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- 1. Bayesian Tip Dating Approach Chi Zhang Institute of Vertebrate Paleontology and Paleoanthropology, Chinese Academy of Sciences Center for Excellence in Life and Paleoenvironment, Chinese Academy of Sciences
- 2. Question • Extinct taxa (fossils: F1, F2) • morphological characters • ages • Extant taxa (living species: S1, S2) • morphological characters • molecular sequences • Unknown (??) • tree topology (τ) • divergence times (t) • evolutionary rates (r) ? ? ? S1 S2 F1 F2
- 3. Bayesian tip-dating approach • Data (D) • Parameters • tree topology (τ) • divergence times (t) • evolutionary rates (r) • others (θ) • Joint inference • P(τ, t, r, θ|D) = P(D|τ, t, r, θ) P(τ, t|θ)P(r|θ)P(θ) / P(D) posterior likelihood priors Taxa Morphological Molecular Age(Ma) F1 000- ????? 40-60 F2 01?0 ????? 100 S1 1010 CATAT 0 S2 1001 GATGT 0
- 4. Likelihood • Substitution models • Mk (Lewis 2001) • Evolutionary rate: r = (k − 1)λ 𝑄 = 𝜆 1 − 𝑘 1 ⋮ 1 1 1 − 𝑘 ⋮ 1 ⋯ ⋯ ⋱ ⋯ 1 1 ⋮ 1 − 𝑘 𝑝!! 𝑡 = 1 𝑘 + 𝑘 − 1 𝑘 e"#$% 𝑝!& 𝑡 = 1 𝑘 − 1 𝑘 e"#$% 𝑃 𝑡 = 𝑝!&(𝑡) = e'% (k = 3)
- 5. Likelihood • t = {t1, t2, t3, t4, t5} • r = {r1, r2, r3, r4, r5, r6} S1 S2 F1 F2 t1 t2 t3 t5 t4 1 1 x0 x2 x1 0 0 r4 r5 r3 r1 r2 r6 Pr 0011 𝜏, 𝐭, 𝐫, 𝜃 = + !! + !" + !# 𝜋!! 𝑃!!" 𝑡1 − 𝑡3, 𝑟1 ×𝑃!!!" (𝑡1 − 𝑡2, 𝑟2)𝑃!"# 𝑡2, 𝑟3 ×𝑃!"!# 𝑡2 − 𝑡4, 𝑟4 𝑃!#" 𝑡4 − 𝑡5, 𝑟5 𝑃!## 𝑡4, 𝑟6 Pr 𝐷|𝜏, 𝐭, 𝐫, 𝜃 = 6 $%# & Pr 𝐷$ 𝜏, 𝐭, 𝐫, 𝜃
- 6. Priors • P(τ, t | θ) : prior for the timetree • Fossilized Birth-Death (FBD) process • λ : speciation (birth) rate • μ : extinction (death) rate • ψ : fossil sampling rate • ρ : extant sampling probability • condition origin time (t0) or root age (t1) (Stadler 2010) S1 S2 F1 F2 t1 t2 t3 t5 t4
- 7. Priors • P(r | θ) : prior for the evolutionary rates • expected number of changes per character per myr • Strict (global) clock • ri’s are equal • Relaxed clock models • independent rates – ri’s follow independent distributions (Drummond et al 2006, Rannala and Yang 2007, Lepage et al. 2007) • auto-correlated rates – ri depends on the rate of its ancestral branch (Kishino et al 2001, Thorne and Kishino 2002) S1 S2 F1 F2 r4 r5 r3 r1 r2 r6
- 8. Software • MrBayes • http://nbisweden.github.io/MrBayes/ • BEAST 2 • http://www.beast2.org/ • RevBayes • https://revbayes.github.io/
- 9. Applications • Birds (Aves) PNAS 2021 2019
- 10. (speciation rate minus extinctioin rate) 0 0.03 0.06 Ma ψ 170 145 100 66 0 Baptornis advenus Gallus Patagopteryx Bohaiornis Yangavis Archaeopteryx Shenqiornis Longipteryx Enaliornis Eoenantiornis Pterygornis Pengornis Longirostravis Qiliania Tianyuornis Cathayornis Schizooura Dromaeosauridae Eoconfuciusornis Dunhuangia Iteravis Archaeornithura Apsaravis Protopteryx Confuciusornis sanctus Longusunguis Jeholornis Monoenantiornis Yanornis Sulcavis Yixianornis Rapaxavis Archaeorhynchus Elsornis Sapeornis Shanweiniao Concornis Jianchangornis Longicrusavis Parabohaiornis Ichthyornis Parahongshanornis Vescornis Piscivorenantiornis Anas Chongmingia Fortunguavis Gansus Confuciusornis dui Neuquenornis Hesperornis Vegavis Parahesperornis Vorona Cruralispennia Linyiornis Eopengornis Eocathayornis Changchengornis Songlingornis Hongshanornis Zhouornis Baptornis varneri Boluochia Eoalulavis Bellulornis Gobipteryx Piscivoravis 0.03 4.5 mean relative rate 1.0 0.5 clade probability Aves Pygostylia Ornithothoraces Ornithuromorpha Enantiornithes 0 5 10 15 20 25 30 relative rate forelimb pelvic girdle pectoral girdle and sternum hindlimb Aves Pygostylia Ornithothoraces Enantiornithines Ornithuromorpha skull axial skeleton
- 11. Acknowledgments • Xing Xu (IVPP) • Min Wang (IVPP) • Yilun Yu (IVPP) • Funding • National Natural Science Foundation of China • Chinese Academy of Sciences