Phenotypic Evolution. and phylogenetic comparative methods. G562 Geometric Morphometrics. Department of Geological Sciences Indiana University

Transcription

1 Phenotypic Evolution and phylogenetic comparative methods

2 Phenotypic Evolution Change in the mean phenotype from generation to generation... Evolution = Mean(genetic variation * selection) + Mean(genetic variation * drift) + Mean(nongenetic variation)

3 Quantitative evolutionary theory Lande s formula for multivariate phenotypic evolution Selection coefficients Random Directional Stabilizing Etc. Δz = βg Change in phenotype Population variance (additive genetic variancecovariance matrix) Lande, R Quantitative genetic analysis of multivariate evolution, applied to brain: body size allometry. Evolution, 33:

4 Phenotypic trait divergence less predictable than genetic divergence Molar shape divergence Mitochondrial DNA divergence Brown, W.M., M. George, Jr., & A.C. Wilson Rapid evolution of animal mitochondrial DNA. PNAS, 76: Polly, P.D Paleophylogeography: the tempo of geographic differentiation in marmots (Marmota). Journal of Mammalogy, 84:

5 Monte Carlo simulation of Brownian Motion properties Monte Carlo is a type of modelling in which you simulate random samples of variables or systems of interest. Here we simulate 1000 random walks to see whether it is true that the average outcome is the same as the starting point and whether the variance and standard deviation of outcomes occur as expected. walks = Table[RandomWalk[100, 1], {1000}]; ListPlot[walks, Joined -> True, Axes -> False, Frame -> True, PlotRange -> All] Histogram[walks[[1 ;;, -1]], Axes -> False] Mean[walks[[1;;, -1]]] Variance[walks[[1;;,-1]]] StandardDeviation[walks[[1;;,-1]]]

6 Random walks 1 random walk 100 random walks

7 Statistics of Brownian motion evolution Random walk evolution: 1. Change at each generation is random in direction and magnitude 2. Direction of change at any point does not depend on previous changes Consequently The most likely endpoint is the starting point 4. The distribution of possible endpoints has a variance that equals the average squared change per generation * number of generations 5. The standard deviation of possible endpoints increases with the square root of number of generations

8 Random Walks in Mathematica In Phylogenetics for Mathematica 1.1: RandomWalk[n, i] where n is the number of generations and i is the rate of change per generation. walk = RandomWalk[100, 1]; ListPlot[walk, Joined -> True, Axes -> False, Frame -> True, PlotRange -> All]

9 Results of Monte Carlo experiment 100 generations, rate of 1.0 per generation, squared rate of 1.0 per generation, 10,000 runs Expected Observed Mean = 0 Mean = Variance = * 100 =100 Variance = SD = Sqrt[1.0 2 * 100] =10 SD = 10.10

10 Two ways to think about phenotypic evolution Phenotype graphs (phenotypic value over time) Divergence graphs (phenotypic change over time)

11 Divergence graphs Plots of divergence against phylogenetic, genetic, or geographic distance Mophometric Divergence (Procrustes distance) Each data point records the differences (morphological and phylogenetic) between two taxa (known as pairwise distances) Phylogenetic or Genetic Distance (time elapsed)

12 Divergence graphs can be constructed from phylogenetic data Difference Divergence Difference Divergence (2x) Polly, P.D Paleontology and the comparative method: ancestral node reconstructions versus observed node values. American Naturalist, 157:

13 Monte Carlo with Divergence Graph ListPlot[Sqrt[walks^2], Joined -> True, Axes -> False, Frame -> True, PlotRange -> All]

14 How does one model evolution of shape? Random walks of landmark coordinates are not realistic because the landmarks are highly correlated in real shapes. Polly, P. D On the simulation of the evolution of morphological shape: multivariate shape under selection and drift. Palaeontologia Electronica, 7.2.7A: 28pp, 2.3MB. palaeo-electronica.org/paleo/2004_2/evo/issue2_04.htm

15 How does one model evolution of shape? Random walks of landmark coordinates are not realistic because the landmarks are highly correlated in real shapes. Polly, P. D On the simulation of the evolution of morphological shape: multivariate shape under selection and drift. Palaeontologia Electronica, 7.2.7A: 28pp, 2.3MB. palaeo-electronica.org/paleo/2004_2/evo/issue2_04.htm

16 Shape evolution can be simulated in morphospace This approach takes covariances in landmarks into account 1. Collect landmarks, calculate covariance matrix 2. Convert covariance matrix to one without correlations by rotating data to principal components 3. Perform simulation in shape space, convert simulated scores back into landmark shape models

17 1 million generations of random selection 100 lineages, 18 dimensional trait Arrangement of cusps (red dots at right) Positions of 100 lineages in first two dimensions of morphospace Divergence graph of 100 lineages Polly, P. D On the simulation of the evolution of morphological shape: multivariate shape under selection and drift. Palaeontologia Electronica, 7.2.7A: 28pp, 2.3MB. palaeo-electronica.org/paleo/2004_2/evo/issue2_04.htm

18 1 million generations of directional selection 100 lineages, 18 dimensional trait Arrangement of cusps (red dots at right) Positions of 100 lineages in first two dimensions of morphospace Divergence graph of 100 lineages Polly, P. D On the simulation of the evolution of morphological shape: multivariate shape under selection and drift. Palaeontologia Electronica, 7.2.7A: 28pp, 2.3MB. palaeo-electronica.org/paleo/2004_2/evo/issue2_04.htm

19 1 million generations of stabilizing selection 100 lineages, 18 dimensional trait Arrangement of cusps (red dots at right) Positions of 100 lineages in first two dimensions of morphospace Divergence graph of 100 lineages Polly, P. D On the simulation of the evolution of morphological shape: multivariate shape under selection and drift. Palaeontologia Electronica, 7.2.7A: 28pp, 2.3MB. palaeo-electronica.org/paleo/2004_2/evo/issue2_04.htm

20 Background Forward: landmarks to scores in shape space proc = Procrustes[landmarks, 10, 2]; consensus = Mean[proc]; resids = # - consensus &/@proc; CM = Covariance[resids]; {eigenvectors, v, w} = SingularValueDecomposition[CM]; eigenvalues = Tr[v, List]; scores = resids.eigenvectors; Backward: scores in shape space to landmarks resids = scores.transpose[eigenvectors]; proc = # + consensus &/@ resids;

21 Random walk in one-dimensional morphospace walk = Transpose[{Table[x,{x,101}], RandomWalk[100, 1]}]; Graphics[Line[walk], Frame -> True, AspectRatio-> 1/GoldenRatio] 0-5 PC Steps

22 Random walk in two dimensions of shape space z1=0; z2=0; walk2d = Table[{t,z1=z1+Random[NormalDistribution[0,1], z2=z2+random[normaldistribution[0,1]},{t,100}]; Graphics3D[Line[walk2d]] PC2 Steps PC1

23 Same 2D random walk shown in two dimensions z1=0; z2=0; walk2d = Table[{z1=z1+Random[NormalDistribution[0,1], z2=z2+random[normaldistribution[0,1]},{t,100}]; Graphics3D[Line[walk2d]] 0-5 PC PC1

24 What rate to choose? Variance = Eigenvalues 0.15 DOG 0.10 OTTER Node PC FOSSA Node 2 Node 3 Node 1 Node LEOPARD HUMAN WALLABY PC 1 Variance of random walk = rate 2 * number of steps rate = Sqrt[Eigenvalues / number of steps]

25 Monte carlo simulation of evolving turtles turtlespace = Graphics[{PointSize[0.02], Black, Point[scores[[1 ;;, {1, 2}]]]}, AspectRatio -> Automatic, Frame -> True]

26 Monte carlo simulation of evolving turtles rates = Sqrt[eigvals[[1 ;; 2]]/100]; walk2d = Transpose[Table[RandomWalk[100, rates[[x]]], {x, Length[rates]}]]; Show[Graphics[{Gray, Line[walk2d]}, Frame -> True], turtlespace, PlotRange -> All]

27 Animated turtle evolution ListAnimate[Table[tpSpline[consensus, (walk2d[[x]].(transpose[eigenvectors][[1 ;; 2]])) + consensus], {x, Length[walk2d]}]]

28 Important notes These simulations are based on the covariances of the taxa, not the covariances of a single population. Therefore they are not a true model of the evolution of a population by means of random selection. The rates used in this simulation are estimated without taking into account phylogenetic relationships among the taxa. The rates estimated using the variance of the taxa will be approximately correct, but one might really want to estimate them by taking into account phylogenetic relationships (e.g., Martins and Hansen, 1993).

29 Reconstructing evolution of shape Brownian motion in reverse Most likely ancestral phenotype is same as descendant, variance in likelihood is proportional time since the ancestor lived Descendant Ancestor?

30 Ancestor of two branches on phylogenetic tree If likelihood of ancestor of one descendant is normal distribution with variance proportional to time, then likelihood of two ancestors is the product of their probabilities. This is the maximum likelihood method for estimating phylogeny, and for reconstructing ancestral phenotypes. (Felsenstein, Descendant 1 Descendant 2 Common ancestor?

31 Phylogenetic tree projected into morphospace Ancestral shape scores reconstructed using maximum likelihood (assuming Brownian motion process of evolution) Ancestors plotted in morphospace Tree branches drawn to connect ancestors and nodes 0.15 DOG 0.10 OTTER 0.05 Node 4 PC FOSSA Node 2 Node 3 Node 1 Node LEOPARD HUMAN WALLABY PC 1

32 (c) 2016, P. David Polly Selection and drift: Lande s adaptive peak model Note: each geographic cell in the simulation has its own adaptive peak. Selection acts on local populations, not entire species. Probability of extinction Local Phenotype (crown height in local population) Direction of Selection Mean Optiumum Local Adaptive Peak (selection on crown height based on local conditions) Change in phenotype Variance Peak Width election coefficients can be: andom irectional tabilizing tc. Parameters Selection coefficients Additive genetic variance covariance matrix Lande, R Evolution, 30: Selection vector = proportional to log slope of adaptive peak at population mean Extirpation probability = chance event with probability that increases with distance from optimum Genetic variance = population variance times heritability Drift (not shown) = chance sampling based on heritable phenotypic variance and local population size

33 Evolution on an adaptive landscape Loosely following Lande (1976) Δz = h 2 *σ 2 * δ ln(w)/δz(t) z mean phenotype h 2 heritability σ 2 phenotypic variance W selective surface (adaptive landscape) δ derivative (slope) Lande, R Natural Selection and random genetic drift in phenotypic evolution. Evolution, 30:

34 Simulating an adaptive landscape from observational data Convert PDF to adaptive landscape and selection coefficients Probability meadow Trait value Ln(Probability) meadow Trait value Derivative Ln(Probability) Trait value Adaptive landscape Fitness Selection Coefficient

35 Evolution on an adaptive landscape Time (generations) Trait value