what do phylogenies tell us about evolution?snuismer/nuismer_lab/548...brownian motion y~n(0, σ2 *...
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What do phylogenies tell us about evolution?
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Comparative Methods
• Connecting quantitative genetics and phylogenetic trees
• What is phylogenetic signal?
• How do we interpret patterns of phylogenetic community structure?
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Comparative Methods
• Connecting quantitative genetics and phylogenetic trees
• What is phylogenetic signal?
• How do we interpret patterns of phylogenetic community structure?
Wednesday, February 16, 2011
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What will happen over multiple generations?
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Brownian Motion
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Brownian Motion
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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
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Brownian Motion: The Model
• Usually called the Wiener process
• A continuous-time stochastic process
• Describes a “random walk” of evolution for continuously-valued characters
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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)
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Parameters of BM
• Brownian motion models have two parameters:
- Θ, the starting value; W(0) = Θ
- σ2, the rate parameter
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Parameters of BM
• Brownian motion models have two parameters:
- Θ, the starting value; W(0) = Θ
- σ2, the rate parameter
σ2 = G/n
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Histogram of x1[100, ]
x1[100, ]
Freq
uenc
y
−50 0 50
05
1015
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Histogram of x2[500, ]
x2[500, ]
Freq
uenc
y
−50 0 50
02
46
810
1214
Histogram of x1[100, ]
x1[100, ]
Freq
uenc
y
−50 0 50
05
1015
20Wednesday, February 16, 2011
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Histogram of x2[500, ]
x2[500, ]
Freq
uenc
y
−50 0 50
02
46
810
1214
Histogram of x1[100, ]
x1[100, ]
Freq
uenc
y
−50 0 50
05
1015
20
Histogram of x3[1000, ]
x3[1000, ]Fr
eque
ncy
−50 0 50
02
46
8Wednesday, February 16, 2011
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Histogram of r4[1000, ]
r4[1000, ]
Freq
uenc
y
−4 −2 0 2 4
050
010
0015
00
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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)
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How do they covary?A
Bt1
t2
t3
var(A)
var(B)
cov(A,B)
σ2
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How do they covary?A
Bt1
t2
t3
var(A)
var(B)
cov(A,B)=σ2(t1+t2)
σ2
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How do they covary?A
Bt1
t2
t3
var(A)
var(B)
cov(A,B)=σ2(t1+t2)
=σ2(t1+t3)
σ2
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How do they covary?A
Bt1
t2
t3
var(A)
var(B)
cov(A,B)=σ2(t1+t2)
=σ2(t1+t3)
=σ2(t1)
σ2
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How do they covary?A
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
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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
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from Revell et al. 2008Wednesday, February 16, 2011
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from Revell et al. 2008Wednesday, February 16, 2011
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from Revell et al. 2008
One can simply draw from this single distribution to simulate BM evolution on a tree
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• Calculating the likelihood for a single character evolving under a BM model
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Likelihood for a single character
tΘ y
Brownian motion
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Likelihood for a single character
tΘ y
Brownian motion
y~N(0, σ2 * t)
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Likelihood for a single character
tΘ y
Brownian motion
y~N(0, σ2 * t)
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Multivariate NormalA
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
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Multivariate NormalA
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
C
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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
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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
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Two dimensions (x, y) correspond to tree with n=2
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Two dimensions (x, y) correspond to tree with n=2
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Two dimensions (x, y) correspond to tree with n=2
More dimensions gets more complicatedEasy to do with computers
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Analytic Solution for MLE
x = vector of trait values, n = number of species, C = coancestry matrix (shared path lengths)
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• Alternative models for continuous character evolution
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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
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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
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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
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Prop
ortio
n of
wei
ght
0.0
0.2
0.4
0.6
0.8
1.0
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
Prop
ortio
n of
wei
ght
0.0
0.2
0.4
0.6
0.8
1.0
Body
sha
pe
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Comparative Methods
• Connecting quantitative genetics and phylogenetic trees
• What is phylogenetic signal?
• How do we interpret patterns of phylogenetic community structure?
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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
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How do they covary?A
Bt1
t2
t3
var(A)
var(B)
cov(A,B)
σ2
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How do they covary?A
Bt1
t2
t3
var(A)
var(B)
cov(A,B)=σ2(t1+t2)
σ2
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How do they covary?A
Bt1
t2
t3
var(A)
var(B)
cov(A,B)=σ2(t1+t2)
=σ2(t1+t3)
σ2
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How do they covary?A
Bt1
t2
t3
var(A)
var(B)
cov(A,B)=σ2(t1+t2)
=σ2(t1+t3)
=σ2(t1)
σ2
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How do they covary?A
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
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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?”
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• But what does “phylogenetic signal” tell you about the process of evolution?
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simulated populations
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simulated populations
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simulated populations
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simulated populations
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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
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• But what does “phylogenetic signal” tell you about the process of evolution?
Very little
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Comparative Methods
• Connecting quantitative genetics and phylogenetic trees
• What is phylogenetic signal?
• How do we interpret patterns of phylogenetic community structure?
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History and Ecology
● Incorporate historical information into ecology● Direct historical knowledge
– uncommon● Historical inferences
– Phylogenies
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How do we use phylogenies?Community 1 Community 2
Trait ATrait B
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Community 1 Community 2
Trait ATrait B
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Explanation 1Community 1 Community 2
Trait ATrait B
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Explanation 2Community 1 Community 2
Trait ATrait B
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History vs. current processes
● Can we use phylogenies to discriminate between these two possibilities?
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History and communities
Case 1
Case 2
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History and communities
Case 1
Case 2
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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?
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Traits and history
● What if ecological sorting is based on species traits?● And...● Closely related species tend to have similar traits?
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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
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Interpretation depends on:
● 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
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Interpretation depends on:
● History:– have species had time to spread evenly through the
geographic area you're studying?– depends on spatial scale, time scale
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Interpreting NRI depends on traits
From Webb et. al 2002 Wednesday, February 16, 2011
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Interpreting NRI depends on traits
From Webb et. al 2002
clusteredrandomoverdispersed
overdispersed
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Interpreting NRI depends on traits
From Webb et. al 2002
clusteredrandomoverdispersed
overdispersed
Historical factors (dispersal limitation)
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Interpreting NRI depends on traits
From Webb et. al 2002
clusteredrandomoverdispersed
overdispersed
Historical factors (dispersal limitation) clustered clustered
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Conclusion
• We can really only understand phylogenetic community ecology if we know something about traits and how they have evolved.
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Comparative Methods
• Connecting quantitative genetics and phylogenetic trees
• What is phylogenetic signal?
• How do we interpret patterns of phylogenetic community structure?
Wednesday, February 16, 2011