Brownian motion (on a phylogeny) - Integrative Biologyib.berkeley.edu/courses/ib200/2016/lect/lect19-slides.pdf · Brownian motion (on a phylogeny) The expected distribution of the

Brownian motion (on a phylogeny)

‘borrowed’  from  Liam  Revell  lecture  notes  

Brownian motion (on a phylogeny)

The expected distribution of the tips & nodes of the tree under Brownian motion is multivariate normal with variance-covariance matrix in which each i,jth term is proportional to the height above the roots for the common ancestor of i and j.

‘borrowed’  from  Liam  Revell  lecture  notes  

Blomberg’s K – measure of phylogenetic signal

Blomberg et al. 2003 Evolution examples from Ackerly 2009 PNAS

K = 0.18 K ~ 1 K = 1.62

low brownian highphylogenetic signal

Data diagnostics

K  >  1  

Brownian motion – assumptions and interpretations

Evolutionary models

Brownian motion – assumptions and interpretations

Evolutionary models

∞

-∞

Ornstein-Uhlenbeck model (OU-1)

Evolutionary models

the math: brownian motion + ‘rubber band effect’ change is unbounded (in theory), but as rubber band gets stronger, bounds are established in practice repeated movement back towards center erases phylogenetic signal, leading to K << 1

0 50 100 150 200 250 300

-15

-50

510

15

time

trait

valu

e

see Hansen 1997 Evolution Butler and King 2004 Amer. Naturalist

Ornstein-Uhlenbeck model (OU-1)

Evolutionary models

the math: brownian motion + ‘rubber band effect’ change is unbounded (in theory), but as rubber band gets stronger, bounds are established in practice repeated movement back towards center erases phylogenetic signal, leading to K << 1

0 50 100 150 200 250 300

-15

-50

510

15

time

trait

valu

e

see Hansen 1997 Evolution Butler and King 2004 Amer. Naturalist

Rates of phenotypic diversification under Brownian motion

time

var(x)

1 felsen = 1 Var(loge(trait)) million yrs

Rates of phenotypic diversification under Brownian motion

time

var(x)

higher rate lower rate

Diversification of height in maples, Ceanothus and silverswords

~30 Ma

~45 Ma

rate = 0.015 felsens 0.10 felsens 0.83 felsens

Ackerly 2009 PNAS

~5.2 Ma

Evolutionary rates

Rates of phenotypic diversification (estimated for Brownian motion model) Ra

te (f

elsen

s)

Leaf size Height

North temperate California Hawai’i

±1 s.e.

Ackerly, PNAS in review