Post on 04-Aug-2018
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
and phylogenetic comparative methodsPhenotypic Evolution
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Phenotypic EvolutionChange in the mean phenotype from generation to generation...
Evolution = Mean(genetic variation * selection) + Mean(genetic variation * drift) +
Mean(nongenetic variation)
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Quantitative evolutionary theory
Gz β=Δ
Change in phenotype
Lande’s formula for multivariate phenotypic evolution
Population variance(additive genetic variance-
covariance matrix)
Selection coefficients•Random•Directional•Stabilizing•Etc.
Lande, R. 1979. Quantitative genetic analysis of multivariate evolution, applied to brain: body
size allometry. Evolution, 33: 402-416.
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Brown, W.M., M. George, Jr., & A.C. Wilson. 1979. Rapid evolution of animal mitochondrial DNA. PNAS, 76: 1967-1971.
Phenotypic trait divergence less predictable than genetic divergence
Mitochondrial DNA divergence
Polly, P.D. 2003. Paleophylogeography: the tempo of geographic differentiation in marmots (Marmota). Journal of
Mammalogy, 84: 369-384.
Molar shape divergence
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
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]]]
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Random walks
1 random walk 100 random walks
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
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....
3. 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 ofgenerations
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Random Walks in MathematicaIn 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]
0 20 40 60 80 100
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0
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Results of Monte Carlo experiment100 generations, rate of 1.0 per generation, squared rate of 1.0 per generation, 10,000 runs
Expected Observed
Mean = 0 Mean = 0.034
Variance = 1.02 * 100 =100 Variance = 100.37
SD = Sqrt[1.02 * 100] =10 SD = 10.10
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Two ways to think about phenotypic evolution
Phenotype graphs(phenotypic value over time)
Divergence graphs(phenotypic change over time)
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Divergence graphs
Plots of divergence against phylogenetic, genetic, or geographic distance
Phylogenetic or Genetic Distance (time elapsed)
Mop
hom
etric
Div
erge
nce
(Pro
crus
tes
dist
ance
)
Each data point records the differences (morphological and phylogenetic) between two taxa (known as pairwise distances)
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Divergence
Divergence
(2x)
Difference
Difference
Polly, P.D. 2001. Paleontology and the comparative method: ancestral node reconstructions versus observed node values.
American Naturalist, 157: 596-609.
Divergence graphs can be constructed from phylogenetic data
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
0 20 40 60 80 1000
5
10
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25
Monte Carlo with Divergence GraphListPlot[Sqrt[walks^2], Joined -> True, Axes -> False, Frame -> True, PlotRange -> All]
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
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. 2004. On the simulation of the evolution of morphological shape: multivariate shape under selection and
drift. Palaeontologia Electronica, 7.2.7A: 28pp, 2.3MB. http://palaeo-electronica.org/paleo/2004_2/evo/issue2_04.htm
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
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. 2004. On the simulation of the evolution of morphological shape: multivariate shape under selection and
drift. Palaeontologia Electronica, 7.2.7A: 28pp, 2.3MB. http://palaeo-electronica.org/paleo/2004_2/evo/issue2_04.htm
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Shape evolution can be simulated in morphospaceThis approach takes covariances in landmarks into account
1. Collect landmarks, calculate covariance matrix2. Convert covariance matrix to one without correlations by rotating data to principal components3. Perform simulation in shape space, convert simulated scores back into landmark shape models
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
1 million generations of random selection100 lineages, 18 dimensional trait
Arrangement of cusps (red dots at right)
Positions of 100 lineages in first two dimensions of
morphospaceDivergence graph of 100 lineages
Polly, P. D. 2004. On the simulation of the evolution of morphological shape: multivariate shape under selection and
drift. Palaeontologia Electronica, 7.2.7A: 28pp, 2.3MB. http://palaeo-electronica.org/paleo/2004_2/evo/issue2_04.htm
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
1 million generations of directional selection100 lineages, 18 dimensional trait
Arrangement of cusps (red dots at right)
Positions of 100 lineages in first two dimensions of
morphospaceDivergence graph of 100 lineages
Polly, P. D. 2004. On the simulation of the evolution of morphological shape: multivariate shape under selection and
drift. Palaeontologia Electronica, 7.2.7A: 28pp, 2.3MB. http://palaeo-electronica.org/paleo/2004_2/evo/issue2_04.htm
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
1 million generations of stabilizing selection100 lineages, 18 dimensional trait
Arrangement of cusps (red dots at right)
Positions of 100 lineages in first two dimensions of
morphospaceDivergence graph of 100 lineages
Polly, P. D. 2004. On the simulation of the evolution of morphological shape: multivariate shape under selection and
drift. Palaeontologia Electronica, 7.2.7A: 28pp, 2.3MB. http://palaeo-electronica.org/paleo/2004_2/evo/issue2_04.htm
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
BackgroundForward: 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;
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Random walk in one-dimensional morphospace
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Steps
PC 1
walk = Transpose[{Table[x,{x,101}], RandomWalk[100, 1]}];Graphics[Line[walk], Frame -> True, AspectRatio-> 1/GoldenRatio]
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Random walk in two dimensions of shape space
Steps PC1
PC2
z1=0; z2=0;
walk2d = Table[{t,z1=z1+Random[NormalDistribution[0,1], z2=z2+Random[NormalDistribution[0,1]},{t,100}];Graphics3D[Line[walk2d]]
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Same 2D random walk shown in two dimensionsz1=0; z2=0;
walk2d = Table[{z1=z1+Random[NormalDistribution[0,1], z2=z2+Random[NormalDistribution[0,1]},{t,100}];Graphics3D[Line[walk2d]]
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PC1
PC2
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
What rate to choose?
WALLABY
LEOPARDHUMAN
OTTER
FOSSA
DOG
Node 0
Node 1
Node 2
Node 3
Node 4
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
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0.00
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0.15
PC 1
PC2
Variance = Eigenvalues
Variance of random walk = rate2 * number of stepsrate = Sqrt[Eigenvalues / number of steps]
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Monte carlo simulation of evolving turtlesturtlespace = Graphics[{PointSize[0.02], Black, Point[scores[[1 ;;, {1, 2}]]]}, AspectRatio -> Automatic, Frame -> True]
-0.05 0.00 0.05
-0.04
-0.02
0.00
0.02
0.04
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Monte carlo simulation of evolving turtlesrates = 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]
-0.10 -0.05 0.00 0.05 0.10
-0.04
-0.02
0.00
0.02
0.04
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Animated turtle evolutionListAnimate[Table[tpSpline[consensus, (walk2d[[x]].(Transpose[eigenvectors][[1 ;; 2]])) + consensus], {x, Length[walk2d]}]]
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
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).
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Reconstructing evolution of shapeBrownian motion in reverse
Most likely ancestral phenotype is same as descendant, variance in likelihood is proportional time since the ancestor lived
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Descendant
Ancestor?
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
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 2
Common ancestor?
Descendant 1
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Phylogenetic tree projected into morphospace
WALLABY
LEOPARDHUMAN
OTTER
FOSSA
DOG
Node 0
Node 1
Node 2
Node 3
Node 4
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
PC 1
PC2
• 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
Department of Geological Sciences | Indiana University(c) 2016, P. David Polly
Change in phenotype
Selection coefficients
Additive genetic variance – covariance matrix
Selection coefficients can be: Random Directional Stabilizing Etc.
Local Adaptive Peak
(selection on crown height
based on local conditions)Local Phenotype
(crown height in local population)
Mean
Variance
Op
tiu
mu
m
Peak Width
Direction of
Selection
Probability of extinction
Selection and drift: Lande’s adaptive peak model
Lande, R. 1976. Evolution, 30: 314-334.
• 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
Parameters
Note: each geographic cell in the simulation has its own adaptive peak. Selection acts on local populations, not entire species.
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Evolution on an adaptive landscape
Loosely following Lande (1976)…
Lande, R. 1976. Natural Selection and random genetic drift in phenotypic evolution. Evolution, 30: 314-334.
Δz = h2*σ2 * δ ln(W)/δz(t) z – mean phenotype h2 –heritability σ2 – phenotypic variance W – selective surface (adaptive landscape) δ – derivative (slope)
Department of Geological Sciences | Indiana University (c) 2012, P. David Polly
G562 Geometric Morphometrics
Simulating an adaptive landscape from observational data
Convert PDF to adaptive landscape and selection coefficients
1.1 1.2 1.3 1.4 1.502468
meadow
Trait value
Prob
abili
ty 1.1 1.2 1.3 1.4 1.5
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0meadow
Trait value
Ln(P
roba
bilit
y)
1.1 1.2 1.3 1.4 1.5
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Trait value
Der
ivat
ive
Ln(P
roba
bilit
y)
Adaptive landscape
Fitness Selection Coefficient