Speed Dating using Least-Squares

Transcription

1 Speed Datng usng Least-Squares Thu Hen TO, Mattheu JUNG, Samantha LYCETT, Olver GASCUEL Bonformatque Evolutve, C3BI USR3756 Insttut Pasteur & CNRS, Pars Insttut de Bologe Computatonnelle, Montpeller France Insttute of Evolutonary Bology, Ednburgh Unted Kngdom Speed Datng usng Least-Squares A deluge of data Fast algorthms are needed We must rely on smple models 1

2 Speed Datng usng Least-Squares A deluge of data Dozens of thousands of vrus sequences (eg 40,000 n the UK HIV database) Orgn of epdemcs, phylodynamcs, resstance mutatons, survellance Datng s essental n all of these tasks Fast algorthms are needed Lnear n tme and space (.e. proportonal to the number of taxa) We must rely on smple models Gaussan, (truncated) normal dstrbuton of the nose Strct molecular clock (SMC), but robust Speed Datng usng Least-Squares A deluge of data Dozens of thousands of vrus sequences (eg 40,000 n the UK HIV database) Orgn of epdemcs, phylodynamcs, resstance mutatons, survellance Datng s essental n all of these tasks Fast algorthms are needed Lnear n tme and space (.e. proportonal to the number of taxa) We must rely on smple models Gaussan, (truncated) normal dstrbuton of the nose Strct molecular clock (SMC), but robust Suprzngly accurate! 2

3 Speed Datng usng Least-Squares Quck survey of datng models and methods The dstance-based approach, root-to-tp regresson and LF model A smple (but robust) Gaussan model Datng usng lnear algebra (LD, unconstraned) Quadratc programmng datng (QPD, temporal constrants) Tree rootng Smulaton results Applcaton to a large H1N1 nfluenza data set Dscusson Quck survey Basc prncple 3

4 Quck survey Basc prncple Much more dffcult than ths wth real data: Phylogenetc uncertanty Non molecular clock (unrooted) trees Several (ncompatble) calbraton ponts Hgh uncertanty dependng on the calbraton pont poston Quck survey Basc prncple 4

5 Quck survey Basc prncple Much more dffcult than ths wth real data: Phylogenetc uncertanty Non molecular clock (unrooted) tree Several (ncompatble) samplng tmes Hgh uncertanty dependng on samplng tmes, tree shape Quck survey Basc prncple 5

6 Quck survey Basc prncple Lader shape stll vsble, but datng s more dffcult Quck survey Basc prncple Lader shape stll vsble, but datng s more dffcult 6

7 Quck survey Input data Sequences/parwse dstances/topology/phylogeny Outgroup/ngroup only Rooted/unrooted phylogeny Internal calbraton ponts/tps sampled through tme Quck survey Man attempts Estmatng the global rate of evoluton Estmatng several rates (before/after treatment) Constrants needed! Estmatng the root poston and ts date Estmatng the dates of all nodes n the tree Estmatng a complete, tme-scaled tree (e.g. BEAST) 7

8 Quck survey Clock models Strct molecular clock: the tme s proportonal to the number of substtutons per ste (plus nose) Uncorrelated rates, wth known dstrbuton (e.g. lognormal, wth mean and varance to be estmated) Correlated under some model (e.g. the mean of daughter branch s drawn from a dstrbuton wth mean equal to mother s rate) Relaxed, correlated clock models Slow Fast 8

9 Relaxed, correlated clock models Quck survey Clock models Strct molecular clock: the tme s proportonal to the number of substtutons per ste (plus nose) Uncorrelated rates, wth known dstrbuton (e.g. lognormal, wth mean and varance to be estmated) Correlated under some model (e.g. the mean of daughter branch s drawn from a dstrbuton wth mean equal to mother s rate) Models of ncreasng complexty, typcally requrng MCMC or ABC algorthms, usually slow and lmted to a few hundred taxa-sequences 9

10 Quck survey Clock models Strct molecular clock: the tme s proportonal to the number of substtutons per ste (plus nose) Uncorrelated rates, wth known dstrbuton (e.g. lognormal, wth mean and varance to be estmated) Correlated under some model (e.g. the mean of daughter branch s drawn from a dstrbuton wth mean equal to mother s rate) No evdence that correlated models are useful for vruses (Drummond et al. 2006) Quck survey Clock models Strct molecular clock: the tme s proportonal to the number of substtutons per ste (plus nose) Uncorrelated rates, wth known dstrbuton (e.g. lognormal, wth mean and varance to be estmated) Correlated under some model (e.g. the mean of daughter branch s drawn from a dstrbuton wth mean equal to mother s rate) No model, just smoothng (e.g. PathD8) 10

11 Dstance-based approach: root-to-tp regresson t r Input: rooted tree, dated tps Strct molecular clock Model: root-to-tp dstances are affected by..d. normal nose Output: rate () and root date Smple and fast (O(n)) Hghly senstve to root poston Evolutonary correlaton not accounted for Dstance-based approach: root-to-tp regresson t r t r ) t r Standard regresson (GLS does not work) Able to select the root poston n O(n 2 ) 11

12 Dstance-based, Langley-Ftch (LF) model - r8s Input: a rooted tree, wth branch lengths and dated tps Output: substtuton rate () and all nodes dates Strct molecular clock Substtutons on each tree branch (, a()) follow a Posson dstrbuton wth mean s(t t a() ) Mult-dmensonal optmsaton of the lkelhood functon, usng the Powell algorthm (r8s, Sanderson 2003) Relatvely fast (but not fast enough for tree rootng) A smple Gaussan approxmaton of LF model The length b of branch (, a()) s normally dstrbuted b ( t t ) N 0, 2 a() t 2 t s 2 b Cs a E b Pseudo-count 12

13 Robust to some volaton of SMC Uncorrelated, normal, relaxed clock model N 0, 2 t t a b t t N0, a s 2 t 2 t b t t N0, a t ta s b s stll normally dstrbuted ts varance s agan an ncreasng functon of b a Least-squares crteron Temporal constrant Log-Lkelhood (Weghted Least Squares) crteron: 1 LL(, t,..., t ) ( b ( t t )) 1 n1 2 a( ) 1 ( b ( t ta() )) b C s 2 2 Precedence constrant for every node/leaf (except the root): t t a 13

14 LD (unconstraned) The unque, optmal (OLS) soluton satsfes a 1 t tl bl tr br ta b a 3 b 1 troot tlroot bl root trroot b rroot 2 A lnear system that s solved n lnear tme (usng bottom-up and top-down tree traversals just as wth parsmony), thus provdng the value of t gven : l b l b r r t c k We use these equaltes n WLS crteron to obtan n lnear tme, and then all dates t LD (unconstraned) The unque, optmal (OLS) soluton satsfes 1 t tl bl tr br ta b a 3 a ak k 1 troot tlroot bl root trroot b rroot 2 A lnear system that s solved n lnear tme (usng bottom-up and top-down tree traversals just as wth parsmony), thus provdng the value of t gven : l t c k t wt v a u We use these equaltes n WLS crteron to obtan n lnear tme, and then all dates t j r 14

15 QPD (wth temporal constrants) Quadratc functon of the (changed) varables: LL ( b ( t t )) leaves a() 2 ( b t ) ( b ) t 2 2 a a nternal for the nternal nodes Subject to: nternal nodes: tree leaves: a t a Unque soluton, obtaned usng an actve set method QPD (wth temporal constrants) Actve set method (summary) 1. Run LD 2. All volated constrants are put n the actve set 3. Compute the optmal soluton x* and the Lagrange multplers correspondng to the actve constrants Use a varant of LD on the collapsed tree b 0 4. If x* s feasble and all constrants are useful, then output x*, else remove the most useless constrant 0 and go to 3 5. If x* s not feasble, add to the actve set the most volated constrant and go to 3 Tme complexty O(n x k) k = # teratons << n (~70 wth ~900 nfluenza strans) t t a 15

16 Tree rootng For any gven edge, we use a slghtly modfed versons of LD and QPD to fnd the best rootng poston on that edge (.e. mnmzng WLS). Run LD or QPD on every edge of the tree, and fnd the best root poston n O(n 2 ) Stll qute fast wth LD Wth QPD, we frst run LD to fnd an ntal soluton, and then run QPD n a hll-clmbng fashon to mprove that soluton (most of the tme LD soluton s best, or nearly best) Smulaton results Brth-death trees wth varous death rates (DR), 70 to 110 taxa Uncorrelated, log-normal relaxed clock model F84+ substtuton model, 500 stes "HIV" parameters (n between Pol and env) nter-host DR = 0.75 ntra-host DR =

17 Rate Low mpact of topologcal errors tmrca No obvous advantage n usng PhyML, rather than FastME (dstance-based) Rate Outgroup-based rootng makes a bg dfference Rate Rooted LF* > QPD* LD* > RTT* tmrca tmrca Rooted LF* QPD* LD* > RTT* Unrooted QPD LD RTT 17

18 Rate tmrca Desappontng results wth BEAST (no outgroup, complete tme-scaled tree) All dates True rooted tree topology BEAST-RMC s consstently best to estmate all node dates (but tny dfferences, and stll some trouble wth rate estm.) 18

19 Computng tmes (n seconds taxa) 1,195 H1N1 nfluenza strans + outgroup Same methods and optons as wth smulated data We also ran BEAST wth fxed rooted PhyML topology 100 bootstrap replcates to obtan confdence ntervals 19

20 Results are mostly consstent wth smulatons Large ntervals wth unrooted nput tree (LD, QPD, RTT) QPD* and LF* are very close, and compatble wth BEAST* LD+PhyML : ~7% of volated temporal constrants (> 1month) QPD has a clear advantage! BEAST + TreeAnnotator : ~2% - BEAST* + TreeAnnotator : 0% 20

21 Computng tmes (wth 100 boostrap rep.) BEAST : PhyML : FastME : RTT, LD, QPD, LF* : QPD* : RTT*, LD* : 5 (*) to 20 days (Beagle, GPU ) 4 days (desktop, not parallelzed) 1 hour 1 hour 2 mn 10 sec. Summary Ablty to deal wth rooted and unrooted trees Provde estmates for the rate and all node dates Smlar accuracy as LF (despte normal approxmaton) and BEAST (stll unexplaned) Fast and already used wth very large datasets Mourad et al. (AIDS 2015), transmsson of resstance mutatons n HIV, 24,000 strans, rooted tree, ~30 mnutes (LF > 2 weeks) PANGEA_HIV consortum to estmate phylodynamcs parameter from rooted/unrooted trees ( 20,000 strans) 21

22 To be done - To be fnshed-publshed Fast confdence ntervals (e.g. based on the second dervatve of the lkelhood functon, parametrc bootstrap ) Extenson to tme calbraton ponts (see also Xa 2011) Analyse the LS resdues (e.g. to check for MC) Extend to correlated rate models (Sanderson 2002) 22