Characteristics of a population Genotype frequency--the relative proportion of different genotypes in a population with respect to a given locus

Genotype Frequencies Define the Population For example, consider the A_ locus We can define a population in terms of the frequencies of the AA, Aa, and aa genotypes. (We often illustrate these concepts with just one locus, since for 2 alleles per locus, there are 3 n possible genotypes--when n=5, there are 243 genotypes).

Gene Frequencies Also Define the Population economy--there are many more genotypes than genes (see previous slide) gene is the stable unit of inheritance-- genes, rather than genotypes, are transmitted from parents to progeny

Gene Frequencies a change in gene frequency is the goal of selection. This is the goal of the breeder, and one of the ways in which we measure progress from selection.

Genotypic Array GenotypeNo.Freq.Aa AA Aa aa

Estimating Gene Frequencies Calculate directly from the number of genes: 130 A alleles/200 = a alleles/200 = 0.35

Estimating Gene Frequencies 2. Estimate from genotype frequencies p 1 (A)=.5 +.5(.3)=0.65 p 2 (a)=.2 +.5(.3)=0.35 or p 2 = = 0.35, since p 1 + p 2 =1.0

Factors affecting gene frequency 1. Population size 2. Migration 3. Mutation 4. Selection 5. Mating system--progeny genotypes constituted by union of gametes in parental generation

Hardy-Weinberg In 1908 British mathematician Hardy and German physician Weinberg showed independently that: For any gene frequency, the following holds true in a large random mating population in the absence of factors which affect gene frequency:

Hardy-Weinberg Given locus A_ in which the frequency of A is p 1 and the frequency of a is p 2, after 1 cycle of random mating, the genotype frequencies of the progeny will be: P 1 2 AA : 2p 1 p 2 Aa : p 2 2 aa

HW Frequencies

Random Mating = Random Union of Gametes MFMF 0.5 A 0.5 a 0.5 A 0.25 AA 0.25 Aa 0.5 a 0.25 Aa 0.25 aa

Hardy-Weinberg Equilibrium 1. Large RM population will reach an equilibrium with respect to gene and genotypic frequencies after only one cycle of RM, regardless of gene freq. in parents. 2. Genotype frequencies of progeny determined solely by gene frequencies of parents.

Hardy-Weinberg Equilibrium 3. In the absence of mutation, migration, selection, and drift, this equilibrium will persist.

HW: Does it Matter to the Plant Breeder? All selection theory that is the underpinning of plant breeding based on HW populations One classic example of an HW population is the Aztec farmer’s field of maize The other classic example of an HW pop is the F 2 generation of a self pollinated crop like wheat or soybean

HW- The OP Maize Field Each plant has the opportunity to mate with any other plant OP varieties were commonly grown before hybrids took hold OP landraces are still common in developing countries

The F 2 Generation has Hardy- Weinberg Frequencies This is harder to grasp, but look at the genotypes and their frequencies in the F 2 AAAaaa The frequency of the A and the a alleles is 0.5 The genotype frequencies in the F 2 are exactly HW

Departure from HW Once we begin to inbreed (non random mating) or select (change gene frequency) we disrupt HW equilibrium But it is useful to know that each breeding population starts off in HW equilibrium

Designation of Inbred Lines The commonly used systems for describing the generations that follow the mating of two parents are the “F” and “S” systems. F refers to Filial, and is most commonly used with self pollinated species and S refers to generations of selfing and is used with cross pollinated species. We will use the F system almost exclusively in this course, but you should be aware of the equivalence of the two systems: F systemS system F2S0 F3S1 F4S2 F5S3

Designation of Inbred Lines One of the conventions we will use in this course is the 2 tier system described by Fehr (p ) in which a line is designated as follows: F x:y in which x is the generation of derivation and y is the current generation. Why is this important? Because there is considerable difference between an F 2 -derived line in the F 6 and an F 4 -derived line in the F 6, yet both could be classified as ‘an F 6 line’. So if we write F 2:4, we are referring to a line that is now in the F4 generation, which came about (was derived) when we harvested a head in the F 2 generation.

Segregation at the A_Locus Under Self Pollination ParentsAAaa F1F1 Aa F2F AA0.5 Aa0.25 aa F3F3 AA AA Aa aa AA aA aa aa Sorted by genotype6/16 AA2/16 Aa2/16 aA6/16 aa Which equals AA0.25 Aa0.375 aa F4F AA0.125 Aa aa The frequency of heterozygotes is estimated by (1/2) G, where G is the number of generations of selfing, i.e., F 2 = (1/2) 1 = 1/2; F 3 = (1/2) 2 = 1/4; F 4 = (1/2) 3 = 1/8.

F 2 – F 8 Generations are Critical If you understand the next two slides, you will have mastered a critical component of plant breeding So Pay Attention!

F2 Bulk Harvest 1000 plants 0.25 AA0.5 Aa0.25 aa F 2:3 Lines250 Lines500 Lines250 Lines Homogeneous AA Like an F 2 population Segregating.25 AA.5 Aa.25 aa Homogeneous aa F 2: 4 LinesHomogeneous AA Like an F 3 population Segregating.375 AA.25 Aa.375 aa Homogeneous aa F 2:5 LinesHomogeneous AA Like an F4 population Segregating.4375 AA.125 Aa.4375 aa Homogeneous aa F 2:6 Lines250 Homogeneous AA Lines Like an F 5 population Segregating AA.0625 Aa aa 250 Homogeneous aa Lines InbredAAAA and aaaa Development of F 2 Derived Lines Without Selection

F 2 Bulk0.25 AA0.5 Aa0.25 aa F 3 Bulk0.375 AA0.25 Aa0.375 aa F 4 Bulk AA0.125 Aa aa Harvest 1000 Individual plants from F4 Bulk 438 Homogeneous AA 125 Segregating Aa Lines438 Homogeneous aa Lines F 4:5 Lines438 Homogeneous AA Lines Like an F 2 population 125 Segregating.25 AA.50 Aa.25 aa Homogeneous aa F 4:6 Lines438 Homogeneous AA Lines Like an F 3 population 125 Segregating.375 AA. 25 Aa.375 aa 438 Homogeneous aa Lines InbredAAAA and aaaa Development of F 4 Derived Lines Without Selection

F 2:6 vs F 4:6 : What are the practical consequences? Note that in the F 4:6 lines 876 of 1000 lines are descendents of homozygous F 2 plants Contrast that with F 2:6 lines, where only 500 of 1000 lines are descendents of homozygous F 2 plants

F 2:6 vs F 4:6 : What are the practical consequences? When you walk the F 2:6 lines, you find much more heterogeneity within a line Concern: will performance be repeatable?

Alleviate this Concern by Delaying Selection (1) Higher proportion of homozygotes in later generations (2) if one observes a superior line in later generations, subsequent performance is more predictable.

Purifying lines for Cultivar Release The first generation of yield testing may be F 4:6 lines which will be somewhat variable If they make the cut for further testing, then purification begins The line that is eventually released as a cultivar may be F 8 derived

Segregation at Multiple Loci with Self-Pollination All F 2 populations will be segregating at more than one locus. This reduces the proportion of completely homozygous individuals compared to the case of single locus segregation.

Segregation at Multiple Loci with Self-Pollination If the probability of homozygosity at the A or B locus is 0.5. The probability of homozygosity at both loci is 0.5 x 0.5 = 0.25.

The Potential Number of Genotypes is Huge Assume parents in a cross have different alleles at n loci : 2 n inbred genotypes can be derived from the cross. If parents differ at 20 loci, then 2 20 or 1,048,576 genotypes are possible

Number of Genotypes in the F 2 is even Bigger Consider though, that 3 20 different genotypes are possible in the F 2 generation--i.e., 3,486,784,401 This fact makes inbreds seem attractive

Larger Populations or More Populations? One of the most vexing plant breeding questions Our F 2 populations typically contain individuals; not 3,486,784,401! This year at Lex we have > 500 F 2 populations and a subset at Princeton

Larger Populations or More Populations? I don’t believe in growing enormous populations in hopes of finding the perfect plant Instead, we try to sample as many populations as possible in hopes of finding a few that meet our many selection criteria

Larger Populations or More Populations? Current issue: are the pop sizes we are using for marker assisted selection big enough??

What are Transgressive Segregates? Progeny that exceed the high parent For example, assume a trait is governed by 20 loci – Inbred Parent 1 contains the superior allele at loci 1 to 11, – Inbred Parent 2 contains the superior allele at loci 12 to 20

What are Transgressive Segregates? Plant breeding progress to date has been built on incremental gains in recovering progeny with additional superior alleles in the homozygous state

How to Predict Success The Binomial Probability Formula: P(x = k) = ( n k )p k (1 - p) n-k Where n = number of loci controlling the trait k = number of loci homozygous for the superior allele p= probability of fixing the superior allele in the homozygous state in the selfing generation F i (i.e., 0.25 in F 2 ; in F 3 ; in F 4 )

How to Predict Success The Binomial Probability Formula: P(x = k) = ( n k )p k (1 - p) n-k the binomial coefficient which estimates the number of ways in which k successes can be chosen from among n trials

How to Predict Success Consider the likelihood of recovering homozygotes for the superior allele at 12 of 20 loci F 2 : p = F 10 : p =

Linkage If favorable alleles are linked in coupling (eg AB), it works in the breeder’s favor Repulsion linkages of favorable alleles (Ab) require recombination to break up Fehr (p 56) gives example of nematode resistance and seed color (r=0.0035)

Recombination – More is Better, Isn’t It? Linkage is a conservative influence which maintains parental arrangements The Debate: Favorable linkage blocks should be maintained vs More genetic variation should be unleashed through intermating

Maize Linkage Blocks