estimating rates and dates: the molecular clock - UCLA · the utility of a molecular clock ... Puma...

estimating rates and dates:the molecular clock

Wednesday, July 27, 2011

is there a molecular clock?


molecular clocks

• most molecular phylogenies

are unrooted (or rooting is determined by prior information)

have branch lengths indicating genetic change

3introduction clocks calibrations clock tests relaxed clocks conclusionWednesday, July 27, 2011

molecular phylogenies

• given

a phylogenetic tree

branch heights (rate * time)

calibration (e.g. a time estimate for a node, T)

• can we estimate dates for the other nodes?

4

T

a b c d

introduction clocks calibrations clock tests relaxed clocks conclusionWednesday, July 27, 2011

• zuckerkandl and pauling (1962) noted that the rate of amino acid replacements in animal haemoglobins was roughly proportional to real time, as judged against the fossil record

support for a molecular clock?

evol

ving

pop

ulat

ions

0

20

40

60

0 100

200

300

400

500

time to common ancestor (myr)

no. s

ubst

itutio

ns

(to h

uman

)

cow

platypus chickencarp

80

100shark



• the molecular clock is particularly striking when compared to the obvious differences in rates of morphological evolution...

evol

ving

pop

ulat

ions

0

20

40

60

0 100

200

300

400

500

time to common ancestor (myr)

no. s

ubst

itutio

ns

(to h

uman

)

cow

platypus chickencarp

80

100shark



• pairwise nucleotide substitutions among 17 mammal species from 7 gene products, plotted against date of divergence

• the strong linear relationship suggests that molecular differences between pairs of species are proportional to the time of their separation

evol

ving

pop

ulat

ions

from AC Wilson, 1976



• 8 HIV-1 patients:


the clock is not a metronome

• if a mutation occurs every million years, with Poisson variance

95% of lineages 15M years old have 8-22 substitutions

a lineage with 8 substitutions could also be <5M years

9

Molecular Systematics, p532.


variation in tick rate


different genes evolve at different rates

% g

enet

ic d

iver

genc

e

Time since divergence (Myr)

100%

50%

75%

25%

1500300 600 900 1200

Fibrinopeptides

Hemoglobin

Cytochrome c

Histone IV

11

• variation in selection?

genes coding for some molecules may be under very strong stabilizing selection

introduction strict clocks rate variation relaxed clocks clock conclusions beastWednesday, July 27, 2011

different nucleotide positions evolve at different rates

evol

ving

pop

ulat

ions

Inter leuk in 2Prolactin

Inter leuk in 6

Inter leuk in 1bThrombomodulin

Lactoferr inInter leuk in 1a

IGF binding protein 1Urok inase-plasminogen activator

A lbuminGrow th hormone

Inter leuk in 7A lkaline phosphatase intestineCor ticotropin-releasing factor

Grow th hormone receptorFibr inogen g

IGF binding protein 3Plasminogen activator inhibi tor

Terminal transferaseTransforming grow th factor b3 recep tor

b-1, 4-galactosy l transferaseNeurophysin IINeurophysin I

Insulin-like grow th factor 2Acid phosphatase type 5

Lu teinising hormone receptorProopiomelanocor tin

A lkaline phosphatase liverTransforming grow th factor b1

Neuroleuk inAcety lcholine recep tor b

Aspar tate aminotransferase cy tosolicHexok inas I

Orni thine decarboxy laseOpsin

Protein d isul fide isomeraseTransforming grow th factor b3

Lactate dehydrogenase AAspar tate aminotransferase m tDNA

Acety lcholine recep tor aInsulin-like grow th factor 1

Dopamine receptor D2Glucose transpor ter

Transforming grow th factor b2ATP synthase a

Myelin proteolip id proteinConnexin

ATP synthase bCarboxypep tidase

0 0.5 1 1.5

Nonsynonymous

Synonymous

12introduction strict clocks rate variation relaxed clocks clock conclusions beastWednesday, July 27, 2011

different organisms evolve at different rates

10 10 10 10 10 10 10 10 10-9 -8 -7 -6 -5 -4 -3 -2 -1

nucleotide substitutions per site per yearpl

ant c

hlor

opla

st d

nam

amm

alia

n nu

clea

r dn

ae

coli

and

salm

onel

la e

nter

ica

dros

ophi

la n

ucle

ar d

na

hum

an t

cell

lym

phot

ropi

c vi

rus

hbv

rna

viru

ses

picornaviridae calciviridaeflaviridaetogaviridaecoronaviridaerhabdoviridaeparamyxoviridaeorthomyxoviridaereoviridaebirnaviridaeretroviridae

evol

ving

pop

ulat

ions


what causes variation in mutation rate?

14

• differences in generation time

• differences in population size

• differences in selective pressure

• differences in metabolic rate

• differences in efficiency of DNA repair

= mutation rate

= probability of fixation


lineage effects and the molecular clock

• substitution rate varies with underlying neutral mutation rate

• three ways for mutation rates to vary between species:

differences in generation time

differences in metabolic rate

differences in efficiency of DNA repair

• these are known as lineage effects:

neutralists believe that lineage effects alone can account for all variation in molecular clock


lineage effects: generation time

16

• at the molecular level, generation time is the time it takes for germ-line DNA to replicate

• the rate of substitution is a function of both µ and g

• the general conclusion from molecular data is that the clock is generation time dependent at silent sites and in non-coding DNA

mut

atio

ns

mutations

generations

long generation time

short generation time

time

one generation

one generation

• synonymous rates for orang-utan, gorilla and chimpanzee are 1.3-, 2.2- and 1.2-fold faster than in humans, which corresponds to proportionally shorter generation times


the metabolic rate hypothesis

17

• in sharks, the rate of silent change is 5-fold to 7-fold lower than in primates and ungulates with similar generation times

• are differences in molecular rate better explained by differences in metabolic rates?

mutagenic effects of oxygen radicals produced by aerobic respiration

organisms with high metabolic rates have higher levels of DNA synthesis


metabolic rate and body size (things can be confounded!)

0.01 0.1 1 10 100 1E3 1E4 1E50.1

1

10

% s

eque

nce

dive

rgen

ce /

Myr

Body mass (kg)

Rodents

GeeseDogs

Primates HorsesBears

WhalesNewtsFrogs

Tortoises

TortoisesSalmon

Sea turtlesSharks

HomeothermPoikilotherm

• mitochondrial DNA evidence for metabolic rate hypothesis:

1. warm-blooded animals have higher mutation rates than cold-blooded animals

2. small bodied animals, which have higher metabolic rates, tend to have higher mutation rates (and shorter generation times!)


• DNA repair may influence mutation rate

highly transcribed genes are more efficiently repaired

silent rates in mammalian genes tends to be gene- rather than species-specific

however, closely related species such as primates, which share very similar repair mechanisms, can exhibit greatly differing substitution rates

DNA repair and mutation

DNA

Directdamage

Replicationerrors

Repair Incorrectlyrepaired

Correctlyrepaired

Mutation


lineage effects and the molecular clock

• substitution rate varies with underlying neutral mutation rate

• three ways for mutation rates to vary between species:

differences in generation time

differences in metabolic rate

differences in efficiency of DNA repair

• these are known as lineage effects:

neutralists believe that lineage effects alone can account for all variation in molecular clock

selectionists believe that genes also show rate variation due to other, selection-driven factors (residue effects)


what causes variation in mutation rate?

21

• differences in generation time

• differences in population size

• differences in selective pressure

• differences in metabolic rate

• differences in efficiency of DNA repair

= mutation rate

= probability of fixation


calibrating the molecular clock


the utility of a molecular clock

• measuring evolutionary time makes it possible to

estimate genetic distance- d = genetic distance

use paleontological data to determine the date of a common ancestor- T = time since divergence

estimate calibration rate (number of genetic changes expected per unit time)- r = dac/2Tac

calculate time of divergence for novel sequences- Tab = dab/2r

23

T

a b c d


calibrating time phylogenies: node calibration

24

time

now

20-25 Ma

5-10 Ma

nodes with point calibrations

contemporary sample, probabilistic calibrations

7 M

a

22 M

a


calibrating time phylogenies: node calibration

25

time

now

95% CI20-30 Ma

95% CI5-15 Ma

nodes with point calibrations

contemporary sample, probabilistic calibrations

7 M

a

22 M

a


Calibrating a node

hum

an

chim

p

gorill

a

uniform[5.0, 1.5]

foss

il A =

1.5

My

foss

il B =

5.0

My


node calibration: sources

27

fossils

introduction clocks calibrations clock tests relaxed clocks conclusion

100

93

83

89

100

100

100

100

100

100

100

100

100

95

100

100

Felis catus

Puma concolor

Lynx rufus

Panthera leo

Panthera uncia

FdPV1

PcPV1

LrPV1

PlpPV1

UuPV1

COPV

PlPV1

biogeography

host-pathogen co-divergence


biogeographic calibration

• the volcanic origin of the Hawaiian islands has produced a chain of islands of increasing geological age

• the phylogenetic relationships of island endemic species reflect this volcanic ‘conveyer belt’

28

for example, the honeycreeper species and fruit flies (Drosophila spp.) from the oldest islands form the deepest branch of the tree, and the younger islands on the tips of the tree.

Hemignathus spp. Drosophila spp.

Fleischer, McIntosh & Tarr (1998)

• a remarkably linear relationship is observed between genetic divergence and time when DNA distance is plotted against island age


• weinstock et al (2005) used the date of the formation of the Isthmus of Panama, which allowed the radiation of horses out of South America, to calibrate an analysis of modern equid species

29

biogeographic calibration


calibrating time phylogenies: tip calibration

30

time (years BP)

0

20000

10000

contemporary sample, no time structure

serial sample, with time structure


calibrating the clock with tips

• 2 major sources:

31

1. ancient DNA

large data sets of radiocarbon-dated specimens

2. RNA viruses

evolve quickly: 10-3 - 10-5 substitutions per site per year.

Measurably evolvin

g

population


• genealogic versus phylogenetic time scales

what is an appropriate calibration?

32

STUDYSPECIES

OUTGROUPSPECIES

Intraspecificcalibration

Extraspecificcalibration

treeModel.rootHeight

0.0 500000.0 1000000.0 1500000.0 2000000.0 2500000.0 3000000.00.0

0.0000025

0.000005

0.0000075

0.00001

0.0000125

root“western” europetmrca(clade1)

0.0 250000.0 500000.0 750000.0 1000000.0 1250000.00.0

0.000005

0.00001

0.000015

0.00002

0.000025

0.00003

fossil calibrationtip calibrationho et al 2008


incorporating sampling time: naive method

sampling time 1t1

sampling time 2t2

observed number of substitutions or genetic divergence

d

substitution rate, µ = d / |t1 - t2|



ancestral

diversity

troot t2 t1


µ = (d1 - d2) / (t1 - t2)

d1

d2

troot t2 t1



linear regression

• can be rearranged:

di = µ (ti - troot)

E[di] = µ . ti - µ . troot

gradient is: µy-intercept is: - µ . troot

x-intercept is: troott2 t1

d1

d3

troot t3

d2

µ = di / (ti - troot)


linear regression (root-to-tip regressions)

Time (months since seroconversion) Time (year)

Roo

t-to

-tip

div

erge

nce

R = 0.672

R = 0.89 2

0 1 2 3 4 5 6 7 8 -1 -2 9 10 12 13 11 1985 1990 1995 1980 1975 1970 1965 1960

Roo

t-to

-tip

div

erge

nce


linear regression (root-to-tip regression)

• estimates

the substitution rate, µ

the time to root (troot)

• requires a rooted tree

• underestimates statistical error, because points are (incorrectly) assumed to be independent

t2 t1troot t3


testing the molecular clock


molecular clock v non-clock

• strict molecular clockZuckerkandl & Pauling (1962) in Horizons in Biochemistry, pp. 189–225

all lineages evolve at the same rate

makes it possible to estimate the root of the tree and the dates of the individual nodes

• unconstrained (unrooted) Felsenstein modelFelsenstein (1981) JME, 17: 368 - 376

each branch has its own rate, independent of all the others

time and rate are confounded, and can only be estimated as a compound parameter (branch lengths)


Two types of tests

• relative rate test

• likelihood ratio/Bayes factor test

41Wednesday, July 27, 2011

the relative rate test

• relative rate test compares the difference between the number of substitutions separating two closely related taxa in comparison to a third, more distantly related, taxon

• if A and B have evolved according to a molecular clock, they should be equidistant from C

null: dAC - dBC = 0

• in order for this test to work, A and B must be closely related, and C cannot be too distantly removed

42

A B C

X


the relative rate test

• Synonymous sites in nine nuclear genes (3520 bp):

d12 = 6.7d13 – d23 = 2.3 ± 0.6

• ψη-globin pseudogene (1827 bp):d12 = 7.9d13 – d23 = 1.5 ± 0.4

• Three introns (3376 bp):d12 = 6.9d13 – d23 = 1.0 ± 0.5

• Two flanking regions (936 bp):d12 = 7.9d13 – d23 = 3.1 ± 1.1

43

1 2 3

Old Worldmonkey Human

New Worldmonkey


Two types of tests

• relative rate test

• likelihood ratio/Bayes factor test

44Wednesday, July 27, 2011

non-clock phylogeny

• unrooted tree

• 2n-3 independent branches

• all of b1-b7 need to be estimated

• maximum likelihood (LRT)

45

P1: IKB

CB502-10 CB502-Salemi & Vandamme CB502-Sample-v3.cls March 18, 2003 13:50 Char Count= 0

268 David Posada

A C

D

E

B

E

C

D

A

B

b6

b4b4

b3b1

b2

b6

b7b5

b8b3 b2

b5b7

b1

unrooted tree2n ! 3 independent branches

rooted treen ! 1 independent branches

All b1, b2, b3, b4, b5, b6, and b7need to be estimated

Only b1, b3, b4, and b6,for example, need to be estimated,because under the molecular clock:

b2 = b1b5 = b1 + b3 ! b6b7 = b6b8 = b4 ! b5 ! b6

ANonclocklike phylogenetic tree

n taxa = 5

BClocklike phylogenetic tree

n taxa = 5

Figure 10.5 Number of free parameters in clock and nonclock trees. Under the free rates model(= nonclock), all the branches need to be estimated (2n ! 3). Under the molecular clock,only n ! 1 branches have to be estimated. The difference in the number of parametersamong a nonclock and a clock model is n ! 2.

Maximum-likelihood methods can estimate the branch lengths of a tree by enforc-ing or not enforcing a molecular clock. In the absence of a molecular clock (thefree-rates model), 2n ! 3 branch lengths must be inferred for a strictly bifurcatingunrooted phylogenetic tree with n taxa (Figure 10.5B). If the molecular clock isenforced, the tree is rooted, and just n ! 1 branch lengths need to be estimated (seeFigure 10.4 and Chapter 1). This should appear obvious considering that under amolecular clock, for any two taxa sharing a common ancestor, only the length of thebranch from the ancestor to one of the taxa needs to be estimated, the other one be-ing the same. Statistically speaking, the molecular clock is the null hypothesis (i.e.,the rate of evolution is equal for all branches of the tree) and represents a specialcase of the more general alternative hypothesis that assumes a specific rate for eachbranch (i.e., free-rates model). Thus, given a tree relating n taxa, the LRT can beused to evaluate whether the taxa have been evolving at the same rate (Felsenstein,1988). In practice, a model of nucleotide (or amino-acid) substitution is chosenand the branch lengths of the tree with and without enforcing the molecular clockare estimated. To assess the significance of this test, the LRT can be compared witha !2 distribution with (2n ! 3) ! (n ! 1) = n ! 2 degrees of freedom, becausethe only difference in parameter estimates is in the number of branch lengths thatneeds to be estimated.

L(τ, v, θ |y1,...,yn) = Pr[yi |τ, v, θ]Πi=1

ΘA C G T

ACG

T

y1

GGTT

y2

AGCC

y2

CCCA

...

v = {b1,...,b7}


clock phylogeny

• rooted tree

• n-1 independent parameters

• only b1, b3, b4 and b6, need to be estimated,

because under the molecular clock

b2 = b1 b5 = b1 + b3 − b6 b7 = b6 b8 = b4 − b5 − b6

46

b

b

b

bb

b

bb

1

23

4

56

7

8

A

B

C

E

D

μaπC

.μdπcμeπc

. μbπG μcπT

.μaπA

.

μdπG μeπT

μfπTμbπA

μcπA μfπG

Q =


t

t

1

2

t3

t4

A

B

C

E

D

clock phylogeny

• rooted tree

• n-1 independent parameters

• only b1, b3, b4 and b6, need to be estimated,

because under the molecular clock

b2 = b1 b5 = b1 + b3 − b6 b7 = b6 b8 = b4 − b5 − b6

47

• t1, t2, t3, t4 are the ‘heights’ of the nodes


t

t

1

2

t3

t4

A

B

C

E

D

clock v non-clock LRT

• complex model (H1)

48

P1: IKB


268 David Posada

A C

D

E

B

E

C

D

A

B

b6

b4b4

b3b1

b2

b6

b7b5

b8b3 b2

b5b7

b1





b2 = b1b5 = b1 + b3 ! b6b7 = b6b8 = b4 ! b5 ! b6


n taxa = 5


n taxa = 5



• null model (H0)

• likelihood ratio test with N-2 degrees of freedom

• models are nested because values of b1-b7 can be specifiedthat give node heights t1-t4

N-1 parameters2N-3 parameters

LR = 2(log L(HaD) - log L(H0D))


Human

Chimp

Gorilla

Orang-utan

Gibbon

Human

Chimp

Gorilla

Orang-utan

Gibbon

log Likelihood = -2660.61 log Likelihood = -2659.18

• The differences in log likelihood can be compared directly (not significantlydifferent in this case - primate mitochondrial DNA)

0 2 4 6 8 10 12

0.00

0.05

0.10

0.15

0.20

0.25

x

dchi

sq(x

, 3)

df = 3

0.050.01

49

clock v non-clock LRT


50introduction clocks calibrations clock tests relaxed clocks conclusion

Model testing using Bayes factors

p(θ|D,M) = p(D|θ,M) p(θ|M)

p(D|M)

p(D|M) = p(D|θ,M) p(θ|M) dθ

B12 = p(D|M1) p(D|M2)

∫θ

• Harmonic mean estimator

• Path sampling (thermodynamic integration)

Newton and Raftery, 1994; Suchard et al., 2003

Gelman, 1998; Ogata, 1989; Lartillot and Philippe, 2006

• posterior

• marginal likelihood

• Bayes factor


relaxing the clock


a relaxed clock model

• the unrooted model of phylogeny and the strict molecular clock are two ends of a continuum, and both are evolutionarily unrealistic

52

• fortunately, the assumption of the strict molecular clock can be relaxed, to allow for variation within a data set


model complexity and the molecular clock

53

P1: IKB


268 David Posada

A C

D

E

B

E

C

D

A

B

b6

b4b4

b3b1

b2

b6

b7b5

b8b3 b2

b5b7

b1





b2 = b1b5 = b1 + b3 ! b6b7 = b6b8 = b4 ! b5 ! b6


n taxa = 5


n taxa = 5



‣ Pybus (2006) Genome Biol. 4, e151


relaxed molecular clock models

• some phylogenetic models allow the rate to vary among branches in a controlled manner

54

Local clock models (PAML, QDate)

Non-parametric rate smoothing (r8s)

Ad hoc heuristic rate smoothing (PAML)

Penalized likelihood (r8s)

Bayesian relaxed-clock methods (multidivtime, PhyloBayes, BEAST)


t

t

1

2

t3

t4

A

B

C

E

D

P1: IKB


268 David Posada

A C

D

E

B

E

C

D

A

B

b6

b4b4

b3b1

b2

b6

b7b5

b8b3 b2

b5b7

b1





b2 = b1b5 = b1 + b3 ! b6b7 = b6b8 = b4 ! b5 ! b6


n taxa = 5


n taxa = 5



• specify H0 beforehand

• problem of identifiability

‣ Yoder and Yang (2000) Mol Biol & Evol 17: 1081-1090.

55

local molecular clocks


• Most relaxed clock models assume inheritance of rates of evolution, resulting in correlation between ancestral lineages and their descendants

- e.g., Thorne & Kashino (1998) Mol Biol Evol, 15: 1647-165

- Descendant branches draw a rate from a distribution with a mean given by the ancestral branch. Distributions can be exponential, gamma, lognormal etc.

- Assume a single fixed tree topology

- Use Bayesian MCMC to sample rates and times given the tree

56

relaxed molecular clocks


• rates for each branch are drawn from a distribution centered on the rate of the ancestor

€

ri ~ LogNormal(rA (i),σ2Δti)

AA

h3

h1

h2

GA AC GC

r6r5

r4r3r2r1

r7

‣ e.g., Thorne JL, Kishino H, Painter IS (1998) Mol Biol & Evol 15: 1647-1657.

?

?

‣ A prior degree ofautocorrelation?

‣ but what is the rate at the root?

57

autocorrelated relaxed clocks


AA

h3

h1

h2

GA AC GC

r6r5

r4r3r2r1€

r ~ Exp(λ) [7]

€

r ~ LogNormal(µ,σ 2) [8]

€

r ~ Gamma(α,β) [9]

0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

x

dlno

rm(x

, 0, 1

)

‣ Drummond et al. (2006) Plos Biology 4: e88.

58

• rates for each branch are drawn independently from an identical distribution

uncorrelated relaxed clocks


AA

h3

h1

h2

GA AC GC

r6r5

r4r3r2r1€

r ~ Exp(λ) [7]

€

r ~ LogNormal(µ,σ 2) [8]

€

r ~ Gamma(α,β) [9]

0 1 2 3 4 5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

x

dlno

rm(x

, 0, 1

)

‣ Drummond et al. (2006) Plos Biology 4: e88.

59

• rates for each branch are drawn independently from an identical distribution

uncorrelated relaxed clocks


60

uncorrelated relaxed clocks: an example


Slow

Ape

s

Fast Rodent

1

=i

00

00

0

1

0

Random local clocks

➡ Rate changes do not necessarily occur regularly or on every branch➡ Small number of significant changes➡ Can we handle the uncertainty in the number and locations of (a

small number of) local clocks?

•three local clocks•two rate changes

➡ How to explore 22n-2 clock models?


0 2 4 6 8 10 12

0.0

0.1

0.2

0.3

0.4

0.5

0.6

# of Rate Changes

Prob

abilit

y

PosteriorPrior

Random local clocks

➡ Using Bayesian stochastic search variable selection: formulate a prior that such that many rate changes (indicators) are 0 but allow the data to determine which ones are required to explain (most of the) rate variation using MCMC

Local Clock Comparison withDouzery (2003)

3 Nuclear Genes from 42 Mammals (GTR + !)

0 2 4 6 8 10 12

0.0

0.1

0.2

0.3

0.4

0.5

0.6

# of Rate Changes

Prob

abilit

y

PosteriorPrior

Consistentresults (5-12local clocks).

RLC model provides an automated approach to discover localclocks and their uncertainty.

PhyloGroup, September 2007 – p.9

➡ Three mtDNA nuclear genes from 42 mammals (Douzery, 2003)

➡ 5-12 local clocks

Drummond and Suchard, 2010.


relaxed clocks: summary

• can be used to estimate phylogenies and divergence times in the face of uncertainty in evolutionary rates and divergence times

• provides a means for measuring the clocklike-ness of data and comparing this measurement between different genes and different taxonomic groups

• allows investigation of autocorrelation between rates

rates of evolution are drawn randomly from some parametric distribution

parameters of substitution (rate and variance) can be estimated


summary

• a molecular clock is a reasonable assumption... sometimes

• molecular clocks make it possible to correlate genetic divergence with time

estimate divergence dates, timing of demogrpahic and phylogenetic events, etc

• clocks need to be calibrated

• various statistical tests have been developed to test the clock-likeness of any particular data set


vicariance versus dispersal: ratite birds


origin of ratites

• non molecular clock maximum likelihood tree

• complete mtDNA

‣ Cooper A et al. (2001) Nature, 409, 704-707.Wednesday, July 27, 2011

vicariance and dispersal

• is ostrich evolving faster?

• did it reach Africa via dispersalrather than vicariance?

• is the tree wrong?

80 My reconstruction


origin of ratites

• strict molecularclock


origin of ratites

• relaxed molecular clock

uncorrelated lognormal

lognormal prior on emu-cassewary

red: fast rate

blue: slow rate


date of origin of (mostly) extant ratites

• estimate of the age of

the root of ratite tree

black - strict clock

blue - relaxed clock

• clock gives the sameestimate with less variance


0.0 50.0 100.0 150.0 200.0 250.00.0

0.01

0.02

0.03

0.04

0.05

0.06

Age (My)


effect of calibration priors

• prior age of emu-cassowary: • posterior age of root:

tmrca(Oz)

20.0 30.0 40.0 50.0 60.0 70.00.0

0.05

0.1

0.15

0.2

0.25

0.3

0.35


0.0 50.0 100.0 150.0 200.0 250.00.0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Age (My)Age (My)


break


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