6 Price equation and Selection in quantitative characters

Transcription

1 6 Price equation and Selection in quantitative characters There are several levels of population description. At the most fundamental level, e describe all genotypes represented in the population. With to alleles at each of L loci, one needs 2 2L dynamic variables to describe the population. In some situations (for example if the population is at Hardy-Weinberg proportions), one can use a simpler description based on 2 L gamete frequencies instead of genotype frequencies. Invoking further assumptions (e.g. linkage equilibrium assumption justified by assuming that selection is eak) one can simplify the approach even more by using allele frequencies. This gives only L dynamic variables. Next, e consider the simplest approach for describing populations based on using a single dynamic variable representing the average value of a quantitative character. 6.1 Standard model for a quantitative character Quantitative traits are phenotypic traits that exhibit continuous variation and are subject to microenvironmental effects. Examples: size, eight etc. Quantitative traits are thought to be controlled by many loci ith small effects. The standard model for a quantitative character is described by equation z = g + e, here z is the trait value, g is the contribution of genotype ( genotypic value ; the average trait value for a group of organisms ith the same genotype), and e is the contribution of microenvironment. First, e ill assume that g and e are independent (that is there is no genotype-environment interaction). Genotypic value g can be thought of as a sum of contributions from many loci: g = (α i + α i), here α i and α i are the contributions of the i-th locus from paternal and maternal gamete, respectively. The above model implies that the trait is additive. Microenvironmental deviation e is usually modeled as a random variable ith zero mean and a constant variance: e =0, var{e} =E. Population state is described by the genotypic distribution, p(g), of g in the population. One can also define the moments: the mean value, g, variance, G, etc.: g = gp(g)dg, G = (g g) 2 p(g)dg. Phenotypic distribution, p(z), has the mean, z, and the phenotypic variance, P : 6.2 Types of viability selection z =g, P var{z} =G + E. In major locus models that e studied before fitness as usually assigned to genotype, e.g. AA is fitness of genotype AA (genotype fitness). With quantitative traits fitness depends on phenotype hich in turn is controlled by genotype and environment (genotype + environment phenotype fitness). 56

2 Examples of phenotypic fitness function, = (z). Directional selection: = a + bz or = e az. Here, fitness function monotonically increases (or decreases) ith z. Stabilizing selection: = 1 sx 2 or = e az2. Here, fitness reaches a maximum at an intermediate value of z. Disruptive selection: = e a(z 1)2 + e a(z+1)2. Here, fitness increases ith deviation from an intermediate value of z. The mean fitness of the population is defined as = (z)p(z)dz = (g)p(g)dg, here the (induced) genotypic fitness is (g) = (g + e)p(e)de. 6.3 Robertson-Price formula If p(z) is the phenotypic distribution before selection, then the phenotypic distribution after selection is p s (z) = (z) p(z). Let ψ = ψ(z) be a function of z. The mean value of ψ before selection is ψ = ψ(z)p(z)dz. After selection ψ s = ψ(z)p s (z)dz = ψ(z) p(z)(z) dz. The change in ψ as a result of selection is ψ ψ s ψ = ψ(z) p(z)(z) [ψ(z)(z)]p(z)dz ψ dz ψ = = ψ ψ. Thus, cov(ψ, ) ψ =, (51) here cov(a, b) is the covariance of a and b. This is the Robertson-Price formula. Note that no specific assumptions about the distributions and selection regimes have been made so far. For example, if ψ = z, then cov(z, ) z =. (52) If ψ = z 2, then z 2 = cov(z2, ). (53) 57

3 The Robertson-Price formula predicts the change in a specific population characteristic as a result of ithingeneration selection. Under some additional assumptions it can be used to describe the changes beteen generations. For example, if the trait z is additive, then recombination and segregation ill not change its mean value. Thus, equation (52) describes the change in the mean trait value beteen to subsequent generations. Homeork: Assume that the distribution of z before selection has mean z, variance P and third moment µ 3 (µ 3 (z z) 3 p(z)dz). Find z for linear and quadratic fitness functions ( lin = a + bz, quad = 1 sz 2 ). 6.4 Lande formula: normal approximation Let us assume that both the distribution of g and the distribution of e are Gaussian: p(g) = 1 ) (g g)2 exp (, 2πG 2G p(e) = 1 ) exp ( e2. 2πE 2E The distribution of z ill be normal as ell: p(z) = 1 2πP exp ( ) (z z)2, 2P here P = G + E, z = g. Differentiating ith respect to g g = (g) p(g) g dg = (g) g g G p(g)dg = 1 [ ] (g)gp(g)dg g (g)p(g)dg. G Thus, 1 g = 1 [ G g (g)p(g) ] dg g = 1 G (g s g) = 1 R, (54) G here R g s g is selection response. In a similar ay, differentiating ith respect to z z = (z) p(z) z dz = (z) z z P p(z)dz = 1 [ ] (z)zp(z)dz z (z)p(z)dz. P Thus, 1 z = 1 [ P here S z s z is selection differential. Combining (54) and (55), z (z)p(z) ] dz z = 1 P (z s z) = 1 S, (55) P R = G P S = h2 S, (56) 58

4 here h 2 = G G + E is heritability (in the broad sense). Heritability characterizes the proportion of heritable genetic variation in the overall phenotypic variation. Equation (56) is kno as the breeders equation. It shos that selection response equals heritability times selection differential. From (54), R = G ln (57) g For an additive trait, segregation and recombination do not change the mean trait value (g = g s, R = g g g). Thus, the change in g beteen generations g = G ln (58) g (Lande, 1976). If genotypic variance G does not change, one can use (58) for predicting the long-term dynamics. Because z = g and ln / g = ln / z, one can also rite z = G ln z. (59) Implications: gradient-type dynamics (evolution toards an equilibrium, no cycles, no chaos, average fitness is maximized). 6.5 Lande formula: multivariate case Let there be n phenotypic traits: z i = g i + e i, i = 1,..., n. Assume that the distribution of the genotypic values g i is multivariate normal ith the mean g = (g 1,..., g n ) T and a n n covariance vector G. Then the change in g in one generation is here the vector of selection gradients ln g For example in the case of to traits x and y ( x y = ( ln g 1 g = G ln g,,..., ln g n ) T (Lande, 1979). ) ( ) ( Gx C ln = x ln C G y y ), (60) here G x and G y are genotypic variances, and C is covariance of x and y. Note that even if a trait, say y, does not affect fitness ( ln / y = 0), it ill change if C 0 (correlated response to selection). 6.6 Lande formula: eak selection approximation The assumption that all relevant distributions stay normal is not easy to justify. Also, the mean fitness of the population can be found easily only in some special cases. Here, e consider an alternative method of the derivation of equations analogous to (58) based on less restrictive assumptions. We start ith the Robertson-Price formula g = cov(g, ). (61) 59

5 Expanding = (g) in a Taylor series at g = g, one gets Using the covariance properties, e find that (g) = (g) + d dg (g g) cov(g, (g)) =cov(g g, (g)) =cov(g g, (g) + d dg (g g) =0 + d dg cov(g g, g g) =G d dg d 2 dg 2 µ , d 2 dg 2 (g g) (62) d 2 dg 2 (g g) ) d 2 dg 2 cov(g g, (g g) here µ 3 = (g g) 3 p(g)dg is the third moment of the distribution of g (hich measures asymmetry of p(g)) and all derivatives are evaluated at g = g. Computing the expectation of both sides of (62), one can see that Thus, hich can be approximated by =(g) + d dg (g g) =(g) Gd2 dg g = G d dg d 2 dg 2 (g g) d 2 dg 2 µ (g) G d2 dg g = G d ln dg, (63) here the derivative is evaluated at g = g. Note that to apply (63) one needs to kno the derivative of the individual fitness rather than the mean fitness of the population. The approximation is good is G d2 dg 2 << (g), µ d 2 3 dg 2 << Gd dg. The former condition is satisfied if differences beteen fitnesses are small (eak selection). The latter condition is satisfied if differences beteen selection gradients, d/dg, are small (eak non-linearity in selection; it (g) is linear, all derivatives higher than first ill be zero). Note that equation (63) can be used even if individual fitness depends on the state of the population (that is ith frequency dependent fitness, e.g. = (z, z)). 60

6 6.7 Coevolution in an exploiter-victim system Consider a system of to coevolving species, X and Y. Species X, the victim, suffers from (possibly indirect) interactions ith species Y, hile species Y, the exploiter, benefits from these interactions. Assume that, ithin each species, individuals differ from one another ith respect to an additive polygenic character, x in species X and y in species Y. Characters x and y are under direct (stabilizing) natural selection and also determine the strength of ithin and beteen species interactions. These interactions ill be incorporated in the model by assuming that fitnesses of individuals ith phenotype x in species X, W x (x, p y ), and phenotype y in species Y, W y (y, p x ), depend on the phenotypic distributions p x and p y, i.e., fitnesses are frequency-dependent. The changes in the mean values beteen generations can be approximated by equations ln W x (x, p y ) x = G x, x (64a) ln W y (y, p x ) y = G y, y (64b) here G x and G y are the corresponding additive genetic variances and the partial derivatives are evaluated at x = x, y = y. Assume that to types of selection (stabilizing natural selection and selection arising from interactions beteen species) operate independently throughout the life span of individuals. This allos one to express the overall fitness as a product of to fitness components: W x (x, p y ) = W x,stab (x) W xy,int (x, p y ), W y (y, p x ) = W y,stab (y) W yx,int (y, p x ), (65a) (65b) here, for example, W x,stab and W xy,int describe direct stabilizing selection and selection on x arising from beteen species interactions. Fitness consequences of direct interactions beteen individuals can be described in the folloing ay. First, one introduces a function α ij (u, v) measuring a fitness component for an individual of species i ith phenotype u interacting ith an individual of species j ith phenotype u. To find the fitness component of phenotype u, one then integrates α ij (u, v) over the phenotypic distribution of species j. For example, the fitness component of phenotype x in species X resulting from interactions ith species Y is W xy,int (x, p y ) = α xy (x, y)p y (y)dy for an appropriate function α xy. If selection is eak, the integral is approximately α xy (x, y) (because α xy (x, y) α xy (x, y) + α xy(x, y)(y y) α xy (x, y)). We ill use a Gaussian form for α s leading to W xy,int (x, p y ) = α xy (x, y) = exp[β x (x y) 2 ], W yx,int (y, p x ) = α yx (y, x) = exp[ β y (y x) 2 ]. (66a) (66b) Here β x > 0 and β y > 0 characterize the victim s loss and the exploiter s gain resulting from beteen species interactions. If species X and Y are a prey and its predator, or a host and its parasite, then x and y can be considered as describing individual size or some other quantitative character. Equation (66) implies that for each predator (or parasite) there is an optimum prey (or host) size (or some other quantitative character). If species X and Y represent a Batesian model-mimic pair, then x and y can be considered as describing coloration patterns. Equation (66) implies that a model loses the least hen it is different from the modal phenotype of the mimic species, hile a mimic gains the most hen it is similar to the modal phenotype of the model species. In analyzing the model dynamics e ill make the standard assumption that additive genetic variances G x and G y are constant (for example, maintained by a balance beteen mutation and selection). This is also implied by the eak selection approximation. It is useful to start ith a model here stabilizing selection is absent. 61

7 a b 2 1 y x x Figure 5: The phase-plane dynamics ith no stabilizing selection. (a) R < 1 (the line of equilibria y = x is unstable). (b) R > 1 (the line of equilibria y = x is stable) No stabilizing selection Let us first assume that stabilizing selection is absent (W x,stab = W y,stab = const). To derive the dynamic equations belo, e approximate the difference equations (64) by the corresponding differential equations and re-scale time to τ = 2β x G x t. The dynamics of coevolution are described by ẋ = x y, ẏ = R(x y), (67a) (67b) here R = (β y /β x )(G y /G x ). The straight line x = y represents a line of equilibria. This line is stable if R > 1 and is unstable if R < 1 (see Fig. 5). Species X ins (i.e., escapes and increases its distance from Y as time goes on) if its loss from interactions is bigger than species Y s gain (β x > β y ) and/or its genetic variance is larger than that of Y (G x > G y ). Otherise, species Y ins and the mean values for both species coincide after some transient time. One can say that in this model a species ith a stronger incentive and/or ability to in ins. 62

8 1 fitness character Figure 6: Comparison of Gaussian fitness function (solid line) and quartic exponential fitness function (dashed line) for s = 1, x 0 = Gaussian stabilizing selection A standard choice of a fitness function describing stabilizing selection is Gaussian: W x,stab = exp[ s x (x x 0 ) 2 ], W y,stab = exp[ s y (y y 0 ) 2 ], here x 0 and y 0 are optimum trait values, and s x and s y are parameters characterizing the strength of stabilizing selection. Introducing ne variables u = x x 0, v = y y 0, the dynamics are described by u = e x u + u v + d, v = R( e y v + u v + d), (68a) (68b) here dimensionless parameters e x = s x /β x, e y = s y /β y characterize the strength of stabilizing selection relative to selection arising from beteen-species interactions, and d = x 0 y 0 is the difference of the optimum values. Because this is a linear system, its dynamics are clear. Homeork: interpret the dynamical regimes of (68) in biological terms Quartic exponential stabilizing selection Stabilizing selection refers to situations hen fitness decreases ith deviation from some optimum value. There are different ays to choose a specific functional form of a fitness function describing stabilizing selection. Although a Gaussian function is a popular choice, it is used primary because of its mathematical convenience. Here, e ill use an exponential function of fourth order polynomials W x,stab = exp[ s x (x x 0 ) 4 ], W y,stab = exp[ s y (y y 0 ) 4 ]. (69a) (69b) Stabilizing selection described by (69) is eaker than Gaussian selection near the optimum phenotype but becomes stronger beyond a certain value (see Fig.6). With stabilizing selection as specified in equation (69), the dynamic 63

9 equations for the deviations of the mean values from x 0 and y 0 are u = 2e x u 3 + u v + d, v = R( 2e y v 3 + u v + d), (70a) (70b) here parameters e x, e y, R and d are defined by the same formulae as above. Homeork: Find the conditions for existence and stability of equilibria of system (70) assuming that d = 0. Represent these conditions using a to-dimensional parameter space (k, R) ith k = (e x /e y ) 1/3. (Hint: see Fig.7.) Use Maple (or your favorite softare) to dra trajectories of (70) corresponding to different dynamical patterns. Interpret the conditions for cycling in biological terms S R 1.0 SUS 0.5 U UUU k Figure 7: Areas in parameter space (k, R) corresponding to different patterns of existence and stability of equilibria in (70). 64

10 6.8 Sexual selection In many species, females prefer mates ith extreme characters that are apparently useless or deleterious for survival, such as bright colors, elaborate ornaments and conspicuous songs. Sexual selection as proposed by Darin to explain the evolution of such traits. The distinction beteen natural and sexual selection is that the former arises from differences in individual survival and/or fecundity hereas the latter arises from differences in mating success. To types of sexual selection are emphasized: intra-sexual selection (competition among individuals of the same sex for access to the other sex) and inter-sexual selection (choice of individuals of one sex by the other) Lande s model (1981, Proc. Natl. Acad. Sci. USA 78: ) We consider to trait: a trait, y, expressed in males only (e.g., plumage in birds) and a trait, x, expressed in females only (preference for y). We assume that there are to components of male fitness: = stab sex, coming from stabilizing selection that acts directly on the trait, stab = exp[ s(y θ) 2 )] and from sexual selection, sex, hich is defined belo. We assume that there is no direct selection in females. Hoever, females ill evolve as a result of correlated selection. We define a preference function ψ(y x) as the probability that male y is chosen by female x. There are several possibilities for choosing a specific function ψ(y x): open ended preference : ψ(y x) = exp(αxy) absolute preference : ψ(y x) = exp[ α(y x) 2 ] relative preference : ψ(y x) = exp[ α(y (y + x)) 2 ] The fitness component sex is defined as as the probability to be chosen: exp[αyx] sex = ψ(y x)p(x)dx = exp[ α(y x) 2 ] exp[ α(y y x) 2 ], here p(x) is the distribution of x in females. The corresponding selection gradients are ln sex y=y = y αx 2α(y x) 2αx and can be represented as α (x ɛy), here ɛ = 0 or 1 and α = α or 2α. Thus, the overall selection gradient is ln sex y=y = s(y θ) + α (x ɛy), y here the first term is coming from stabilizing natural selection. The change in y in one generation is y = G y ln sex / y y=y, here G y is the variance of y in males. The correlated change in y in one generation is x = C ln sex / y y=y, here C is covariance of x and y (coming from pleiotropy and/or linkage disequilibrium; see equation 60). Using the differential approximation, the dynamic equations become ẋ = C[ s(y θ) + α (x ɛy)], ẏ = G y [ s(y θ) + α (x ɛy)]. 65

11 Re-scaling time to τ = tg y α, these equations can be reritten as ẋ = r[x (ɛ + s α )y + s α θ], ẏ = x (ɛ + s α )y + s α θ. (71a) (71b) here r = C/G y. The analysis of (71) is simple (this is a linear system). There is a line of equilibria x = (ɛ + s α )y s α θ, hich is unstable if r > ɛ + s α and is stable if r < ɛ + s α. Thus, there are to different dynamical regimes: runaay evolution toards infinite trait values or evolution toards the line of equilibria. In the first regime, sexual selection drives the evolution of exaggerated male characters. This regime is promoted if stabilizing selection is eaker than sexual selection (s << α ) or if genetic covariance of male and female traits is large (r >> 1). In the second regime, sexual selection leads to a line of equilibria. This regime is promoted if stabilizing selection is stronger than sexual selection (s >> α ) or if genetic covariance of male and female traits is small (r << 1). Note that finite populations ill diverge along the line of equilibria by random genetic drift Sexual conflict (see the Nature paper) 66