BI3010H07

1

BI3010H07 Halliburton chapter 13

A quantitative trait shows a continuous range of phenotypes. This is due to the contribution from many polymorphic loci and their genotypic combinations, as well as a modifying influence from the environment. A typical quantitative genetic trait is body height in humans; although there is clearly a genetic basis for the observed variability, clear-cut, discrete genotypes cannot be identified in the way we can for qualitative, single-locus traits.

There are two principal types of quantitative traits;1. True quantitative traits which shows continuous phenotypic distributions2. Meristic traits (e.g. vertebra count) which shows discrete phenotypes (integer values), but which has a quantitative basis.

Closely related to these are the "Threshold traits", where individuals are classified as either having the trait or not, although the underlying basis is quantitative. Many of these are classification traits, like e.g. high blood pressure or obesity. For all three types of quantitative traits, the analytical methods for analysing them are the same.

2


On the second page of this chapter (p. 526), Halliburton presents a list of six questions concerning quantitative traits which he sets out to answer underway:

1. What is the genetic basis of quantitative traits; that is, are they subject to the rules of Mendelian inheritance?2. How do we separate genetic effects from environmental effects on a quantitative trait?3. How many loci affect quantitative traits, and how large are their effects?4. How much genetic variation for quantitative traits is there in natural populations?5. How is this genetic variation maintained?6. How important are mutation, linkage, dominance, epistasis, and pleiotrophy in the evolution of quantitative traits?

During his treatment of various topics in this chapter, Halliburton remains true to this task. You will notice in many places that he presents the stuff in such a way as to show the connection between qualitative and quantitative genetics (for example, by showing how population means are tied to allele- and genotype frequencies).

3


It is commonly observed that tall parents tend to have tall children and vice versa.Actually, the expectation is that the offspring performs intermediate between the parents for additive, quantitative traits.

So, how many polymorphic loci are behind a quantitative trait? The answer is usually that the number is not known, but probably high. On the other hand a relatively small set of loci (<10) can create a quite smooth, continuous distribution of phenotypes (cf Fig. 13.2 with explanation p.530-531). An important notice is that the contribution from each locus needs not, and probably rearly does, contribute equally to the quantitative phenotype.

The contribution from environment may vary widely in space and time, and hence also the proportion of the total phenotypic variation that is genetic. Despite the fact that the phenotypic variation is variable and continuous, it is firmly established that quantitative genetic variation as a phenomenon is consistent with Mendel's laws, and Fisher (1918) demonstrated mathematically that the inheritance of quantitative traits can be explained by Mendelian inheritance at multiple polymorphic loci.

4


In order to quantify the genetic part of total phenotypic variation, we must find ways to separate the genetic part from the environmental effects (Fisher helped us out here). Furthermore, for understanding evolution, we would want to know how quantitative genetic variation is maintained in the populations. This involves knowledge about the importance of factors like mutation, linkage, epistasis, and pleiotrophy, and their role in the evolution of quantitative traits. Many of these questions are analogues of those asked for qualitative traits in previous chapters.

13.1 Genetic and Environmental Effects on Quantitative TraitsIn an early experiment studying parent-offspring relations for seed weight in inbred lines of beans (NB! this was before Fisher's (1918) work), Johannsen (1903) noted a clear parent-offspring relation among inbred lines, but no consistent parent-offspring relation within lines (Fig. 13.1 a,b). This experiment thus demonstrated that there was a genetic factor G (variation among genetically different lines), as well as an environmental factor E (variation among offspring within lines, independent of parental performance).

P = G + E

5


13.2 The Genetics of Quantitative CharactersConsider a simple polymorphism and its observed phenotype performance

Genotype Mean weight (Phenotypic value)

A1A1 14 g

A1A2 12 g

A2A2 6g

The midpoint between highest (A1A1) and lowest (A2A2) weight is 10g. Let a and -a be the deviations from this value for A1A1 and A2A2.Let d be a measure of dominance for the heterozygote, measuring the deviation from the mid-value of 10 for the heterozygote. (cf Fig. 13.5 next slide). Now;

Genotype Value Frequency

A1A1 a ( = 4 ) p2

A1A2 d ( = 2 ) 2pq

A2A2 -a ( = - 4 ) q2

6


Illustration of genetic effectsHalliburton Fig. 13.5 p. 533

7


With this information, the mean value ( ) of the entire population can be found as the weighted average of a, d and -a (weighted by the genotypic frequencies):

µ = p2a + 2pqd + q2(-a), i.e.µ = a(p-q) + 2pqd (13.2) Having found the population mean µ, the genotypic value G of each genotype can be expressed by its deviation from the this value.

G11 = a - µ G12 = d - µG22 = -a - µ

In combination with (13.2) this gives:

G11 = 2qa - 2pqd (13.3)G12 = a(q-p) + d(1-2pq) (13.4)G22 = -2pa - 2pqd (13.5)

The genotypic values G are measures of how superior or inferior the average of each genotype is compared to the population mean. Therefore these values depend on the allele frequencies.

!! Note that the genotypic value is not the same as the breeding value (next slide).

8


Decomposition of the PhenotypeThe phenotype of an individual is determined to some degree by its genotype. However, in sexual reproduction, the genotypes of the parents are broken up during the Mendelian segregation, so that each parent transmits only one allele to the offspring. Therefore, we need some measure of the average value of an allele when it combines at random with other alleles in the population (its breeding value).

Consider allele A1. Under random union of gametes it will combine with another A1 with a frequency of p. Likewise, it will combine with an A2 with a frequency of q. The resulting genotypes A1A1 and A1A2 have values of a and d, respectively, and the average value of A1-containing genotypes will be:

µ1 = pa + qd which, expressed as (⍺), the deviation from the population mean is⍺1 = µ1 - µ = [pa + qd] - [ a(p-q) + 2pqd]⍺1 = q[a + d(q-p)] (13.6)

⍺1 is called the average effect of A1.

Sililarly, the average effect of A2 is:⍺2 = - p[a + d(q-p)] ( 13.7)

9


If we call the quantity in the brackets (a + d(q-p)), which are the same for both alleles, for ⍺ (the deviation from the population mean)the deviation from the population mean)

⍺ = (a + d(q-p)) (13.8)

without a subscript, we can express ⍺1 and ⍺2 as:

⍺1 = q⍺ and⍺2 = -p⍺

We now define the breeding value (BV) of an individual as the sum of the average effects of the two alleles it carries

BV11 = ⍺1 + ⍺1 = * 2q⍺ BV12 = ⍺1 + ⍺2 = * (q-p)⍺BV22 = ⍺2 + ⍺2 = * -2p⍺

Because the breeding value of an individual is the sum of the average effect of the alleles, it is often called the additive effect and symbolized A, with subscripts to indicate the genotype. Thus, the additive effects of the three genotypes are, also

A11 = ⍺1 + ⍺1 = * 2q⍺ (13.9 )A12 = ⍺1 + ⍺2 = * (q-p) ⍺ (13.10)A22 = ⍺2 + ⍺2 = * -2p ⍺ (13.11)

10


The breeding values (13.9–11) are not the same as the genotypic values unless d=0 (i.e. no dominance). Expressing 13.3-13.5 in terms of ⍺ the effect of allele interaction (dominance, D) for the three genotypes can be singled out as

D11 = -2q2dD12 = 2pqdD22 = -2p2d

These D's are called the dominance deviations. Thus, if dominance is involved in addition to additive effects, the genotypic values of individuals can be partitioned into

G11 = A11 + D11

G12 = A12 + D12

G22 = A22 + D22

All these quantities are expressed as deviations from the population meandeviations from the population mean. In breeding, only the additive effects can be utilized. If the trait is influenced by many polymorphic loci, there is a possibility of epistasis (interaction of genes at different loci) between them, symbolised by I. In the equiation on slide 3 (P = G + E), we can now add the following detail:

P = (A + D + I) + E (13.20)

Thus, the phenotype of an individual is due to a genetic component and an environmental component. The genetic component can be partitioned into an additive effect from the actual alleles, a dominance effect, and an epistasis effect. How can we make practical use of this? In order to make progress we must study variation within the population.

11


"Statistical genes"Before embarking on the description of mathematical tools in quantitative genetics, it must be emphasized that its treatment of phenotypic variation is based on a set of important assumptions. First, it is more or less an axiom that each trait is controlled by a large number of unlinked loci, each of which has a small effect on the phenotype. If so, we can, via the central limit central limit theoremtheorem, assume that the trait is approximately normally distributed. We also assume that the environmental effects are normally distributed with a mean environmental effects are normally distributed with a mean effect of zeroeffect of zero, so that the phenotypic mean will be equal to the genotypic mean.

Under these assumptions we can analyse quantitative characteres with the pure statistical tools of the attractive normal distribution, and ignore the underlying genetical complexities.

Variance components for quantitative traitsIf we describe the phenotype of an individual by P = G + E, the variance in the population is described byVP= VG + VE + 2cov(G,E)Under the (strict) assumption that there is no (genotype x environment) interaction, the covariance term is zero, and the phenotypic variance is divided in a genotypic and an environmental component. The total phenotypic variance is easily obtained by measurements, but we need some way to estimate how much of this variation is due to genetic variation in the population. This is, of course, important in breeding programs.

12


Focusing first on the genetic component (VG) for a single locus trait (thus epistasis is excluded):VG = VA + VD

We recall the variance of a random variable X is calculated as:Var(X) = Σ fi (Xi – µ)2

where fi is the frequency of the ith values of X.

Thus, if the X'es are genotypes, the genotypic variance depends on the genotypic distribution at the locus. Under HW equilibrium we can express genotypes by allele frequencies, and we can write:VG = p2(G11)2 + 2pq(G12)2 + q2(G22)2 Using our former expressions for G (eq. 13.12-13.14) in this equiation, and using ⍺, the deviations from the population mean (µ) we arrive after a much algebraic simplification at:VG = 2pq⍺2 + (2pqd)2 where ⍺ stands for additive effects and d for dominance effectsThus, we define:-----------------------------------------------------------------------------------------------------------------------------------VA = 2pq⍺2 (the additive genetic variance) 13.21andVD = (2pqd)2 (the dominance variance) 13.22-----------------------------------------------------------------------------------------------------------------------------------

If considering more than one locus for a trait, epistatic effects (VI) must be incuded, so that:

VG = VA + VD + VI (still assuming all covariances =0)

Finally, adding the environmental effects (VE), the total phenotypic variance can be decomposed into:

VVGG = V = VAA + V + VDD + V + VI I + V+ VEE (assuming all covariances=0, and no genotype x environment interaction).

13


Heritability

After having partitioned the total pheotypic variance into a genetic and an environmental component, we can determine the proportion due to genetic variation. This ratio is called the

"Broad Sense Heritability":

H2 = VG / VP

Similarly, the ratio of the additive genetic variance to the total phenotypic variability is called the

NNarrow sense heritabilityarrow sense heritability::

hh22 = V = VA A / V/ VPP

The narrow sense heritability, in which dominance and interaction effects are not included, is useful in animal breeding programs. In plant breeding, the broad sense heritability is useful because there, entire genotypes including dominance and interaction effects can be replicated by asexual propagation.

The exponential (2) in the heritability measures is historical and reflects that the heritabilty is a ratio of variances (which in turn are based on squared deviations from the mean).

14


BOX 1. Repetition of how to compute a simple linear regression

Variances and covariances, which are heavily used in quantitative genetics, are also basis for a very common tool for assessing the relations between variables, known as regression analysis. In its simplest form, linear regression, the computation steps are straightforward. Note that in regression, the value of one of the variables (Y) are dependent on the value of the other (X). [If no such dependence we perform correlation analysis].

Consider two variables X and Y for which we have a list of 4 corresponding data points. We want the

constants a and b in the formula for the best-fit straight line through these data point: Y = a + bX. --------- >

X-values: 1 2 3 4-------------------------------------------------------------------------------------------Y-values: 3.2 7.4 8.7 12.1

We first calculate the basic terms:

n (# of data points) = 4Σ X = 10 Σ Y = 31.4 Σ XY=92.5Xmean = 2.5 Ymean = 7.85 Σ X2 = 30 Σ (X)2/n = 100/4=25

then fill in the computational formula; lowercase x and y are respective deviations from mean X and mean Y

Σ x2 = Σ X2 - Σ (X)2/n = 30 – 25 = 5Σ xy = Σ XY – (Σ X Σ Y)/n = 92.5 – (10x31.4)/4 = 14

b = Σxy / Σx2 = 14/5 = 2.8a = Ymean - b Xmean = 7.85 – (2.8x2.5) = 0.85

Thus the equation is: Y = 0.85 + 2.8X ------------------------------------------------------------ >

1

6

11

1 2 3 4

a = Y interceptb = slope of line

15


Covariance between relatives

A parent gives half of its alleles to each offspring. Hence full-sibs share ½ of their alleles with each other, half-sibs share 1/4, and grandparents and grandchildren also ¼. For quantitative characters which are heritable, we would thus expect close relatives to be more similar than distant relatives or unrelated individuals in the population.

The genetic similarity among relatives is fundamental in quantitative genetics and breeding genetics. Similarity is expressed as a covariance. cov(X,Y) = Σ fi (Xi – EX)(Yi – EY) for all XY pairswhere fi is the frequency of the ith XY pair, and E = expected value = (weighted) mean value = µ------------------------Under some strict assumptions (i.e. that environmental effects are random and independent of genotype), the phenotypic covariance equals the genotypic covariance:cov(XP,YP) = cov(XG,YG)

The covariances between relatives are simplefractions of VA and VD. (Table 13.5 ---->).

The degree of phenotypic resemblence betweenrelatives can be used to estimate the heritability h2

of a trait. Remember from Box 1 above that the magnitude of the covariance term between the variables determined the slope in the linear regression.

In a regression of e.g. offspring on mid-parent the slope of the regression line is an estimate of the heritability h2

(Fig. 13.6 on next slide).

16


Hence, in setting up an experiment to estimate heritability of a trait, we can utilize our knowledge of the genetic similarities and expected covariance between relatives (Table 13.5).

For example, we can perform a largenumber of crossings between pairs of parents with known individual performance for a specific trait.

From each crossing we then measure the performance of the pertinent offspring (i.e. the family groups) for the trait under study.

In a regression of offspring (O) performance on mid-parent (P)

(cov(O,P)= ½ VA) performance, the slope of the regression line estimates the narrow sense heritability h2 of the trait under the given environmental conditions (Fig. 13.6 ---->).

17


13.3 Artificial selection on a quantitative trait.

Truncation selectionis based on a ranking of individuals based on their measured phenotypic performance. Phenotypes above a chosen truncation point are selected to be parents of the next generation. (See graphs in figures 13.7 and 13.8).( --------------------------------------------->).

Whether this will lead to a genetic improvement of the breeding stock depends, e.g., on the heritability of the trait; h2 (page 543); the ratio of additive genetic variance to phenotypic variance VA/VP.

An equivalent meaning of the heritbility is the regression of breeding value (A) on phenotypic value (P):

h2 = bAP

(if we use knowledge of genetic similarities among full- or half-sibs as basis for estimating heritability of traits, we speak about correlation anaysis, not regression analysis.

18


BOX 2. Heritability By regarding the heritability as the regresssion of breeding value (VA; the additive variance component) on phenotypic value we see that an individual's estimated breeding value is the product of its phenotypic value and the heritability:

A(expected) = h2P

where breeding values and phenotypic values are both reckoned as deviations from the population mean. The heritability enters into almost every formula connected with breeding methods, and many practical desicions about procedures depend on its magnitude. The determination of heritability is one of the first objectives in the genetic study of a metric character .

It is important to realize that the heritability is a property not only of a character but also of the population and of the environmental circumstances to which the individuals are subjected. Since the value of the heritability depends on the magnitude of all the components of variance, a change in any one of these will affect it. All the genetic components are affected by gene frequencies and may therefore differ from one population to another, according to the past history of the population. In particular, small populations maintained long enough for an appreciable amount of fixations to have taken place are expected to show lower heritabilities than large populations. The environmental variance is dependent on the conditions of culture or management: more variable conditions reduce the heritability; more uniform increase it. So, whenever a value is stated for the heritability of a given character it must be understood to refer to a particular population under particular conditions. Empiri has shown that in general, traits important for fitness have lower heritabilities than other, less important traits.

19


Some definitions:(cf Fig. 13.7 --->)

Selection differential (S) = the difference between the mean of the parental population and the mean of the individuals selected for breeding.

Selection intensity (I) = S measured in SD units.

Selection response (R) = the difference between the mean of the parental generation and the mean of the offspring from the individuals selected for breeding.

Breeder's equation:The regression of offspring value on midparent value in Fig. 8 is basis for a reasoning of the author from which he arrives at some very important connections between S and R above, e.g. that the heritability is equal to the ratio R/S, and from there to what is known as the breeder's equiation:R = h2S because it can be used to predict the response to directional selection on a quantitative trait.

20


Using the breeder's equiation: R = h2S

Commonly in quantitative genetics, traits are normally distributed and subgroup characteristics measured as deviations from population mean. (in SD units).

In practical implementations of the breeder's equiation, Falconer's Table A is widely used.

It contains a compilation of values of p (the percentage of the population above a specific truncation point T), x, the value of T in SD units, and i; the mean value (in SD units) of individuals with values exceeding T.

For example, if the truncation point T is chosen so that group selected for breeding is the best 20% of the distribution, its corresponding value in SD units (x) is 0.842, and the mean value of the individuals exceeding T is 1.4 SD units.

cont'd next slide

21


breeder's equiation cont'd

In the breeder's equiation R = h2S, (NB! one generation) the selection differential S can be written asS = I * SDP,, so that the breeder's equiation can also be written asR = h2 * I * SDP

where I = the selection intensity (the difference in mean value between population mean and the selected group, measured in SD units (of population mean), and SDP is the standard deviation of the population's phenotypic mean value.

A practical example:Consider a farmed salmon breeding population in which a breeding program for increased body growth is started. Assume that the mean weight of the two-years old salmon constituting the parental population is 1kg with an SD=0.4 kg, and let the heritability for this trait be h2=0.30. The best 2.4% of the population is used to produce the next generation. What is the expected increased mean weight among the offspring when they reach two years of age?

The truncation T point which corresponds to the upper 2.4% of the population lies ~1.98 SD units above the population mean (Table A), that is (1.98x0.4kg) 0.79 kg above. Hence all individuals heavier than 1.79 kg were used as a brood stock. These individuals have a mean weight which according to Table A is 2.35 SD units above the population mean; i.e. (2.35x0.4) = 0.94 kg above. Thus the selection differential (S) is 0.94 kg, and R = h2*S = 0.28

The expected mean weight of two years old individuals in the offspring will be: 1.28 kg.

22


Response to repeated selection(general observations)

Short term response:1. Response constant short term2. Thereafter a plateau3. When relaxed, the gain is reduced4. Response up/down assymmetrical5. Viability/fertility reduced over time

Long term response:1. plateau reached (Fig. 13.9c -->)2. Causes: Genetic drift and selection changes variability and allele freq.

The genetic drift effect: Ht = H0 [1 - 1/(2N) ] t

an analogous formula for additive genetic variance is

VVA(t)A(t) = V = VA(0)A(0) [1 - 1/(2N) ][1 - 1/(2N) ] t t

23


Response to repeated selection(cont'd)

Results from short-term selection(5 generations) for increased cholesterol level in mice (Table 13.6 and Fig. 13.10).

The slope of the regression line in Fig. 13.10 is an estimate of the realised heritability. The respons is fairly constant over 5 generations.

24


Correlated response to selection:Frequently, selection for one trait can affect another trait, because of genetic correlations between traits. Genetic correlations can be due to pleiotrophy (one gene affects more than one trait) or gametic disequilibrium (alleles at a locus affecting one trait is in gametic disequilibrium with alleles at a locus affecting another trait e.g. due to linkage).

The correlation coefficient (r) between two random variables is defined as: r = cov(XY) / (σXσY ), where σX and σY are the standard deviations (SD) of X and Y. 13.42

Genetic correlations (covariance) have two components, a genetic correlation and an environmental.The environmental correlation is due to milieu factors affecting both traits simultaneously. Hence

rP = hXhYrA + eXeYrE (derived in Box 13.2) 13.43

The response in Y (RY) when selection is on trait X in the same individuals can be predicted by:

RY = rAhXhYSσPY /σPX or equivalently because, since selection intensity i = S/ σPX , 13.44

RY = rAhXhYiσPY 13.45

The term rAhXhY is called the coheritability coheritability

The rA is difficult to estimate, but a reasonable assumption is that rA ~ rP

NB! Correlated responses (genetic correlations) may be the main reason for reduced fitness of populations under artificial selection.

25


REMINDERS:

Pleiotrophy = when a single gene influences multiple phenotypic traits

Epistasis = when the effect of a gene at one locus is modified by genes at one or several other loci

26


13.4 Natural Selection on Quantitative TraitsIn nature, many quantitative traits (weight, growth rate, age at maturation, fecundity, etc) affect fitness, and thus are almost certainly subject to natural selection.

Kinds of natural selection (cf Fig. 13.11 in Halliburton)These are the same as treated for single locus, qualitative genetic characters:1. Directional selection (probably affects many fitness-related traits)2. stabilizing selection (e.g. birth weight in humans)3. Disruptive selection (importance in nat. pop. unclear)

Antagonistic pleiotrophy (Drosophila; development time vs high fecundity) can be due to negative correlations between fitness components (p. 557).

Natural selection on correlated traitsHow can we determine if natural selection acts directly on a trait, or via selection on a correlated trait? Lande & Arnold (1983) showed how this can be done by multiple linear regression (p. 558-562). This method has been widely used; Price et al. (1984) showed that body weight and beak depth in Darwin's finches were under strong natural selection compared to a set of correlated morphological traits.

The strength of natural selectionNatural selection can take a wide range of values, and overlaps extensively with the intensities overlaps extensively with the intensities applied in artificial selection experimentsapplied in artificial selection experiments (Endler 1983). Whether directional or stabilizing selection is the most common form is under debate. (NB! Note an error in Table 13.10; in the table heading subtext, "directional" should be changed to "disruptive").

27


13.5 Quatitative Trait Loci – "QTL" (loci that affect a quantitative trait)A QTL is a relatively small region on a chromosome. It does not necessarily correspond to asingle gene; it can also consist of several tightly linked genes which are inhereted as one unit.

1. The number of loci affecting a QTL can be estimated under certain circumstances, using the so-called Castle-Wright estimator.

2. Also, it is sometimes possible to map the genes controlling a quantitative trait (a QTL) to specific regions on a chromosome by using a linked marker gene.

28


1. How Many Loci Affect a Quantitative Trait?

The Castle-Wright estimator is based on the phenotypic difference between two inbred lines (fixed for alternative alleles at loci affecting a quantitative trait). The inbred lines are crossed, and then the F1 are crossed with each other to produce an F2.

The mean phenotypic values for the two inbred lines are denoted by M1 and M2. The variance in the F2 will be higher than in either inbred line or the F1.The excess variance is called the segregational variance Vseg = VF2 – VF1.

Castle (1921) and Wright (1968) showed that an estimate of the minimum number (ne ) of loci affecting the trait is

ne = (M1 – M2 )2 / ( 8Vseg ) (also called the effective number of loci affecting the trait)

29


Box 3. The Castle-Wright estimator:

ne = (M1 – M2 )2 / ( 8Vseg )

Castle (1921) and Wright (1968) deviced a method for estimating the number of (polymorphic) loci affecting a quantitative trait.

Basically, the method utilizes the increase in phenotypic variance due to segregation.

The starting point is two inbed lines (fixed for different alleles). These are crossed to get an F1 generation (all heterozygotes). Then an F1xF1 crossis performed to get all three genotypes segregated in the F2 generation.

The segregation in all three genotypes leads to increased phenotypic variance for the trait in F2, proportional to the number of segregating loci affecting the trait.

30


Numeric example:

Phenotypic mean value for inbred line A: 40.0 (all homozygotes)Phenotypic mean value for inbred line B: 20.0 (all homozygotes)

Cross AxB to produce F1Variance in F1: 5.0 (all heterozygotes)

Cross F1xF1 to produce F2Variance in F2: 10.0 (segregating into all three genotypes)

ne = (40.0 – 20.0)2 / 8(10.0 – 5.0) = 400 / 40 = 10 loci

31


Test your understanding:

Assume two populations P1 and P2 of a species, and a normally distributed trait, say body weight at 2 years of age, which is determined by genotypes at a QTL. P1 and P2 have been kept isolated from each other and selected for high and low weight, respectively, for many generations. They are both assumed to be completely inbred, but fixed for different alleles at all the loci included in the QTL.

How would you proceed to estimate the number of polymorphic loci affecting the body weight at age 2 years?

32


2. Mapping Quantitative Trait LociThe approach for estimaing the number of loci affecting a quantitative trait (above) does not reveal the location of QTLs on the chromosome. For this purpose, a genetic mapping using marker loci with known positions can be used. Most commonly, two different inbred lines, which are different in their phenotypic value for a trait, are crossed. Then the resulting F1 progeny are crossed with each other to produce F2. The segregation for a marker locus and the phenotypic values in F2 can be used to identify the location of a QTL (in terms of number of crossing-over units from the marker). When lines that differ in both marker and QTL are crossed, linkage disequilibrium (D') is generated between the loci; the magnitude of D' it is then used to detect the presence and location of the QTL. The simplest approach is the "single-marker analysis":

Single-marker analysisAssume two inbred lines that are fixed for different alleles at both a marker locus M with alleles M1 and M2, and a quantitative trait locus Q with alleles Q1 and Q2.

Line Genotype Genotypic value

L1 Q1M1 a

L2 Q2M2 -a

The two inbred lines are crossed to produce F1, which are all of the same genotype Q1M1 / Q2M2, i.e. in complete gametic disequilibrium. Then the F1 are crossed with each other to produce F2. If the marker and the QTL are completely linked (no recombination; r = 0), the F2 combined genotypes and their expected frequencies will be:

Genotype Frequency Marker genotype Genotypic value

Q1M1 / Q1M1 0.25 M1M1 a

Q1M1 / Q2M2 0.50 M1M2 d

Q2M2 / Q2M2 0.25 M2M2 -a

33

BI3010H07 Halliburton chapter 13Numeric example:The data of Sax (1923) on seed weight in beans can be used as an example. Sax crossed two inbred lines of beans which differed in both pigmentation and mean seed weight, and produced an F1 generation. The F1 were then crossed with each other to produce F2 (which has Mendelian segregation of alleles into different genotypes). The pigmentation locus is here assigned two alleles P and p, with genotypes PP, Pp, and pp. The mean seed weight in the marker genotypes of the inbred parental lines and the F2 are shown in the table below.

Group Mean weight PP mean weight Pp mean weight pp mean weight

Parental 1 48.0 (+a)

Parental 2 21.0 (- a)

F2 30.7 28.3 26.4

The difference in mean seed weight between marker homozygotic genotypes (30.7-26.4 = 4.3) accounts for only 16% of the difference (48-21 = 27) between the parental lines but is statistically significant. Hence there is some degree of chromosomal association (linkage) between the marker locus and some gene (QTL) affecting the quantitative trait seed weight. On the other hand the effect is not complete, so some degree of recombination (0< r <0.5 ) must have occurred in the gamete formation (meiosis) in F1, which broke up the initial disequilibrium between Q1 and M1, and Q2 and M2. The expected (E) difference between the homozygous marker genotypes in F2 is :

E(M1M1) – E(M2M2) = 2a (1 – 2r) (where r denotes the recombination rate between marker and QTL) 13.49

The equiation implies that with complete linkage (r=0) , the homozygote differences in F2 is the same as that between the parental inbred lines. If r=0.5 (unlinked loci), the F2 homozygotes do not differ for the QTL trait, i.e. the marker alleles are randomly associated with the QTL alleles.In the bean data of Sax (above), the observed difference in mean weight between parental lines was (48-21) = 27 cg. The difference between marker homozygotes was not nearly as big (30.7 - 26.4) = 4.3 cg, suggesting that the marker locus is not very tightly linked to the QTL. Putting in the F2 marker homozygote difference (4.3) on the left-hand side in (13.49), and the parental line difference (27) in for 2a on the right-hand side yields r = 0.42, i.e. the marker locus and the QTL are 42 map units (µ), or 42 cM, apart from each other on the chromosome (see centiMorgan definition next page) .

34


Definition of centiMorgan:One centiMorgan is defined as the genetic distance between two loci with a statistically corrected recombination frequency of 1%; the genetic distance in centiMorgans is numerically equal to the recombination frequency expressed as a percentage. Symbol, cM. The centimorgan is now more commonly called a “map unit” (symbol, mu) or Locus Map Unit (symbol, LMU).

The qualification “statistically corrected” is necessary because at "genetic distances" greater than about 7 cM, the relationship between recombination frequency and genetic distance is no longer linear. Researchers have developed mathematical models that can correct for this difficulty.

The centimorgan is not a measure of physical distance, but typically a genetic distance of 1 cM corresponds to a physical distance of roughly one million base pairs. Attempts to assign a physical length to the centimorgan have led to an estimate that it is roughly about 0.003 millimeters.

35


The backcross methodIn QTL mapping, an alternative to the crossing of F1 with each other, is to cross F1 with one of the inbred parental lines. This procedure saves one pure parental chromosome all the way to the F2 generation, and is the simplest one available. The statistical methods used, and the statistical issues that arise, are largely the same for all the different types of crosses. However, the backcross has the advantage of simplicity; at each locus in the genome, the backcross progeny have one of only two possible genotypes (genetic composition). The procedure is outlined in the figure to the right.

36


QTL in Humans and Natural PopulationsThe study of completely inbred lines outlined above is not possible in humans and natural populations, hence alternative approaches must be used. In humans the necessary information about gametic stage disequilibrium must be inferred from pedigree data, which is too slim in many families. Also, sample sizes are usually small, resulting in low statistical power. Lynch & Walsh (1998) review techniques developed for use on humans and other outbred populations.

13.6 Evolutionary Quantitative GeneticsQuantitative genetic descriptions and predictions in natural populations must consider long-term evolutionary processes. This means that the many simplifying assumptions which was reasonable when exploring short-term effect, cannot be made for long-term processes. Thus, effects from dominance, pleiotrophy, epistasis and mutations cannot be overlooked, they may be important players on the evolutionary scene. For answering the question "How much genetic variation for quantitative traits is there in natural populations?" one must examine the predicted effects of those evolutionary forces that are responsible for genetic differentiation, namely:1. natural selection2. mutation3. genetic drift

The effect of natural selection on genetic variationStabilising selection (in the multilocus sense) will decrease the phenotypic variance, but it is less certain whether this means a decrease in the genotypic variance. (cf text page 573 upper part). Roff (1977) reported that most models suggest that additive genetic variance will decrease in response to stabilising selection.Directional selection will also tend to reduce additive genetic variance. Disruprive selection is more uncertain in this respect. Most models suggest that genetic variance will increase over the short-term, but the long-term effect is unclear (cf Fig. 13.19 w/text).

Presently, the consensus is that most forms of natural selection should cause a long-term decrease in genetic variance; not unsimilar to the expectations in artificial selection regimes.

37


Heritabilities in natural populationsIt is widely believed that fitness-related traits (viability, fertility) are under stronger natural selection than e.g. morphological traits, and hence that life history characters should have less additive genetic variation (i.e. lower heritability) than morphological characters. This prediction has been confirmed (cf Table 13.14) p. 574), but life history trait heritabilities are not zero! Furthermore, laboratory experiment estimates of heritability may be overestimates relative to those in natural populations because of a lower environmental component in captive populations. Also, possible (genotype x environment) interaction in nature may be underestimated in the laboratory. Looking away from this, however, heritability estimates from laboratory and natural populations are not significantly different. Hence heritabilities stemming from laboratory estimates may be good proxies for those in natural populations (but see below).

Heritability vs additive genetic varianceEstimated heritability may not always be a good surrogate for additive genetic variance, because other genetic variance components ("residual variance”; cf slide 12) than VA may be substantial. Houle (1992) suggested that the Coefficient of additive genetic variation CVA is better for comparing variation between traits.

CVA = SQRT[VA ] / µ x 100, andCVR = SQRT[VP – VA ] / µ x 100 (R denotes residual and µ =population mean )

CVA and CVR has been shown to be higher for fitness-related traits than for morpholocical traits. Also, h2 and CVR appear to be negatively correlated. This suggest that the lower heritabilities of fitness-related traits may be due to high amounts of residual genetic variance, which is different from the view that it is due to lower amounts of additive genetic variance.

Mutation rates for quantitative charactersThere is currently uncertainty concerning mutation rates for quantitative characters, due to apparent overestimates from available models. Cf text p. 579 ff.

38


Effects of mutation and genetic drift on genetic varianceMutations create new alleles each generation, which will increase the genetic variance for a trait. The amount of genetic variance created by mutations each generation is called mutational variance (VM) and will add to the additive genetic variation each generation by:

VA(t+1) = VA(t) + VM

The value of VM depends on the number of loci mutating, the mutation rate per locus, and the phenotypic effect of a mutation.

Joint effects of mutation and natural selection on genetic varianceTheoretical models for exploring this have not yielded consistent results. Very much, the conclusions depend on the validity of assumptions made for the models. Caballero & Keightly (1994) analyzed some complex models including dominance and pleiotrophy effects of mutations on viability, and various distributions of the effects of mutation on the quantitative trait and fitness.The main 6 conclusions they drew are liste in Halliburton p. 583-584.

How inportant are dominance, linkage, epistasis and pleiotrophy?Recent advances in molecular and statistical techniques have allowed the examination of fundamental assumptions of the basic quantitative genetics model. Shortly told, all of these assumptions are violated to a greater or lesser degree. This may have little effect on short-term predictions, but demands a re-examination of long-term predictions. Mackay (2001) gives an overview of the status on this topic.

What maintains genetic variation for quantitative traits?Contrary to most theoretical predictions, genetic variation for quantitative traits is common in natural populations. There is a long list of possible explanations, but we do not know which is correct. Recent approaches using molecular marker loci and powerful statistical techniques may suggest a rapid progress in this scientific field.

39


40


BI3010H07

Documents

true quantitative traits

quantitative basis

quantitative genetic

quantitative genetics

qualitative traits

meristic traits

classification traits

threshold traits