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Basic Modelling in Quantitative Genetics

Basic Modelling in Quantitative Genetics


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I am pretty bad in thinking quantitative genetics models. I am trying to get some basic understanding of modelling the evolution of a quantitative trait. I am therefore asking for help to analyze a very simple model. I welcome any explanation of some other classic model of the evolution of quantitative traits.

Scenario

Consider a haploid population of constant size $N$. The fitness of the individuals is determined exclusively by a single quantitative trait $z$. $l$ loci codes for this trait. The genetic value at each locus adds up to give the quantitative trait $z$. The mutation rate at each locus is $mu$. A mutation changes the genetic value at any given locus by $+ Delta z$ with probability $frac{1}{2}$ and by $- Delta z$ with probability $frac{1}{2}$. If you prefer to consider the effects of a mutation to be drawn from a normal distribution with mean $=0$ and variance $= sigma^2$, please feel free to do so. The fitness $w(z)$ of an individual is given as a gaussian function of its quantitative trait $z$

$$w(z)=expigl(-Vz^2igr),$$

where $V$ is the strength of selection. The optimal phenotype is $0$, that is $w(0)=1$. The larger is $V$, the more severe it is to be at a given distance to the optimal phenotype. If you want to assume that the fitness $w(z)$ is some other function of the trait $z$, such as the even more simple $w(z)=z$, please feel free to do so. For simplicity, we assume no environmental variance on the quantitative trait.

Questions

What is the equilibrium mean fitness $ar w$ of the population?

What is the equilibrium genetic variance (=phenotypic variance) for fitness $w$ and for phenotype $z$?


Trait z is represented by k genes: z1… zk (I am using k instead of l because the former is visually differentiable from 1 ). For simplicity let's assume that there is only one mutable site in a gene. So a mutation can impart a change of $pm Delta z$. Starting from initial state of z at zero, the system will proceed to equilibrium where the rate of forward mutation would be same as that of backward mutation; since the rates are same the deterministic steady state of z should be zero.

A mutation event can also be considered a binomially distributed RV. After n events, the mean number of forward mutations would be 0.5×n.

 Mean fitness = mean forward mutation - mean backward mutations = 0 

You can also model the mutation events as a simple random walk. The mean of that is 0 and the variance is n. But what about large n; I am not sure. You should read about random walks or ask in CrossValidated. With $t o infty$, you would get a lot of variants but variants would be filtered out by selection (as the optimum is at 0). I think selection can be incorporated in the random walk with death rate being proportional to the distance from 0. But again I do not have much experience with such hybrid models.

I tried a simulation with n=10000000 (using Monte Carlo); different colours denote different runs (or in another words different loci). Y-axis denotes the value of z.


The simplest way to obtain Brownian evolution of characters is when evolutionary change is neutral, with traits changing only due to genetic drift (e.g. Lande 1976) . To show this, we will create a simple model. We will assume that a character is influenced by many genes, each of small effect, and that the value of the character does not affect fitness. Finally, we assume that mutations are random and have small effects on the character, as specified below. These assumptions probably seem unrealistic, especially if you are thinking of a trait like the body size of a lizard! But we will see later that we can also derive Brownian motion under other models, some of which involve selection.

Consider the mean value of this trait, $ar$, in a population with an effective population size of Ne (this is technically the variance effective population ) 2 . Since there is no selection, the phenotypic character will change due only to mutations and genetic drift. We can model this process in a number of ways, but the simplest uses an "infinite alleles" model. Under this model, mutations occur randomly and have random phenotypic effects. We assume that mutations are drawn at random from a distribution with mean 0 and mutational variance &sigmam 2 . This model assumes that the number of alleles is so large that there is effectively no chance of mutations happening to the same allele more than once - hence, "infinite alleles." The alleles in the population then change in frequency through time due to genetic drift. Drift and mutation together, then, determine the dynamics of the mean trait through time.

If we were to simulate this infinite alleles model many times, we would have a set of evolved populations. These populations would, on average, have the same mean trait value, but would differ from each other. Let&rsquos try to derive how, exactly, these populations 3 evolve.

If we consider a population evolving under this model, it is not difficult to show that the expected population phenotype after any amount of time is equal to the starting phenotype. This is because the phenotypes don&rsquot matter for survival or reproduction, and mutations are assumed to be random and symmetrical. Thus,

Note that this equation already matches the first property of Brownian motion.

Next, we need to also consider the variance of these mean phenotypes, which we will call the between-population phenotypic variance (&sigmaB 2 ). Importantly, &sigmaB 2 is the same quantity we earlier described as the &ldquovariance&rdquo of traits over time &ndash that is, the variance of mean trait values across many independent &ldquoruns&rdquo of evolutionary change over a certain time period.

To calculate &sigmaB 2 , we need to consider variation within our model populations. Because of our simplifying assumptions, we can focus solely on additive genetic variance within each population at some time t, which we can denote as &sigmaa 2 . Additive genetic variance measures the total amount of genetic variation that acts additively (i.e. the contributions of each allele add together to predict the final phenotype). This excludes genetic variation involving interacions between alleles, such as dominance and epistasis (see Lynch and Walsh 1998 for a more detailed discussion) . Additive genetic variance in a population will change over time due to genetic drift (which tends to decrease &sigmaa 2 ) and mutational input (which tends to increase &sigmaa 2 ). We can model the expected value of &sigmaa 2 from one generation to the next as (Clayton and Robertson 1955 Lande 1979, 1980) :

where t is the elapsed time in generations, Ne is the effective population size, and &sigmam 2 is the mutational variance. There are two parts to this equation. The first, ((1-frac<1><2 N_e>)E[sigma_a^2 (t)]), shows the decrease in additive genetic variance each generation due to genetic drift. The rate of decrease depends on effective population size, Ne, and the current level of additive variation. The second part of the equation describes how additive genetic variance increases due to new mutations (&sigmam 2 ) each generation.

If we assume that we know the starting value at time 0, &sigmaaStart 2 , we can calculate the expected additive genetic variance at any time t as:

$ E[sigma_a^2 (t)]=<(1-frac<1><2 N_e>)>^t [sigma_^2 - 2 N_e sigma_m^2 ]+ 2 N_e sigma_m^2 label<3.3>$

Note that the first term in the above equation, (<(1-frac<1><2 N_e>)>^t), goes to zero as t becomes large. This means that additive genetic variation in the evolving populations will eventually reach an equilibrium between genetic drift and new mutations, so that additive genetic variation stops changing from one generation to the next. We can find this equilibrium by taking the limit of Equation ef <3.3>as t becomes large.

Thus the equilibrium genetic variance depends on both population size and mutational input.

We can now derive the between-population phenotypic variance at time t, &sigmaB 2 (t). We will assume that &sigmaa 2 is at equilibrium and thus constant (equation 3.4). Mean trait values in independently evolving populations will diverge from one another. Skipping some calculus, after some time period t has elapsed, the expected among-population variance will be (from Lande 1976) :

Substituting the equilibrium value of &sigmaa 2 from equation 3.4 into equation 3.5 gives (Lande 1979, 1980) :

Thie equation states that the variation among two diverging populations depends on twice the time since they have diverged and the rate of mutational input. Notice that for this model, the amount of variation among populations is independent of both the starting state of the populations and their effective population size. This model predicts, then, that long-term rates of evolution are dominated by the supply of new mutations to a population.

Even though we had to make particular specific assumptions for that derivation, Lynch and Hill (1986) show that Equation ef <3.6>is a general result that holds under a range of models, even those that include dominance, linkage, nonrandom mating, and other processes. Equation ef <3.6>is somewhat useful, but we cannot often measure the mutational variance &sigmam 2 for any natural populations (but see Turelli 1984) . By contrast, we sometimes do know the heritability of a particular trait. Heritability describes the proportion of total phenotypic variation within a population (&sigmaw 2 ) that is due to additive genetic effects (&sigmaa 2 ):

We can calculate the expected trait heritability for the infinite alleles model at mutational equilibrium. Substituting Equation ef<3.4>, we find that:

Here, h 2 is heritability, Ne the effective population size, and &sigmaw 2 the within-population phenotypic variance, which differs from &sigmaa 2 because it includes all sources of variation within populations, including both non-additive genetic effects and environmental effects. Substituting this expression for &sigmaw 2 into Equation ef<3.6>, we have:

So, after some time interval t, the mean phenotype of a population has an expected value equal to the starting value, and a variance that depends positively on time, heritability, and trait variance, and negatively on effective population size.

To derive this result, we had to make particular assumptions about normality of new mutations that might seem quite unrealistic. It is worth noting that if phenotypes are affected by enough mutations, the central limit theorem guarantees that the distribution of phenotypes within populations will be normal &ndash no matter what the underlying distribution of those mutations might be. We also had to assume that traits are neutral, a more dubious assumption that we relax below - where we will also show that there are other ways to get Brownian motion evolution than just genetic drift!

Note, finally, that this quantitative genetics model predicts that traits will evolve under a Brownian motion model. Thus, our quantitative genetics model has the same statistical properties of Brownian motion. We only need to translate one parameter: &sigma 2  = h 2 &sigmaw 2 /Ne 4 .


Lecture 2: Basic Population and Quantitative Genetics - PowerPoint PPT Presentation

Lecture 2: Basic Population and Quantitative Genetics Allele and Genotype Frequencies Hardy-Weinberg Gametes and Gamete Frequencies Linkage disequilibrium . &ndash PowerPoint PPT presentation

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Macroevolution, Quantitative Genetics and

Abstract

Quantitative genetics provides theory of the evolutionary processes that govern the evolution of phenotypic traits within and between populations. In principle, the same theory should govern the long-term evolution of traits in diverging species. However, non-population processes such as punctuated equilibrium, species selection, and evolutionary constraints may shape macroevolutionary patterns more than population-level processes do. This article reviews the ways in which the predictions from quantitative genetics have been used to test macroevolutionary patterns and explores questions that remain to be addressed about the relationship between these two evolutionary scales.


The Genetic Model for Quantitative (Polygenic) Traits

Making genetic progress in the meat goat business requires the breeder to identify genotypes of individuals for loci of interest and select those individuals with the most favorable genotypes. In selecting for polygenic traits such as weaning weight, growth rate, internal parasite tolerance, identifying specific genotypes is out of the question.

The goat breeder must therefore try to identify breeding values of individuals for traits of importance and to select those individuals with the best breeding values. This is not as straight forward as it may seem because a breeding value is an abstract, mathematical idea. It can never be measured directly, and because it is a relative concept its numerical value depends upon the breeding values of all other individuals in the population, e.g. herd or breed.

To understand breeding values we need a conceptual framework or a model in order to grasp definitions in a logical and consistent way. This framework is designed to be used with quantitative traits: traits in which phenotypes show continuous and numerical expression. The basic genetic model for quantitative traits can be written as follows:

  • Phenotypic value is an individual performance record. It is the measure of a meat goat’s own performance for some specific trait.
  • Genotypic value refers to the effect of the goat’s genes --- singly and in combinations, on its performance for the trait. Unlike phenotypic value, it is not directly measurable.
  • Environmental effect is comprised of all non-genetic factors influencing an individual’s performance for a particular trait and is not directly measurable.

Those of you who are somewhat familiar with this concept may not understand the purpose of including the population mean. The purpose for adding the mean is to emphasize that in goat breeding, genotypic values and environmental effects, and all the other elements of the framework or model are relative — relative to that population being considered. They are not absolutes. Their numerical values depend on the average performance of the population, and they are therefore expressed as deviations for the population mean. Although we might put numerical values in hypothetical situations, in reality we cannot know an individual’s genotypic value or environmental effect. All that can be measured directly is the phenotypic value.

Example 1. A young buck weighed 40.5 pounds at weaning. The population mean was 45 pounds. Therefore the phenotypic value deviation is a minus 4.5 pounds, or 10 percent below the mean…..not good. This might prevent the buck from being selected to produce offspring, which would be appropriate if he is genetically inferior. It is conceivable however that the genotypic value of this buck is 2.5 pounds above average, but his actual performance is below average due to a very poor environment which put a 16 percent drag on the buck. Perhaps his mother developed mastitis in one half of her udder. In that case we might have overlooked a genetically superior sire because of poor animal husbandry.

Example 2. Another buck was in a population in which mean postweaning average daily gain equals .50 pounds per day. This guy recorded an ADG of .45 lb per day. The environmental effect accounted for .04 lbs of this negative deviation and his genotypic value accounted for .01 pounds. In the end this buck should not be selected for breeding because although he experienced worse than average environmental effect, he also had a lower than average genotypic value for ADG.

Example 3. Our final buck weighed 100 pounds at seven months of age which was about 20 percent above the population mean of 83 pounds. Obviously this buck should be considered as herd sire material and be put in the “for sale pen” of higher priced bucks. What we do not know at this point is that the genotypic value of this buck is only 6 percent above the population mean, and that a positive environmental effect added an extra 14 percent to allow him to reach the 20 percent superiority. Although he has a higher than average genotypic value, the rate of growth of his offspring may be less than expected.

The breeding value of an individual is contained in the genotypic value, which includes independent gene effects (additive effects or breeding value) which are passed along to the next generation, and gene combination effects which are not passed because the law of recombination breaks up the combinations on the way to the next generation.

One can think of genotypic value as the value of an individual’s genes to its own performance, and breeding value as the value of an individual’s genes to its progeny’s (offspring) performance.

Fundamental genetics for understanding what is happening in cells

During the late 19th and early 20th century, the Austrian monk Gregor Mendel and later on several of his successors identified the fundamental mechanisms and rules of inheritance. This knowledge allows you to better comprehend the important concepts of goat breeding. These concepts apply generally to meat goats like they do to other livestock. There is no evidence that goats are an exception to any of the basic biological principals. Appendix B provides the reader with detailed information on these basic concepts, and acquaint the learner with the useful jargon of genetics.


Short course in EVOLUTIONARY QUANTITATIVE GENETICS

The course:
This course will give a comprehensive review of modern concepts in Evolutionary Quantitative Genetics. The contents of the course are basic statistics, population genetics, quantitative genetics, evolutionary response in quantitative traits, estimating the fitness of traits and mixed models and their extensions.

Course dates:
31 Oct – 4 Nov 2016.

Location:
The Roslin Institute, The University of Edinburgh, Scotland, UK.

Instructor:
Dr Bruce Walsh, Department of Ecology & Evolutionary Biology, University of Arizona.

Brief programme:
Monday, Oct 31- Background: Basic stats, population genetics.
Tuesday, Nov 1- Basic Quantitative Genetics.
Wednesday, Nov 02- Evolutionary response in quantitative traits.
Thursday, Nov 03- Estimating the fitness of traits.
Friday, Nov 04- Mixed Models and their extensions.

A more detailed syllabus can be downloaded here.

Participants:
Participation at the course is open to all.

Course fees:
Level 1 - £300 – Internal University of Edinburgh students and staff.
Level 2 - £400 – External academics.
Level 3 - £500 – Industry.

The course fee includes tuition, lunch and light refreshments.

Registration:
Pre-registration is now open, please email Maria Sanchez subject Walsh course EQG.

Hosts:
John Hickey, Division of Genetics and Genomics, The Roslin Institute, University of Edinburgh.
Josephine Pemberton, Institute of Evolutionary Biology, University of Edinburgh, University of Edinburgh.
Chris Haley, MRC Human Genetics Unit, MRC IGMM, University of Edinburgh.


NASA’s Biological and Physical Sciences presents: Quantitative Genomics of Space Biology Workshop, Wednesday, May 26, 2021 11 – 3:30 pm EST

Exploration of deep space by humans requires a better understanding of how multiple environmental factors encountered during spaceflight (e.g., radiation, microgravity, altered atmospheric composition) impact biological systems including astronauts, plants, and relevant microbiomes.

Quantitative Genetics represents a collection of approaches to assess the genetic basis of phenotypic variation of individuals within a population, often with the focus of how a population is adapted to a specialized environment. While these approaches have been indispensable to terrestrial research in genetic variation, they have not been applied to questions related to adaptation to the novel environment of spaceflight.

Who’s appearing at the workshop:
Thought leaders in spaceflight research and Quantitative Genetics.

In this workshop, we will bring together scientists that have expertise in spaceflight genomics with those that have experience in terrestrial Quantitative Genetics to explore the feasibility of Quantitative Genetic approaches to address critical questions in space biology. We ask, “Can we make better use of the spaceflight environment in order to understand the underlying nature of the complex traits that help terrestrial organisms adjust to this novel environment?”

The goal of this workshop is to explore how tools of Quantitative Genetics can be applied to elucidating traits important to the physiological adaptation of terrestrial biology to exploration-relevant environments. We aim to develop a roadmap for leveraging these approaches through a combination of large-scale ground studies and targeted flight study follow-up. A comprehensive plan for the successful application of such an approach will contribute to the next decade of space biology research.

Speakers include:

· Dr Sharmila Bhattacharya, Program Scientist for Space Biology in the Biological and Physical Science Division at NASA Headquarters (NASA Ames Research Center)

· Dr Trudy FC Mackay, Director for the Center for Human Genetics Professor of Genetics and Biochemistry (Clemson University)

· Dr Ralph J Greenspan, Co-Director of Cal-BRAIN Associate Director for the Kavli Institute for the Brain and Mind (UC San Diego)

· Dr Michael M Weil, Professor at the Department of Environmental and Radiological Health Sciences (Colorado State University)

· Dr Martha Hotz Vitaterna, Research Professor at the Department of Neurobiology (Northwestern University)

· Dr Abraham Palmer, Professor & Vice Chair for Basic Research, Department of Psychiatry (UC San Diego)

· Dr Leah Solberg Woods, Professor at the Department of Internal Medicine, Molecular Medicine (Wake Forest University)

· Dr Robert Ferl, Distinguished Professor and Assistant Vice President for Research Program in Plant Molecular and Cellular Biology Horticultural Sciences and the Genetics Institute (University of Florida)

· Dr Wolfgang Busch, Professor of Plant Molecular and Cellular Biology Hess Chair in Plant Science (Salk Institute of Biological Sciences)

· Dr Marcio Resende, Lead of UF Corn Genomics and Breeding Lab Assistant Professor at the Department of Horticultural Sciences, University of Florida (University of Florida)


Gregor Mendel and the Study of Genetics

Genetics is the study of heredity, or the passing of traits from parents to offspring. Gregor Johann Mendel set the framework for genetics long before chromosomes or genes had been identified, at a time when meiosis was not well understood. For his work, Mendel is often referred to as the &ldquofather of modern genetics. &rdquo Mendel selected a simple biological system, garden peas, and conducted methodical, quantitative analyses using large sample sizes.

Figure (PageIndex<1>): Gregor Mendel: Gregor Johann Mendel was a German-speaking Moravian scientist and Augustinian friar who gained posthumous fame as the founder of the modern science of genetics.

Mendel entered the Augustinian St. Thomas&rsquos Abbey and began his training as a priest. He began studying heredity using mice, but his bishop did not like one of his friars studying animal sex, so he switched to plants. Mendel grew and studied around 29,000 garden pea plants in a monastery&rsquos garden, where he analyzed seven characteristics of the garden pea plants: flower color (purple or white), seed texture (wrinkled or round), seed color (yellow or green), stem length (long or short), pod color (yellow or green), pod texture (inflated or constricted), and flower position (axial or terminal). Based on the appearance, or phenotypes, of the seven traits, Mendel developed genotypes for those traits.

Figure (PageIndex<1>): Appearance and genetic makeup of garden pea plant flowers: Based on Mendel&rsquos experiments, the genotype of the pea flowers could be determined from the phenotypes of the flowers.

Because of Mendel&rsquos work, the fundamental principles of heredity were revealed, which are often referred to as Mendel&rsquos Laws of Inheritance. We now know that genes, carried on chromosomes, are the basic functional units of heredity with the capability to be replicated, expressed, or mutated. Today, the postulates put forth by Mendel form the basis of classical, or Mendelian, genetics. Not all genes are transmitted from parents to offspring according to Mendelian genetics, but Mendel&rsquos experiments serve as an excellent starting point for thinking about inheritance.

Mendel made all of his observations and findings crossing individual plants. We can now view a human karyotype of all of the chromosomes in an individual to visualize chromosomal abnormalities in offspring, even before birth. Shortly after Mendel proposed that traits were determined by what are now known as genes, other researchers observed that different traits were often inherited together, and thereby deduced that the genes were physically linked by being located on the same chromosome. Mendel&rsquos work was the beginning of many of the advances in molecular biology over the years.


Quantitative and Computational Biology

The Program in Quantitative and Computational Biology (QCB) is intended to facilitate graduate education at Princeton at the interface of biology, the more quantitative sciences, and computation. Administered from The Lewis-Sigler Institute for Integrative Genomics, QCB is a collaboration in multidisciplinary graduate education among faculty in the Institute and the Departments of Chemistry, Chemical and Biological Engineering, Computer Science, Ecology and Evolutionary Biology, Molecular Biology, and Physics. The program covers the fields of genomics, computational biology, systems biology, biophysics, quantitative genetics, molecular evolution, and microbial interactions.

An Outstanding Tradition: Chartered in 1746, Princeton University has long been considered among the world’s most outstanding institutions of higher education, with particular strength in mathematics and the quantitative sciences. Building upon the legacies of greats such as Compton, Feynman, and Einstein, Princeton established the Lewis-Sigler Institute of Integrative Genomics in 1999 to carry this tradition of quantitative science into the realm of biology.

World Class Research: The Lewis-Sigler Institute and the QCB program focus on attacking problems of great fundamental significance using a mixture of theory and experimentation. To maximize the chances of paradigm shifting advances, there is an emphasis on studying fundamental processes in biology, such as transcription and metabolism, in tractable model organisms including bacteria, yeasts, worms, and fruit flies.

World Class Faculty: The research efforts are led by the QCB program’s 40+ faculty, who include a Nobel Laureate, 8 members of the National Academy of Sciences, 4 Howard Hughes Investigators, and over a dozen early career faculty who have received major national research awards (e.g., NSF CAREER or NIH Innovator).

Personalized Education: A hallmark of any Princeton education is personal attention. The QCB program is no exception. Lab sizes are generally modest, typically 6 – 16 researchers, and all students have extensive direct contact with their faculty mentors. Many students choose to work at the interface of two different labs, enabling them to build close intellectual relationships with multiple principal investigators.

Stimulating Environment: The physical heart of the QCB program is the Carl Icahn Laboratory, an architectural landmark located adjacent to physics, biology, chemistry, neuroscience, and mathematics on Princeton’s main campus. Students have access to a wealth of resources, both intellectual and tangible, such as world-leading capabilities in DNA sequencing, mass spectrometry, and microscopy. They also benefit from the friendly atmosphere of the program, which includes tea and cookies every afternoon. When not busy doing science, students can partake in an active campus social scene and world class arts and theater events on campus.


The Genetic Architecture of Quantitative Traits

AbstractPhenotypic variation for quantitative traits results from the segregation of alleles at multiple quantitative trait loci (QTL) with effects that are sensitive to the genetic, sexual, and external environments. Major challenges for biology in the post-genome era are to map the molecular polymorphisms responsible for variation in medically, agriculturally, and evolutionarily important complex traits and to determine their gene frequencies and their homozygous, heterozygous, epistatic, and pleiotropic effects in multiple environments. The ease with which QTL can be mapped to genomic intervals bounded by molecular markers belies the difficulty in matching the QTL to a genetic locus. The latter requires high-resolution recombination or linkage disequilibrium mapping to nominate putative candidate genes, followed by genetic and/or functional complementation and gene expression analyses. Complete genome sequences and improved technologies for polymorphism detection will greatly advance the genetic dissection of quantitative traits in model organisms, which will open avenues for exploration of homologous QTL in related taxa.


Watch the video: Introduction to Quantitative Traits (May 2022).