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Tópicos Avançados em Ecologia Filogenética e Funcional Modelos evolutivos, sinal filogenético, conservação de nicho. José Alexandre Felizola Diniz-Filho Departamento de Ecologia , UFG. Modelos evolutivos, sinal filogenético, conservação de nicho. Introdução (programas de pesquisa)

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slide1

Tópicos Avançados em Ecologia Filogenética e Funcional

Modelos evolutivos, sinal filogenético, conservação de nicho

José AlexandreFelizolaDiniz-Filho

Departamentode Ecologia, UFG

slide2

Modelos evolutivos, sinal filogenético, conservação de nicho

  • Introdução (programas de pesquisa)
  • Filogenias e matrizes de relação entre taxa
  • Modelos de Evolução
  • 3.1 . Conceitos gerais
  • 3.2. Métodos Estatisticos
  • 3.3. Abordagens baseadas em modelos de evolução
  • 3.4. Comparação de métodos
  • 4. Conservação de nicho
      • 4.1. Conceitos gerais
      • 4.2. Sinal filogenético e conservação de nicho
slide3

1. Introduction: ontheresearchtraditions...

Phylogenetic

Comparative Methods

Paul Harvey

(1980’s)

Phylogenetic

Diversity

Community

Phylogenetics

Campbell Webb

(2002)

Dan Faith (1992)

slide4

Marc Cadotte

(University of Toronto)

slide5

Traits

Ecophylogenetics

Assemblages

slide7

TRAITS

Phylogenetic Signal

Traits

Correlated Evolution

slide9

Pairwise (patristic) distances

>primcor <- cophenetic(primtree)

>

slide11

((((homo: 0.22,pongo: 0.22): 0.25,macaca:0.47):0.14,ateles: 0.62): 0.38,galago: 1.00): 0.00;

1.00 0.78 0.53 0.38 0.00

0.78 1.00 0.53 0.38 0.00

0.53 0.53 1.00 0.38 0.00

0.38 0.38 0.38 1.00 0.00

0.00 0.00 0.00 0.00 1.00

>primcor <- vcv.phylo(primtree, cor=TRUE)

>

slide12

Phylogenetic variance-covariance (vcv) matrix ( )

Thisisanultrametrictree...distancefrom root to TIP isconstant for allspecies

Main diagonal

slide13

PHYLOGENETIC CORRELATION = Standardized Variance-Covariance =

Shared proportion of branch lenght

This ultrametric tree has a total lenght of 1.0

slide15

The species“covary”, but in termsof “what”?

PHENOTYPES!

So, thephylogeneticvcvmatrixgives na EXPECTED covariancebasedontraitsspecies (whichisactuallysimilarityofmeanvalues) amongthespecies...

slide17

The same phylogeny can generate different OBSERVED vcv matrices, for different traits, for example...

EVOLUTIONARY MODELS

slide18

3. EVOLUTIONARY MODELS

Mechanisms (selection, drift, mutations…)

Evolutionary models

Interspecific data

slide20

Mechanisms (selection, drift, mutations…)

?

The path from evolutionary mechanisms (selection, drift, mutation and so on) to interspecific variation is a conceptual idea, but it may be hard (or even impossible) to reverse it and actually recover such processes from empirical data...

Evolutionary models

Interspecific data

slide21

I = selection intensity

R = response

T = time

h2 = heritability

Vp = phenotypic variance

‘Mechanistic’ versus phenomenological evolutionary models

slide22

Statistical models that “capture” the expectation of alternative evolutionary processes or mechanisms

slide23

BROWNIAN MOTION

  • After Robert Brown (1827)
  • Simplest continuous-time stochastic process

Simple discrete Random walks...

slide24

UNDERSTANDING BROWNIAN MOTION

In Excel, when A1=0...

=A1+(ALEATÓRIO()-0.5)

Uniform distribution (0-1)

15 replications of the same process through time

slide26

WHAT ABOUT PHYLOGENY?

50 time-steps

50 time-steps

Speciation

50 time-steps

slide27

100 time-steps

50 time-steps

100 time-steps

50 time-steps

100 time-steps

50 time-steps

50 time-steps

50 time-steps

Expected VCV matrix

slide29

Here we assumed that species are INDEPENDENT (the started all at the root)

Here species are PHYLOGENETICALLY STRUCTURED

slide31

Each line is a simulation that gives Y values for each species...

Calculate a Pearson

(or covariance) matrix among

Taxa (in “R mode”)

“Observed” matrix (10000 “traits”)

slide32

ape

> rTraitCont(phy, model = "BM", sigma = 0.1, alpha = 1, theta = 0, ancestor = FALSE, root.value = 0, ...)

ntimes=100nsp=5simbw <- matrix(data=NA,nrow=ntimes,ncol=nsp) for(i in 1:ntimes){

simbw[ i, ]<-rTraitCont(primtree)

}

slide33

100 time-steps

95 time-steps

100 time-steps

5 time-steps

100 time-steps

95 time-steps

75 time-steps

25 time-steps

Expected VCV (standardized) matrix

slide34

Expected VCV (standardized) matrix

r = 0.991!!!!

Observed matrix (10000 “traits”)

slide35

PropertiesorBrownianmotion in comparativeanalysis

  • Normal distributionofphenotypes (tips)
  • Meanconstantthrough time (absenceoftrends)
  • Varianceincreaseslinearlywithtime (butrememberthatwe do notknowtheabsoluteexpectedvariance)
  • The evolutionaryinterpretationofBrownianmotion
  • Geneticdrift + Mutation = Neutral (sensu Kimura) evolution
  • Stochasticadaptation in eachlineageateach time step (multipleindependentadaptive forces)
slide36

ConstrainedBrownianmotion: Ornstein-Uhlenbeck (O-U) process

…The Ornstein–Uhlenbeck (O-U) process (named after Leonard Ornstein and George Eugene Uhlenbeck), is a stochastic process that, roughly speaking, describes the velocity of a massive Brownian particle under the influence of friction.

Stabilizing selection...

slide42

Creatingalternativemodelsbywarpingthebranchlenghts...

The tipisto move from a “real” phylogeny (thesequenceofbranchingevents in time) to a “trait” or “model” phylogeneticstructurethat must beused in thestatisticalanalyses....

slide43

Severaloptions to transformbranchlenghts in GEIGER

deltaTree(phy, delta, rescale = T)

lambdaTree(phy, lambda)

kappaTree(phy, kappa)

ouTree(phy, alpha)

tworateTree(phy, breakPoint, endRate)

linearchangeTree(phy, endRate=NULL, slope=NULL)

exponentialchangeTree(phy, endRate=NULL, a=NULL)

speciationalTree(phy)

rescaleTree(phy, totalDepth)

BM

OU

> primtreeOU <-ouTree(primtree,2.5)

> plot(primtreeOU)

slide44

>primcorOU <-vcv.phylo(primtreeOU,cor=TRUE)

> write.table(primcorOU, file="primcorOU.txt")

This is theexpectedvcvunder OU processwith = 2.5!

OU

BM

slide45

“COMPARATIVE” versus “NON-COMPARATIVE” ANALYSIS: The “STAR-PHYLOGENY”

  • This is actuallywhatyou assume whenyousaythatdidnot use comparativemethods (so, theyactually use, butwith a particular vcvmatrix)
  • Doing a standard regressionorcorrelation is a particular formofcomparativeanalysesassuming a Star-Phylogeny
  • - Thisassumptionindicatesthatthetraithas no pattern (the interspecific variation is random in respect to phylogeny)
  • This does notindicatethatthere is no phylogenetic relationshipsamongspecies, ofcourse, onlythatthe processes drivingtraitvariationoccurred in such a waythatthepatterns is completelylost.
slide46

PHYLOGENETIC SIGNAL: BASIC CONCEPTS

  • Relationship between species’ similarity for a trait and phylogenetic distance
  • phylogenetic pattern;
  • phylogenetic component;
  • phylogenetic signal;
  • phylogenetic correlation;
  • phylogenetic inertia

Patternsand processes...

slide47

Measuring Phylogenetic Signal

Statistical

?

Metrics

Model Based

slide48

Moran’sI coefficient for phylogenetic autocorrelation

MatrixW withweights

Numberofspp

Speciestrait Z centered for thespecies i e j

Phylogenetic covariance

variance

Sumofweights in W

slide50

CORRELOGRAMS IN POPULATION GENETICS

Robert Sokal (1924-2012)

Sokal, R. R. & Oden, N. L. 1978. Spatialautocorrelation in biology:

1. methodology

2. Some biologicalimplicationsand four applications ofevolutionaryandecologicalinterest

BiologicalJournalofLinneanSociety 10: 199-249.

slide53

Patristicdistances

Matriz W (1/Dij)

Sumof W = 10.38333

slide54

W

Z

ZijWij

SumZijWij = 8.400781

slide55

-1.0 < Moran’sI < 1.0

Moran’s I

Numeratorphylogenetic covariance = 8.400781 / 10.3833 = 0.809

Denominatorvariance= 23.375 / 8 = 2.984

I = 0.809 / 2.984 = 0.276

Maximumandminimum are a functionofeigenvaluesofW (seeLichstein et al. 2002)

slide57

The W matriz: “inverting” therelationshipbetweenWandD

Gittlemanusedsomethinglikethis, butthis is empirical...

  • Wij = 1 / dij

W

Phylogenetic distance

slide58

Wij = 1/ Dij

Wij = 1/ (Dij ^ 2)

slide59

Wij = 1 / Dij2

I de Moran = 0.72

slide60

Otherpossiblefunctonslinking W andD

  • Wij= 1 / dij
  • Wij= 1 / dij2
  • Wij = e (- dij)

W

Phylogenetic distance

Orwecan use directlyany VCV matrix, previouslydefined...!!!!

slide61

The Rmatrix (sharedbranchlenghtswhen root age is 1.0) is already a Wmatrixthatcanbeuseddirectlyin Moran’s I

slide64

Testingsignificance: theanalytical solution...

Standard normal deviate, (SND, or Z) assuming normal distributionofthestatistics

If| Z | > 1.96, thenMoran’s I is significantat P < 0.05

slide65

Permutation test

Randomizethetipvalues in thephylogeny...

andrecalculateMoran’smany times...

TheP-value (Type I error) is givenbyhowmany times theMoran’s I washigherthantherandomizedvalues

slide66

The PRIMATE example (Lynch 1991):

BodyweightandLongevity (log-scale)

Let’s use R as a weightingmatrix

1.00 0.78 0.53 0.38 0.00

0.78 1.00 0.53 0.38 0.00

0.53 0.53 1.00 0.38 0.00

0.38 0.38 0.38 1.00 0.00

0.00 0.00 0.00 0.00 1.00

slide67

Moran’s I results

Bodyweight:

I = 0.200 ± 0.217;

E(I) = (-1/(n-1) = -0.25

Z = 2.07

P = 0.038

Longevity:

I = -0.121 ± 0.209;

E(I) = (-1/(n-1) = -0.25

Z = 0.617

P = 0.537

Significant phylogenetic signal...

Notsignificant phylogenetic signal...

> primlog <- read.table("primlog.txt",header=TRUE,row.name="spp")

> primtree <-readtree("primtree.txt")

> primcor <-vcv.phylo(primtree)

> diag(Rprim) <-0

> Moran.I(primlog[,c(1)],primcor)

The matriz W is

wronglydefined

in Paradis’ book

slide68

This is the “null” distribution for 1000 random normal values (close to theoreticalinferreddistribution).

Mean = -0.2506

Median = -0.287

ntimes<-5000

I <- numeric(ntimes)

for(i in 1:ntimes){

rnd_vec <- as.numeric(5)

rnd_vec <-rnorm(5,0,1)

diag(primcor) <- 0

a<-Moran.I(rnd_vec,primcor)

I[i]<- a$observed

}

hist(I)

mean(I)

median(I)

slide69

This is thedistributionrandomizing BW 1000 times

Out of 1000 randomized I, nonewaslargerthantheobserved 0.2009, so P = 1/1000 = 0.001

vetor <- primlog[,c(1)]

ntimes<-5000

I <- numeric(ntimes)

for(i in 1:ntimes){

vec <- vetor[sample(length(vetor))]

diag(primcor) <- 0

a<-Moran.I(vec,primcor)

I[i]<- a$observed

}

hist(I)

mean(I)

median(I)

obs

Mean = -0.256

Median = -0.318

In ade4...

>gearymoran(primcor,primlog[,c(1)])

slide70

Moran’ I Correlograms

Moran`s I

Allows evaluation of more complex structures in the matrix W ofphylogenetic relationships…

Time slices

slide72

A

B

C

D

E

1.0

2.0

4.0

1.5

slide73

Distances 0 - 2

Distances > 2

W1

W2

Moran’s I for thesecondclass

Moran’s I for thefirstclass

slide74

Diniz-Filho & Torres (2002, Evol.Ecol. 16: 351-367)

70 species of Carnivora in New World

Body size, geographic range size

Supertree

slide75

CORRELOGRAM

Strong signal for body size

Weak signal for geographic range size

slide76

Autocorrelationstatisticssuch as Moran’s I test for randomnessoftraitvariation in thephylogeny. ButwhatabouttheEVOLUTIONARY MODELS?

slide77

Os correlogramas filogeneticos respondem bem à mudanças nos modelos evolutivos…

(Diniz-Filho 2001 Evolution 55: 1104-1109)

slide78

Partition Methods

Phylogenetic Component P

Total variation T

SpecificComponentS

T = P + S

slide80

Pure autoregressive model

The Y values are a function of all other Ys value, “weighted” by the relationship in W matrix (i.e., ancestrality)

Y1 = Y2*W12+Y3*W13+Y4*W14+...Yn*W1n

slide81

> chev209 <-compar.cheverud(bs209,r209b)

> 1-var(chev209$residuals)/var(bs209)

slide83

Diniz-Filho`s et al. (1998) Phylogenetic eigenVector Regression (PVR) (Evolution 52: 1247-1262.)

Phylogeny

Eigenvectors

(V)

Double centering

Phylogenetic distances

Multiple regression

Estimated values

Regression residuals

S

P

Y

X

R2

slide84

Diniz-Filho`s et al. (1998) phylogenetic eigenvector regression (PVR)

Phylogeny

-Eigenvalues

+

Phylogenetic eigenvectors represent linearly different cuts of phylogeny, allowing evaluation of phylogenetic effects at different `scales`

Eigenvectors

(V)

slide87

Principal coordinateanalysisoftruncatedgeographicdistancesW(PCNM)

Pierre Legendre

Eigenvectorsofdoublecenteredbinary (0/1)connectivitymatrix

Daniel Griffith

slide88

Diniz-Filho & Torres (2002, Evol.Ecol. 16: 351-367)

70 species of Carnivora in New World

Body size, geographic range size

Supertree (12 first eigenvectors)

slide89

Body size

R2 = 0.75 (P << 0.01)

PVR

Geographic range

R2 = 0.28 (P = 0.06)

slide102

Bodymass

Geographic range size

slide104

Velociraptor

Rinchenia

slide105

The PVR/PSR Package (Functions:PVRdecomp, PVR, PSR, VarPartplot)

Santos et al. (in prep)

Nullexpectation

Brownianexpectation

slide107

Measuring Phylogenetic Signal

Statistical

?

Metrics

Model Based

slide110

This is thevarianceofthetrait in respect to ancestral states

This is thephylogeneticallycorrectedvariance (var ofPICs)

slide111

Original Phylogeny (“time”)

Traitwill evolve likethis, butwillbeanalyzedusingthe “known” (time) phylogeny

D-transform 0.25(“time”)

Traitwill evolve likethis, butwillbeanalyzedusingthe “known” (time) phylogeny

OU (alpha 2.5)

slide112

MSE = 0.996

K = 1.018 ± 0.388

MSE = 0.541

K = 1.258 ± 0.442

MSE = 1.727

K = 0.810 ± 0.332

slide113

K = 1.018437 ± 0.388

ntimes<-1000K<- numeric(ntimes)  for(i in 1:ntimes){  trait<-rTraitCont(primtree)  K[i]<-Kcalc(trait,primtree) }K

hist(K)

mean(K)

sd(K)

> bw <-data.frame(primlog[,c(1)])

> multiPhylosignal(bw,primtree)

slide114

Blomberg’s K

Bodyweight:

K = 0.728

K(null) = 0.796 ± 0.391

P(K=0) = 0.001

Longevity:

K = 0.200

K(null) = 0.775 ± 0.327

P(K=0) = 0.422

There is a significant phylogenetic signal

Phylogenetic signal is notsignificant...

KOBS <- Kcalc(primlog[,c(1)],primtree)

vetor <- primlog[,c(1)]

ntimes<-1000

K <- numeric(ntimes)

for(i in 1:ntimes){

vec <- vetor[sample(length(vetor))]

K[i]<-Kcalc(vec,primtree)

}

hist(K)

mean(K)

sd(K)

P1 <- ((sum(K > KOBS[1,1]))+1)/ntimes

slide120

FITTING GENERAL MODELS OF TRAIT EVOLUTION USING PGLS

>library(motmot)

Rob Freckleton

Gavin Thomas

  • >primbw <-as.matrix(primlog[,c(1)])
  • >likTraitPhylo(primbw,primtree)
  • >transformPhylo.ML(primbw,primtree,model="OU")
  • > transformPhylo.ll(primbw,primtree,model="OU",alpha=2)

Getthemaximumlikelihoodoftraitgiventhetree (thetreecanbetransformedintotreesreflectingothermodels (in GEIGER), or...

It canfindtheparameteralphathat maximize thelikelihood

Givesthelikelihood for a modelandparameter

slide121

Severalmodels, including lambda...

>library(motmot)

LONG

BW

primbw <-as.matrix(primlog[,c(1)])

pglsfit <- numeric(10)

lambda <- seq(0.000001,1,0.1)

for(i in 1:length(lambda)){

primll <- transformPhylo.ll(primbw,primtree,model="lambda",lambda=lambda[i])

pglsfit[i] <- primll$logLikelihood[1,1]

}

plot(lambda, pglsfit)

slide124

Measuring Phylogenetic Signal

Statistical

?

Metrics

Model Based

slide127

What are thedifferentmetrics “capturing” in traitevolution?

Lambda = 1 (Brownian)

Lambda = 0.5

Lambda = 0.1

slide128

Blomberg’s K

PVR’s R2

Type I error (correlationamongTIPs)

Moran’s I

slide136

Whatistherelationshipbetweenphylogeneticsignalandnicheconservatism?Whatistherelationshipbetweenphylogeneticsignalandnicheconservatism?

The short answer: NONE

The longanswer: depends, it iscomplicate...

No signalcanindicatestrongconservatism

Brownianmotioncanindicatestrongconservatismwithreductedvariance (use QuantitativeGeneticModels?)

UnderLosos’ / Wiens’ reasoning:

- Fit BM,OU, and “whitenoise” (random) models– nicheconservatismisbettersupportedby OU (actually a balance between shift/conservatism)

slide137

Brownianmotionwithvariable rates

These two  patterns are very different...

slide138

NicheConservatismunder PSR Curve...

Multiplepeak OU

Standard (single peak) OU

slide142

“A portion of the phylogenetic variation of the trait may be related to ecology. This portion is called ‘‘phylogenetic niche conservatism’’, and we propose a method of variation partitioning that allows users to quantify this portion of the variation,called the ‘‘phylogenetically structured environmental variation.’’

slide143

First, compute the following regressions:

(1) Y and XE (“environmental variables”); R2 = [a]+[b]

(2) Y and P (eigenvectors); R2 = [b]+[c]

(3) Y = f (XE, P); R2 = [a]+[b]+[c]

The individual values of a, b, and c can be obtained by subtraction from the previous results:

[a] = R2 (step 3) - R2 (step 2) or ([a]+[b]+[c]) – ([b]+[c])

[b] = R2 (step 1) + R2 (step 2) – R2 (step 3)

[c] = R2 (step 3) - R2 (step 1)

[d] = 1-(a+b+c)