Concepts and Methods in Molecular Divergence Time Estimation 26 November 2012 Prashant P. Sharma American Museum of Natural History
Overview 1. Why do we date trees? 2. The molecular clock 3. Local clocks and autocorrelated rates 4. Bayesian inference using uncorrelated rates 5. Fossil calibrations and uncertainty 6. Innovations in molecular dating
Why do we need dates? Biogeographic hypothesis testing Crisp et al. (2011) Trends Ecol Evol 26: 66-72.
Why do we need dates? Testing hypotheses of co- diversivication Cruaud et al. (2012) Syst Biol 61: 1029-1047.
Why do we need dates? Quantifying and characterizing rates of diversivication Rouse et al. (2013) Mol Phylogenet Evol 66: 161-181.
Why do we need dates? Inferring the age of important evolutionary events Timetree of Life Project (there s an app for that)
Why do we need molecular dates? The fossil record is incomplete The fossil record is biased taxonomically and taphonomically Fossil dates may not be precise At best, fossil dates are minimum age estimates
Zuckerkandl and Pauling (1965) J Theor Biol 8: 357-366.
K = number of substitutions per site T = Time R= rate Ancestor R t = K / (2T) Time Descendent 1 Descendent 2
Strict molecular clocks Phylogeny with branch lengths One or more node estimates Time We can predict dates of other nodes in the tree Time for divergence of novel sequences T ij = d ij / 2r Assumption: Probability of substitutions is constant over time
The theory of the molecular clock Rate constancy is an extension of neutral theory of molecular evolution (Kimura, King and Jukes, 1968-1969) Strongly invluenced early models of molecular evolution (Jukes- Cantor, 1969) A tree under a molecular clock does not have to be exactly ultrametric; only the probability of mutation per unit time is constant How do we test if a tree is ultrametric enough?
Testing the molecular clock 1. Likelihood ratio test 2. Relative rates test
Likelihood ratio test Procedure: Estimate a molecular phylogeny with, and without, a molecular clock Calculate 2[log (L 1 ) log (L 2 )] X 2 test with df = (n 2), where n = number of terminals
Relative rates test T0 T1 T2 T3
Relative rates test T0 K 01 K 02 T1 T2 T3 H 0 : K 01 = K 02 or H 0 : K 01 K 02 = 0
Relative rates test T0 K 01 K 02 T1 T2 T3 K 13 = K 01 + K 03 K 23 = K 02 + K 03 K 12 = K 01 + K 02
Relative rates test T0 K 01 K 02 T1 T2 T3 K 01 = (K 13 + K 12 K 23 )/2 K 02 = (K 12 + K 23 K 13 )/2 K 03 = (K 13 + K 23 K 12 )/2
Relative rates test T0 K 01 K 02 T1 T2 T3 K 01 K 02 = K 13 K 23
Relative rates test It can be shown that this statistic is normally distributed for large samples Calculate for all triplets Calculate Z score Z < 1.96
ConVidence limits for molecular clocks Substitutions occur as a linear function of time Probability of substitution per unit time is constant Rate variation must have a Poisson distribution Time Substitutions
Sources of rate heterogeneity Physiology/life history: generation time Demography: genetic drift affects small populations more strongly than large ones Selection/relaxation: increase or decrease in evolutionary rate Gene duplication: neofunctionalized paralogs evolve faster than copies retaining ancestral functions (Assis and Bachtrog, 2013)
Local clocks O huigin and Li (1992) J Mol Evol 35: 377-384.
Local clocks K s (mouse- rat) = 18.0% K s (mouse- hamster) = 30.3% K s (rat- hamster) = 31.3% Hamsters diverged 1.7 times earlier than mouse- rat divergence There is a molecular clock for rodents Mouse Rat Hamster But substitution rates are higher in rodents than in primates O huigin and Li (1992) J Mol Evol 35: 377-384.
Local clocks Multiple molecular clocks occur in different parts of a tree Rate 2 Rate 1 Rate autocorrelation: substitution rates are heritable Descendent nodes inherit the substitution rate of their ancestor nodes Rate 3
Methods using many rates, assuming rate autocorrelation 1. Non- parametric rate smoothing (Sanderson, 1997) 2. Penalized likelihood (Sanderson, 2002)
Non- parametric rate smoothing b 0 i b 1 b 2 Measure of rate roughness at node i: R i = (r b0 r b1 ) 2 + (r b0 r b2 ) 2 Sanderson (1997) Mol Biol Evol 14: 1218 1231
Non- parametric rate smoothing b 0 i b 1 b 2 Adjust branching times in order to minimize overall roughness, ΣR i
Non- parametric rate smoothing b 0 i b 1 b 2 Drawbacks: 1. Assumes branch lengths are known with complete certainty 2. Attributes differences in sister branches exclusively to variation in rate of evolution
Penalized likelihood Find the set of branch lengths and rates that minimizes the function: Log(L) λp where: λ is a user- devined smoothing parameter P is a penalty function Sanderson (2002) Mol Biol Evol 19: 101 109
Penalized likelihood Log(L) λp Penalty function, P In Sanderson s formulation, the quadratic roughness function R i was used Alternative penalty functions: 1. Lognormal 2. Exponential 3. Ornstein- Uhlenbeck process Sanderson (2002) Mol Biol Evol 19: 101 109
Penalized likelihood Log(L) λp Smoothing parameter, λ Determined empirically through cross- validation procedure Drawback: Computationally expensive and difvicult to implement Sanderson (2002) Mol Biol Evol 19: 101 109
Relaxed clock methods What happens if evolutionary rates are not autocorrelated? Uncorrelated clock methods implement evolutionary rates as prior distributions
Bayesian Evolutionary Inference of Species Trees (BEAST) Implements strict clocks, autocorrelated rate models, and uncorrelated rate models MCMC procedure to derive posterior distributions of Tree topology Rates Divergence times Calibration points can be distributions, not point estimates Drummond et al. (2006) PLoS Biol 4: e55
BEAST f (g, Θ, Φ, Ω D) = (1/Z) Pr {D g, Φ, Ω) f G (g Φ) f ΘΦΩ (Θ, Φ, Ω) Φ : parameters of the relaxed clock model Ω: parameters of the substitution model Θ: hyperparameters of the tree prior Pr {D g, Φ, Ω): standard term for likelihood, where g is a tree with branch lengths in time units f G (g Φ): the tree prior (Yule, birth- death, or coalescent- based) Drummond et al. (2006) PLoS Biol 4: e55
Properties of uncorrelated clock models Strict clocks and rate autocorrelation are special cases of uncorrelated rate models Uncorrelated lognormal distributions better account for cases where evolution is clock- like Uncorrelated exponential distributions have high variance (2 10x higher than uncorrelated lognormal) Drummond et al. (2006) PLoS Biol 4: e55
Uncertainty is inherent to molecular dating Tree topology Branch lengths Rate variation Fossil calibration
Fossil calibrations How old is the fossil? Where does the fossil Vit in the tree? What does the placement of the fossil mean for the calibration?
Fossil calibrations Fossil age Uncertainty Fossil taxon Extant taxa Past a: Time of fossil lineage s divergence b: Time of fossil lineage s extinction a b Present
Fossil calibrations Past a Present Alternative 1: Constrain preceding node using fossil Alternative 2: Constrain subsequent node using fossil b
Fossil calibrations Past a b Present a is never observed in molecular dating For this reason, fossil calibrations yield mimimum age estimates
Closing the rocks and clocks gap Realistic priors for fossil calibrations Exponential and lognormal distribution priors Interval estimates on fossil ages Increasing sampling of fossils used for calibration Cross- validation of fossil calibrations
Total evidence molecular dating 161 morphological characters RAG- 1 (2652 bp) Pyron (2011) Syst Biol 60: 466-481
Problems with integrating morphology in dating Model- based approaches to dating require a model for morphological data partitions The Lewis (2001) model Abundant evidence for rate heterogeneity in morphological evolution
Improving precision in molecular dating After Shih and Matzke (2013) Proc Natl Acad Sci USA 110: 12355-12360
Shih and Matzke (2013) Proc Natl Acad Sci USA 110: 12355-12360
14-26% reduction in size of convidence intervals Shih and Matzke (2013) Proc Natl Acad Sci USA 110: 12355-12360
Summary 1. Molecular dating is a matter of quantifying uncertainty a. Tree topology b. Branch length c. Rate variation d. Fossil age calibration e. Fossil placement 2. Implement with caution, interpret with skepticism