Evolutionary quantitative genetics and one-locus population genetics READING: Hedrick pp. 57 63, 587 596 Most evolutionary problems involve questions about phenotypic means Goal: determine how selection causes evolutionary change in the mean of a quantitative character. Aside: Quantitative vs. Qualitative Traits Back to our main story... selection on a trait such as body size body size of individual is denoted z mean body size in a population: z Suppose selection alters allele frequencies at loci that affect z Consider the effect on body size of one specific locus, i In general, to determine how much z changes, one needs to determine: 1) how selection changes p ( i ) at all loci affecting z 2) how changes in p ( i ) combine to change z. body size z ( i ) ( Δz i ) Δp ( i) 0 1 p ( i ) We'll focus on the effects of one of the loci that affects z The phenotype of an individual depends on: 1) its genotype at locus i 2) its genotype at other loci that affect the trait 3) its environmental experience VI-1
lump 2 & 3 into a single factor, c Can think of an individual's phenotype as z = a jk + c a 22 z = c + a 12 a 11 A 1 A 1 A 1 A 2 A 2 A 2 The average phenotype can similarly be broken down into the average contribution of locus i and the average over other loci and environmental effects: z = a + c Consider the effect of a change in p ( i ) on z (holding all other loci constant) ( ( ) ) " z p ( i)!z ( i) = z p ( i) +!p i ( ) dp ( i )!p ( i ) (drop i) dp " % $ 1 d ln w 2 p( 1! p) ' $ 14 243 dp variance 12 3 ' # & selection (term in brackets is just our old friend) " 1 2 p( 1! p) dp # $ dp d ln w % &' (using the "chain rule") or!z ( i) 1 2 pq 2! # $ d ln w " dp% (form emphasizes the dependency of fitness w on phenotype z) What is dp? VI-2
dp = d dp ( a + c ) = d ( dp p 2 a 1 1 + 2pqa 1 2 + q 2 a 2 2 + c ) (assuming H- W) = 2pa 1 1 + 2qa 1 2! 2 pa 1 2! 2qa 2 2 + 0 So, putting all the pieces together: or!z ( i) " 1 2 # pq # 4( pa 1 1 + qa 1 2 $ pa 1 2 $ qa 2 2 ) 2 d ln w!z ( i) " G ( i) # where G ( i ) = 2pq pa 1 1 + qa 1 2 [( )! ( pa 1 2 + qa 2 2 )] 2 and! = d ln w. What is!? Called the selection gradient It is a measure of the strength of selection acting on a trait: What is G ( i )? It is an extremely important quantity called the additive genetic variance contributed by locus i Why "additive"? Dominance Variance Consider the total amount of variation in z due to variation at locus i: var ( z ( i) ) = var ( a jk + c) = var ( a jk ) since we have fixed c. [ ] 2 So, var ( z ( i) 2 ) = var ( a jk ) = E( a jk )! E( a jk ) (assuming H-W equilibrium) = p 2 a 2 1 1 + 2 pqa 2 + q 2 a 2 1 2 2 2 ( )! ( p 2 a 1 1 + 2pqa 1 2 + q 2 a 2 2 ) 2 VI-3
(after a lot of tedious algebra) = G ( i ) & + 2 pq " a 1 2! a + a 1 1 2 2 $ ) ' * ( # 2 % + = G ( i ) + D ( i ) 2 where D ( i ) & = 2 pq " a 1 2! a + a 1 1 2 2$ ) ' * ( # 2 % + 2. What is D ( i )? It's called the dominance variance contributed by locus i. Note: D ( i ) is never negative and has units of z 2 Why "dominance"? Relationship between G ( i ) and D ( i ) Case 1: No Dominance Case 2: Symmetric Overdominance What about other loci that contribute to the traits? Generally difficult (recall complications of 2-locus population genetics) to understand; Assuming linkage equilibrium for all loci contributing to a trait:!z =!z ( 1) +!z ( 2) +!z ( 3) +L = [ G ( 1) + G ( 2 ) + G ( 3) +L]" n " = G ( i ) %! # $ i =1 &' ( or (recall that! reflects how a change in mean phenotype changes mean population fitness: it doesn't care what the cause of the change in phenotype is.)!z = G" VI-4
This is the central equation of quantitative genetics Note its similarity to the equation for selection at one locus. Note that it's written in several other ways Important Message: It's not the total phenotypic variance nor even the total genotypic variance that determines how fast a population mean evolve in response to selection Total genotypic variance = G + D (where D = D ( 1) + D ( 2) +L=! D ( i) ) In general, there's between-locus genetic "interaction" variance as well. Total phenotypic variance = G + D + E (also written V P = V A + V D + V E ) E is the variance that's due to environmental effects. n i =1 A few quibbles with heritability Selection on a single character (!z = G" ): Other ways of determining G We now have an algebraic definition of G in terms of a jk ' s and p's at all loci There are, however, other ways to get G We'll skip the tedious algebra (see Falconer), but the methods are important since they show how G can actually be measured! (most of Falconer's book concerns estimating G in agricultural or laboratory settings. See, e.g., Simms & Rausher for methods of measuring G in natural populations) (1) Breeding Values Measure the type of offspring an individual tends to produce in a particular reference population breeding values were originally used by plant and animal breeders for assessing the value of an individuals for mating. Definition: g = z + 2( z o! z ) VI-5
note that an individual's own phenotype doesn't enter in this - except through z z o = average value of the trait among the individual's offspring when she/he is mated to a large # of randomly chosen individuals alternatively, z o may be thought of as the expected phenotype the individual's offspring were it to mate with a randomly chosen individual in the population. Why the 2? Only 1/2 of an offspring phenotype is attributable to that individual. The other 1/2 comes from its mate. Important note: the breeding value of an individual refers to a particular population as well as the individual. That is, the same individual could have a different breeding value were it moved to a different population. Uses of breeding value Suppose a farmer (or nature) chooses a set of individuals to reproduce The mean value of the offspring will equal the average breeding value of the selected individuals Breeding values can be estimated In principle, one could 1) mate an individual to many others 2) raise and measure the offspring 3) estimate the breeding value using the average offspring phenotype This is actually done for important breeding stock, dairy cattle, race horses, etc. What about variation in breeding values? It's obviously related to a population's genetic variation can actually compute it under our earlier assumptions It turns out that var ( g) = G VI-6
Significance Statistically, can think of the phenotype of an individual as z = g + d + e g and d are inherited from its parents, but the individual will only pass on g By the way g and d are defined, d is acquired by an offspring independently of g These results suggest a way to measure G without ever knowing the p's and a ij 's! could experimentally determine breeding values of individuals in a population, then compute their variance this is, essentially, what a "half-sib" breeding design does Drawback: this method is rather tedious, expensive, and labor intensive Turns out there's an easier way... (2) Parent-offspring regression The response to selection, "z, depends on the relationship between parents and offspring diagrammed here. "z depends on how strongly selection acts on the mean (s) the orientation of the cloud of points note that the orientation depends on inheritance e.g., if there's no genetic variance, the regression line would be flat. This orientation is related to genetics. It can be show (see H&C or Falconer) that, under our previous assumptions: G = 2cov ( z o,z p ) This is measurable (and, once again, doesn't require estimating the p's and a ij 's). It's one of the major methods for estimating G Selection on a single character (!z = G" ): The selection gradient revisited We'll now consider two definitions of the selection gradient VI-7
(1)! = change in the logarithm of a population's mean fitness that would result if the population's mean phenotype was increased by one "unit" Mathematically, ln w ( z +!) i.e.,! = d ln w [ ] " ln[ w ( z )] # d ln w ( ) $ ln w ( z ) " ln w z + # #! = $!,! measures steepness of mean population fitness function this is the relevant measure of the strength of selection acting on the mean. The units of! are 1/char. The relationship between w( z), w, and! : w( z) : individual fitness function describes the fitness of an individual as a function of it's phenotype w : population mean fitness it's the average fitness of individuals in the population i.e., it's the average of w( z) with respect to the distribution of phenotypes z in the population The relationship between w( z) and w : Suppose that ecologists/evolutionary biologists determine the following fitness function a) If P = 0 (i.e., there is no phenotypic variation => all individuals are identical) then, w = w z Note: w = w ( z ): ( ) = w( z ) since z = z. b) If there's lots of phenotypic variation (P >> 0): In general, w is, roughly, a smooth version of w( z). "weak selection":! is approximately independent of P VI-8
or, w( z) changes little over the distribution of z refers to stabilizing or disruptive component of selection note that the directional component of selection may still be strong (2)! = regression of relative fitness onto the value of the character Mathematically,! = cov ( w, z ) var ( z) this indicates how! can be measured! Effects of selection within a generation: the "selection differential" The selection differential measures the within-generation impact of selection on a population's mean phenotype. Mathematically, the selection differential is defined as s = z *! z Another mathematically equivalent definition (due to Robertson & Price): s = cov ( w, z) The relationship between the selection gradient and selection differential: s = P!! = s P Note: for a single trait, there seems little difference between s and!. However, when the simultaneous evolution of several quantitative traits is being considered, the selection gradient have many advantages over the selection differential (both mathematically and in biological interpretation). Traditional form for the response to selection:!z = G" = G s P = G P s = h 2 s where h 2 = G P is the "narrow-sense" heritability of z. Comments on heritability: There are two types: "broad-sense" and "narrow-sense" Broad-sense heritability is the fraction of phenotypic variability that is due to all forms genetic variation = (G + D + "epistatic variance")/p. VI-9
Narrow-sense heritability (h 2 ) is the fraction of phenotypic variability due to additive-genetic variance. h 2 and s are relatively uninformative regarding rates of evolution (compared to G and! ): without changing G and! (and hence the rate of evolution!z = G" ), one can s simply by P (=> h 2 ). Breeders like s because it is easier to control. Evolutionary Equilibrium under selection:!z = G"!z = 0 for two possible reasons: (1)! = 0 (2) G = 0. Possibility (1) corresponds to a balance of ecological forces ˆ z (the equilibrium mean of the population) is either a maximum of minimum of w the former is stable, the latter is unstable If selection is sufficiently weak, z will evolve in such a way that mean fitness will. Note: fitness refers to overall fitness, not to antagonistic components of selection that may occur within a generation e.g., Darwin's finches: Price found that small size is favored in juveniles (possibly due to energy conservation) but large size is favored in adults (easier to crack seeds). Possibility (2) corresponds to a lack of evolutionary "material" This situation correponds to what are commonly called "developmental" or "genetic" constraints. additive variance can reappear in several ways mutation & migration VI-10
recombination between loci linkage disequilibrium affects the amount of variance expressed. e.g., a population with only (+ -) (- +) gametes has no phenotypic variance despite having copious amounts of genetic variance. recombination can create additive genetic variance recombination can destroy variation as well (e.g., (+ +) (- -) population.) VI-11