1. Molecular Clocks
Rose Hoberman

2. The Holy Grail
Fossil evidence is sparse and imprecise (or nonexistent)
Predict divergence times by comparing molecular data

3. Given Can we date other nodes in the tree?M
Given
a phylogenetic tree
branch lengths (rt)
a time estimate for one (or more) node
110 MYA
Can we date other nodes in the tree?
Yes... if the rate of molecular change is constant across all branches

4. Rate Constancy?
Page & Holmes p240

5. Protein Variability Protein structures & functions differ
Proportion of neutral sites differ
Rate constancy does not hold across different protein types
However...
Each protein does appear to have a characteristic rate of evolution

6. Evidence for Rate Constancy in Hemoglobin
Large carniverous marsupial
Page and Holmes
p229

7. The Molecular Clock Hypothesis
Amount of genetic difference between sequences is a function of time since separation.
Rate of molecular change is constant (enough) to predict times of divergence

8. Outline Methods for estimating time under a molecular clock
Estimating genetic distance
Determining and using calibration points
Sources of error
Rate heterogeneity
reasons for variation
how its taken into account when estimating times
Reliability of time estimates
Estimating gene duplication times

9. Measuring Evolutionary time with a molecular clock
Estimate genetic distance
d = number amino acid replacements
Use paleontological data to determine date of common ancestor
T = time since divergence
Estimate calibration rate (number of genetic changes expected per unit time)
r = d / 2T
Calculate time of divergence for novel sequences
T_ij = d_ij / 2r

10. Estimating Genetic Differences
If all nt equally likely, observed difference would plateau at 0.75
Simply counting differences underestimates distances
Fails to count for multiple hits
(Page & Holmes p148)
graph on p157 felsenstein better

11. Estimating Genetic Distance with a Substitution Model
accounts for relative frequency of different types of substitutions
allows variation in substitution rates between sites
given learned parameter values
nucleotide frequencies
transition/transversion bias
alpha parameter of gamma distribution
can infer branch length from differences
Simple Poisson correction:
d = -ln(1 – n/L) where d = subst. per site

12. Distances from Gamma-Distributed Rates
rate variation among sites
“fast/variable” sites
3rd codon positions
codons on surface of globular protein
“slow/invariant” sites
Trytophan (1 codon) structurally required
1st or 2nd codon position when di-sulfide bond needed
alpha parameter of gamma distribution describes degree of variation of rates across positions
modeling rate variation changes branch length/ sequence differences curve

13. Gamma Corrected Distances
high rate sites saturate quickly
sequence difference rises much more slowly as the low-rate sites gradually accumulate differences
Felsenstein Inferring Phylogenies p219

14. The ‘Sloppy’ Clock ‘Ticks’ are stochastic, not deterministic
Mutations happen randomly according to a Poisson distribution.
Many divergence times can result in the same number of mutations
Actually over-dispersed Poisson
Correlations due to structural constraints

15. Poisson Variance (Assuming A Pefect Molecular Clock)
If mutation every MY
Poisson variance
95% lineages 15 MYA old have 8-22 substitutions
8 substitutions also could be 5 MYA
Molecular Systematics p532

16. Need for Calibrations Changes = rate*time
Can explain any observed branch length
Fast rate, short time
Slow rate, long time
Suppose 16 changes along a branch
Could be 2 * 8 or 8 * 2
No way to distinguish
If told time = 8, then rate = 2
Assume rate=2 along all branches
Can infer all times

17. Estimating Calibration Rate
Calculate separate rate for each data set (species/genes) using known date of divergence (from fossil, biogeography)
One calibration point
Rate = d/2T
More than one calibration point
use regression
use generative model that constrains time estimates (more later)

18. Calibration Complexities
Cannot date fossils perfectly
Fossils usually not direct ancestors
branched off tree before (after?) splitting event.
Impossible to pinpoint the age of last common ancestor of a group of living species

19. Linear Regression Fix intercept at (0,0)
Fit line between divergence estimates and calibration times
Calculate regression and prediction confidence limits
Molecular Systematics p536

20. Molecular Dating Sources of Error
Both X and Y values only estimates
substitution model could be incorrect
tree could be incorrect
errors in orthology assignment
Poisson variance is large
Pairwise divergences correlated (Systematics p534?)
inflates correlation between divergence & time
Sometimes calibrations correlated
if using derived calibration points
Error in inferring slope
Confidence interval for predictions much larger than confidence interval for slope

21. Rate Heterogeneity Rate of molecular evolution can differ between
nucleotide positions
genes
genomic regions
genomes (nuclear vs organelle), species
species
over time
If not considered, introduces bias into time estimates

22. Rate Heterogeneity among Lineages
Cause
Reason
Repair equipment
e.g. RNA viruses have
error-prone polymerases
Metabolic rate
More free radicals
Generation time
Copies DNA more frequently
Population size
Effects mutation fixation rate

23. Local Clocks?
Closely related species often share similar properties, likely to have similar rates
For example
murid rodents on average 2-6 times faster than apes and humans (Graur & Li p150)
mouse and rat rates are nearly equal (Graur & Li p146)

24. Rate Changes within a Lineage
Cause
Reason
Population size changes
Genetic drift more likely to fix neutral alleles in small population
Strength of selection changes over time
new role/environment
gene duplication
change in another gene

25. Working Around Rate Heterogeneity
Identify lineages that deviate and remove them
Quantify degree of rate variation to put limits on possible divergence dates
requires several calibration dates, not always available
gives very conservative estimates of molecular dates
Explicity model rate variation

26. Search for Genes with Uniform Rate across Taxa
Many ‘clock’ tests:
Relative rates tests
compares rates of sister nodes using an outgroup
Tajima test
Number of sites in which character shared by outgroup and only one of two ingroups should be equal for both ingroups
Branch length test
deviation of distance from root to leaf compared to average distance
Likelihood ratio test
identifies deviance from clock but not the deviant sequences

27. Likelihood Ratio Test
estimate a phylogeny under molecular clock and without it
e.g. root-to-tip distances must be equal
difference in likelihood ~ 2*Chi^2 with n-2 degrees of freedom
asymptotically
when models are nested
when nested parameters aren’t set to boundary

28. Relative Rates Tests
Tests whether distance between two taxa and an outgroup are equal (or average rate of two clades vs an outgroup)
need to compute expected variance
many triples to consider, and not independent
Lacks power, esp
short sequences
low rates of change
Given length and number of variable sites in typical sequences used for dating, (Bronham et al 2000) says:
unlikely to detect moderate variation between lineages (1.5-4x)
likely to result in substantial error in date estimates

29. Modeling Rate Variation Relaxing the Molecular Clock
A B C
D E F M N R
Modeling Rate Variation Relaxing the Molecular Clock
Learn rates and times, not just branch lengths
Assume root-to-tip times equal
Allow different rates on different branches
Rates of descendants correlate with that of common acnestor
Restricts choice of rates, but still too much flexibility to choose rates well

30. Relaxing the Molecular Clock
Likelihood analysis
Assign each branch a rate parameter
explosion of parameters, not realistic
User can partition branches based on domain knowledge
Rates of partitions are independent
Nonparametric methods
smooth rates along tree
Bayesian approach
stochastic model of evolutionary change
prior distribution of rates
Bayes theorem
MCMC

31. Parsimonious Approaches
Sanderson 1997, 2002
infer branch lengths via parsimony
fit divergence times to minimize difference between rates in successive branches
(unique solution?)
Cutler 2000
rates drawn from a normal distribution (negative rates set to zero)

32. Bayesian Approaches Learn rates, times, and substitution parameters simultaneously
Devise model of relationship between rates
Thorne/Kishino et al
Assigns new rates to descendant lineages from a lognormal distribution with mean equal to ancestral rate and variance increasing with branch length
Huelsenbeck et al
Poisson process generates random rate changes along tree
new rate is current rate * gamma-distributed random variable
Use MCMC to explore space, estimate posterior

33. Comparison of Likelihood & Bayesan Approaches for Estimating Divergence Times (Yang & Yoder 2003)
Analyzed two mitochondrial genes
each codon position treated separately
tested different model assumptions
used
7 calibration points
Neither model reliable when
using only one codon position
using a single model for all positions
Results similar for both methods
using the most complex model
use separate parameters for each codon position (could use codon model?)

34. Sources of Error/Variance
Lack of rate constancy (due to lineage, population size or selection effects)
Wrong assumptions in evolutionary model
Errors in orthology assignment
Incorrect tree
Stochastic variability
Imprecision of calibration points
Imprecision of regression
Human sloppiness in analysis
self-fulfilling prophecies

35. Reading the entrails of chickens (Graur and Martin 2004)
single calibration point
error bars removed from calibration points
standard error bars instead of 95% confidence intervals
secondary/tertiary calibration points treated as reliable and precise
based on incorrect initial estimates
variance increases with distance from original estimate
few proteins used

36. Multiple Gene Loci
“Trying to estimate time of divergence from one protein is like trying to estimate the average height of humans by measuring one human”
--Molecular Systematics p539
Use multiple genes!
(and multiple calibration points)

37. Even so... Be Very Wary Of Molecular Times
Point estimates are absurd
Sample errors often based only on the difference between estimates in the same study
Even estimates with confidence intervals unlikely to really capture all sources of variance

38. McLysaght, Hokamp, Wolfe 2002 Dating Human Gene Duplications
[758] Trees generated (ML method using PAM matrix)
[602] Alpha parameter for gamma distribution learned
(Gu and Zhang 1997) faster than ML, more accurate than parsimony
Thrown out if variance > mean. Why would this happen?
“May be problematic to apply this model for gene family evolution because of the possible functional divergence among paralogous genes”
[481] NJ trees built from Gamma-corrected distances
Family kept only if worm/fly group together
[191] Two-cluster test of rate constancy (Takezaki et al 1995)

39. Blanc, Hokamp, Wolfe Dating Arabadopsis Duplications
Create nucleotide alignments
Estimate “Level of” Synonymous substitutions (Yang’s ML method)
per site? per synonymous site?
Ks values > 10 ignored (Yang; Anisimova)
Why used different method than for human?
How reliable is ranking of Ks values? How much variance expected?

40. Ks > 10 unreliable ?
Yang (abstract) calculates effect of evolutionary rate on accuracy of phylogenic reconstruction
Anisimova calculates accuracy and power of LRT in detecting adaptive molecular evolution
Neither seems to give any cutoff regarding dS > 10.

41. Future Improvements
Calculate accurate confidence intervals taking into account multiple sources of variance
Novel models that account for variation in rates between taxa
Build explicit models that predict rates based on an understanding of the underlying processes that generate differences in substitutions rates

42. General References Reviews/Critiques
Bronham and Penny. The modern molecular clock, Nature review in genetics?, 2003.
Graur and Martin. Reading the entrails of chickens...the illusion of precision. Trends in Genetics, 2004.
Textbooks:
Molecular Systematics. 2nd edition. Edited by Hillis, Moritz, and Mable.
Inferring Phylogenies. Felsenstein.
Molecular Evolution, a phylogenetic approach. Page and Holmes.

43. Rate Heterogeneity References
Dealing with Rate Heterogeneity
Yang and Yoder. Comparison of likelihood and bayesian methods for estimating divergence times... Syst. Biol, 2003.
Kishino, Thorne, and Bruno. Performance of a divergence time estimation method under a probabilistic model of rate evolution. Mol. Biol. Evol, 2001.
Huelsenbeck, Larget, and Swofford. A compound poisson process for relaxing the molecular clock. Genetics, 2000.
Testing for Rate heterogeneity
Takezaki, Rzhetsky and Nei. Phylogenetic test of the molecular clock and linearized trees. Mol. Bio. Evol., 1995.
Bronham, Penny, Rambaut, and Hendy. The power of relative rates test depends on the data. J Mol Evol, 2000.

44. Dating Duplications References
McLysaght, Hokamp, and Wolfe. Extensive genomic duplication during early chordate evolution. Nature Genetics?, 2002.
Blanc, Hokamp, and Wolfe. Recent polyploidy superimposed on older large-scale duplications in the Arabidopsis genome. Genome Research, 2003.
Reference used for dating duplications in above papers
Gu and Zhang. A simple method for estimating the parameter of substitution rate variation among sites. Mol. Biol. Evol., 1997.
Yang Z. On the best evolutionary rate for phylogenetic analysis. Syst. Biol, 1998.
Anisimova, Bielawski, Yang. Accuracy and power of the likelihood ratio test in detecting adaptive molecular evolution. Mol. Biol. Evol., 2001.

45. Relative vs Absolute Rates
M. Systematics p540
“Differences in rates of divergence among lineages detract only from methods of analysis that require clocklike behavior of molecules, and alternative methods of analysis exist for all applications of molecular systematics except for the absolute estimation of time.”
t1 = 2 * t2 still requires clocklike behavior?

46. Synonymous vs Nonsynonymous Distance
Syn sites are sites where a nt change does not cause an AA change
only ~25% of sites, so become saturated more quickly
Between proteins
more variation in non-synonymous rates
Within same protein
more variation in synonymous rates
Which are used? What is effect?

47. Two-cluster Test Takezaki, Rzhetsky and Nei (1995?)
estimate tree
for each nonroot interior node:
calculate average “rate” for both descendant clades
test equality of rates (using variance & covariance of branch lengths) [doesn’t appear to correct for multiple testing]
move up from leaves, eliminating a cluster if not equal
finally, linear tree created
reestimate branch lengths under clock constraint

48. Neutral Hypothesis
Most mutations have no influence on fitness of the organism
Advantageous mutations rare
Deleterious mutations rapidly removed
Greatest proportion of mutations have no effect on protein function
Rate of change is thus affected only by mutation rate, and so should be relatively constant within a species
Variation in rate among genes b/c differences in selective constraints

49. Mutation Rate in Nuclear Genes of Mammals (Yang & Nielsen 1997)
dS (P)
dS (R)
dN (P)
dN(R)
Acid phosphotase
0.354
0.680
0.028
0.049
Myelin Proteolipid
0.033
0.117
0.009
0.000
Interleukin 6
0.100
0.566
0.191
0.373
IGF binding 1
0.307
0.667
0.109
0.084
Thrombomodulin
0.414
1.337
0.092
0.108
Average
0.190
0.525
0.039
0.066

50. Perfect Molecular Clock
Change linear function time (substitutions ~ Poisson)
Rates constant (positions/lineages)
Tree perfect
Molecular distance estimated perfectly
Calibration dates without error
Regression (time vs substitutions) without error

51. Yang, effect of evol. rate abstract
Yang calculates effect of evolutionary rate on accuracy of phylogenic reconstruction
simulation study
branch length = “expected total number nt substitutions per site” (not synonymous?)
estimates proportion of correctly recovered branch partitions
“optimum levels of sequence divergence were even higher than previously suggested for saturation of substitutions, indicating that the problem of saturation may have been exaggerated”

52. Bayesian parametric estimation
Density function for x, given the training data set
From the definition of conditional probability densities
The first factor is independent of X(n) since it just our assumed form for parameterized density.
Therefore

53. Bayesian parametric estimation
Instead of choosing a specific value , the Bayesian approach performs a weighted average over all values of
If the weighting factor , which is a posterior of peaks very sharply about some value we obtain
Thus the optimal estimator is the most likely value of given the data and the prior of

54. The Holy Grail
Fossil evidence is sparse and imprecise (or nonexistent)
Predict divergence times by comparing molecular data