Evolutionary processes that can lead to change or stasis over time
Pattern:
Change
Stasis
Adaptive
Directional
selection
• Stabilizing selection
• Fluctuating selection (noise
with no trend)
Nonadaptive
Mutation
Genetic drift
• Lack of genetic variation
• Constraint (?)
• Antagonistic correlations
among traits under selection
• Swamping by gene flow
Blomberg’s K – measure of phylogenetic signal
phylogenetic signal
low
K = 0.18
Data diagnostics
brownian
K~1
high
K = 1.62
Blomberg et al. 2003 Evolution
examples from Ackerly 2009 PNAS
K>1
Brownian motion – assumptions and interpretations
Evolutionary models
Brownian motion – assumptions and interpretations
∞
-∞
Evolutionary models
Ornstein-Uhlenbeck model (OU-1)
the math:
brownian motion + ‘rubber band effect’
change is unbounded (in theory), but as rubber band gets stronger,
bounds are established in practice
5
0
-5
-15
trait value
10
15
repeated movement back towards center erases phylogenetic signal,
leading to K << 1
0
50
100
150
200
250
300
time
Evolutionary models
see Hansen 1997 Evolution
Butler and King 2004 Amer. Naturalist
Ornstein-Uhlenbeck model (OU-1)
the math:
brownian motion + ‘rubber band effect’
change is unbounded (in theory), but as rubber band gets stronger,
bounds are established in practice
5
0
-5
-15
trait value
10
15
repeated movement back towards center erases phylogenetic signal,
leading to K << 1
0
50
100
150
200
250
300
time
Evolutionary models
see Hansen 1997 Evolution
Butler and King 2004 Amer. Naturalist
Ornstein-Uhlenbeck model (OU-2+)
the math:
brownian motion + ‘rubber band effect’ with different optimal trait values
for clades in different selective regimes
Balance of stabilizing selection within clades vs. how different the optima
are can lead to strong or weak phylogenetic signal
This example would be VERY strong signal
Evolutionary models
see Hansen 1997 Evolution
Butler and King 2004 Amer. Naturalist
Early-burst model
the math:
brownian motion with a declining rate parameter
change is unbounded (in theory), but divergence happens rapidly at first
and then rates decline and lineages change little
divergence among major clades creates high signal: K >> 1
Evolutionary models
Harmon et al. 2010
Assign proportional weighting of
alternative models that best fit data
Harmon et al. 2010
Rates of phenotypic diversification under Brownian
motion
1 felsen =
1 Var(loge(trait))
million yrs
time
var(x)
Rates of phenotypic diversification under Brownian
motion
higher rate
lower rate
time
var(x)
Diversification of height in maples, Ceanothus and silverswords
rate = 0.015 felsens
0.10 felsens
0.83 felsens
~5.2 Ma
~30 Ma
~45 Ma
Evolutionary rates
Ackerly 2009 PNAS
Rates of phenotypic diversification (estimated for Brownian motion
model)
Rate (felsens)
North temperate
California
Hawai’i
Height
Leaf size
±1 s.e.
Ackerly, PNAS in review
time
var(x)
Linear parsimony
0
Squared change
parsimony = ML with BL
=1
2.44
0.08
2
1.32
1
0.52
0
0.24
0
0.12
0
ML with BL as shown
0.096
2.54
1.6
0.96
0.67
0.56
ML with BL as shown
C
F
Node
ML
estimate
Lower
95% CI
Upper
95% CI
A
0.56
-0.77
1.89
B
0.67
-0.43
1.78
C
0.096
-0.61
0.81
D
0.96
0
1.95
E
1.6
0.76
2.45
F
2.54
1.86
3.2
E
D
B
A
Oakley and Cunningham 2000
Polly 2001 Am Nat
Independent contrasts
a
2
8
6
16
1
6
b
R = 0.74
11
14
8.5
9
11.5
11
19
18
13
12
a
2
8
6
16
1
6
11
14
8.5
9
11.5
11
10
8
4
8
4
12
19
18
13
12
3
2
6
10
-6
-6
10
10
16
15
2
-2
6
5
5
11
13
17
8
6
9
14
b
R = 0.74
c
R = 0.92
Oakley and Cunningham 2000
Oakley and Cunningham 2000
A21223
Fig. 2
1) Assume bivariate normal
distribution of variables with  = 0
Distribution of correlation coefficients
(R) under null hypothesis
2) Draw samples of 22 and
calculate correlation coefficient
Crit(R,  = 0.05, df = 20) is 0.423
3) Repeat 100,000 times!
N < -0.423 = 2519; N > 0.423 = 2551
Type I error = 0.051
1) Assume bivariate normal
distribution of variables with  = 0.5
Crit(R,  = 0.05, df = 20) is 0.423
2) Draw samples of 22 and
calculate correlation coefficient
N < -0.423 = 5
N > 0.423 = 68858
3) Repeat 100,000 times!
Power = 0.69
1) Assume bivariate normal
brownian motion evolution along a
phylogeny, with  ~ 0.0
2) Calculate R using normal
correlation coefficient
3) Repeat 10,000 times!
Crit(R,  = 0.05, df = 20) is 0.423
N < -0.423 = 1050
N > 0.423 = 1044
Type I error = 0.21
1) Assume bivariate normal
brownian motion evolution along a
phylogeny, with  ~ 0.0
2) Calculate R using independent
contrasts
3) Repeat 10,000 times!
Crit(R,  = 0.05, df = 20) is 0.423
N < -0.423 = 246
N > 0.423 = 236
Type I error = 0.048
From Ackerly, 2000, Evolution