1.8 The site frequency spectrum

Transcription

1 .8 The site frequency spectrum The simplest sttistic for smple under the infinitely mny sites muttion model is the number of segregting sites, whose distribution we discussed in.5, but one cn lso sk for more detiled informtion. Definition.3 Site frequency spectrum) For smple of size k under the infinitely mny sites muttion model, write M j k) for the number of sites t which exctly j individuls crry muttion. The vector M k),m k),...,m k k)) is clled the site frequency spectrum of the smple. PICTURE Lemm.4 If the genelogy of the smple is determined by the Kingmn colescent, then under the infinitely mny sites model we hve E[M j k)] = θ j. 5) Note: the θ here is s before, so muttions occur t rte θ/ long ech ncestrl linege. Proof of Lemm.4 We use the dulity between the Kingmn colescent nd the Morn model. Suppose tht muttion rose t time t tht is t before the present) nd denote individuls in our smple crrying tht muttion s type. For the dul Morn model with popultion size k), we think of the smple s the whole popultion nd so the Morn popultion hs nothing to do with the popultion from which we re smpling. From the point of view of the Morn model, the probbility tht we see j type individuls in the smple is the probbility tht muttion rising on single individul t time zero is crried by j individuls t time t lter. We write X t for the number of type individuls t time t nd pt,i,j) = P[X t = j X = i]. In this nottion, the probbility tht t time there re exctly j type individuls in the smple is pt,,j). Since ech muttion occurs t different point on the genome nd muttions occur t rte θ/ per individul nd the popultion size is k), the expected totl number of sites t which we see muttion crried by exctly j individuls is then just E[M j k)] = k θ pt,,j)dt. 6) Now Gi,j) pt,i,j)dt is just the expected totl time tht the process {X t } t spends in site j if it strted from i nd our next tsk is to clculte this. Note tht if X s = i, then it moves to new vlue t rte ik i) which is just the number of the k ) wys of smpling pir from the popultion in which the two individuls smpled re of different types) nd when it does move, it is eqully likely to move to i or i +. Let T i = inf{t > : X t = i} 7

2 denote the first hitting time of site i. Then since is trp for the process we hve G,j) = P[T j < T X = ] Gj,j). Now, becuse it is just timechnge of simple rndom wlk,for i j, nd similrly, for j l k, P[T < T j X = i] = j i, j P[T k < T j X = l] = l j k j. Thus, prtitioning on whether the first jump out of j is to j or to j +, we find tht if it is currently t j, the probbility tht this is the lst visit tht X t mkes to j is ρ = j + k j = k jk j). In other words, if we strt from j, the number of visits to j including the current one) before either the llele is fixed in the popultion or lost is geometric with prmeter ρ. Ech visit lsts n exponentilly distributed time with men jk j). Thus G,j) = j Gj,j) = j ρ jk j) = kj. Substituting into 6) completes the proof..9 The lookdown process The consistency of the k-colescents for different vlues of k N llowed us to recover ll of them s projections of single stochstic process, Kingmn s colescent. Since genelogicl trees for the Morn model re precisely governed by the Kingmn colescent, it is resonble to hope tht we cn lso construct Morn models corresponding to different popultion sizes s projections of single stochstic process. This is t the hert of the powerful Donnelly & Kurtz lookdown process. To see how it works, we exploit the connection with the Kingmn colescent. Suppose tht the popultion t the present time is lbelled {,,...,N}. Recll tht the full description of the Kingmn colescent is s process tking vlues mong the set of equivlence reltions on {,,...,N}, with ech ncestrl linege corresponding to single equivlence clss. Now suppose tht we lbel ech equivlence clss by its smllest element. If blocks with lbels i < j colesce, then fter the colescence the new block is necessrily lbelled i. In our grphicl representtion of the Morn model, this just dicttes the direction of the rrow corresponding to tht colescence event; it will lwys be the individul with the smller lbel tht gve birth. Bckwrds in time, our process is equivlent to one 8

3 in which, s before, t the points of rte one Poisson process π i,j) rrows re drwn joining the lbels i nd j, but now the rrows re lwys in the sme direction upwrds with our convention). The genelogies re still determined by the Kingmn colescent, we hve simply chosen convenient lbelling, nd so in prticulr they re precisely those of the Morn model. But wht bout forwrds in time? Wht we sw bckwrds in time ws tht choosing the direction of the rrows corresponded to choosing prticulr lbelling of the popultion. If the distribution of the popultion is exchngeble, tht is it doesn t depend on the lbelling, then forwrds in time too we should not hve chnged the distribution in our popultion. Our next tsk will be to check this, but first we need forml definition. Definition.5 The N-prticle lookdown process) The N-prticle lookdown process will be denoted by the vector ζ t),...,ζ N t)). Ech index is thought of s representing level, with ζ i t) denoting the llelic type of the individul t level i t time t. The evolution of the process is described s follows. The individul t level k is equipped with n exponentil clock with rte k ), independent of ll other individuls. At the times determined by the corresponding Poisson process it selects level uniformly t rndom from {,,...,k } nd dopts the current type of the individul t tht level. The levels of the individuls involved in the event do not chnge. Remrk.6 Becuse of our convention over the interprettion of rrows, it is not t ll cler from the bove why one should cll this the lookdown process. The explntion is tht t rte k ) the kth individul looks down to level chosen uniformly t rndom from those below nd dopts the type of the individul t tht level. To see tht the lookdown process nd the Morn model produce the sme distribution of types in the popultion, provided we strt from n exchngeble initil condition, we exmine their infinitesiml genertors. Recll the definition of the genertor of continuous time Mrkov process. Definition.7 Genertor of continuous time Mrkov process) Let {X t } t be rel-vlued continuous time Mrkov process. For simplicity suppose tht it is time homogeneous. For function f : R R define E[fX δt ) fx) X = x] Lfx) = lim δt δt if the limit exists. We ll cll the set DL) of functions for which the limit exists the domin of L nd the opertor L cting on DL) the infinitesiml genertor of {X t } t. If we know L, then we cn write down differentil eqution for the wy tht E[fX t )] evolves with time. If Lf is defined for sufficiently mny different functions then this completely chrcterises the distribution of {X t } t. We suppose tht the types of individuls re smpled from some type spce E. The Morn model for popultion of size N is then simply continuous time Mrkov chin on E N nd its infinitesiml genertor, K N, evluted on function f : E N R, is given by N K N fx,x,...,x N ) = A i fx,x,...,x N ) + N N [Φ ij fx,...,x N ) fx,...,x N )], 7) i= 9 i= j=

4 where Φ ij fx,...,x N ) is the function obtined from f by replcing x j by x i. The opertor A i is the genertor of the muttion process, A, cting on the ith coordinte. Recll tht in the Morn model muttion ws superposed s Mrkov process long lineges.) The genertor of the N-prticle lookdown process, L N is given by L N f x,x,...,x N ) = N A i f x,x,...,x N ) + i= i<j N [Φ ij f x,x...,x N ) f x,x...,x N )]. 8) Assuming tht we strt both processes from the sme exchngeble initil condition, we should like to show tht, ζ t),ζ t),...,ζ N t)) nd the types under the originl Morn process which we denote Z t),z t),...,z N t)) hve the sme distribution for ech fixed t >, even though the processes re mnifestly different. Following Dwson 993), we check tht the genertors of the two processes gree on symmetric functions. Observe first tht ny symmetric function, f, stisfies f x,x,...,x N ) = f ) x N! π),x π),...,x πn, π where the sum is over ll permuttions of {,,...,N}. Substituting this expression for f into eqution 8), we recover precisely eqution 7). In other words, the genertors of ζ,ζ,...,ζ N ) nd Z,Z,...,Z N ) gree on symmetric functions s required. We re implicitly ssuming uniqueness of the distribution on symmetric functions corresponding to this genertor. It follows from dulity with the N-colescent, but we don t llow tht to detin us here.) The key observtion now is tht our Nth lookdown process is simply the first N levels of the N + k)th lookdown process for ny k. The infinite lookdown process cn then be constructed s projective limit. Theorem.8 Donnelly & Kurtz 996) There is n infinite exchngeble prticle system {W i,i N} such tht for ech N, W,W,...,W N ) D = ζ,ζ,...,ζ N ), where ζ,ζ,...,ζ N is the N-prticle lookdown process. Remrk.9 In fct more is true. It is known tht the sequence of empiricl mesures N N i= δ Z i t) converges to Fleming-Viot superprocess s N. Donnelly & Kurtz lso show tht Y = lim N N N δ Wi, is Fleming-Viot superprocess. Rther thn introduce the generl Fleming-Viot superprocess, which tkes its vlues mong probbility mesures on the type spce E, in. we shll consider wht this limit looks like in the specil cse when E is two-point set representing two lleles nd A in which cse it is enough to specify the evolution of the proportion of type individuls in the popultion. i=

5 Since the genelogy of smple of size k from the Morn model is k-colescent nd since we ve seen tht the genelogy of the first k levels in the lookdown process is lso k-colescent, with this lbelling we hve nice consistent wy of smpling from Morn model of rbitrry size. The genelogy of the smple is tht of the first k levels in the lookdown process. And the evolution of those levels does not depend on the popultion size - becuse we only ever look down we don t see the popultion size N t ll.. A more simplistic limit Rther thn discussing generl Fleming-Viot superprocesses which would llow us to consider essentilly rbitrry type spces) we now turn to identifying the limiting model for llele frequencies when our popultion is subdivided into just two types which, s usul, we lbel nd A. Just s in our discussion of the rescled Wright-Fisher model, we consider the proportion, p t, of individuls of type t time t. The only possible muttions re between the two types. We suppose tht ech type individul muttes to type A t rte ν nd ech type A individul muttes to type t rte ν. Recll tht for the Morn model we re lredy in the timescle of the Kingmn colescent nd so we should think of ν i = Nµ i where µ nd µ re the true muttion rtes. Remrk.3 The ide tht we cn mutte bckwrds nd forwrds between types my seem t odds with our discussion of muttions in.3. Models of this type were introduced long before biologists knew bout nd hd ccess to DNA sequences. Clssiclly one might imgine smll number of lleles defined through phenotype, for exmple colour. In modern terms one cn justify the model by pooling sequences into clsses ccording to the corresponding phenotype. The genertor for the Morn model for popultion of size N is then ) N L N fp) = p p) fp + N ) ) N ) fp) + p p) fp N ) ) fp) + Nν p fp N ) fp) ) + Nν p) fp + N ) fp) ). To see this, note tht the reproduction events in the Morn model tke plce t the points of Poisson process with rte N ) nd t the time of such trnsition, if the current proportion of lleles is p, then p p + N with probbility p p), p p N with probbility p p) nd there is no chnge with probbility p p). The chnce tht we see reproduction event in time intervl of length δt is ) N δt + Oδt) )

6 nd the probbility of seeing more thn one trnsition is Oδt) ). For muttion events, t totl rte Npν, one of the Np type individuls will mutte to type A, resulting in reduction of p by /N nd t totl rte N p)ν one of the N p) type A individuls will mutte to type. Putting ll this together gives tht for f : [,] R nd p = i N for some i {,,...,N} L N fp) = ) N p p) fp + N ) ) fp) + ) N p p) fp N ) ) fp) + Npν fp + N ) fp) ) + Npν fp N ) fp) ). To see wht our popultion process will look like for lrge N we tke f to be twice continuously differentible nd use Tylor s Theorem to find n pproximtion for Lf. Thus ) N L N fp) = p p) fp) + N f p) + N f p) + O ) N 3) fp) ) N + p p) fp) N f p) + N f p) + O ) N 3) fp) Npν fp) + N f p) + O ) N + N p)ν fp) N f p) + O ) N = p p)f p) + p)ν pν )f p) + O N ). So s N, L N L where Lfp) = d dt E[fp t) p = p] = t= p p)f p) + ν ν + ν )p)f p). 9) In prticulr, if we set ν = ν = we obtin Lfp) = p p)f p) which is exctly the genertor tht we obtined in the lrge popultion limit from our Wright-Fisher model. It is not hrd to extend the work tht we did there to include muttions nd recover the full genertor 9). Wht we hve written down is the genertor of one-dimensionl diffusion. We should like to be ble to use the convergence of genertors tht we hve verified to justify using the corresponding one-dimensionl diffusion s n pproximtion for the Morn, Wright-Fisher nd Cnnings models on suitble timescles). Theorem.3 Let E be metric spce. Suppose tht for ech N N, {X N) t } t is n E-vlued Mrkov processes with genertor L N nd tht X is n E-vlued Mrkov process with genertor L. If, for every f DL), lim N LN fx) = Lfx), uniformly for x E,

7 then the finite-dimensionl distributions of X N) converge to those of X. Tht is, for every finite set of times t < t < < t n, X N) t ),...,X N) t n )) d Xt ),...,Xt n )) s N. In fct we hve used slightly more thn this s our Wright-Fisher model ws in discrete time. For tht Ethier & Kurtz, Chpter, Theorem 6.5 is exctly wht we need. Remrk.3 This sort of convergence is enough to justify using our limiting Wright-Fisher diffusion to pproximte things like time to fixtion nd fixtion probbilities. However, if we re relly interested in the genelogies of popultions, then we need more. For our Morn models, the Donnelly-Kurtz lookdown construction gve us much stronger result. In generl we must be creful. It is possible to rrive t the sme diffusion for llele frequencies from mny different individul bsed models for our popultion, nd it is not lwys the cse tht the genelogies converge to the sme limit. Before we cn exploit Theorem.3 we need to know tht there is Mrkov process with genertor 9) nd tht we cn ctully clculte quntities of interest for it. Hppily both re true.. Diffusions In this section we re going to remind ourselves of some useful fcts bout one-dimensionl diffusions. We strt in firly generl setting. Definition.33 One-dimensionl diffusion) A one-dimensionl diffusion process {X t } t is strong Mrkov process on R which trces out continuous pth s time evolves. At ny instnt in time, X t is continuous rndom vrible but lso ny relistion of {X t } t is continuous function of time. Its rnge need not be the whole of R nd indeed for the most prt we ll be interested in diffusions on,). For the time being let us tke the stte spce to be n intervl,b) possibly infinite). The genertor of the diffusion tkes the form Lfx) = σ x) d f df x) + µx) x). ) dx dx Evidently for this to be defined f must be twice continuously differentible on,b). Depending on the behviour of the diffusion close to the boundries of its domin, f my lso hve to stisfy boundry conditions t nd b. We ll specify these precisely in Theorem.45, but for now ssume tht if we pply the genertor to function then it is in the domin. To void pthologies, we mke the following ssumptions:. For ny compct intervl I,b), there exists ɛ > such tht σ x) > ɛ for ll x I,. the coefficients µx) nd σ x) re continuous functions of x,b). 3

8 Note tht crucilly for pplictions in genetics) we do llow σ x) to vnish t the boundry points {, b}. This is more thn) enough to gurntee tht the diffusion hs trnsition density function, denoted pt,x,y) see Knight 98 Theorem for more generl result). Definition.34 The trnsition density function of {X t } t is the function p : R + R R R + for which P[X t A X = x] P x [X t A] = pt,x,y)dy for ny subset A R. Let us write h Xt) = X t+h X t, then tking f x) = x in the genertor nd using the Mrkov property) we see tht Lf X t ) = lim h h E[ hxt) X t ] = µx t ) nd so A E[ h Xt) X t ] = hµx t ) + oh) s h. ) Now observe tht we cn write X t+h X t ) = X t+h X t X tx t+h X t ) nd so, tking f x) = x, which yields This motivtes the stndrd terminology. Lf X t ) X t Lf X t ) = lim h h E[ hxt)) Xt ] = σ X t ) E[ h Xt)) X t ] = hσ X t ) + oh) s h. ) Definition.35 Infinitesiml drift nd vrince) The coefficients µx) nd σ x) re clled the infinitesiml) drift nd vrince of the diffusion {X t } t. In fct if strong Mrkov process {X t } t is càdlàg tht is its pths re right continuous with left limits) nd stisfies ), ) nd the dditionl condition lim h h E[ hxt) p X t = x] = for some p > where the convergence is uniform in x,t) on compct subsets of,b) R +, then {X t } t is necessrily diffusion see Krlin & Tylor, 5., Lemm.). The cnonicl exmple of one-dimensionl diffusion is one-dimensionl Brownin motion which hs genertor L B fx) = d f dx x) 4

9 nd trnsition density function pt,x,y) = ) x y) exp. πt t Brownin motion cn be thought of s building block from which other one-dimensionl diffusions re constructed. One pproch is to observe tht the diffusion corresponding to the genertor L of eqution ) cn be expressed s the solution of stochstic differentil eqution driven by Brownin motion with pproprite boundry conditions) dx t = µx t )dt + σx t )db t. Remrk.36 Mthemticl drift versus genetic drift) We hve lredy encountered the Wright- Fisher diffusion severl times, corresponding to the solution of the stochstic differentil eqution dp t = ν p t ) ν p t )dt + p t p t )db t. It is n unfortunte ccident of history, tht the stndrd terminology for the stochstic term driven by Brownin motion) is genetic drift, wheres to mthemticin it is the deterministic muttion term tht corresponds to drift... Speed nd scle Our pproch to constructing one-dimensionl diffusions from Brownin motion will not be vi stochstic differentil equtions, but rther through the theory of speed nd scle. A nice feture of one dimensionl diffusions is tht mny quntities cn be clculted explicitly. This is becuse except t certin singulr points which will only ever be t or b under our conditions) ll one-dimensionl diffusions cn be trnsformed into Brownin motion first by chnge of spce vrible through the so-clled scle function) nd then timechnge through wht is known s the speed mesure). To see how this works, we first investigte wht hppens to diffusion when we chnge the timescle. Suppose tht diffusion {Z t } t hs genertor L Z. We define new process {Y t } t by Y t = Z τt) where τt) = t βy s )ds, for some function βx) which we ssume to be bounded, continuous nd strictly positive. So if Y = Z, then the increment of Y t over n infinitesiml time intervl,dt) is tht of Z t over the intervl,dτt)) =,βy )dt). In our previous nottion, E[ h Y ) Y )] = βy )hµ Z Z ) + oh) = βy )µ Z Y )h + oh), nd E[ h Y ) Y ] = βy )hσ Z Z ) + oh) = βy )σ Y )h + oh). 5

10 In other words, L Y fx) = βx)l Z fx). We re now in position to understnd speed nd scle. Let {X t } t be governed by the genertor ). Suppose now tht Sx) is strictly incresing function on,b) nd consider the new process Z t = SX t ). Then the genertor L Z of Z cn be clculted s L Z fx) = d dt E[fZ t) Z = x] t= = d dt E[fSX t)) SX ) = x] t= = σ S x)) d dx fx) + µs x)) d dx fx) = { σ S x)) S x)) d f dx x) + S x) df } dx x) + µs x))s x) df dx x) = σ S x))s S x)) d d dxfx) + LSx) fx). 3) dx Now if we cn find strictly incresing function S tht stisfies LS, then the drift term in the mthemticl sense) in 3) will vnish nd so Z t will just be time chnge of Brownin motion on the intervl S),Sb)). Such n S is provided by the scle function of the diffusion. Definition.37 Scle function) For diffusion X t on,b) with drift µ nd vrince σ, the scle function is defined by x y ) µz) Sx) = exp x η σ z) dz dy, where x, η re points fixed rbitrrily) in,b). The scle chnge resulted in timechnged Brownin motion. The chnge of time required to trnsform this into stndrd Brownin motion is dictted by the speed mesure. is the density of the speed me- Definition.38 Speed mesure) The function mξ) = sure or just the speed density of the process X t. We write Mx) = x x mξ)dξ. σ ξ)s ξ) Remrk.39 The function m plys the rôle of β before. Notice tht x x mξ)dξ = Sx) Sx ) ms y)) S S y)) dy = Sx) Sx ) σ S y)) S S y)) ) dy. The dditionl S y) in the genertor 3) hs been bsorbed since our time chnge is pplied to diffusion on S),Sb)). 6

11 In summry, we hve the following. Lemm.4 Denoting the scle function nd the speed mesure by S nd M respectively we hve Lf = d f dm/ds ds = ) d df. dm ds Proof d dm ) df ds since S solves LS = ) s required. = ) d df dm/dx dx ds/dx dx = σ x)s x) d ) df dx S x) dx = σ x) d f dx σ x)s x) S x) S x)) df dx = σ x) d f df + µx) dx dx.. Hitting probbilities nd Feller s boundry clssifiction Before going further, let s see how we might pply this. Suppose tht diffusion process on, ) represents the frequency of n llele, sy, in popultion nd tht zero nd one re trps for the process. One question tht we should like to nswer is Wht is the probbility tht the -llele is eventully lost from the popultion? In other words, wht is the probbility tht the diffusion hits zero before one? To prove generl result we need first to be ble to nswer this question for Brownin motion. Lemm.4 Let {B t } t be stndrd Brownin motion on the line. For ech y R, let T y denote the rndom time t which it hits y for the first time. Then for < x < b, P[T < T b B = x] = b x b. Sketch of Proof Let ux) = P[T < T b B = x] nd choose h smll enough tht P[T T b < h B = x] = oh). We suppose tht u is twice differentible, then ux) = E[uB h ) B = x] + oh) = E[ux) + B h x)u x) + B h x) u x)] + oh) = ux) + hu x) + oh). 7

12 Subtrcting ux) from ech side, dividing by h nd letting h tend to zero, we obtin u x) =. We lso hve the boundry conditions u) = nd ub) =. This is esily solved to give ux) = b x b s required. Of course this reflects the corresponding result for simple rndom wlk tht we used in the proof of Lemm??. In generl we cn reduce the corresponding question for {X t } t to solution of the eqution Lux) = with u) = nd ub) =, but in fct we hve lredy done ll the work we need. We hve the following result. Lemm.4 Hitting probbilities) Let {X t } t be one-dimensionl diffusion on,b) with infinitesiml drift µx) nd vrince σ x) stisfying the conditions bove. If < < x < b < b then P[T < T b X = x] = Sb ) SX ) Sb ) S ), where S is the scle function for the diffusion. Remrk.43 Notice tht η cncels in the rtio nd x in the difference, so tht this rtio is welldefined. Proof Evidently it is enough to consider the corresponding hitting probbilities for the process Z t = SX t ), where S is the scle function. The process Z t is time chnged Brownin motion, but since we only cre bout where not when the process exits the intervl S ),Sb )), then we need only determine the hitting probbilities for Brownin motion nd the result follows immeditely from Lemm.4. Before continuing to clculte quntities of interest, we fill in gp left erlier when we filed to completely specify the domin of the genertors of our one-dimensionl diffusions. Whether or not functions in the domin must stisfy boundry conditions t nd b is determined by the nture of those boundries from the perspective of the diffusion. More precisely, we hve the following clssifiction. Definition.44 Feller s boundry clssifiction) Define The boundry b is sid to be ux) = x x MdS, vx) = x x SdM. regulr if ub) < nd vb) < exit if ub) < nd vb) = entrnce if ub) = nd vb) < nturl if ub) = nd vb) = 8

13 with symmetric definitions t. Regulr nd exit boundries re sid to be ccessible while entrnce nd nturl boundries re clled inccessible. Theorem.45 The domin of the genertor ) is continuous functions f on [, b] which re twice continuously differentible on the interior nd for which. if nd b re inccessible there re no further conditions,. if b resp. ) is n exit boundry, then lim x b ) resp. lim Lfx) = x. If b resp. ) is regulr boundry, then for ech q [,] we get different process by restricting f in the domin to stisfy ) q lim Lfx) = q) lim S x)f x) resp. q lim Lfx) = q) lim S x)f x). x b x b x x For more creful discussion see Ethier & Kurtz 986), Chpter Green s functions Lemm.4 tells us the probbility tht we exit,b) for the first time through, but cn we glen some informtion bout how long we must wit for X t to exit the intervl,b) either through or b) or, more generlly, writing T for the first exit time of,b), cn we sy nything bout E[ T gx s )ds X = p]? Putting g = this gives the men exit time.) Let us write T wp) = E[ gx S )ds X = p] nd we ll derive the differentil eqution stisfied by w. We ssume tht g is continuous. First note tht w) = wb) =. Now consider smll intervl of time of length h. We re going to split the integrl into the contribution up to time h nd fter time h. Becuse {X t } t is Mrkov process, T T E[ gx s )ds X h = z] = E[ gx s )ds X = z] = wz) h nd so for < p < b wp) = E[ h gx s )ds X = p] + E[wX h ) X = p]. 4) 9

14 Since g is continuous nd the pths of X re continuous we hve the pproximtion E[ h gx s )ds X = p] = hgp) + Oh ). 5) Now subtrct wp) from both sides of 4), divide by h nd let h to obtin µp)w p) + σ p)w p) = gp), w) = = wb). 6) Let us now turn to solving this eqution. Using Lemm.4 we hve ) d dp S p) w p) = gp)mp) nd so S p) w p) = p gξ)mξ)dξ + β where β is constnt. Multiplying by S p) nd integrting gives wp) = p S ξ) ξ gη)mη)dηdξ + βsp) S)) + α for constnts α, β. Since w) =, we immeditely hve tht α =. Reversing the order of integrtion, nd wb) = now gives Finlly then wp) = = wp) = = β = p p p η S ξ)dξgη)mη)dη + βsp) S)) Sp) Sη))gη)mη)dη + βsp) S)) Sb) S) { Sp) S)) Sb) S) { Sp) S)) Sb) S) b b b Sb) Sη))gη)mη)dη. Sb) Sη))gη)mη)dη Sb) S)) p Sb) Sη))gη)mη)dη + Sb) Sp)) where the lst line is obtined by splitting the first integrl into b = b p + p. 3 p } Sp) Sη))gη)mη)dη } Sη) S))gη)mη)dη

15 Theorem.46 For continuous function g, where for < p < b we hve Gp,ξ) = E[ T gx s )ds X = p] = b Gp, ξ)gξ)dξ, { Sp) S)) Sb) S)) Sb) Sξ))mξ), for p < ξ < b Sξ) S))mξ), for < ξ < p, Sb) Sp)) Sb) S)) with S the scle function given in Definition.37 nd mξ) = mesure. σ ξ)s ξ) Definition.47 The function Gp,ξ) is clled the Green s function of the process X t., the density of the speed By tking g to pproximte x,x we see tht x x Gp,ξ)dξ is the men time spent by the process in x,x ) before exiting,b) if initilly X = p. Sometimes, the Green s function is clled the sojourn density. Exmple.48 Consider the Wright-Fisher diffusion with genertor Lfp) = p p)f p). Notice tht since it hs no drift term µ = ) it is lredy in nturl scle, Sx) = x up to n rbitrry dditive constnt). Wht bout E[T ]? Using Theorem.46 with g = we hve E p [T ] = E[ = = p T p p ds X = p] = p ξ) ξ ξ) dξ + dξ + p) ξ p Gp, ξ)dξ p ξ dξ = {p log p + p)log p)}. p)ξ ξ ξ) dξ In our Morn model, t lest if the popultion is lrge, then we expect tht if the current proportion of lleles is p, the time until either the llele or the A llele is fixed in the popultion hs men pproximtely {p log p + p)log p)}. 7) 3

16 In fct by conditioning on whether the proportion of -lleles increses or decreses t the first reproduction event, one obtins recurrence reltion for the number of jumps until the process first hits either zero or one. This recurrence reltion cn be solved explicitly nd since jumps ocur t independent exponentilly distributed times with men / N ), it is esy to verify tht 7) is indeed good pproximtion. For the Wright-Fisher model, in its originl timescle, there is no explicit expression for the expected time to fixtion, tp). However, since chnges in p over single genertion re typiclly smll, one cn expnd tp) in Tylor series, in just the wy we did to derive eqution ) nd thus verify tht for lrge popultion, This is redily solved to give just s predicted by our diffusion pproximtion. p p)t p) = N, t) = = t). tp) = N {p log p + p)log p)},..4 Sttionry distributions nd reversibility Before moving on to models in which gene is llowed to hve more thn two lleles, we consider one lst quntity for our one-dimensonil diffusions. First generl definition. Definition.49 Sttionry distribution) Let {X t } t be Mrkov process on the spce E. A sttionry distribution for {X t } t is probbility distribution ψ on E such tht if X hs distribution ψ, then X t hs distribution ψ for ll t. In prticulr this definition tells us tht if ψ is sttionry distribution for {X t } t, then d dt E [fx t) X ψ] =, where we hve used X ψ to indicte tht X is distributed ccording to ψ. In other words d E [fx t ) X = x]ψdx) =. dt Evluting the time derivtive t t = gives Lfx)ψdx) =. E E Sometimes this llows us to find n explicit expression for ψdx). Let {X t } t be one-dimensionl diffusion on, b) with genertor given by ). We re going to suppose tht there is sttionry distribution which is bsolutely continuous with respect to Lebesgue mesure. Let us buse nottion 3

17 little by using ψx) to denote the density of ψdx) on,b). Then, integrting by prts, we hve tht for ll f DL), = = b b { } σ x) d f df x) + µx) dx dx x) ψx)dx { d fx) σ dx x)ψx) ) d } dx µx)ψx)) dx + boundry terms. This eqution must hold for ll f in the domin of L nd so, in prticulr, Integrting once gives d σ dx x)ψx) ) d µx)ψx)) =. dx d σ x)ψx) ) µx)ψx)) = C, dx for some constnt C nd then using S x) s in integrting fctor we obtin from which d S y)σ y)ψy) ) = C S y), dy Sx) ψx) = C S x)σ x) + C S x)σ x) = mx)[c Sx) + C ]. If cn rrnge constnts so tht ψ nd b ψξ)dξ = then the sttionry density exists nd equls ψ. In prticulr, if b my)dy <, then ψx) = mx) b my)dy is sttionry mesure for the diffusion. Exmple.5 Recll the genertor of the Wright-Fisher diffusion with muttion, Lfx) = x f ) df x)d dx + ν x) ν x dx. 33