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Bayesian Divergence Time Estimation – Workshop Lecture

The document discusses divergence time estimation using relaxed clock models and Bayesian inference, highlighting the importance of accounting for variable substitution rates across lineages. It also emphasizes the necessity of using fossils for calibrating absolute node ages and presents various models that accommodate lineage-specific rate variations. Key concepts include estimating ages of internal nodes, node-age priors, and the implications of specifying appropriate calibration densities in Bayesian analyses.

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B D T E Tracy Heath Ecology, Evolution, & Organismal Biology Iowa State University @trayc7 http://phyloworks.org 2016 Workshop on Molecular Evolution Woods Hole, MA USA
O Overview of divergence time estimation • Relaxed clock models – accounting for variation in substitution rates among lineages • Tree models – lineage diversification and sampling break BEAST v2 Tutorial — Divergence-time estimation under birth-death processes • Dating Bear Divergence Times with the Fossilized Birth-Death Process dinner
A T -S E Phylogenies with branch lengths proportional to time provide more information about evolutionary history than unrooted trees with branch lengths in units of substitutions/site. 0.03 substitutions/site Time (My) 20 0 Oligocene Miocene Plio Pleis (silhouette images from http://phylopic.org)
A T -S E Phylogenetic divergence-time estimation • What was the spacial and climatic environment of ancient angiosperms? • How has mammalian body-size changed over time? • How has the infection rate of HCV in Egypt changed over time? • Is diversification in Caribbean anoles correlated with ecological opportunity? • How has the rate of molecular evolution changed across the Tree of Life? Epidemiology (Antonelli & Sanmartin. Syst. Biol. 2011) (Lartillot & Delsuc. Evolution 2012) (Stadler et al. PNAS 2013) (Mahler, Revell, Glor, & Losos. Evolution 2010) (Nabholz, Glemin, Galtier. MBE 2008) Historical biogeography Molecular evolution Trait evolution Diversification Anolis fowleri (image by L. Mahler) Understanding Evolutionary Processes
D T E Goal: Estimate the ages of interior nodes to understand the timing and rates of evolutionary processes Model how rates are distributed across the tree Describe the distribution of speciation events over time External calibration information for estimates of absolute node times calibrated node 100 0.020.040.060.080.0 Equus Rhinoceros Bos Hippopotamus Balaenoptera Physeter Ursus Canis Felis Homo Pan Gorilla Pongo Macaca Callithrix Loris Galago Daubentonia Varecia Eulemur Lemur Hapalemur Propithecus Lepilemur Mirza M. murinus M. griseorufus M. myoxinus M. berthae M. rufus1 M. tavaratra M. rufus2 M. sambiranensis M. ravelobensis Cheirogaleus Simiiformes Microcebus Cretaceous Paleogene Neogene Q Time (Millions of years)
T G M C Assume that the rate of evolutionary change is constant over time (branch lengths equal percent sequence divergence) 10% 400 My 200 My A B C 20% 10% 10% (Based on slides by Je Thorne; http://statgen.ncsu.edu/thorne/compmolevo.html)
T G M C We can date the tree if we know the rate of change is 1% divergence per 10 My A B C 20% 10% 10% 10% 200 My 400 My 200 My (Based on slides by Je Thorne; http://statgen.ncsu.edu/thorne/compmolevo.html)
T G M C If we found a fossil of the MRCA of B and C, we can use it to calculate the rate of change & date the root of the tree A B C 20% 10% 10% 10% 200 My 400 My (Based on slides by Je Thorne; http://statgen.ncsu.edu/thorne/compmolevo.html)
R G M C Rates of evolution vary across lineages and over time Mutation rate: Variation in • metabolic rate • generation time • DNA repair Fixation rate: Variation in • strength and targets of selection • population sizes 10% 400 My 200 My A B C 20% 10% 10%
U A Sequence data provide information about branch lengths In units of the expected # of substitutions per site branch length = rate ⇥ time 0.2 expected substitutions/site PhylogeneticRelationshipsSequence Data
R T The sequence data provide information about branch length for any possible rate, there’s a time that fits the branch length perfectly 0 1 2 3 4 5 0 1 2 3 4 5 BranchRate Branch Time time = 0.8 rate = 0.625 branch length = 0.5 Methods for dating species divergences estimate the substitution rate and time separately (based on Thorne & Kishino, 2005)
B D T E length = rate length = time R = (r1,r2,r3,...,r2N 2) A = (a1,a2,a3,...,aN 1) N = number of tips
B D T E length = rate length = time R = (r1,r2,r3,...,r2N 2) A = (a1,a2,a3,...,aN 1) N = number of tips
B D T E Posterior probability f (R,A, R, A, s | D, ) R Vector of rates on branches A Vector of internal node ages R, A, s Model parameters D Sequence data Tree topology
B D T E f(R,A, R, A, s | D) = f (D | R,A, s) f(R | R) f(A | A) f( s) f(D) f(D | R,A, R, A, s) Likelihood f(R | R) Prior on rates f(A | A) Prior on node ages f( s) Prior on substitution parameters f(D) Marginal probability of the data
B D T E Estimating divergence times relies on 2 main elements: • Branch-specific rates: f (R | R) • Node ages & Topology: f (A | A) DNA Data Rate Matrix Site Rates Branch RatesTree
B D T E Estimating divergence times relies on 2 main elements: • Branch-specific rates: f (R | R) • Node ages & Topology: f (A | A) DNA Data Rate Matrix Site Rates Branch RatesTree
M R V Some models describing lineage-specific substitution rate variation: • Global molecular clock (Zuckerkandl & Pauling, 1962) • Local molecular clocks (Hasegawa, Kishino & Yano 1989; Kishino & Hasegawa 1990; Yoder & Yang 2000; Yang & Yoder 2003, Drummond and Suchard 2010) • Punctuated rate change model (Huelsenbeck, Larget and Swo ord 2000) • Log-normally distributed autocorrelated rates (Thorne, Kishino & Painter 1998; Kishino, Thorne & Bruno 2001; Thorne & Kishino 2002) • Uncorrelated/independent rates models (Drummond et al. 2006; Rannala & Yang 2007; Lepage et al. 2007) • Mixture models on branch rates (Heath, Holder, Huelsenbeck 2012) Models of Lineage-specific Rate Variation
G M C The substitution rate is constant over time All lineages share the same rate branch length = substitution rate low high Models of Lineage-specific Rate Variation (Zuckerkandl & Pauling, 1962)
R -C M To accommodate variation in substitution rates ‘relaxed-clock’ models estimate lineage-specific substitution rates • Local molecular clocks • Punctuated rate change model • Log-normally distributed autocorrelated rates • Uncorrelated/independent rates models • Mixture models on branch rates
L M C Rate shifts occur infrequently over the tree Closely related lineages have equivalent rates (clustered by sub-clades) low high branch length = substitution rate Models of Lineage-specific Rate Variation (Yang & Yoder 2003, Drummond and Suchard 2010)
L M C Most methods for estimating local clocks required specifying the number and locations of rate changes a priori Drummond and Suchard (2010) introduced a Bayesian method that samples over a broad range of possible random local clocks low high branch length = substitution rate Models of Lineage-specific Rate Variation (Yang & Yoder 2003, Drummond and Suchard 2010)
A R Substitution rates evolve gradually over time – closely related lineages have similar rates The rate at a node is drawn from a lognormal distribution with a mean equal to the parent rate (geometric brownian motion) low high branch length = substitution rate Models of Lineage-specific Rate Variation (Thorne, Kishino & Painter 1998; Kishino, Thorne & Bruno 2001)
P R C Rate changes occur along lineages according to a point process At rate-change events, the new rate is a product of the parent’s rate and a -distributed multiplier low high branch length = substitution rate Models of Lineage-specific Rate Variation (Huelsenbeck, Larget and Swo ord 2000)
I /U R Lineage-specific rates are uncorrelated when the rate assigned to each branch is independently drawn from an underlying distribution low high branch length = substitution rate Models of Lineage-specific Rate Variation (Drummond et al. 2006; Rannala & Yang 2007; Lepage et al. 2007)
I /U R Lineage-specific rates are uncorrelated when the rate assigned to each branch is independently drawn from an underlying distribution Models of Lineage-specific Rate Variation (Drummond et al. 2006; Rannala & Yang 2007; Lepage et al. 2007)
I M M Dirichlet process prior: Branches are partitioned into distinct rate categories The number of rate categories and assignment of branches to categories are random variables under the model branch length = substitution rate c5 c4 c3 c2 substitution rate classes c1 Models of Lineage-specific Rate Variation (Heath, Holder, Huelsenbeck. 2012 MBE)
M R V These are only a subset of the available models for branch-rate variation • Global molecular clock • Local molecular clocks • Punctuated rate change model • Log-normally distributed autocorrelated rates • Uncorrelated/independent rates models • Dirchlet process prior Models of Lineage-specific Rate Variation
M R V Are our models appropriate across all data sets? cave bear American black bear sloth bear Asian black bear brown bear polar bear American giant short-faced bear giant panda sun bear harbor seal spectacled bear 4.08 5.39 5.66 12.86 2.75 5.05 19.09 35.7 0.88 4.58 [3.11–5.27] [4.26–7.34] [9.77–16.58] [3.9–6.48] [0.66–1.17] [4.2–6.86] [2.1–3.57] [14.38–24.79] [3.51–5.89] 14.32 [9.77–16.58] 95% CI mean age (Ma) t2 t3 t4 t6 t7 t5 t8 t9 t10 tx node MP•MLu•MLp•Bayesian 100•100•100•1.00 100•100•100•1.00 85•93•93•1.00 76•94•97•1.00 99•97•94•1.00 100•100•100•1.00 100•100•100•1.00 100•100•100•1.00 t1 Eocene Oligocene Miocene Plio Plei Hol 34 5.3 1.823.8 0.01 Epochs Ma Global expansion of C4 biomass Major temperature drop and increasing seasonality Faunal turnover Krause et al., 2008. Mitochondrial genomes reveal an explosive radiation of extinct and extant bears near the Miocene-Pliocene boundary. BMC Evol. Biol. 8. Taxa 1 5 10 50 100 500 1000 5000 10000 20000 0100200300 MYA Ophidiiformes Percomorpha Beryciformes Lampriformes Zeiforms Polymixiiformes Percopsif. + Gadiif. Aulopiformes Myctophiformes Argentiniformes Stomiiformes Osmeriformes Galaxiiformes Salmoniformes Esociformes Characiformes Siluriformes Gymnotiformes Cypriniformes Gonorynchiformes Denticipidae Clupeomorpha Osteoglossomorpha Elopomorpha Holostei Chondrostei Polypteriformes Clade r ε ΔAIC 1. 0.041 0.0017 25.3 2. 0.081 * 25.5 3. 0.067 0.37 45.1 4. 0 * 3.1 Bg. 0.011 0.0011 OstariophysiAcanthomorpha Teleostei Santini et al., 2009. Did genome duplication drive the origin of teleosts? A comparative study of diversification in ray-finned fishes. BMC Evol. Biol. 9.
M R V These are only a subset of the available models for branch-rate variation • Global molecular clock • Local molecular clocks • Punctuated rate change model • Log-normally distributed autocorrelated rates • Uncorrelated/independent rates models • Dirchlet process prior Considering model selection, uncertainty, & plausibility is very important for Bayesian divergence time analysis Models of Lineage-specific Rate Variation
B D T E Estimating divergence times relies on 2 main elements: • Branch-specific rates: f (R | R) • Node ages: f (A | A) DNA Data Rate Matrix Site Rates Branch RatesTree
P T N A Relaxed clock Bayesian analyses require a prior distribution on time trees Di erent node-age priors make di erent assumptions about the timing of divergence events Tree Priors
S B P Node-age priors based on stochastic models of lineage diversification Yule process: assumes a constant rate of speciation, across lineages A pure birth process—every node leaves extant descendants (no extinction) Tree Priors
S B P Node-age priors based on stochastic models of lineage diversification Constant-rate birth-death process: at any point in time a lineage can speciate at rate or go extinct with a rate of Tree Priors
S B P Node-age priors based on stochastic models of lineage diversification Constant-rate birth-death process: at any point in time a lineage can speciate at rate or go extinct with a rate of Tree Priors
S B P Di erent values of and lead to di erent trees Bayesian inference under these models can be very sensitive to the values of these parameters Using hyperpriors on and (or d and r) accounts for uncertainty in these hyperparameters Tree Priors
S B P Node-age priors based on stochastic models of lineage diversification Birth-death-sampling process: an extension of the constant-rate birth-death model that accounts for random sampling of tips Conditions on a probability of sampling a tip, Tree Priors
P N T Sequence data are only informative on relative rates & times Node-time priors cannot give precise estimates of absolute node ages We need external information (like fossils) to provide absolute time scale Node Age Priors
C D T Fossils (or other data) are necessary to estimate absolute node ages There is no information in the sequence data for absolute time Uncertainty in the placement of fossils A B C 20% 10% 10% 10% 200 My 400 My
C D Bayesian inference is well suited to accommodating uncertainty in the age of the calibration node Divergence times are calibrated by placing parametric densities on internal nodes o set by age estimates from the fossil record A B C 200 My Density Age
A F C Misplaced fossils can a ect node age estimates throughout the tree – if the fossil is older than its presumed MRCA Calibrating the Tree (figure from Benton & Donoghue Mol. Biol. Evol. 2007)
F C Age estimates from fossils can provide minimum time constraints for internal nodes Reliable maximum bounds are typically unavailable Minimum age Time (My) Calibrating Divergence Times
P D C N Common practice in Bayesian divergence-time estimation: Parametric distributions are typically o -set by the age of the oldest fossil assigned to a clade These prior densities do not (necessarily) require specification of maximum bounds Uniform (min, max) Exponential (λ) Gamma (α, β) Log Normal (µ, σ2) Time (My)Minimum age Calibrating Divergence Times
P D C N Describe the waiting time between the divergence event and the age of the oldest fossil Minimum age Time (My) Calibrating Divergence Times
P D C N Overly informative priors can bias node age estimates to be too young Minimum age Exponential (λ) Time (My) Calibrating Divergence Times
P D C N Uncertainty in the age of the MRCA of the clade relative to the age of the fossil may be better captured by vague prior densities Minimum age Exponential (λ) Time (My) Calibrating Divergence Times
P D C N Common practice in Bayesian divergence-time estimation: Estimates of absolute node ages are driven primarily by the calibration density Specifying appropriate densities is a challenge for most molecular biologists Uniform (min, max) Exponential (λ) Gamma (α, β) Log Normal (µ, σ2 ) Time (My)Minimum age Calibration Density Approach
I F C We would prefer to eliminate the need for ad hoc calibration prior densities Calibration densities do not account for diversification of fossils Domestic dog Spotted seal Giant panda Spectacled bear Sun bear Am. black bear Asian black bear Brown bear Polar bear Sloth bear Zaragocyon daamsi Ballusia elmensis Ursavus brevihinus Ailurarctos lufengensis Ursavus primaevus Agriarctos spp. Kretzoiarctos beatrix Indarctos vireti Indarctos arctoides Indarctos punjabiensis Giant short-faced bear Cave bear Fossil and Extant Bears (Krause et al. BMC Evol. Biol. 2008; Abella et al. PLoS ONE 2012)
I F C We want to use all of the available fossils Example: Bears 12 fossils are reduced to 4 calibration ages with calibration density methods Domestic dog Spotted seal Giant panda Spectacled bear Sun bear Am. black bear Asian black bear Brown bear Polar bear Sloth bear Zaragocyon daamsi Ballusia elmensis Ursavus brevihinus Ailurarctos lufengensis Ursavus primaevus Agriarctos spp. Kretzoiarctos beatrix Indarctos vireti Indarctos arctoides Indarctos punjabiensis Giant short-faced bear Cave bear Fossil and Extant Bears (Krause et al. BMC Evol. Biol. 2008; Abella et al. PLoS ONE 2012)
I F C We want to use all of the available fossils Example: Bears 12 fossils are reduced to 4 calibration ages with calibration density methods Domestic dog Spotted seal Giant panda Spectacled bear Sun bear Am. black bear Asian black bear Brown bear Polar bear Sloth bear Zaragocyon daamsi Ballusia elmensis Ursavus brevihinus Ailurarctos lufengensis Ursavus primaevus Agriarctos spp. Kretzoiarctos beatrix Indarctos vireti Indarctos arctoides Indarctos punjabiensis Giant short-faced bear Cave bear Fossil and Extant Bears (Krause et al. BMC Evol. Biol. 2008; Abella et al. PLoS ONE 2012)
I F C Because fossils are part of the diversification process, we can combine fossil calibration with birth-death models Domestic dog Spotted seal Giant panda Spectacled bear Sun bear Am. black bear Asian black bear Brown bear Polar bear Sloth bear Zaragocyon daamsi Ballusia elmensis Ursavus brevihinus Ailurarctos lufengensis Ursavus primaevus Agriarctos spp. Kretzoiarctos beatrix Indarctos vireti Indarctos arctoides Indarctos punjabiensis Giant short-faced bear Cave bear Fossil and Extant Bears (Krause et al. BMC Evol. Biol. 2008; Abella et al. PLoS ONE 2012)
I F C This relies on a branching model that accounts for speciation, extinction, and rates of fossilization, preservation, and recovery Domestic dog Spotted seal Giant panda Spectacled bear Sun bear Am. black bear Asian black bear Brown bear Polar bear Sloth bear Zaragocyon daamsi Ballusia elmensis Ursavus brevihinus Ailurarctos lufengensis Ursavus primaevus Agriarctos spp. Kretzoiarctos beatrix Indarctos vireti Indarctos arctoides Indarctos punjabiensis Giant short-faced bear Cave bear Fossil and Extant Bears (Krause et al. BMC Evol. Biol. 2008; Abella et al. PLoS ONE 2012)
P N “Except during the interlude of the [Modern] Synthesis, there has been limited communication historically among the disciplines of evolutionary biology, particularly between students of evolutionary history (paleontologists and systematists) and those of molecular, population, and organismal biology. There has been increasing realization that barriers between these subfields must be overcome if a complete theory of evolution and systematics is to be forged.”. Reaka-Kudla & Colwell: in Dudley (ed.), The Unity of Evolutionary Biology: Proceedings of the Fourth International Congress of Systematic & Evolutionary Biology, Discorides Press, Portland, OR, p. 16. (1994) Two Separate Fields, Same Goals
P N Two Separate Fields, Same Goals
C F E D Statistical methods provide a way to integrate paleontological & neontological data Two Separate Fields, Same Goals
C F E D Combine models for sequence evolution, morphological change, & fossil recovery to jointly estimate the tree topology, divergence times, & lineage diversification rates DNA Data Substitution Model Site Rate Model Branch Rate Model Morphological Data Substitution Model Site Rate Model Branch Rate Model Time Tree Model Fossil Occurrence Time Data
C F E D Until recently, analyses combining fossil & extant taxa used simple or inappropriate models to describe the tree and fossil ages DNA Data Substitution Model Site Rate Model Branch Rate Model Morphological Data Substitution Model Site Rate Model Branch Rate Model Time Tree Model Fossil Occurrence Time Data
M T O T Stadler (2010) introduced a generating model for a serially sampled time tree — this is the fossilized birth-death process. (Stadler. Journal of Theoretical Biology 2010)
P FBD This graph shows the conditional dependence structure of the FBD model, which is a generating process for a sampled, dated time tree and fossil occurrences T F µ time tree ⇢x0origin time sampling probability fossil occurrence times fossil recovery ratespeciation rate extinction rate
P FBD We re-parameterize the model so that we are directly estimating the diversification rate, turnonver and fossil sampling proportion T F µ d r s time tree ⇢x0origin time sampling probability fossil occurrence times fossil recovery rate speciation rate extinction rate diversification rate turnover fossil sampling proportion = d 1 r = rd 1 r = s 1 s rd 1 r
T F B -D P (FBD) Improving statistical inference of absolute node ages Eliminates the need to specify arbitrary calibration densities Useful for ‘total-evidence’ analyses Better capture our statistical uncertainty in species divergence dates All reliable fossils associated with a clade are used 150 100 50 0 Time (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) Recovered fossil specimens provide historical observations of the diversification process that generated the tree of extant species 150 100 50 0 Time Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) The probability of the tree and fossil observations under a birth-death model with rate parameters: = speciation = extinction = fossilization/recovery 150 100 50 0 Time Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) The occurrence time of the fossil indicates an observation of the birth-death process before the present 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) The fossil must attach to the tree at some time and to some branch: F 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) If it is the descendant of an unobserved lineage, then there is a speciation event at time F 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) MCMC is used to propose new topological placements for the fossil 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) Using rjMCMC, we can propose F = , which means that the fossil is a “sampled ancestor” 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) Using rjMCMC, we can propose F = , which means that the fossil is a “sampled ancestor” 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) The probability of any realization of the diversification process is conditional on: , , and 0250 50100150200 Time (My) N Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) Using MCMC, we can sample the age of the MRCA • and the placement and time of the fossil lineage 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) Under the FBD, multiple fossils are considered, even if they are descended from the same MRCA node in the extant tree 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) Given F and , the new fossil can attach to the tree via speciation along either branch in the extant tree at time F 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) Given F and , the new fossil can attach to the tree via speciation along either branch in the extant tree at time F 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) Given F and , the new fossil can attach to the tree via speciation along either branch in the extant tree at time F 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) Or the unobserved branch leading to the other fossil 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) If F = , then the new fossil lies directly on a branch in the extant tree 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) If F = , then the new fossil lies directly on a branch in the extant tree 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) Or it is an ancestor of the other sampled fossil 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) The probability of this realization of the diversification process is conditional on: , , and N 0250 50100150200 Time (My) N Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
T F B -D P (FBD) Using MCMC, we can sample the age of the MRCA • and the placement and time of all fossil lineages 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
S A Sampled lineages with sampled descendants 150 100 50 0 Time There is a non-zero probability of sampling ancestor-descendant relationships from the fossil record
S A Complete FBD Tree Reconstructed FBD Tree Because fossils & living taxa are assumed to come from a single diversification process, there is a non-zero probability of sampled ancestors
S A Complete FBD Tree No Sampled Ancestor Tree If all fossils are forced to be on separate lineages, this induces additional speciation events and will, in turn, influence rate & node-age estimates.
C F E D DNA Data Substitution Model Site Rate Model Branch Rate Model Morphological Data Substitution Model Site Rate Model Branch Rate Model Time Tree Model Fossil Occurrence Time Data 0001010111001 0101111110114 0011100110005 1??010000???? ???110211???? CATTATTCGCATTA CATCATTCGAATCA AATTATCCGAGTAA ?????????????? ?????????????? Geological Time (turtle tree image by M. Landis)
M M C C 0001010111001 0101111110114 0011100110005 1??010000???? ???110211???? Geological Time (turtle tree image by M. Landis)
M M C C The Lewis Mk model Assumes a character can take k states Transition rates between states are equal Q = 2 6 6 6 4 1 k 1 ... 1 ... 1 k ... 1 ... ... ... ... 1 1 ... 1 k 3 7 7 7 5 0 0 1 2 2 1 1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 T1 T2 T3 T4 T5 T6 T7 0 0 0 0 0 0 0 (Lewis. Systematic Biology 2001)
M M C C The Lewis Mkv model Accounts for the ascertainment bias in morphological datasets by conditioning the likelihood on the fact that invariant characters are not sampled Q = 2 6 6 6 4 1 k 1 ... 1 ... 1 k ... 1 ... ... ... ... 1 1 ... 1 k 3 7 7 7 5 2 2 2 2 2 2 2 1 1 1 1 1 1 1 T1 T2 T3 T4 T5 T6 T7 0 0 0 0 0 0 0 0 0 1 2 2 1 1 (Lewis. Systematic Biology 2001)
“T -E ” A Integrating models of molecular and morphological evolution with improved tree priors enables joint inference of the tree topology (extant & extinct) and divergence times DNA Data Substitution Model Site Rate Model Branch Rate Model Morphological Data Substitution Model Site Rate Model Branch Rate Model Time Tree Model Fossil Occurrence Time Data

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Bayesian Divergence Time Estimation – Workshop Lecture

  • 1.
    B D T E TracyHeath Ecology, Evolution, & Organismal Biology Iowa State University @trayc7 http://phyloworks.org 2016 Workshop on Molecular Evolution Woods Hole, MA USA
  • 2.
    O Overview of divergencetime estimation • Relaxed clock models – accounting for variation in substitution rates among lineages • Tree models – lineage diversification and sampling break BEAST v2 Tutorial — Divergence-time estimation under birth-death processes • Dating Bear Divergence Times with the Fossilized Birth-Death Process dinner
  • 3.
    A T -SE Phylogenies with branch lengths proportional to time provide more information about evolutionary history than unrooted trees with branch lengths in units of substitutions/site. 0.03 substitutions/site Time (My) 20 0 Oligocene Miocene Plio Pleis (silhouette images from http://phylopic.org)
  • 4.
    A T -SE Phylogenetic divergence-time estimation • What was the spacial and climatic environment of ancient angiosperms? • How has mammalian body-size changed over time? • How has the infection rate of HCV in Egypt changed over time? • Is diversification in Caribbean anoles correlated with ecological opportunity? • How has the rate of molecular evolution changed across the Tree of Life? Epidemiology (Antonelli & Sanmartin. Syst. Biol. 2011) (Lartillot & Delsuc. Evolution 2012) (Stadler et al. PNAS 2013) (Mahler, Revell, Glor, & Losos. Evolution 2010) (Nabholz, Glemin, Galtier. MBE 2008) Historical biogeography Molecular evolution Trait evolution Diversification Anolis fowleri (image by L. Mahler) Understanding Evolutionary Processes
  • 5.
    D T E Goal:Estimate the ages of interior nodes to understand the timing and rates of evolutionary processes Model how rates are distributed across the tree Describe the distribution of speciation events over time External calibration information for estimates of absolute node times calibrated node 100 0.020.040.060.080.0 Equus Rhinoceros Bos Hippopotamus Balaenoptera Physeter Ursus Canis Felis Homo Pan Gorilla Pongo Macaca Callithrix Loris Galago Daubentonia Varecia Eulemur Lemur Hapalemur Propithecus Lepilemur Mirza M. murinus M. griseorufus M. myoxinus M. berthae M. rufus1 M. tavaratra M. rufus2 M. sambiranensis M. ravelobensis Cheirogaleus Simiiformes Microcebus Cretaceous Paleogene Neogene Q Time (Millions of years)
  • 6.
    T G MC Assume that the rate of evolutionary change is constant over time (branch lengths equal percent sequence divergence) 10% 400 My 200 My A B C 20% 10% 10% (Based on slides by Je Thorne; http://statgen.ncsu.edu/thorne/compmolevo.html)
  • 7.
    T G MC We can date the tree if we know the rate of change is 1% divergence per 10 My A B C 20% 10% 10% 10% 200 My 400 My 200 My (Based on slides by Je Thorne; http://statgen.ncsu.edu/thorne/compmolevo.html)
  • 8.
    T G MC If we found a fossil of the MRCA of B and C, we can use it to calculate the rate of change & date the root of the tree A B C 20% 10% 10% 10% 200 My 400 My (Based on slides by Je Thorne; http://statgen.ncsu.edu/thorne/compmolevo.html)
  • 9.
    R G MC Rates of evolution vary across lineages and over time Mutation rate: Variation in • metabolic rate • generation time • DNA repair Fixation rate: Variation in • strength and targets of selection • population sizes 10% 400 My 200 My A B C 20% 10% 10%
  • 10.
    U A Sequence dataprovide information about branch lengths In units of the expected # of substitutions per site branch length = rate ⇥ time 0.2 expected substitutions/site PhylogeneticRelationshipsSequence Data
  • 11.
    R T The sequencedata provide information about branch length for any possible rate, there’s a time that fits the branch length perfectly 0 1 2 3 4 5 0 1 2 3 4 5 BranchRate Branch Time time = 0.8 rate = 0.625 branch length = 0.5 Methods for dating species divergences estimate the substitution rate and time separately (based on Thorne & Kishino, 2005)
  • 12.
    B D TE length = rate length = time R = (r1,r2,r3,...,r2N 2) A = (a1,a2,a3,...,aN 1) N = number of tips
  • 13.
    B D TE length = rate length = time R = (r1,r2,r3,...,r2N 2) A = (a1,a2,a3,...,aN 1) N = number of tips
  • 14.
    B D TE Posterior probability f (R,A, R, A, s | D, ) R Vector of rates on branches A Vector of internal node ages R, A, s Model parameters D Sequence data Tree topology
  • 15.
    B D TE f(R,A, R, A, s | D) = f (D | R,A, s) f(R | R) f(A | A) f( s) f(D) f(D | R,A, R, A, s) Likelihood f(R | R) Prior on rates f(A | A) Prior on node ages f( s) Prior on substitution parameters f(D) Marginal probability of the data
  • 16.
    B D TE Estimating divergence times relies on 2 main elements: • Branch-specific rates: f (R | R) • Node ages & Topology: f (A | A) DNA Data Rate Matrix Site Rates Branch RatesTree
  • 17.
    B D TE Estimating divergence times relies on 2 main elements: • Branch-specific rates: f (R | R) • Node ages & Topology: f (A | A) DNA Data Rate Matrix Site Rates Branch RatesTree
  • 18.
    M R V Somemodels describing lineage-specific substitution rate variation: • Global molecular clock (Zuckerkandl & Pauling, 1962) • Local molecular clocks (Hasegawa, Kishino & Yano 1989; Kishino & Hasegawa 1990; Yoder & Yang 2000; Yang & Yoder 2003, Drummond and Suchard 2010) • Punctuated rate change model (Huelsenbeck, Larget and Swo ord 2000) • Log-normally distributed autocorrelated rates (Thorne, Kishino & Painter 1998; Kishino, Thorne & Bruno 2001; Thorne & Kishino 2002) • Uncorrelated/independent rates models (Drummond et al. 2006; Rannala & Yang 2007; Lepage et al. 2007) • Mixture models on branch rates (Heath, Holder, Huelsenbeck 2012) Models of Lineage-specific Rate Variation
  • 19.
    G M C Thesubstitution rate is constant over time All lineages share the same rate branch length = substitution rate low high Models of Lineage-specific Rate Variation (Zuckerkandl & Pauling, 1962)
  • 20.
    R -C M Toaccommodate variation in substitution rates ‘relaxed-clock’ models estimate lineage-specific substitution rates • Local molecular clocks • Punctuated rate change model • Log-normally distributed autocorrelated rates • Uncorrelated/independent rates models • Mixture models on branch rates
  • 21.
    L M C Rateshifts occur infrequently over the tree Closely related lineages have equivalent rates (clustered by sub-clades) low high branch length = substitution rate Models of Lineage-specific Rate Variation (Yang & Yoder 2003, Drummond and Suchard 2010)
  • 22.
    L M C Mostmethods for estimating local clocks required specifying the number and locations of rate changes a priori Drummond and Suchard (2010) introduced a Bayesian method that samples over a broad range of possible random local clocks low high branch length = substitution rate Models of Lineage-specific Rate Variation (Yang & Yoder 2003, Drummond and Suchard 2010)
  • 23.
    A R Substitution ratesevolve gradually over time – closely related lineages have similar rates The rate at a node is drawn from a lognormal distribution with a mean equal to the parent rate (geometric brownian motion) low high branch length = substitution rate Models of Lineage-specific Rate Variation (Thorne, Kishino & Painter 1998; Kishino, Thorne & Bruno 2001)
  • 24.
    P R C Ratechanges occur along lineages according to a point process At rate-change events, the new rate is a product of the parent’s rate and a -distributed multiplier low high branch length = substitution rate Models of Lineage-specific Rate Variation (Huelsenbeck, Larget and Swo ord 2000)
  • 25.
    I /U R Lineage-specificrates are uncorrelated when the rate assigned to each branch is independently drawn from an underlying distribution low high branch length = substitution rate Models of Lineage-specific Rate Variation (Drummond et al. 2006; Rannala & Yang 2007; Lepage et al. 2007)
  • 26.
    I /U R Lineage-specificrates are uncorrelated when the rate assigned to each branch is independently drawn from an underlying distribution Models of Lineage-specific Rate Variation (Drummond et al. 2006; Rannala & Yang 2007; Lepage et al. 2007)
  • 27.
    I M M Dirichletprocess prior: Branches are partitioned into distinct rate categories The number of rate categories and assignment of branches to categories are random variables under the model branch length = substitution rate c5 c4 c3 c2 substitution rate classes c1 Models of Lineage-specific Rate Variation (Heath, Holder, Huelsenbeck. 2012 MBE)
  • 28.
    M R V Theseare only a subset of the available models for branch-rate variation • Global molecular clock • Local molecular clocks • Punctuated rate change model • Log-normally distributed autocorrelated rates • Uncorrelated/independent rates models • Dirchlet process prior Models of Lineage-specific Rate Variation
  • 29.
    M R V Areour models appropriate across all data sets? cave bear American black bear sloth bear Asian black bear brown bear polar bear American giant short-faced bear giant panda sun bear harbor seal spectacled bear 4.08 5.39 5.66 12.86 2.75 5.05 19.09 35.7 0.88 4.58 [3.11–5.27] [4.26–7.34] [9.77–16.58] [3.9–6.48] [0.66–1.17] [4.2–6.86] [2.1–3.57] [14.38–24.79] [3.51–5.89] 14.32 [9.77–16.58] 95% CI mean age (Ma) t2 t3 t4 t6 t7 t5 t8 t9 t10 tx node MP•MLu•MLp•Bayesian 100•100•100•1.00 100•100•100•1.00 85•93•93•1.00 76•94•97•1.00 99•97•94•1.00 100•100•100•1.00 100•100•100•1.00 100•100•100•1.00 t1 Eocene Oligocene Miocene Plio Plei Hol 34 5.3 1.823.8 0.01 Epochs Ma Global expansion of C4 biomass Major temperature drop and increasing seasonality Faunal turnover Krause et al., 2008. Mitochondrial genomes reveal an explosive radiation of extinct and extant bears near the Miocene-Pliocene boundary. BMC Evol. Biol. 8. Taxa 1 5 10 50 100 500 1000 5000 10000 20000 0100200300 MYA Ophidiiformes Percomorpha Beryciformes Lampriformes Zeiforms Polymixiiformes Percopsif. + Gadiif. Aulopiformes Myctophiformes Argentiniformes Stomiiformes Osmeriformes Galaxiiformes Salmoniformes Esociformes Characiformes Siluriformes Gymnotiformes Cypriniformes Gonorynchiformes Denticipidae Clupeomorpha Osteoglossomorpha Elopomorpha Holostei Chondrostei Polypteriformes Clade r ε ΔAIC 1. 0.041 0.0017 25.3 2. 0.081 * 25.5 3. 0.067 0.37 45.1 4. 0 * 3.1 Bg. 0.011 0.0011 OstariophysiAcanthomorpha Teleostei Santini et al., 2009. Did genome duplication drive the origin of teleosts? A comparative study of diversification in ray-finned fishes. BMC Evol. Biol. 9.
  • 30.
    M R V Theseare only a subset of the available models for branch-rate variation • Global molecular clock • Local molecular clocks • Punctuated rate change model • Log-normally distributed autocorrelated rates • Uncorrelated/independent rates models • Dirchlet process prior Considering model selection, uncertainty, & plausibility is very important for Bayesian divergence time analysis Models of Lineage-specific Rate Variation
  • 31.
    B D TE Estimating divergence times relies on 2 main elements: • Branch-specific rates: f (R | R) • Node ages: f (A | A) DNA Data Rate Matrix Site Rates Branch RatesTree
  • 32.
    P T NA Relaxed clock Bayesian analyses require a prior distribution on time trees Di erent node-age priors make di erent assumptions about the timing of divergence events Tree Priors
  • 33.
    S B P Node-agepriors based on stochastic models of lineage diversification Yule process: assumes a constant rate of speciation, across lineages A pure birth process—every node leaves extant descendants (no extinction) Tree Priors
  • 34.
    S B P Node-agepriors based on stochastic models of lineage diversification Constant-rate birth-death process: at any point in time a lineage can speciate at rate or go extinct with a rate of Tree Priors
  • 35.
    S B P Node-agepriors based on stochastic models of lineage diversification Constant-rate birth-death process: at any point in time a lineage can speciate at rate or go extinct with a rate of Tree Priors
  • 36.
    S B P Dierent values of and lead to di erent trees Bayesian inference under these models can be very sensitive to the values of these parameters Using hyperpriors on and (or d and r) accounts for uncertainty in these hyperparameters Tree Priors
  • 37.
    S B P Node-agepriors based on stochastic models of lineage diversification Birth-death-sampling process: an extension of the constant-rate birth-death model that accounts for random sampling of tips Conditions on a probability of sampling a tip, Tree Priors
  • 38.
    P N T Sequencedata are only informative on relative rates & times Node-time priors cannot give precise estimates of absolute node ages We need external information (like fossils) to provide absolute time scale Node Age Priors
  • 39.
    C D T Fossils(or other data) are necessary to estimate absolute node ages There is no information in the sequence data for absolute time Uncertainty in the placement of fossils A B C 20% 10% 10% 10% 200 My 400 My
  • 40.
    C D Bayesian inferenceis well suited to accommodating uncertainty in the age of the calibration node Divergence times are calibrated by placing parametric densities on internal nodes o set by age estimates from the fossil record A B C 200 My Density Age
  • 41.
    A F C Misplacedfossils can a ect node age estimates throughout the tree – if the fossil is older than its presumed MRCA Calibrating the Tree (figure from Benton & Donoghue Mol. Biol. Evol. 2007)
  • 42.
    F C Age estimatesfrom fossils can provide minimum time constraints for internal nodes Reliable maximum bounds are typically unavailable Minimum age Time (My) Calibrating Divergence Times
  • 43.
    P D CN Common practice in Bayesian divergence-time estimation: Parametric distributions are typically o -set by the age of the oldest fossil assigned to a clade These prior densities do not (necessarily) require specification of maximum bounds Uniform (min, max) Exponential (λ) Gamma (α, β) Log Normal (µ, σ2) Time (My)Minimum age Calibrating Divergence Times
  • 44.
    P D CN Describe the waiting time between the divergence event and the age of the oldest fossil Minimum age Time (My) Calibrating Divergence Times
  • 45.
    P D CN Overly informative priors can bias node age estimates to be too young Minimum age Exponential (λ) Time (My) Calibrating Divergence Times
  • 46.
    P D CN Uncertainty in the age of the MRCA of the clade relative to the age of the fossil may be better captured by vague prior densities Minimum age Exponential (λ) Time (My) Calibrating Divergence Times
  • 47.
    P D CN Common practice in Bayesian divergence-time estimation: Estimates of absolute node ages are driven primarily by the calibration density Specifying appropriate densities is a challenge for most molecular biologists Uniform (min, max) Exponential (λ) Gamma (α, β) Log Normal (µ, σ2 ) Time (My)Minimum age Calibration Density Approach
  • 48.
    I F C Wewould prefer to eliminate the need for ad hoc calibration prior densities Calibration densities do not account for diversification of fossils Domestic dog Spotted seal Giant panda Spectacled bear Sun bear Am. black bear Asian black bear Brown bear Polar bear Sloth bear Zaragocyon daamsi Ballusia elmensis Ursavus brevihinus Ailurarctos lufengensis Ursavus primaevus Agriarctos spp. Kretzoiarctos beatrix Indarctos vireti Indarctos arctoides Indarctos punjabiensis Giant short-faced bear Cave bear Fossil and Extant Bears (Krause et al. BMC Evol. Biol. 2008; Abella et al. PLoS ONE 2012)
  • 49.
    I F C Wewant to use all of the available fossils Example: Bears 12 fossils are reduced to 4 calibration ages with calibration density methods Domestic dog Spotted seal Giant panda Spectacled bear Sun bear Am. black bear Asian black bear Brown bear Polar bear Sloth bear Zaragocyon daamsi Ballusia elmensis Ursavus brevihinus Ailurarctos lufengensis Ursavus primaevus Agriarctos spp. Kretzoiarctos beatrix Indarctos vireti Indarctos arctoides Indarctos punjabiensis Giant short-faced bear Cave bear Fossil and Extant Bears (Krause et al. BMC Evol. Biol. 2008; Abella et al. PLoS ONE 2012)
  • 50.
    I F C Wewant to use all of the available fossils Example: Bears 12 fossils are reduced to 4 calibration ages with calibration density methods Domestic dog Spotted seal Giant panda Spectacled bear Sun bear Am. black bear Asian black bear Brown bear Polar bear Sloth bear Zaragocyon daamsi Ballusia elmensis Ursavus brevihinus Ailurarctos lufengensis Ursavus primaevus Agriarctos spp. Kretzoiarctos beatrix Indarctos vireti Indarctos arctoides Indarctos punjabiensis Giant short-faced bear Cave bear Fossil and Extant Bears (Krause et al. BMC Evol. Biol. 2008; Abella et al. PLoS ONE 2012)
  • 51.
    I F C Becausefossils are part of the diversification process, we can combine fossil calibration with birth-death models Domestic dog Spotted seal Giant panda Spectacled bear Sun bear Am. black bear Asian black bear Brown bear Polar bear Sloth bear Zaragocyon daamsi Ballusia elmensis Ursavus brevihinus Ailurarctos lufengensis Ursavus primaevus Agriarctos spp. Kretzoiarctos beatrix Indarctos vireti Indarctos arctoides Indarctos punjabiensis Giant short-faced bear Cave bear Fossil and Extant Bears (Krause et al. BMC Evol. Biol. 2008; Abella et al. PLoS ONE 2012)
  • 52.
    I F C Thisrelies on a branching model that accounts for speciation, extinction, and rates of fossilization, preservation, and recovery Domestic dog Spotted seal Giant panda Spectacled bear Sun bear Am. black bear Asian black bear Brown bear Polar bear Sloth bear Zaragocyon daamsi Ballusia elmensis Ursavus brevihinus Ailurarctos lufengensis Ursavus primaevus Agriarctos spp. Kretzoiarctos beatrix Indarctos vireti Indarctos arctoides Indarctos punjabiensis Giant short-faced bear Cave bear Fossil and Extant Bears (Krause et al. BMC Evol. Biol. 2008; Abella et al. PLoS ONE 2012)
  • 53.
    P N “Except duringthe interlude of the [Modern] Synthesis, there has been limited communication historically among the disciplines of evolutionary biology, particularly between students of evolutionary history (paleontologists and systematists) and those of molecular, population, and organismal biology. There has been increasing realization that barriers between these subfields must be overcome if a complete theory of evolution and systematics is to be forged.”. Reaka-Kudla & Colwell: in Dudley (ed.), The Unity of Evolutionary Biology: Proceedings of the Fourth International Congress of Systematic & Evolutionary Biology, Discorides Press, Portland, OR, p. 16. (1994) Two Separate Fields, Same Goals
  • 54.
    P N Two SeparateFields, Same Goals
  • 55.
    C F ED Statistical methods provide a way to integrate paleontological & neontological data Two Separate Fields, Same Goals
  • 56.
    C F ED Combine models for sequence evolution, morphological change, & fossil recovery to jointly estimate the tree topology, divergence times, & lineage diversification rates DNA Data Substitution Model Site Rate Model Branch Rate Model Morphological Data Substitution Model Site Rate Model Branch Rate Model Time Tree Model Fossil Occurrence Time Data
  • 57.
    C F ED Until recently, analyses combining fossil & extant taxa used simple or inappropriate models to describe the tree and fossil ages DNA Data Substitution Model Site Rate Model Branch Rate Model Morphological Data Substitution Model Site Rate Model Branch Rate Model Time Tree Model Fossil Occurrence Time Data
  • 58.
    M T OT Stadler (2010) introduced a generating model for a serially sampled time tree — this is the fossilized birth-death process. (Stadler. Journal of Theoretical Biology 2010)
  • 59.
    P FBD This graphshows the conditional dependence structure of the FBD model, which is a generating process for a sampled, dated time tree and fossil occurrences T F µ time tree ⇢x0origin time sampling probability fossil occurrence times fossil recovery ratespeciation rate extinction rate
  • 60.
    P FBD We re-parameterizethe model so that we are directly estimating the diversification rate, turnonver and fossil sampling proportion T F µ d r s time tree ⇢x0origin time sampling probability fossil occurrence times fossil recovery rate speciation rate extinction rate diversification rate turnover fossil sampling proportion = d 1 r = rd 1 r = s 1 s rd 1 r
  • 61.
    T F B-D P (FBD) Improving statistical inference of absolute node ages Eliminates the need to specify arbitrary calibration densities Useful for ‘total-evidence’ analyses Better capture our statistical uncertainty in species divergence dates All reliable fossils associated with a clade are used 150 100 50 0 Time (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 62.
    T F B-D P (FBD) Recovered fossil specimens provide historical observations of the diversification process that generated the tree of extant species 150 100 50 0 Time Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 63.
    T F B-D P (FBD) The probability of the tree and fossil observations under a birth-death model with rate parameters: = speciation = extinction = fossilization/recovery 150 100 50 0 Time Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 64.
    T F B-D P (FBD) The occurrence time of the fossil indicates an observation of the birth-death process before the present 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 65.
    T F B-D P (FBD) The fossil must attach to the tree at some time and to some branch: F 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 66.
    T F B-D P (FBD) If it is the descendant of an unobserved lineage, then there is a speciation event at time F 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 67.
    T F B-D P (FBD) MCMC is used to propose new topological placements for the fossil 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 68.
    T F B-D P (FBD) Using rjMCMC, we can propose F = , which means that the fossil is a “sampled ancestor” 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 69.
    T F B-D P (FBD) Using rjMCMC, we can propose F = , which means that the fossil is a “sampled ancestor” 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 70.
    T F B-D P (FBD) The probability of any realization of the diversification process is conditional on: , , and 0250 50100150200 Time (My) N Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 71.
    T F B-D P (FBD) Using MCMC, we can sample the age of the MRCA • and the placement and time of the fossil lineage 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 72.
    T F B-D P (FBD) Under the FBD, multiple fossils are considered, even if they are descended from the same MRCA node in the extant tree 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 73.
    T F B-D P (FBD) Given F and , the new fossil can attach to the tree via speciation along either branch in the extant tree at time F 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 74.
    T F B-D P (FBD) Given F and , the new fossil can attach to the tree via speciation along either branch in the extant tree at time F 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 75.
    T F B-D P (FBD) Given F and , the new fossil can attach to the tree via speciation along either branch in the extant tree at time F 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 76.
    T F B-D P (FBD) Or the unobserved branch leading to the other fossil 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 77.
    T F B-D P (FBD) If F = , then the new fossil lies directly on a branch in the extant tree 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 78.
    T F B-D P (FBD) If F = , then the new fossil lies directly on a branch in the extant tree 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 79.
    T F B-D P (FBD) Or it is an ancestor of the other sampled fossil 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 80.
    T F B-D P (FBD) The probability of this realization of the diversification process is conditional on: , , and N 0250 50100150200 Time (My) N Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 81.
    T F B-D P (FBD) Using MCMC, we can sample the age of the MRCA • and the placement and time of all fossil lineages 0250 50100150200 Time (My) Diversification of Fossil & Extant Lineages (Heath, Huelsenbeck, Stadler. PNAS 2014)
  • 82.
    S A Sampled lineageswith sampled descendants 150 100 50 0 Time There is a non-zero probability of sampling ancestor-descendant relationships from the fossil record
  • 83.
    S A Complete FBDTree Reconstructed FBD Tree Because fossils & living taxa are assumed to come from a single diversification process, there is a non-zero probability of sampled ancestors
  • 84.
    S A Complete FBDTree No Sampled Ancestor Tree If all fossils are forced to be on separate lineages, this induces additional speciation events and will, in turn, influence rate & node-age estimates.
  • 85.
    C F ED DNA Data Substitution Model Site Rate Model Branch Rate Model Morphological Data Substitution Model Site Rate Model Branch Rate Model Time Tree Model Fossil Occurrence Time Data 0001010111001 0101111110114 0011100110005 1??010000???? ???110211???? CATTATTCGCATTA CATCATTCGAATCA AATTATCCGAGTAA ?????????????? ?????????????? Geological Time (turtle tree image by M. Landis)
  • 86.
    M M CC 0001010111001 0101111110114 0011100110005 1??010000???? ???110211???? Geological Time (turtle tree image by M. Landis)
  • 87.
    M M CC The Lewis Mk model Assumes a character can take k states Transition rates between states are equal Q = 2 6 6 6 4 1 k 1 ... 1 ... 1 k ... 1 ... ... ... ... 1 1 ... 1 k 3 7 7 7 5 0 0 1 2 2 1 1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 T1 T2 T3 T4 T5 T6 T7 0 0 0 0 0 0 0 (Lewis. Systematic Biology 2001)
  • 88.
    M M CC The Lewis Mkv model Accounts for the ascertainment bias in morphological datasets by conditioning the likelihood on the fact that invariant characters are not sampled Q = 2 6 6 6 4 1 k 1 ... 1 ... 1 k ... 1 ... ... ... ... 1 1 ... 1 k 3 7 7 7 5 2 2 2 2 2 2 2 1 1 1 1 1 1 1 T1 T2 T3 T4 T5 T6 T7 0 0 0 0 0 0 0 0 0 1 2 2 1 1 (Lewis. Systematic Biology 2001)
  • 89.
    “T -E ”A Integrating models of molecular and morphological evolution with improved tree priors enables joint inference of the tree topology (extant & extinct) and divergence times DNA Data Substitution Model Site Rate Model Branch Rate Model Morphological Data Substitution Model Site Rate Model Branch Rate Model Time Tree Model Fossil Occurrence Time Data