Introduction to molecular dating methods

Principles

• Ultrametricity: All descendants of any node are equidistant from that node

• For extant species, branches, in units of time, are ultrametric

F

BA

10090 80 70 60 50 40 30 20 10

CD

E

Evolutionary branch length

• Expected number of substitutions/site = rate of change x branch duration

• Rate = 0.001 sub/site/Ma

• “True” length = 0.02

• Actual length ≈ 0.02

20 Ma

What is a “molecular clock”?

a) All internodes have equal duration

b) All branches have equal rate of substitution

c) All tips are the same number of time units from the root

d) The expected number of substitutions per site is the same for all branches

e) The observed number of substitutions is the same for all descendants of a given node

The molecular clock idea

• First proposed by Zuckerkandl and Pauling (1965) based on haemoglobin data

• If there is the same rate for all branches there will be a linear relationship between sequence distance and time since divergence

O

B

A

If you know one divergence date then you can calculate others

Per

cent

seq

uenc

e di

verg

ence

Time since divergence

x

y

If you know one divergence date then you can calculate others

Time since divergence

x

z

Per

cent

seq

uenc

e di

verg

ence

Issue 1: There will be error around the estimates

Per

cent

seq

uenc

e di

verg

ence

Time since divergence

x

z

Uncertainty in dating

Stochastic rate variation

Inferred age

Range

Issue 2: You need to correct for multiple hits

Per

cent

seq

uenc

e di

verg

ence

x

z

Assumed re

lationship

Actual relationship

Inferred ageActual age

Issue 3: Is evolution clock-like?

Issue 3: Is evolution clock-like?

Local clock: clade-specific rates

Issue 3: Is evolution clock-like?

No clock: rates vary greatly

Why should we expect a clock?

• Under neutral evolution: but that is too fast for most (all?) data sets

• If there is reasonable constancy of population size, mutation rate, and patterns of selection

• We can hope that rates of evolution change slowly and/or rarely

The likelihood approach

• Consider two models of evolution– The usual model– The same model but

• A root is specified

• The summed branch lengths from any node to all descendants of that node are the same

• Do a likelihood ratio test

Which is the simpler model?

How many degrees of freedom?

• Depends on the number of taxa (n)

• Branch length parameters in the non-clock model = 2n - 3

• Branch length parameters in the clock model = n - 1

• Difference = (2n - 3) - (n - 1) = n - 2

If a clock model is not rejected• Calculate rates and then extrapolate from known

to unknown pairwise distances

DOA = 0.4 ; DAB = 0.1TOA = 90 ; TAB = (0.1/0.4) x 90 = 22.5 Ma

O A B

0.05 0.05

0.195

0.2

90

22.5

Should obtain confidence intervals around date estimates• Look at the curvature of the likelihood

surface (can be done with PAML)

• Use bootstrapping (parametric or non-parametric)– Generate multiple pseudoreplicate data sets– For each data set calculate relative nodal ages– Discard the upper and lower 2.5%

Calibrating the tree

• How does one attach a date to an internal node? How old is the fossil? Where does a fossil fit on the tree?

Calibrating the tree

• How does one attach a date to an internal node? How old is the fossil? Where does a fossil fit on the tree?

F (90 Ma)

What does that tell us?

O

B

AF (90 Ma)

This node is at least 90 Ma

What else?

O

B

AF

This node is at least 90 Ma

This node is more than 90 Ma

The lineage leading to F could have been missed

O

B

AF

This node is at least 90 Ma

General issues

• Fossils generally provide only minimal ages• The age is attached to the node below the

lowest place on the tree that the fossil could attach

• Maximal or absolute ages can only be asserted when there are lots of fossil data

• Geological events can sometimes be used to obtain minimal ages

What if a clock is rejected?

• Until recently three (bad) choices– Give-up on molecular dating– Go ahead and use molecular dating anyway– Delete extra-fast or extra-slow taxa

• Now we have other options– Assume local clocks– Relaxed clock methods

Local clocks

Can use likelihood ratio tests to compare to strict clock and non-clock models

How many parameters?

Non-Parametric Rate-Smoothing(NPRS: Sanderson 1998)

a

d1

d2

The rate of branch a = ra= La/Ta

(L = branch length; T = time duration)

^

Node k

Non-Parametric Rate-Smoothing(NPRS: Sanderson 1998)

a

d1

d2

Measure of rate roughness = Rk = (ra - rd1)2 + (ra - rd2) 2^ ^ ^ ^

Node k

Non-Parametric Rate-Smoothing(NPRS: Sanderson 1998)

a

d1

d2

Adjust times so as to minimize overall roughness:

€

kRk=1

k= n−1

∑

NPRS

• Uses branch lengths only (ignores raw data)

• Quick and easy to do

• Assumes rate change is smooth

Penalized Likelihood(Sanderson 2001)

• Semi-parametric likelihood approach

• Uses raw data but penalizes the likelihood score by the roughness score, , weighted by a smoothness parameter ()

• Selects optimal value of using cross-validation (pick the value that minimizes the errors made in predicting branch lengths)

€

kRk=1

k= n−1

∑

Penalized Likelihood

• Uses more data than NPRS - more accurate

• More difficult to implement