Inférence en génétique des populations IV.
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1 Inférence en génétique des populations IV. François Rousset & Raphaël Leblois M2 Biostatistiques FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
2 Modeling and analysis of covariances in allele frequencies Spatial Intra-family FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
3 Modeling and analysis of covariances in allele frequencies Modeling: important parameters in selection processes FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
4 Modeling and analysis of covariances in allele frequencies Modeling: important parameters in selection processes Artificial selection, or how to increase milk production by selecting bulls Wright s shifting balance theory General formal theory of selection with interactions between relatives FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
5 Modeling and analysis of covariances in allele frequencies Modeling: important parameters in selection processes Artificial selection, or how to increase milk production by selecting bulls Wright s shifting balance theory General formal theory of selection with interactions between relatives Statistical analysis: relatively simple (and routine) methods of inference FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
6 Modeling and analysis of covariances in allele frequencies Modeling: important parameters in selection processes Artificial selection, or how to increase milk production by selecting bulls Wright s shifting balance theory General formal theory of selection with interactions between relatives Statistical analysis: relatively simple (and routine) methods of inference Two main spatial models (Wright): island model, and isolation-by-distance Estimate Nm or the neighborhood size 4Dπσ 2 A few statistically desirable properties Analytical understanding FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
7 Modeling and analysis of covariances in allele frequencies Modeling: important parameters in selection processes Artificial selection, or how to increase milk production by selecting bulls Wright s shifting balance theory General formal theory of selection with interactions between relatives Statistical analysis: relatively simple (and routine) methods of inference Two main spatial models (Wright): island model, and isolation-by-distance Estimate Nm or the neighborhood size 4Dπσ 2 A few statistically desirable properties Analytical understanding Huge literature, often based on very naive ideas FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
8 Outline An example of the role of spatial structure in evolution Modeling of the role of spatial structure Formal modelling of covariances: diffusion vs. coalescence Genealogical interpretations of covariances and regressions An example of statistical inference based on covariances FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
9 Outline An example of the role of spatial structure in evolution Modeling of the role of spatial structure Formal modelling of covariances: diffusion vs. coalescence Genealogical interpretations of covariances and regressions An example of statistical inference based on covariances FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
10 Example: evolution of social behaviour Siderophore production as public good D Pseudomonas aeruginosa C D A Defector that does not produce the public good still benefits from those produced by the Cooperators. Public good production should be counter-selected. FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
11 Biologists test theories about the importance of dispersal Controlling bacterial dispersal Different agar concentrations: FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
12 Public goods production mean relative mutant fitness ± s.e. (v) % 0.25% 0.5% 1% agar concentration Viscosity Siderophore-defective strain wins Siderophore-producing strain wins Kümmerli et al, PRSB, 2009 FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
13 The effect of dispersal almost explained FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
14 The effect of dispersal almost explained FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
15 The effect of dispersal almost explained FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
16 The effect of dispersal almost explained FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
17 The effect of dispersal almost explained FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
18 The effect of dispersal almost explained FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
19 The effect of dispersal almost explained FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
20 The effect of dispersal almost explained FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
21 The effect of dispersal almost explained FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
22 The effect of dispersal almost explained FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
23 The effect of dispersal almost explained Same logic important for virtually any trait evolving in a population spatially subdivided in (small) subpopulations. FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
24 Outline An example of the role of spatial structure in evolution Modeling of the role of spatial structure Formal modelling of covariances: diffusion vs. coalescence Genealogical interpretations of covariances and regressions An example of statistical inference based on covariances FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
25 Formulation in terms of partial regression We have seen that (for deterministic models) the change in allele frequency can be written in terms of allelic fitnesses w and w, as p p =(w w )p (1 p ) =β w,1 Var(1 ) = Cov[w i, 1 (i))] where β w,1 is the regression coefficient of an individual s allelic fitness w i on her genotype 1 (i) FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
26 Formulation in terms of partial regression We have seen that (for deterministic models) the change in allele frequency can be written in terms of allelic fitnesses w and w, as p p =(w w )p (1 p ) =β w,1 Var(1 ) = Cov[w i, 1 (i))] where β w,1 is the regression coefficient of an individual s allelic fitness w i on her genotype 1 (i) We can apply the standard recursive relationship for regression coefficients between a response variable W and two predictors F and N: β W,F = β W,F.N + β N,F β W,N.F. here for F := 1, the indicator variable for the (focal) individual s genotype, and N := the mean value of the indicator variable for a group of social partners FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
27 Formulation in terms of partial regression We have seen that (for deterministic models) the change in allele frequency can be written in terms of allelic fitnesses w and w, as p p =(w w )p (1 p ) =β w,1 Var(1 ) = Cov[w i, 1 (i))] where β w,1 is the regression coefficient of an individual s allelic fitness w i on her genotype 1 (i) This entails the two-predictor regression expression for the change in allele frequency: p p = (β W,F.N + β N,F β W,N.F ) Var(1 ) where β N,F is the regression coefficient of the allele frequency among social partners to allele presence in the focal individual. β N,F is (one definition of) relatedness in the genetic literature. FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
28 Outline An example of the role of spatial structure in evolution Modeling of the role of spatial structure Formal modelling of covariances: diffusion vs. coalescence Genealogical interpretations of covariances and regressions An example of statistical inference based on covariances FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
29 Probability of identity Probability that two gene copies are of same allelic state = Identity in state FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
30 Probability of identity Probability that two gene copies are of same allelic state = Identity in state Cf Wright-Fisher model with two alleles a u A v FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
31 Probability of identity Probability that two gene copies are of same allelic state = Identity in state Cf Wright-Fisher model with two alleles a u A v pdf of allele frequency f (p) =Beta(α = 2Nu, β = 2Nv) FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
32 Probability of identity Probability that two gene copies are of same allelic state = Identity in state Cf Wright-Fisher model with two alleles a u A v pdf of allele frequency f (p) =Beta(α = 2Nu, β = 2Nv) 1 P(identity) = 0 f (x) dx FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
33 Probability of identity Probability that two gene copies are of same allelic state = Identity in state Cf Wright-Fisher model with two alleles a u A v pdf of allele frequency f (p) =Beta(α = 2Nu, β = 2Nv) 1 P(identity) = [x 2 + (1 x) 2 ]f (x) dx = u + 2Nu2 + v + 2Nv 2 0 (u + v)[1 + 2N(u + v)] Nu if u = v FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
34 Probability of identity Probability that two gene copies are of same allelic state = Identity in state Cf Wright-Fisher model with two alleles a u A v pdf of allele frequency f (p) =Beta(α = 2Nu, β = 2Nv) 1 P(identity) = [x 2 + (1 x) 2 ]f (x) dx = u + 2Nu2 + v + 2Nv 2 0 (u + v)[1 + 2N(u + v)] Nu if u = v Mathematical overkill! FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
35 Probability of identity: coalescent argument Infinite allele model identity by descent FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
36 Probability of identity: coalescent argument Infinite allele model identity by descent For c t the probability that coalescence of a a given pair of genes occurred t generations ago, P(identity) = Q = c t...? t=0 FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
37 Probability of identity: coalescent argument Infinite allele model identity by descent For c t the probability that coalescence of a a given pair of genes occurred t generations ago, P(identity) = Q = c t (1 µ) 2t = t=0 c t γ t for γ (1 µ) 2 t=0 FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
38 Probability of identity: coalescent argument Infinite allele model identity by descent For c t the probability that coalescence of a a given pair of genes occurred t generations ago, P(identity) = Q = c t (1 µ) 2t = t=0 c t γ t for γ (1 µ) 2 t=0 Generating function FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
39 Probability of identity: coalescent argument Infinite allele model identity by descent For c t the probability that coalescence of a a given pair of genes occurred t generations ago, P(identity) = Q = Generating function c t (1 µ) 2t = t=0 For Wright-Fisher model: Q = 1 c t γ t for γ (1 µ) 2 t=0 ( ) t 1 γ t γ = N N γ + N(1 γ) Nµ when N and µ 0 FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
40 Infinite island model; Wright s F ST A very simple, easily solved model Immigration m Frequency of some focal allele locally, p = p i for given island i, and globally, p. FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
41 Infinite island model; Wright s F ST Diffusion approximation for frequency in each island i: P i Const p 2Nm p 1 i (1 p i ) 2Nm(1 p) 1 =Beta(2Nm p, 2Nm(1 p)) FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
42 Infinite island model; Wright s F ST Diffusion approximation for frequency in each island i: P i Const p 2Nm p 1 i (1 p i ) 2Nm(1 p) 1 =Beta(2Nm p, 2Nm(1 p)) We have seen that the stationarity distribution satisfies: 0 = f (p, t) t a(p)f (p, t) = + 2 b(p)f (p, t) p 2 p 2 f (p) exp ( 2 p a(x)/b(x) dx ) b(p) FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
43 Infinite island model; Wright s F ST Diffusion approximation for frequency in each island i: P i Const p 2Nm p 1 i (1 p i ) 2Nm(1 p) 1 =Beta(2Nm p, 2Nm(1 p)) Wright-Fisher model: b(p) = p(1 p) and a(p) = N( vp + u(1 p)) = N[(u + v)(p u/(u + v))] Frequency P Const p 2Nu 1 (1 p) 2Nv 1 =Beta(α = 2Nu, β = 2Nv) FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
44 Infinite island model; Wright s F ST Diffusion approximation for frequency in each island i: P i Const p 2Nm p 1 i (1 p i ) 2Nm(1 p) 1 =Beta(2Nm p, 2Nm(1 p)) Wright-Fisher model: b(p) = p(1 p) and a(p) = N( vp + u(1 p)) = N[(u + v)(p u/(u + v))] Frequency P Const p 2Nu 1 (1 p) 2Nv 1 =Beta(α = 2Nu, β = 2Nv) Island model b(p i ) = p i (1 p i ) and a(p i ) = N(m(p i p)) Frequency P Beta[α = 2Nm p, β = 2Nm(1 p)] FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
45 Infinite island model; Wright s F ST Diffusion approximation for frequency in each island i: P i Const p 2Nm p 1 i (1 p i ) 2Nm(1 p) 1 =Beta(2Nm p, 2Nm(1 p)) F ST Var p i p(1 p) = E(p2 i ) p2 p(1 p) = Nm More generally variances and covariances of allele frequencies can be expressed in terms of probabilities of identity of pair of genes. Here locally, P( p i ; p) = F ST p i + (1 F ST ) p 2 (with mutation at a low enough rate: proven in next slides); hence globally. P( ; p) = F ST p + (1 F ST ) p 2 FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
46 Outline An example of the role of spatial structure in evolution Modeling of the role of spatial structure Formal modelling of covariances: diffusion vs. coalescence Genealogical interpretations of covariances and regressions Infinite island model F ST, separation of time scales Finite island model coalescence times, separation of time scales Isolation by distance separation of time scales Robustness of regression coefficients An example of statistical inference based on covariances FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
47 Genealogical interpretation of F ST R R R R R R R R R R R R R R R R R R R R R R R R R FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
48 Genealogical interpretation of F ST past generations -2-1 R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
49 Genealogical interpretation of F ST past generations p (or p i ) -2-1 R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
50 Genealogical interpretation of F ST past generations p (or p i ) p R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
51 Genealogical interpretation of F ST past generations F ST p (or p i ) (1 F ST ) p R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
52 Wright s F ST redefined (For any genetic marker) One can describe the correlation of allele types within some given sub-population relative to the allele types of random genes in the total population as F ST = Q w Q b 1 Q b in terms of probabilities of identity Q w and Q b within and between subpopulations. This leads to relatively simple analysis. FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
53 General facts about Wright s F -statistics ˆF ST = ˆQ w ˆQ b 1 ˆQ b F IS = Q 0 Q w 1 Q w FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
54 General facts about Wright s F -statistics Naive but straightforward estimators ˆF ST = ˆQ w ˆQ b 1 ˆQ b F IS = Q 0 Q w 1 Q w FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
55 General facts about Wright s F -statistics ˆF ST = ˆQ w ˆQ b 1 ˆQ b F IS = Q 0 Q w 1 Q w Relationship with average coalescence times Q = c t γ t for γ (1 µ) 2 = 1 2µ E(T ) + O(µ 2 ) t=0 F ST E(T b) E(T w ) E(T b ) when µ 0. FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
56 Modelling in terms of probabilities of identity Q Example: finite island model Relationship of Q w with Q w, Q b one generation earlier? where and FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
57 Modelling in terms of probabilities of identity Q Example: finite island model Relationship of Q w with Q w, Q b one generation earlier? where Q(t + 1) = γa[q(t) + c(t)] and FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
58 Modelling in terms of probabilities of identity Q Example: finite island model Relationship of Q w with Q w, Q b one generation earlier? Q(t + 1) = γa[q(t) + c(t)] where and Q(t) = ( Qw (t) Q b (t) ) and c(t) = ( 1 Qw(t) N 0 ) FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
59 Modelling in terms of probabilities of identity Q Example: finite island model Relationship of Q w with Q w, Q b one generation earlier? where and Q(t) = A = ( Q(t + 1) = γa[q(t) + c(t)] ( Qw (t) Q b (t) ) and c(t) = ( 1 Qw(t) N 0 ) a 11 = (1 m) 2 + m2 n d 1 1 a 11 1 a 11 a 11 n d 1 n d 1 is the transition matrix for the random walk of two gene lineages between the state within same deme and the complementary state. ) FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
60 The stationary solution in an informative form Q = γ(i γa) 1 Ac Eigensystem of A λ 1 = 1, e 1 = (1, 1) ( ) 2, λ 2 = 1 m n d n d 1 e2 = (n d 1, 1) ( 1 Qw ) c = N = 1 Q w 1 (e e 2 ) N n d FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
61 The stationary solution in an informative form Q = γ(i γa) 1 Ac Eigensystem of A λ 1 = 1, e 1 = (1, 1) ( ) 2, λ 2 = 1 m n d n d 1 e2 = (n d 1, 1) ( 1 Qw ) c = N = 1 Q w 1 (e e 2 ) N n d Q = 1 Q w N 1 n d ( γ 1 γ e 1 + γλ ) 2 e 2 1 γλ 2 FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
62 The stationary solution in an informative form Q = γ(i γa) 1 Ac Eigensystem of A λ 1 = 1, e 1 = (1, 1) ( ) 2, λ 2 = 1 m n d n d 1 e2 = (n d 1, 1) ( 1 Qw ) c = N = 1 Q w 1 (e e 2 ) N n d Q = 1 Q w N 1 n d ( γ 1 γ e 1 + γλ ) 2 e 2 1 γλ 2 Q w Q b 1 Q w = 1 N γλ 2 1 γλ 2 FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
63 The stationary solution in an informative form Q = γ(i γa) 1 Ac Eigensystem of A λ 1 = 1, e 1 = (1, 1) ( ) 2, λ 2 = 1 m n d n d 1 e2 = (n d 1, 1) ( 1 Qw ) c = N = 1 Q w 1 (e e 2 ) N n d Q = 1 Q w N 1 n d ( γ 1 γ e 1 + γλ ) 2 e 2 1 γλ 2 F ST Q w Q b 1 Q b = γλ 2 γλ 2 + N(1 γλ 2 ) Nm FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
64 The stationary solution in another informative form in terms of eigenvalues of another matrix G: Q = γa(q + c) = γ[gq + (A G)1] = γ[gq + (I G)1] where G is the matrix whose ijth element is the coefficient of the jth element of Q in the ith element of A[Q + c]. Denote l j and u j the eigenvalues and eigenvectors of G; and write (I G)1 as j x ju j. Then FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
65 The stationary solution in another informative form in terms of eigenvalues of another matrix G: Q = γa(q + c) = γ[gq + (A G)1] = γ[gq + (I G)1] where G is the matrix whose ijth element is the coefficient of the jth element of Q in the ith element of A[Q + c]. Denote l j and u j the eigenvalues and eigenvectors of G; and write (I G)1 as j x ju j. Then Q = γ(i γg) 1 j x j u j = j γx j u j = 1 γl j t γ t j l t 1 j x j u j i.e. for ith element of Q, c i,t = j lt 1 j x j u ji. FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
66 Genealogical interpretation of F -statistics (N = 100, m = 10 3, n d = 10) Probability of coalescence FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
67 Genealogical interpretation of F -statistics Interpretation of F ST from comparison of two distributions of coalescence times FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
68 Genealogical interpretation of F -statistics robustness: ancient events have no effect on F ST FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
69 Genealogical interpretation of F -statistics (N = 100, m = 10 3, n d = 10) Probability of coalescence FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
70 Genealogical interpretation of F -statistics (N = 100, m = 10 3, n d = 1000) Probability of coalescence FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
71 Genealogical interpretation of F -statistics (N = 100, m = 10 3, n d = ) Probability of coalescence FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
72 Genealogical interpretation of F ST past generations F ST p (1 F ST ) p R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
73 Island model with self-fertilization ( selfing ) Three eigenvalues of G = different time scales dependent on magnitude of n d N vs. N vs selfing rate. FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
74 Isolation by distance Pattern of isolation by distance Genealogical interpretation FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
75 Isolation by distance: a quantitative approximation Generating function of one-lineage dispersal ψ(z) := k m kz k ; FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
76 Isolation by distance: a quantitative approximation Generating function of one-lineage dispersal ψ(z) := k m kz k ; Generating function of two-lineage increments in distance dispersal ψ 2 (z); FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
77 Isolation by distance: a quantitative approximation Generating function of one-lineage dispersal ψ(z) := k m kz k ; Generating function of two-lineage increments in distance dispersal ψ 2 (z); In terms of the eigensystem of A for any distribution of dispersal distance: Q = 1 1 Q w Nn d j γλ j 1 γλ j e j λ k = ψ 2 ( e ı2πk/nx ) and e k = (cos(2πki/n x )) for (i = 0,..., n x 1); FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
78 Isolation by distance: a quantitative approximation Generating function of one-lineage dispersal ψ(z) := k m kz k ; Generating function of two-lineage increments in distance dispersal ψ 2 (z); In terms of the eigensystem of A for any distribution of dispersal distance: Q = 1 1 Q w Nn d j γλ j 1 γλ j e j λ k = ψ 2 ( e ı2πk/nx ) and e k = (cos(2πki/n x )) for (i = 0,..., n x 1); Two dimensional λ kl = ψ 2 ( e ı2πk/nx ) ψ 2 ( e ı2πl/ny ) ; FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
79 Isolation by distance: a quantitative approximation Q w Q ij 1 Q w = 1 Nn d 1 Nπ 2 k l π π Q w Q d 1 Q w log(d) 2Nπσ 2 + constant 0 0 γλ kl 1 γλ kl (e kl,0 e kl,ij ) γψ 2 (e ıx ) ψ 2 (e ıy ) 1 γψ 2 (e ıx ) ψ 2 (e ıy [1 cos(kx) cos(ly)] dx dy ) FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
80 Isolation by distance: a quantitative approximation Q w Q ij 1 Q w = 1 Nn d 1 Nπ 2 k l π π Q w Q d 1 Q w log(d) 2Nπσ 2 + constant 0 0 γλ kl 1 γλ kl (e kl,0 e kl,ij ) γψ 2 (e ıx ) ψ 2 (e ıy ) 1 γψ 2 (e ıx ) ψ 2 (e ıy [1 cos(kx) cos(ly)] dx dy ) 2Nπσ 2 Q w Q d 1 Q w FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
81 Outline An example of the role of spatial structure in evolution Modeling of the role of spatial structure Formal modelling of covariances: diffusion vs. coalescence Genealogical interpretations of covariances and regressions An example of statistical inference based on covariances FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
82 A simple inference under isolation by distance 2Nπσ 2 Q w Q d 1 Q w regress ˆQ w ˆQ d 1 ˆQ w against log(distance) to obtain an estimate of 1/2Nπσ 2. FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
83 A simple inference under isolation by distance 2Nπσ 2 Q w Q d 1 Q w regress ˆQ w ˆQ d 1 ˆQ w against log(distance) to obtain an estimate of 1/2Nπσ 2. FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
84 Statistical inference: an illustration FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
85 Statistical inference: an illustration FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
86 Statistical inference: an illustration FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
87 Statistical inference: an illustration 1/regression slope = estimate of neighborhood size Nb 4Dπσ 2 Here one finds Nb ˆ=753 with CI obtained by bootstrapping over loci. (mark-recapture experiments yielded Nb ˆ=555) FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
88 Moment vs. likelihood method Nb Regression PAC likelihood FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
89 Summary Various evolutionary processes can be understood in terms of covariances in allele frequencies; Formal analysis in terms of probabilities of identity and distributions of coalescence times; Wright s F ST and its variants can characterize recent events; They can provide some reasonable data analyses, although not the most precise ones. FR & RL Inférence en génétique des populations IV. M2 Biostatistiques / 33
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