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G562 Geometric Morphometrics
Phenotypic Evolution
and phylogenetic comparative methods
Department of Geological Sciences | Indiana University
(c) 2012, P. David Polly
G562 Geometric Morphometrics
Phenotypic Evolution
Change 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
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. 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
Phenotypic trait divergence less predictable
than genetic divergence
Molar shape divergence
Mitochondrial DNA divergence
Brown, W.M., M. George, Jr., & A.C. Wilson. 1979. Rapid
evolution of animal mitochondrial DNA. PNAS, 76: 1967-1971.
Polly, P.D. 2003. Paleophylogeography: the tempo of
geographic differentiation in marmots (Marmota). Journal of
Mammalogy, 84: 369-384.
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
Department of Geological Sciences | Indiana University
100 random walks
(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 of
generations
Department of Geological Sciences | Indiana University
(c) 2012, P. David Polly
G562 Geometric Morphometrics
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]
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Department of Geological Sciences | Indiana University
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(c) 2012, P. David Polly
G562 Geometric Morphometrics
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 = 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
Mophometric Divergence
(Procrustes distance)
Plots of divergence against phylogenetic,
genetic, or geographic distance
Each data point records the differences
(morphological and phylogenetic) between
two taxa (known as pairwise distances)
Phylogenetic or Genetic Distance (time elapsed)
Department of Geological Sciences | Indiana University
(c) 2012, P. David Polly
G562 Geometric Morphometrics
Divergence graphs can be constructed from phylogenetic data
Difference
Divergence
Difference
Divergence
(2x)
Polly, P.D. 2001. Paleontology and the comparative method:
ancestral node reconstructions versus observed node values.
American Naturalist, 157: 596-609.
Department of Geological Sciences | Indiana University
(c) 2012, P. David Polly
G562 Geometric Morphometrics
Monte Carlo with Divergence Graph
ListPlot[Sqrt[walks^2], Joined -> True, Axes -> False, Frame -> True, PlotRange -> All]
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Department of Geological Sciences | Indiana University
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(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.
Department of Geological Sciences | Indiana University
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
(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.
Department of Geological Sciences | Indiana University
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
(c) 2012, P. David Polly
G562 Geometric Morphometrics
Shape evolution can be simulated in morphospace
This approach takes covariances in landmarks into account
1.
2.
3.
Collect landmarks, calculate covariance matrix
Convert covariance matrix to one without correlations by rotating data to principal components
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 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. 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 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. 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 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. 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
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;
Department of Geological Sciences | Indiana University
(c) 2012, P. David Polly
G562 Geometric Morphometrics
Random walk in one-dimensional morphospace
walk = Transpose[{Table[x,{x,101}], RandomWalk[100, 1]}];
Graphics[Line[walk], Frame -> True, AspectRatio-> 1/GoldenRatio]
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Steps
Department of Geological Sciences | Indiana University
(c) 2012, P. David Polly
G562 Geometric Morphometrics
Random walk in two dimensions of shape space
z1=0; z2=0;
Step
Department of Geological Sciences | Indiana University
s
PC1
PC2
walk2d = Table[{t,z1=z1+Random[NormalDistribution[0,1],
z2=z2+Random[NormalDistribution[0,1]},{t,100}];
Graphics3D[Line[walk2d]]
(c) 2012, P. David Polly
G562 Geometric Morphometrics
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]]
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PC1
Department of Geological Sciences | Indiana University
(c) 2012, P. David Polly
G562 Geometric Morphometrics
Variance =
Eigenvalues
What rate to choose?
0.15
DOG
0.10
OTTER
Node 4
PC 2
0.05
FOSSA
0.00
Node 2
Node 3 Node 1
Node 0
HUMAN
-0.05
LEOPARD
-0.10
WALLABY
-0.15
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-0.2
-0.1
0.0
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PC 1
Variance of random walk = rate2 * number of steps
rate = Sqrt[Eigenvalues / number of steps]
Department of Geological Sciences | Indiana University
(c) 2012, P. David Polly
G562 Geometric Morphometrics
Monte carlo simulation of evolving turtles
turtlespace =
Graphics[{PointSize[0.02], Black, Point[scores[[1 ;;, {1, 2}]]]},
AspectRatio -> Automatic, Frame -> True]
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-0.05
Department of Geological Sciences | Indiana University
0.00
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(c) 2012, P. David Polly
G562 Geometric Morphometrics
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]
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Department of Geological Sciences | Indiana University
-0.05
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(c) 2012, P. David Polly
G562 Geometric Morphometrics
Animated turtle evolution
ListAnimate[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 shape
Brownian motion in reverse
Most likely ancestral phenotype is
same as descendant, variance in
likelihood is proportional time since
the ancestor lived
Descendant
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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.
Descendant 1
Descendant 2
This is the maximum
likelihood method for
estimating phylogeny, and
for reconstructing
ancestral phenotypes.
(Felsenstein,
Common ancestor?
Department of Geological Sciences | Indiana University
(c) 2012, P. David Polly
G562 Geometric Morphometrics
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
Node 4
PC 2
0.05
FOSSA
0.00
Node 2
Node 3 Node 1
Node 0
HUMAN
-0.05
LEOPARD
-0.10
WALLABY
-0.15
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
PC 1
Department of Geological Sciences | Indiana University
(c) 2012, P. David Polly
Department of Geological Sciences | Indiana University
(c) 2016, P. David Polly
Selection and drift: Lande’s adaptive peak model
Probability of extinction
Local Phenotype
Mean
Direction of
Selection
Optiumum
Note: each geographic cell in the simulation
has its own adaptive peak. Selection acts on
local populations, not entire species.
Local Adaptive Peak
(selection on crown height
based on local conditions)
(crown height in local population)
Variance
Change in phenotype
Peak Width
Additive genetic variance –
covariance matrix
Selection coefficients can be:
Random
Directional
Stabilizing
Etc.
Selection coefficients
Lande, R. 1976. Evolution, 30: 314-334.
Parameters
•
•
•
•
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
G562 Geometric Morphometrics
Evolution on an adaptive landscape
Loosely following Lande (1976)…
Δz = h2*σ2 * δ ln(W)/δz(t)
z – mean phenotype
h2 –heritability
σ2 – phenotypic variance
W – selective surface (adaptive landscape)
δ – derivative (slope)
Lande, R. 1976. Natural Selection and random genetic drift
in phenotypic evolution. Evolution, 30: 314-334.
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
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meadow
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1.1
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1.3
1.4
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1.1
1.2
1.3
Trait value
Adaptive
landscape
Department of Geological Sciences | Indiana University
1.4
1.5
Trait value
Fitness
1.5
Derivative
Ln(Probability)
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6
4
2
0
Ln(Probability)
Probability
meadow
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1.1
1.2
1.3
1.4
1.5
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-20
Trait value
Selection
Coefficient
(c) 2012, P. David Polly
G562 Geometric Morphometrics
Evolution on an adaptive landscape
Time (generations)
Trait value
Department of Geological Sciences | Indiana University
(c) 2012, P. David Polly