Lecture 22: Quantitative Traits II Date: 11/07/02 Review: quantitative genetics terms Simple tests for QTL localization Review project 1

Quantitative Genetics Terminology Definition Under HWE additive effect least-squares regression coefficients a1 = p1a a2 = -p2a average excess difference in mean genotypic values of carriers of at least one copy of allele and random individual breeding value sum of additive effects of an individual’s genes 2a1 a1+a2 2a2

Average Genotypic Value of Offspring Consider a random mating population and consider the average genotypic value of offspring of each genotype. Subtract mG B1B1 B1B2 B2B2 0p1 + a(1+k)p2 0.5a(1+k)p2 + 0.5a(1+k)p1 + ap2 a(1+k)p1 + 2ap2 a1 0.5(a1+ a2) a2

Breeding Value and Random Mating Consider the expected genotypic values of progeny produced by parental genotypes. Genotype Breeding Value (A) Progeny Expected Genotypic Value Deviation (A/2) B1B1 2a1 ap2(1+k) a1 B1B2 a1+a2 a[p2 + (1+k)/2] 0.5(a1+a2) B2B2 2a2 a[2p2 + p1(1+k)] a2

Breeding Value Estimation In random mating populations, therefore, one can estimate the breeding value of an individual as twice the difference between their offspring average genotypic value and the population genotypic value. The factor of two results because a parent can only pass on ½ of their genetic information (half of their total breeding value) to their offspring.

Multiple Allele Average Excess The average excess of allele i is given by When random mating can be assumed, then

Multiple Allele Additive Effect Define the multiple regression for multiple alleles: Then, the additive effects are the regression coefficients that minimize the mean squared error.

Finding Regression Coefficients The general solution is involved. However, when random mating is assumed, recall that the additive effects (or these regression coefficients that we seek) are equal to average excess. Therefore,

Variance Components First, we consider the total variance: The model component of variance is given by: Additive genetic variance is the model component:

Covariance of Genotypic Value and Gene Content We seek an expression for the s(G, Ni). To find it, we need E(Ni), E(G), and E(NiG). E(G) = mG. E(Ni) = 2pi.

Covariance of Genotypic Value and Gene Content (cont) Putting the results of the previous slide together, we have But that implies And under random mating, gives mean square additive effect factor accounts for diploidy

Multiple Allele Breeding Value The breeding value remains unchanged. It is just the sum of the additive effects of the two alleles of the parent. In general, then, we can write and the variance can be written as Under random mating, the additive genetic effect is the variance of breeding values in the population.

Summary a homozygous effect intrinsic properties of alleles k dominance coefficient ai additive effect properties of allele in population ai* average excess A breeding value property of individual in population additive genetic variance property of a population

Basic Overview of QTL Hunt Genotype individuals at markers and classify them into groups based on marker genotype. The average phenotypic values within each group can be taken as estimates of genotypic values. They are averaged over environmental effects. Estimate the total genotypic variance that is associated with the marker locus. From this we get an idea of how close the marker is to the QTL.

Basic Overview of QTL Hunt (cont) D QTL Single marker: No locus order required, but no precise QTL locations obtained. Used when QTL detection, not location, is key. Interval mapping: Require locus order and do provide estimates of QTL location.

Interval Mapping Variants Interval mapping or flanking-marker analysis considers each pair of markers in turn. Slightly higher power and more precise estimates of location and QTL effect. Composite interval mapping uses two markers plus a few other selected markers. Multipoint mapping considers all linked markers at once.

Inbred vs. Outbred QTL mapping also varies depending on the population you study. Inbred line crosses are controlled crosses (F2, backcross and the like). Outbred populations are natural populations (such as humans) where the experimenter cannot control the cross.

Single Locus – General Model Let yi be the trait value for the ith individual in your sample. Let mi be the marker genotype for the ith individual in your sample. Then, it is generally assumed that

Single Locus – Analysis The analysis can proceed in multiple ways t-test (when there are two marker classes) analysis of variance linear regression likelihood estimation and testing

Backcross Model – Setup A and Q are in coupling phase Marker Genotype Count Marg. Freq. QTL Genotype Trait Value QQ Qq AA n1 0.5 0.5(1-q) 0.5q Aa n2 1-q q qg (1-q)g

Backcross – t-test H0: mean genotypic values are the same for both marker classes.

Backcross – Interpreting Significant Result Find the expected value under HA: In this case, g = (1+k)a or g = (1-k)a A significant result therefore implies |k| ¹ 1 OR a ¹ 0 OR q ¹ 0.5 |k| ¹ 1 a ¹ 0 q ¹ 0.5

Backcross – Linear Model A linear model for this scenario is The linear model yields a total sum of squares between the observations and mean: for m marker classes

Backcross – ANOVA ANOVA partitions the total sum of squares SST into between marker class and within marker class components:

Backcross – ANOVA (cont) We now seek expressions for the expected sums of squares: where N = n1 + n2 +...+nm

Backcross – ANOVA (cont) And for the expectation of the marker sum of squares: where we use with to obtain

Backcross – ANOVA Table Factor df SS MS E(MS) B/W marker class 2-1 SSM MSM W/I marker class N-2 SSe MSe Total N-1 SST MST

Backcross – Estimation of Variance Components Using method of moments, we can obtain estimates of the variance components.

Backcross – ANOVA Statistical Test The F statistic for the ANOVA table can be calculated and used to test the null hypothesis of no linkage.

Backcross – Regression Model In the regression framework, a significant squared correlation r2 indicates that the marker and QTL are linked. Recall that r2 is the trait variance that is explained/predicted by marker genotype.

Backcross – Summary The two explanations for a significant results confound QTL location and QTL effect. Genetic effect is estimated with bias. The QTL location cannot be identified with accuracy. The method has low power when linkage is loose. Some power gained in other methods, but more-or-less minimal. This impacts power calculations.

Project #1 Genotype Observed Count AABB 5 AaBB 1 aaBB AABb 11 AaBb 7 AABb 11 AaBb 7 aaBb 3 AAbb Aabb aabb 4

Coupling or Repulsion?? Codominant loci means the log likelihoods are symmetric about 0.5. Assume coupling. Use EM algorithm to estimate q mle. Estimate: 0.35.

Significant Linkage? Calculate the G statistic to determine if there is significant linkage. Statistic: 3.06. The test for linkage is a one-sided test. Thus, the critical value is 2.71 and we reject H0: q=0.5 with a significance value of 0.05.

Do we have enough power? Calculate the expected G statistic: EG = 13.5. The power to detect linkage assuming all assumptions are met is given by the probability that a chi-square with noncentrality parameter EG exceeds the central chi-squared critical value.

Segregation Analysis Genotype Observed Count Marginal Counts AABB 5 1 aaBB AA = 16 (0.53) AABb 11 Aa = 13 (0.43) AaBb 7 aa = 7 (0.23) aaBb 3 AAbb Chi-squared: 23.6 Aabb aabb 4

The Actual Simulation Conditions The project 1 dataset was generated assuming the following parameter values: 0.28, 0.39, 0.19, 0.22, 0.19, 0.28, 0.39, 0.40 0.21, 0.32, 0.18, 0.21, 0.18, 0.25, 0.39, 0.26

Accounting for Segregation Distortion Gen. Expected Freq. P(R|G) P( |G) AABB AaBB aaBB AABb AaBb aaBb AAbb Aabb aabb