Molecular Clocks Rose Hoberman The Holy Grail Fossil evidence is sparse and imprecise (or nonexistent) Predict divergence times by comparing molecular data Given a phylogenetic tree branch lengths (rt) a time estimate for one (or more) node C D R M 110 MYA H Rate Constancy? Can we date other nodes in the tree? Yes... if the rate of molecular change is constant across all branches Page & Holmes p240 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 Evidence for Rate Constancy in Hemoglobin Large carniverous marsupial Page and Holmes p229 1
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 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 Measuring Evolutionary time with a molecular clock 1. Estimate genetic distance d = number amino acid replacements 2. Use paleontological data to determine date of common ancestor T = time since divergence 3. Estimate calibration rate (number of genetic changes expected per unit time) r = d / 2T 4. Calculate time of divergence for novel sequences T_ij = d_ij / 2r 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) 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 Distances from Gamma-Distributed Rates rate variation among sites fast/variable sites 3 rd codon positions codons on surface of globular protein slow/invariant sites Trytophan (1 codon) structurally required 1 st or 2 nd 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 2
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 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 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 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 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) 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 3
Linear Regression Fix intercept at (0,0) Fit line between divergence estimates and calibration times Calculate regression and prediction confidence limits Molecular Systematics p536 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 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 Rate Heterogeneity among Lineages Cause Repair equipment Metabolic rate Generation time Population size Reason e.g. RNA viruses have error-prone polymerases More free radicals Copies DNA more frequently Effects mutation fixation rate 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) Rate Changes within a Lineage Cause Population size changes Strength of selection changes over time Reason Genetic drift more likely to fix neutral alleles in small population 1. new role/environment 2. gene duplication 3. change in another gene 4
Working Around Rate Heterogeneity 1. Identify lineages that deviate and remove them 2. 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 3. Explicity model rate variation 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 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 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 Modeling Rate Variation Relaxing the Molecular Clock D E F Learn rates and times, not just M branch lengths A B C 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 R N 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 5
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 infer branch lengths via parsimony rates drawn from a normal distribution (negative rates set to zero) 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 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?) 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 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 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) 6
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 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) 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? 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. 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 General References Reviews/Critiques 1. Bronham and Penny. The modern molecular clock, Nature review in genetics?, 2003. 2. Graur and Martin. Reading the entrails of chickens...the illusion of precision. Trends in Genetics, 2004. Textbooks: 1. Molecular Systematics. 2 nd edition. Edited by Hillis, Moritz, and Mable. 2. Inferring Phylogenies. Felsenstein. 3. Molecular Evolution, a phylogenetic approach. Page and Holmes. 7
Rate Heterogeneity References Dealing with Rate Heterogeneity 1. Yang and Yoder. Comparison of likelihood and bayesian methods for estimating divergence times... Syst. Biol, 2003. 2. Kishino, Thorne, and Bruno. Performance of a divergence time estimation method under a probabilistic model of rate evolution. Mol. Biol. Evol, 2001. 3. Huelsenbeck, Larget, and Swofford. A compound poisson process for relaxing the molecular clock. Genetics, 2000. Testing for Rate heterogeneity 1. Takezaki, Rzhetsky and Nei. Phylogenetic test of the molecular clock and linearized trees. Mol. Bio. Evol., 1995. 2. Bronham, Penny, Rambaut, and Hendy. The power of relative rates test depends on the data. J Mol Evol, 2000. Dating Duplications References Dating duplications: 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. 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? 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? 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 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 8
Mutation Rate in Nuclear Genes of Mammals (Yang & Nielsen 1997) Acid phosphotase Myelin Proteolipid Interleukin 6 IGF binding 1 Thrombomodulin Average ds (P) 0.354 0.033 0.100 0.307 0.414 0.190 ds (R) 0.680 0.117 0.566 0.667 1.337 0.525 dn (P) 0.028 0.009 0.191 0.109 0.092 0.039 dn(r) 0.049 0.000 0.373 0.084 0.108 0.066 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 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 Bayesian parametric estimation Density function for x, given the training data set ( n) 1 N X = x x {,..., } ( ) ( ) ( n n p x X ) p( x, θ X ) d From the definition of conditional probability densities p X p X p X ( x, θ ) = ( x θ, ) ( θ ). ( ) ( ) ( ) The first factor is independent of X (n) since it just our ( n) assumed form p( x θ, X ) p( x θ ) for parameterized density. Therefore ( ) ( ) ( n n p x X ) p( x ) p( X ) d = = θ θ θ θ Bayesian parametric estimation Instead of choosing a specific value θ, the Bayesian approach performs a weighted average over all values of θ. ( n) If the weighting factor p( θ X ), which is a posterior of θ peaks very sharply about some value we obtain ( n) $θ p( x X ) p( x $ θ ). Thus the optimal estimator is the most likely value of θ given the data and the prior of θ. The Holy Grail Fossil evidence is sparse and imprecise (or nonexistent) Predict divergence times by comparing molecular data 9