A stochastic model for phylogenetic trees

Transcription

1 A stochastic model for phylogeetic trees by Thomas M. Liggett ad Rialdo B. Schiazi Uiversity of Califoria at Los Ageles, ad Uiversity of Colorado at Colorado Sprigs October 10, 2008 Abstract We propose the followig simple stochastic model for phylogeetic trees. New types are bor ad die accordig to a birth ad death chai. At each birth we associate a fitess to the ew type sampled from a fixed distributio. At each death the type with the smallest fitess is killed. We show that if the birth (i.e. mutatio) rate is subcritical we get a phylogeetic tree cosistet with a iflueza tree (few types at ay give time ad oe domiatig type lastig a log time). Whe the birth rate is supercritical we get a phylogeetic tree cosistet with a HIV tree (may types at ay give time, oe lastig very log). 1 Itroductio The iflueza phylogeetic tree is peculiar i that it is very skiy: oe type domiates for a log time ad ay other type that arises quickly dies out. The the domiatig type suddely dies out ad is immediately replaced by a ew domiatig type. The models proposed so far are very complex ad make may assumptios. See for istace Koelle et al. (2006) ad va Nimwege (2006). We would like to use a simple stochastic model to model such a tree. The other motivatio for this work comes from the compariso betwee iflueza ad HIV phylogeetic trees. A HIV tree is characterized by a radial spread outward from a acestral ode, i sharp cotrast with a iflueza tree. Moreover, Korber et al. (2001) ote that the iflueza virus is less diverse worldwide tha the HIV virus is i Amsterdam aloe. However, both types of trees are supposed to be produced by the same basic mechaism: mutatios. Ca the same mathematical model produce two trees that are so differet? Our simple stochastic model will show a strikig differece i behavior depedig o the mutatio rate. We propose a model that has a birth ad death compoet ad a fitess compoet. For the death ad birth compoet we do the followig. If we have 1 types at a certai time t the there is creatio of a ew type (by mutatio) at rate λ. If there are 2 types the there is Partially supported by NSF grat DMS Partially supported by NSF grat DMS Key words ad phrases: phylogeetic tree, iflueza, HIV, stochastic model 2000 Mathematics Subject Classificatio: 60K35 1

2 death of oe type at rate. If oly oe type is left it caot die. That is, + 1 at rate λ 1 at rate if 2 Moreover, the i-th type that appears is associated to a radom variable F i (the fitess of type i). The radom variables F i are i.i.d. Every time there is a death evet the the type that is killed is the oe with the smallest F i. Sice all that matters is the raks of the fitesses, we might as well take their distributio to be uiform o [0, 1]. We give o specific rule o how to attach a ew type after a birth to existig types (i order to costruct a tree). Our results do ot deped o such a rule. Two atural possibilities are to either attach the ew type to the type which has the maximum fitess or to a type take at radom. Theorem 1. Take α (0, 1). If λ 1, the lim P(maximal types at times αt ad t are the same) = α, t while if λ > 1, the this limit is 0. We see that if λ < 1, the domiatig type (i.e. the fittest type) at time t has likely bee preset for a time of order t ad at ay give time there will ot be may types. This is cosistet with the observed structure of a iflueza tree. O the other had, if λ > 1, the the domiatig type at time t has likely bee preset for a time of order shorter tha t ad at ay give time there will be may types. This is cosistet with a HIV tree. 2 Proof of Theorem 1 The proof divides ito three cases, depedig o whether the birth ad death chai is positive recurret, ull recurret, or trasiet. We preset them i order of difficulty. 2.1 Case λ < 1 Let F 0,F 1,... be i.i.d. rvs with the type distributio, τ 1,τ 2,... be the (cotiuous) times betwee successive visits of the chai to 1, T = τ τ, σ 1,σ 2,... be the umber of ew types itroduced i cycles betwee successive visits to 1, ad S = 1 + σ σ. Note that the τ s ad σ s are ot idepedet of each other, but the sequece (τ 1,σ 1 ),(τ 2,σ 2 ),... is i.i.d. ad idepedet of the F sequece. Defie the usual reewal process N(t) correspodig to the τ s by {N(t) = } = {T t < T +1 }. For 0 < s < t, recallig that T N(t) t < T N(t)+1, ad otig that the maximal type is icreasig i time, we see that lies betwee P(maximal types at times s ad t are the same,n(s) < N(t)) (1) P(maximal types at times T N(s) ad T N(t)+1 are the same,n(s) < N(t)) 2

3 ad P(maximal types at times T N(s)+1 ad T N(t) are the same,n(s) < N(t)). Let F be the σ-algebra geerated by (τ 1,σ 1 ),(τ 2,σ 2 ),... The for k l, sice the F sequece is i.i.d. ad idepedet of F, P(maximal types at times T k ad T l are the same F) = S k S l. Therefore, sice N(s) ad N(t) are F measurable, (1) lies betwee [ ] [ ] SN(s) SN(s)+1 E,N(s) < N(t) ad E,N(s) < N(t). (2) S N(t)+1 S N(t) If λ < 1, the Eτ <, ad the reewal theorem gives N(s)/s 1/Eτ a.s., while the strog law of large umbers gives S N(s) /N(s) Eσ a.s., so that S N(s) /s Eσ/Eτ a.s. It follows that for α < 1, by the bouded covergece theorem, lim P(maximal types at real times αt ad t are the same) = α. (3) t This completes the proof of Theorem 1 i the subcritical case. 2.2 Case λ > 1. Defie the τ s ad σ s as above, except that ow, the cycles used are betwee the successive times the chai reaches a ew high. I other words, T is the hittig time of + 1, ad S is the umber of ew types see up to time T. Of course, the σ s ad τ s are o loger idetically distributed. However, (τ 1,σ 1 ),(τ 2,σ 2 ),... are idepedet. Let (Z i ) i 1 be a discrete time radom walk startig at 0 that goes to the right with probability λ/(λ + 1) ad to the left with probability 1/(λ + 1). For every 1, let Z i, be a discrete time radom walk startig at 0 with the same rules of evolutio as Z i except that the radom walk Z i, has a reflectig barrier at + 1. For every 1, the two radom walks Z i ad Z i, are coupled so that they move together util (if ever) they hit + 1 ad thereafter we still couple them so that Z i Z i, for every i 0. Let U ad U be the hittig times of 1 for the radom walks Z i ad Z i,, respectively. Let X 1,X 2,... be a sequece of i.i.d. uit expoetial radom variables that are idepedet of the radom walks Z i ad Z i,. Defie L = U i=1 X i λ + 1. Lemma 2. The first two momets of τ coverge to the correspodig momets of L. Proof of Lemma 2. Usig the fact if X is expoetially distributed with rate 1 the X/λ is expoetially distributed with rate λ, we have that τ has the same distributio as Y = U i=1 X i ( + Z i, )(λ + 1). 3

4 Note that sice our claim is about the distributio of τ oly, we may use the same sequece X i for every 1. Let χ A deote the idicator fuctio of the evet A. Take 0 < ǫ < 1 ad defie A = A(,ǫ) = {Z i < ǫ for some i 1}. Note that o A c, Z i, = Z i for all i 1 ad therefore U = U. Hece, Therefore, Y L χ A c = U X i λ Z i χ A c. i=1 Y L χ A c ǫ 1 ǫ L provided that ǫ( + 2) 1. (This restrictio is eeded to cover the case i = U.) O the other had Y L χ A (Y + L)χ A ( + 1)Lχ A, where the secod iequality comes from U U. We the have ( ) 2 ǫ E Y L ( 1 ǫ )2 E(L 2 ) + ( + 1) 2 E(L 2 χ A ). By the Cauchy-Schwarz iequality (all momets of L exist) we have E(L 2 χ A ) E(L 4 ) 1/2 P(A) 1/2. Sice for fixed ǫ > 0, P(A(,ǫ)) coverges to 0 expoetially fast as a fuctio of, ( ) 2 ǫ lim sup E Y L ( 1 ǫ )2 E(L 2 ). ( 2 We ow let ǫ go to 0 ad this proves that E Y L) coverges to 0. Hece, the first two momets of Y coverge to the correspodig momets of L. This last claim oly ivolves the distributio of Y so the first two momets of τ also coverge to the correspodig momets of L. Lemma 3. Assume that λ > 1. If c > λ 1 the exp( ct)n(t) coverges a.s. to 0. If c < λ 1 the exp( ct)n(t) coverges a.s. to. Proof of Lemma 3. By Lemma 1 the variace of τ is asymptotic to V ar(l)/ 2. This implies the a.s. covergece of the radom series (τ Eτ ) (see for istace Corollary 47.3 i Port (1994)). Therefore the partial sums coverge a.s. ad so T E(T ) log coverges a.s. to 0. Sice E(τ ) coverges to E(L) we have that E(T )/log coverges to E(L) ad therefore T /log coverges to E(L) a.s. Now use {N(t) } = {T t} to show that exp( ct)n(t) 4

5 coverges a.s. to 0 if c > 1/E(L) ad coverges to ifiity if c < 1/E(L). We ow compute E(L). By Wald s idetity E(L) = E(U) λ + 1. We use that E(U) = λ + 1 λ 1 (see (13.22) i Bhattacharya ad Waymire (1990), for istace) to get that E(L) = 1/(λ 1). We are ow ready to complete the proof of Theorem 1 i the supercritical case. First ote that a ew type appears every time there is a birth. Therefore, σ is the umber of steps to the right of the radom walk Z i, stopped at 1. That is, σ is (1 + U )/2. We ow show that U coverges a.s to U. Let δ > 0 we have P( U U > δ) P(U > U ) P(Z i = + 1 for some i 1). As oted before the last probability decays expoetially with. Therefore, P( U U > δ) <. 1 A easy applicatio of Borel-Catelli Lemma implies that U coverges a.s. to U. Sice U U the Domiated Covergece Theorem implies that, for every k 1 the kth momet of σ coverges to the kth momet of (1 + U)/2. I particular, V ar(σ ) is a bouded sequece. This is eough to prove that 1 (σ i E(σ i )) coverges a.s. to 0, i=1 see for istace Propositio i Port (1994). Sice E(σ ) is a coverget sequece we get that S / coverges a.s. to the limit of E(σ ). Sice N(t) a.s., this strog law of large umbers gives that S N(t) /N(t) coverges to the limitig expectatio of σ. This together with Lemma 2 shows that the two terms i (2) coverge to 0 whe we let s = αt ad t goes to ifiity. The proof of Theorem 1 i the supercritical case is complete. 2.3 Case λ = 1. Let φ k (u) be the Laplace trasform of the hittig time of 1 for the cotiuous time birth ad death chai defied i Sectio 1 ad startig at k 1. (By the Laplace trasform of a oegative radom variable W, we mea Ee uw,u 0.) The φ 1 1, ad for k 2, φ k (u) = 2k [ 1 2k + u 2 φ k 1(u) + 1 ] 2 φ k+1(u) (4) sice the holdig time at k is expoetial of rate 2k, ad the the chai moves to k 1 or k + 1 with probabilities 1 2 each. The Laplace trasform of τ i from part I above is give by Ee uτ i = u φ 2(u). 5

6 Let ψ k (u) be the Laplace trasform of the hittig time of k 1 for the cotiuous time birth ad death chai startig at k. The φ (u) = ψ k (u). (5) k=2 Now let V be the hittig time of 1 for the discrete time simple symmetric radom walk Z k startig at 0, ad let X i be i.i.d. uit expoetials. The coditioed o the Z k process, the hittig time of 1 startig at for the cotiuous time chai is distributed as V 1 k=0 Therefore, sice Ee ux k = 1 1+u e u, 1 X k 1 V 1 + Z k k=0 X k. ψ (u) Ee uv/ = ψ(u/), (6) where ψ is the Laplace trasform of V. By page 155 of Durrett (2004), P(V > 2) 1/ π. By a Tauberia theorem (see Theorem 2 i Sectio XIII.5 of Feller (1971)) this implies ψ(u) 1 c u as u 0 for some costat c. It follows that by adjustig the costat, we ca assume that ψ(u) e c u for 0 u 1. Combiig this with (5) ad (6) gives (agai for a ew value of c, possibly i each lie) ad φ (u) Now rewrite (4) i the form ad the sum to get [ ψ (u) exp c k=2 u ], 0 u, (7) [ ] u exp c e c u, 0 u 2. (8) k [φ k 1 (u) φ k (u)] [φ k (u) φ k+1 (u)] = u k φ k(u), 1 φ 2 (u) = u k=2 φ k (u) k + [φ (u) φ +1 (u)], 2. (9) Takig u 0 ad a/u, igorig the last term i (9) (which is oegative) ad usig (8), we see that 1 φ 2 (u) lim if u 0 ulog u a e c. Sice this holds for all a > 0, it follows that lim if u 0 1 φ 2 (u) ulog u 1. 6

7 We ow tur to the lim sup. Observe that By (7) φ (u) φ +1 (u) = φ (u)(1 ψ +1 (u)) 1 ψ +1 (u). φ (u) φ +1 (u) 1 e c u/(+1) c u/( + 1). We use this last iequality i (9) as well as φ k (u) 1 to get Takig u 0 ad 1/u, 1 φ 2 (u) u lim sup u 0 k=2 1 k + c u/( + 1). 1 φ 2 (u) ulog u 1. Therefore φ 2 (u), ad hece Ee uτ i, are asymptotic to 1 + ulog u as u 0. It follows that [ lim E exp u T ] = e u, log so that T log 1 i probability. Sice the evets N(t) ad T t are the same, it follows that N(t) log t t 1 (10) i probability as t. Now, S / 2 coverges i distributio to a oe sided stable law of idex 1 2 (see Theorem (7.7) i Durrett (2004)). By (10), it follows that S N(t) /N(t) 2 also has this distributioal limit. (Note that idepedece betwee the σ s ad τ s is ot required here, which is good sice they are highly depedet. All that is eeded is that the limit i (10) is costat ad that both S ad N(t) are mootoe.) So, the limit i (3) is E(Y α /Y 1 ) for the correspodig stable process. This is α. To see this, it is eough to check it for ratioal α. If α = m/, this boils dow to the simple fact that if V 1,...,V are i.i.d. ad positive, the V i E = 1 V V. Refereces R.N. Bhattacharya ad E.C. Waymire (1990) Stochastic Processes with Applicatios. Wiley. R. Durrett (2004) Probability: Theory ad Examples (3rd editio). Duxbury press. 7

8 W. Feller (1971) A Itroductio to Probability Theory ad its Applicatios, Volume 2 (secod editio). Wiley. B. Korber, B. Gasche, K. Yusim et al. (2001) Evolutioary ad immuological implicatios of cotemporary HIV-1 variatio. British Medical Bulleti 58, K.Koelle, S. Cobey, B. Grefell ad M. Pascual (2006) Epochal evolutio shapes the phylodyamics of iterpademic iflueza A (H3N2) i Humas. Sciece vol. 314, E. va Nimwege (2006). Iflueza escapes immuity alog eutral etworks. Sciece vol. 314, S. C. Port (1994). Theoretical Probability for Applicatios. Wiley. 8