What do phylogenies tell us about evolution?snuismer/Nuismer_Lab/548...Brownian motion y~N(0, σ2 *...

What do phylogenies tell us about evolution?

Wednesday, February 16, 2011

Comparative Methods

• Connecting quantitative genetics and phylogenetic trees

• What is phylogenetic signal?

• How do we interpret patterns of phylogenetic community structure?


What will happen over multiple generations?


Brownian Motion


Brownian Motion

• A model for the evolution of continuously-valued characters

• States change continuously through time

• After some time, expected character states follow a normal distribution


Brownian Motion: The Model

• Usually called the Wiener process

• A continuous-time stochastic process

• Describes a “random walk” of evolution for continuously-valued characters


Three Facts Describe Brownian Motion

• Let W(t) be the value of the character at time t. Then:

- E[W(t)] = W(0)

- Successive steps are independent

- W(t)∼N(W(0),σ2t)


Parameters of BM

• Brownian motion models have two parameters:

- Θ, the starting value; W(0) = Θ

- σ2, the rate parameter


Parameters of BM

• Brownian motion models have two parameters:

- Θ, the starting value; W(0) = Θ

- σ2, the rate parameter

σ2 = G/n


Histogram of x1[100, ]

x1[100, ]

Freq

uenc

y

−50 0 50

05

1015

20Wednesday, February 16, 2011


x2[500, ]

Freq

uenc

y

−50 0 50

02

46

810

1214


x1[100, ]

Freq

uenc

y

−50 0 50

05

1015



x2[500, ]

Freq

uenc

y

−50 0 50

02

46

810

1214


x1[100, ]

Freq

uenc

y

−50 0 50

05

1015

20


x3[1000, ]Fr

eque

ncy

−50 0 50

02

46


Histogram of r4[1000, ]

r4[1000, ]

Freq

uenc

y

−4 −2 0 2 4

050

010

0015

00


Properties of BM on trees

• Variance increases with both σ2 and t

• Expected (mean) value of any tip is always Θ

• Closely related species tend to be similar (they covary)


How do they covary?A

Bt1

t2

t3

var(A)

var(B)

cov(A,B)

σ2



Bt1

t2

t3

var(A)

var(B)

cov(A,B)=σ2(t1+t2)

σ2



Bt1

t2

t3

var(A)

var(B)

cov(A,B)=σ2(t1+t2)

=σ2(t1+t3)

σ2



Bt1

t2

t3

var(A)

var(B)

cov(A,B)=σ2(t1+t2)

=σ2(t1+t3)

=σ2(t1)

σ2



Bt1

t2

t3

var(A)

var(B)

cov(A,B)=σ2(t1+t2)

t1+t2

t1+t3t1

t1

variance-covariance matrix

=σ2(t1+t3)

=σ2(t1)

σ2


General form

• Tip data follow a multivariate normal distribution with mean vector Θ and variance-covariance matrix where

• var(i) = σ2(di); di =distance from root to tip i

• cov(i,j) = σ2(ci,j); ci,j =shared path of tip i and j


from Revell et al. 2008Wednesday, February 16, 2011

from Revell et al. 2008

One can simply draw from this single distribution to simulate BM evolution on a tree


• Calculating the likelihood for a single character evolving under a BM model


Likelihood for a single character

tΘ y

Brownian motion


Likelihood for a single character

tΘ y

Brownian motion

y~N(0, σ2 * t)


Multivariate NormalA

Bt1

t2

t3

var(A)

var(B)

cov(A,B)=σ2(t1+t2)

t1+t2

t1+t3t1

t1


=σ2(t1+t3)

=σ2(t1)

σ2


Multivariate NormalA

Bt1

t2

t3

var(A)

var(B)

cov(A,B)=σ2(t1+t2)

t1+t2

t1+t3t1

t1


=σ2(t1+t3)

=σ2(t1)

σ2

C


Likelihood for continuous characters on trees

• Given phylogeny, measurements of character y for each tip (yi)

• Choose a rate parameter σ2 and mean Θ

• Calculate the phylogenetic variance-covariance matrix for the tree V


Likelihood for continuous characters on trees

• yi~MVN(Θ, σ2V)

• Determine the probability of drawing the vector of yi from the MVN distribution with mean Θ and vcv σ2V


Two dimensions (x, y) correspond to tree with n=2


Two dimensions (x, y) correspond to tree with n=2

More dimensions gets more complicatedEasy to do with computers


Analytic Solution for MLE

x = vector of trait values, n = number of species, C = coancestry matrix (shared path lengths)


• Alternative models for continuous character evolution


Evolution might approximate BM...

• Genetic drift

• Random punctuated change

• Selection that is weak relative to the time interval considered

• Selection that changes randomly through time


Early Burst Model (EB)

• Rate of evolution slows through time

• Highest rate at the root of the tree

• Three parameters: starting value (Θ), starting rate (σ2o), and rate change (r)

i

jsij


Ornstein-Uhlenbeck Model

• Evolution has a tendency to move towards some medial value

• “Brownian motion with a spring”

• Three parameters: starting value (Θ), rate (σ2), and constraint parameter (α)

i

jsij

T = total tree depthWednesday, February 16, 2011

Prop

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ght

0.0

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0.8

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BMCCEBNA

Squam

ates

Birds

Fish Insec

ts

Mammals

Amphibia

nsBo

dy s

ize

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

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Body

sha

pe


Comparative Methods





Phylogenetic Signal

• phylogenetic signal is “the tendency for related species to resemble each other more than they resemble species drawn at random from the [phylogenetic] tree”

Blomberg and Garland 2002



Bt1

t2

t3

var(A)

var(B)

cov(A,B)

σ2



Bt1

t2

t3

var(A)

var(B)

cov(A,B)=σ2(t1+t2)

σ2



Bt1

t2

t3

var(A)

var(B)

cov(A,B)=σ2(t1+t2)

=σ2(t1+t3)

σ2



Bt1

t2

t3

var(A)

var(B)

cov(A,B)=σ2(t1+t2)

=σ2(t1+t3)

=σ2(t1)

σ2



Bt1

t2

t3

var(A)

var(B)

cov(A,B)=σ2(t1+t2)

t1+t2

t1+t3t1

t1


=σ2(t1+t3)

=σ2(t1)

σ2


Phylogenetic Signal

• Phylogenetic signal is often viewed as a constraint

• “How much of the variation in this trait can be explained by the phylogenetic tree?”


• But what does “phylogenetic signal” tell you about the process of evolution?


simulated populations


Phylogenetic signal

• We get phylogenetic signal from unconstrained Brownian motion, which can result from a number of processes

• Real constraints on trait evolution can result in a lack of signal


• But what does “phylogenetic signal” tell you about the process of evolution?

Very little


Comparative Methods





History and Ecology

● Incorporate historical information into ecology● Direct historical knowledge

– uncommon● Historical inferences

– Phylogenies


How do we use phylogenies?Community 1 Community 2

Trait ATrait B


Community 1 Community 2

Trait ATrait B


Explanation 1Community 1 Community 2

Trait ATrait B


Explanation 2Community 1 Community 2

Trait ATrait B


History vs. current processes

● Can we use phylogenies to discriminate between these two possibilities?


History and communities

Case 1

Case 2


History vs. current processes

● First case– Ecological sorting– species in each community random with respect to the

phylogeny● Second case

– Evolutionary history important– groups in the phylogeny match groups found together

in communities– Is this the only way to get this pattern?


Traits and history

● What if ecological sorting is based on species traits?● And...● Closely related species tend to have similar traits?


Interpretation depends on:

● Traits:– phylogenetically conserved = closely related species

have similar traits– phylogenetically convergent = distantly related species

are more similar than closely related ones

conserved convergent



● Ecology:– Habitat filtering = similar species tend to occur in the

same habitats, and thus together– Competitive exclusion = similar species compete;

different species likely to occur together



● History:– have species had time to spread evenly through the

geographic area you're studying?– depends on spatial scale, time scale


Interpreting NRI depends on traits

From Webb et. al 2002 Wednesday, February 16, 2011


From Webb et. al 2002

clusteredrandomoverdispersed

overdispersed





overdispersed

Historical factors (dispersal limitation)





overdispersed

Historical factors (dispersal limitation) clustered clustered


Conclusion

• We can really only understand phylogenetic community ecology if we know something about traits and how they have evolved.


Comparative Methods





What do phylogenies tell us about evolution?snuismer/Nuismer_Lab/548...Brownian motion y~N(0, σ2 *...

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