheritance of quantitative traits

Figure 4 The diameter of Gaillardia pulchella flowers is a quantitative trait. Photo by DanielCD; CC-SA 3.0 via Wikimedia Commons.

Inheritance of quantitative traits involves two or more nonallelic genes (multiple genes or polygenes); the combined action of these genes, as influenced by the environment, produces the phenotype. The effect of individual genes on the trait is not apparent. However, early in the 1900s it was discovered that the inheritance of the individual genes contributing to the phenotype of quantitative traits do indeed follow the same Mendelian inheritance principles as simply-inherited genes.

Figure 5 Herman Nilsson-Ehle. Photo licensed under Public Domain via Wikimedia Commons.

In 1909, Herman Nilsson-Ehle, a Swedish geneticist and wheat breeder, conducted some of the classic studies on quantitatively inherited traits in wheat. He developed what is known as the “Multiple Factor” or multi-factorial theory of genetic transmission. A key observation made by Nilsson-Ehle was that although a spectrum of continuous variation in kernel color (a quantitatively inherited trait influenced by environmental factors) could be observed in segregating generations, he was able to determine that segregation for these genes fit a model that each separate contributing gene followed a pattern of Mendelian inheritance.

Figure 6 Wheat kernels. Photo by zandland; licensed under CC-SA 3.0 via Wikimedia Commons.

Bread wheat is a hexaploid — allopolyploid that contains three slightly different, but similar ancestral genomes (referred to as A, B, and D) in its genome (AABBDD). Depending on the cultivars that Nilsson-Ehle studied, each genome had a single gene that affected kernel color, and each of these loci has a red allele (R) and a white allele (r). Alleles at each locus varied slightly in their effect on kernel color, and will be designated in this example by different superscripts, e.g., R¹ or r³.

He crossed two cultivars of wheat that varied in kernel color, one with dark red seeds (homozygous dominant genotype R¹R¹R²R²R³R³, based on the symbols designating the ancestral genomes) and another with white kernels (homozygous recessive r¹r¹r²r²r³r³). He noted that the F₁ of a cross between these parents (heterozygote R¹r¹R²r²R³r³) was intermediate in color (light red), but the F₂ generation could be grouped into seven classes, ranging in color from dark red to white. He explained the distribution on the basis of three pairs of genes segregating independently, with each dominant allele contributing to the intensity of the red color.

Figure 7

Table 1 Kernel color in F₂ progenies from a wheat cross.

F2 genotypes			Color	Number of dominant alleles	Number of plants out of 64
	R^1R^1R^2R^2R^3R^3		dark red	6	1
R^1R^1R^2R^2R^3r^3	R^1r^1R^2R^2R^3R^3	R^1R^1R^2r^2R^3R^3	moderately dark red	5	6
R^1R^1R^2R^2r^3r^3 r^1r^1R^2R^2R^3R^3	R^1R^1r^2r^2R^3R^3 R^1r^1R^2r^2R^3R^3	R^1r^1R^2R^2R^3r^3 R^1R^1R^2r^2R^3r^3	red	4	15
R^1R^1R^2r^2r^3r^3 ​R^1R^1r^2r^2R^3r^3	R^1r^1R^2r^2R^3r^3 ​R^1r^1R^2R^2r^3r^3 R^1r^1r^2r^2R^3R^3	r^1r^1R^2R^2R^3r^3 r^1r^1R^2r^2R^3R^3	light red	3	20
R^1R^1r^2r^2r^3r^3 r^1r^1R^2R^2r^3r^3	r^1r^1r^2r^2R^3R^3 R^1r^1R^2r^2r^3r^3	R^1r^1r^2r^2R^3r^3 r^1r^1R^2r^2R^3r^3	pink	2	15
R^1r^1r^2r^2r^3r^3	r^1r^1R^2r^2r^3r^3	r^1r^1r^2r^2R^3r^3	light pink	1	6
	r^1r^1r^2r^2r^3r^3		white	0	1

With three independent pairs of genes segregating, each with two alleles, as well as environmental effects acting on kernel color, the F₂ progeny would contain 63 plants with varying shades of red kernels and one with white kernels. Linkage among the genes restricts independent assortment, so that the required size of the F₂ population becomes larger.

Figure 8 Range of wheat kernel color in an F₂ generation. (a) Kernel color depicted by seven discrete classes modeled on segregation of three contributing genes, each exhibiting partial dominance. (b) Kernel color depicted by continuous variation in all seven color classes.

Characteristics of Quantitative Traits

Figure 9 Based on a random sample from a genetically mixed population, the distribution of a quantitative trait’s expression approximates a normal or bell curve.

Phenotypic values for the specific trait, resulting from simultaneous segregation of multiple genes, exhibit continuous variation, rather than distinct classes. In general, the distribution of values of quantitatively inherited traits in a population follows a normal distribution (also called a Gaussian distribution or bell curve). These curves are generally characterized by two parameters, the mean and the variance or standard deviation. In the figure below, the mean of the random sample is depicted by the symbol and the standard deviation by the symbol s. If the reference is made to a population instead of a sample from a population, the population mean is usually symbolized by μ and the population standard deviation by σ.

There are three general types of traits that are quantitatively inherited: continuous, meristic, and threshold. An example of the first type, a continuous trait, is fruit width of pineapple. The second type, a meristic character, is a countable trait that can take on integer values only, e.g., number of tillers of maize or branches of a rose bush. The third type of quantitative trait is known as a threshold character or “all-or-none” trait. Such traits are typically ranked simply as presence or absence, e.g., Downy Mildew disease in soybean.

Threshold Traits

Although they have only two phenotypes, threshold traits are considered to be quantitatively inherited because their expression depends on a liability (such as disease susceptibility or tolerance of nicotine levels) that varies continuously. Heritability of these traits is a function of the incidence of the trait in the population, so it is difficult to determine the importance of genetic factors in different environments or in different populations that differ in incidence. Threshold traits are assumed to be represented by an underlying normally distributed “liability trait” that is the sum of the independent genetic and environmental components of the distribution. A disease would have to be present before you could determine if certain genotypes were susceptible or not. For example, plants might be able to tolerate low to moderate levels of nicotine in their tissues until a threshold was crossed, above which the high level of nicotine present would be lethal.

Threshold characters exhibit only two phenotypes — the trait is either present or absent — but the susceptibility to the trait varies continuously and environmental components of the distribution. A disease would have to be present before you could determine if certain genotypes were susceptible or not. For example, plants might be able to tolerate low to moderate levels of nicotine in their tissues until a threshold was crossed, above which the high level of nicotine present would be lethal.

Threshold characters exhibit only two phenotypes — the trait is either present or absent — but the susceptibility to the trait varies continuously.

Environment has a large influence on the trait’s phenotype. That is, for the particular trait, the relative responses of plants change when grown under different environmental conditions.

Distinct segregation ratios of individual nonallelic genes are not observed. Recombination and segregation patterns are based on the combined effect of the polygenes on the trait. The more loci controlling the character, the greater the complexity.

 

Statistical Terms

The study of quantitative traits is sometimes referred to as “statistical genetics” because of its reliance on statistical methods. In order to understand the inheritance of quantitative characters and the methods applied to these characters, it is essential that you become familiar with some of the fundamentals of descriptive statistics. A basic review is provided here.

When using symbols to represent these population parameters, it is important to distinguish between information about the population and that concerning a sample representing the population.

Similar Statistical Symbols

As a shorthand, statistics commonly use symbols to convey concepts. Often there are several symbols that relate to very similar, but slightly differing concepts. Here’s a list of symbols related to means, variance, and standard deviation that you will encounter in this lesson. The latter two parameters describe the variability or dispersion about the mean of the population or sample derived from a population.

Although the differences between these are important from a statistical perspective, they are commonly used synonymously.

Parameter	For a population	For a fixed or selected sample of a population
Mean	or M
Variance	V or
Standard deviation		s

Descriptive Statistics

Range — the lowest and highest phenotypic values in the population or sample for the character.

Mean (µ or M for population; for sample) — describes the average performance of a random sample from a population for a trait. The mean is a measure of central tendency — it does not tell anything about the distribution of individual observations. Mean equals the sum of the trait values of each individual divided by the number of samples (n):

\[\bar{x} = \frac{\Sigma x}{n}\]

Variance (V or σ² for population; s² for sample) — a measure of the scatter or dispersion of phenotypic values. The greater the variability among individuals, the greater the variance. Two populations with the same mean for the same character could differ greatly in their respective variance for that character.

Average the squared deviations from the mean, (X – X)²:

\[V = \sigma^{2} = \frac{\Sigma [(x - \bar{x})^{2}]}{n - 1}\]

Standard deviation (σ for population; s for sample) — also a measure of dispersion around the mean. Standard deviation expresses the dispersion in the same unit as the mean.

Standard deviation is the square root of the variance.

\[\sigma = \sqrt{V} \text{ or } \sqrt{\sigma^{2}}\]

A small variance and a small standard deviation tell us that the phenotypic values are near the mean value. In contrast, a large variance and standard deviation indicate that the trait values have a wide range.

Samples in which the observations are clustered closely around the mean (red) have a smaller variance and standard deviation than observations dispersed widely (blue).

Coefficient of variation (CV) — the standard deviation as a percentage of the mean. Because the units cancel out, CV is a unitless measure.

Divide the standard deviation by the mean and multiply by 100:

\[CV = \frac{\sigma}{\bar{X}} \times 100\]

A CV of about 10% or less is desirable in assessing biological systems. When the CV is greater than 10%, the variability in the sample or the population may be too great to sort out the factors contributing to that variability.

The mean and the variance are used to describe an individual characteristic, but plant breeders are frequently interested in more than one trait simultaneously. Two or more characteristics can vary together, and are thus not independent of one another.

Covariance provides measure of the strength of the correlation or dependent relationship between two or more sets of variables. When two traits are correlated, a change in one trait is likely to be associated with a change in the other trait. Variance is a special case of covariance that is the covariance of a variable with itself. Correlations between characteristics are measured by a correlation coefficient (r).

Covariance — the mean value of the product of the deviations of two random variables from their respective means. For example, the covariance of two random variables, x and y is expressed as:

\[Cov_{x,y} = \frac{\Sigma (x_{i} - \bar{x})(y_{i} - \bar{y})}{n - 1}\]

Covariance of variables x and y

Correlation coefficient (r) — measures the interdependence of two or more variables and is obtained by dividing the covariance of x and y by the product of the standard deviations of x and y. The correlation coefficient can range from -1 to +1.

\[r = \frac{Cov_{x,y}}{S_{x}S_{y}}\]

Correlation is a “scaled” version of covariance and the two parameters always have the same sign—positive, negative, or zero (0). When the sign is positive, the variables are positively correlated; when it’s negative, they are said to be negatively correlated, and when it’s zero, the variable are described as being uncorrelated. But it is important to note that a correlation between variables indicates that they are associated, but it does not imply a cause-and-effect relationship.

References

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Clausen, J., D. D. Keck, and W. M. Hiesey. 1948. Experimental studies on the nature of species. III. environmental responses of climactic races of Achillea. Carnegie Inst. Wash. Publi. 581.

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Falconer, D.S., and T.F.C. Mackay. 1996. Introduction to Quantitative Genetics. 4th edition. Longman Publ. Group, San Francisco, CA.

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Rausher, M.D.. 2005. Example of Clausen, Keck, and Hiesey Experiment, Lecture 1-Lineages, Populations, and Genetic Variation. Online lecture notes from course on Principles of Evolution.

Department of Crop Science, University of Illinois at Urbana-Champaign. Values obtained for protein in the strains selected for oil and the values for oil obtained for the strains selected for protein each generation (1896-2004). 2007.