Population bottleneck : dramatic reduction of population size followed by rapid expansion,
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1 Selection We hve defined nucleotide diversity denoted by π s the proportion of nucleotides tht differ between two rndomly chosen sequences. We hve shown tht E[π] = θ = 4 e µ where µ cn be estimted directly. Therefore there is mens of estimting e. For exmple it ppers tht diversity in the popultion of Drosophil is bout ten times greter thn diversity in humn popultion. bundnt species hve more genetic diversity thn less bundnt species but the reltionship is not liner. Therefore we hve to consider other phenomen : Popultion bottleneck : drmtic reduction of popultion size followed by rpid expnsion turl selection.. Wright-Fisher Model with selection Definition.. The fitness of n individul is the number of offsprings it leves. The fitness of gene is the number of copies it leves. The fitness of n llele is the verge fitness of genes of tht llelic type. Definition. Wright-Fisher Model with selection. In pnmictic hploïd popultion of constnt size where individuls re of types nd if genertion t time t consists of k individuls of type nd k of type then ccording to the Wright-Fisher Model with selection the genertion t time t+ is formed by smpling independntly with replcement with P smpled = k + s k + s + k. s is clled selection coefficient. We sy tht hve reltive fitness + s :. If s > 0 : is sid to be beneficil If s < 0 : is sid to be deterious. Biologists think of n infinite pool of potentil offsrings from which new genertion re smpled proportions being dictted by. The next step is to dd muttions to this model : proportion µ of the pool of gmetes mutte to. Conversely proportion µ of gmetes mutte to. This leds to the following definition : Definition.3 Wright-Fisher Model with selection nd muttion. If there re k individuls of type mong prents nd k individuls of type then proportion of the potentil offspring tht re of type fter selection nd muttion is umber of offsprings is then Bin ψ k. ψ k = k + s µ k + s + k + kµ k + s + k. In order to get more mngeble model we hve to pss to diffusion pproximtion. To obtin non-trivil limit we suppose tht α = s ν = µ ν = µ nd we count in units of size.
2 Lemm.. s rescled Wright-Fisher with selection nd muttion converges to one-dimensionl diffusion with drift µp = αp p ν p + ν p nd vrince σ p = p p. Proof. Let δ t = be the time between two genertions in rescled time. s in neutrl cse for ll k 3 Ep / p k p = O. If current proportion of lleles is p the the ctul number is k p. We hve then : Ep / p p = ψ k k. But ψ k k = k + α ν + αk = = + αk + αk + kν + αk k + α ν k + kν k αk αk ν k + ν ν k α k αν k = αp ν p + ν ν p αp + o = αp p ν p ν p + o. Since ψ k = k + O E p / p p = ψ k ψ k + O. = p p + O ow for u : [0 ] R sufficiently differentible we hve by Tylor s Theorem : d dt E[up t p 0 = p] t=0 E [ up / up p 0 = p ] { = u pe [ p / p p 0 = p ] + u pe [ p / p p 0 = p ] + O = u pαp p ν p + ν p + u pp p + O. The lst term converges s to u pαp p ν p + ν p+ u pp p. Definition.4. We cll this diffusion the wek solution limit. Lemm.. Suppose there is no muttion ν = ν = 0. If initil proportion of lleles is p the probbility p fix tht eventully fixes in the popultion is } p fix = { exp α exp α if α 0 p if α = 0. 3 Proof. We hve previously seen tht p fix = Sp S0 S S0
3 where S is the scle function corresponding to the diffusion found in lemm. : x y µz Sx = exp x 0 σ z dz dy 4 nd µz = αz z σ z = z z. Tht leds us to : Sx = C exp αx C for some constnts C nd C independnt of x. The result follows. η Specil cses : Deterious lleles : s < 0. If s << nd s >> p fix s exp s. Beneficil lleles : s > 0 s << s >> then p fix s lmost independnt of popultion size. erly neutrl lleles : if s << then is nerly neutrl nd p fix. Summry : Most lleles beneficil or deterious re lost deterious muttions re more likely to fix in smll popultions fitness differences tht re too smll to be mesured in lbortory s << cn still hve evolutionnry impct if s >>. We hve here concentrted on genic selection. More generlly in diploïd popultions different forms of selection cn led to µp = sp p p.. The ncestrl selection grph To understnd how it works we use here the Morn model. Definition.5. In the Morn model for hploïd popultion of size rte pir of individuls selected t rndom one dies the other reproduces. To incorporte selection t dditionl rte s nother pir is chosen ; if both re the sme nothing hppens ; if one is nd the other is dies nd split in two. s plys the role of selection coefficient. Muttions re dded s Poisson process long the lineges. Lemm.3. s rescled Morn model with selection converges to the sme diffusion s in the Wright-Fisher model with selection nd muttion. Proof. For fixed the corresponding genertor of Morn model with selection is given by : L fx = p p fp + fp + p p fp fp + ν p fp fp + ν p fp + fp + s p p fp + fp. 5 3
4 Tht is to sy the genertor corresponding to the neutrl Morn model plus n extr term corresponding to the selection. We tke f to be twice continuously differentible. By Tylor s theorem the lst term of the right hnd side of 5 is equl to αp pf p + O. s L fx converges to Lfx where : Lfx = p pf p + ν ν + ν pf p + αp pf p. 6 To construct ncestry of smple we trce bck neutrl rrows ffecting two individuls in ncestry result in colescences s we hve seen before. Potentil selective events hitting individuls in ncestry led to bench in the ncestry. Ech individul re in pirs ech hit by potentil selective event t rte s = α k. So if there is currently k ncestrl lineges we go from k to k t rte nd we go from k to k + t rte αk. Definition.6. The system of brnching nd colescing lineges described here is clled the ncestrl selection grph. Lemm.4. There is with probbility finite rndom time when the number of lineges is for the first time. Remrk.. The corresponding individul is clled the ultimte ncestor. Proof. If we denote by X t the number of lineges t time t the corresponding embedded Mrkov Chin Y n hs trnsition mtrix P given by : P = Pk k + = α α+k Pk k = k α+k k. For k {0} let T k := inf {n Y n = k}. We wnt to prove tht P k T < = for ll k. We hve P k T < = P k {T < T } = lim P k T < T. For fixed {0} denoting u k = P k T < T nd pplying Mrkov property we hve : nd u k = k α + k u α k + α + k u k+ for k u = u = 0. Then for ll k u k = β k...β where β is given by u l = β l u l l. It is cler tht β k s. So u k s. If muttion rtes ν ν re strictly positive then diffusion describing llele frequencies hs sttionnry distribution. Indeed in this prticulr cse the density m of the speed mesure is given by : mx = Ce αx x ν x ν. In prticulr 0 mxdx <. Therefore ψxdx := mx 0 mydydx is sttionry mesure for the diffusion. In order to resolve the potentil solution events we smple the type of the ultimte ncestor from the sttionnry distribution nd work bck through the ncestrl selection grph. Following euhuser nd Krone we ssume tht muttions rtes re equl : ν = ν. 4
5 Figure : n exmple of n ncestrl selection grph ssuming the U is of type. 5
6 Figure : n exmple of n ncestrl selection grph ssuming the U is of type. 6
7 .3 dding structure to the colescent. In the Kingmn s Colescent we wnt to discrd the ssumption tht popultion size is constnt. Two things led to Kingmn s colescent : the probbility tht two individuls hve common prent is for lrge the probbility tht two distinct pirs of individuls hving commion prents nd the probbility tht three or more individuls hving common prents re both O. We suppose now tht the size of the popultion t genertion in the pst is t. Then the chnce p tht two lineges hve not colesced by time t is such tht : t p = s s= t = exp log exp s= t s= s s for s lrge. We suppose tht s is lrge nd tht we mesure it in unit of size for exmple M = 0 nd tht M Ms M ρs for some nice continuous function ρ. Then the probbility q tht two lineges hve not colesced by Mt is : Mt q = exp s s= Mt exp exp s= t 0 Mρs/M ρs ds. Exctly s in derivtion of Kingmn s colescent we don t see two lineges colescing in single genertion s M. So the genelogy when popultion chnge with time in units of size M is exctly like Kingmn s colescent except tht ech pir colesces not t rte but t instntneous rte ρs i.e. we get time chnge of Kingmn s colescent. Secondly we wnt to discrd the ssumption tht popultion is pnmictic. We suppose here tht popultion is subdivided into two llelic types with muttions between both. We recll here the Wright-Fisher model with muttion : if there re currently proportion p of lleles then the next genertion is smpled from n infinite pool of potentil offsprings of which proportion p µ + pµ re of type nd p µ + pµ re of type. We now suppose tht we smple n individuls of type nd n individuls of type. The chnce tht two of the n individuls hve common prent n is p µ +µ p = n colescence of two of the n individuls of type is proportion p + O. In the sme wy the chnce of p +O. Of the type gmetes µ p p µ +µ p = ν p p + O rises through muttion from gmetes. 7
8 ν Similrly proportion p p + O of type gmetes rises through muttion from type. Following single linege of type we trce bck rndom time which in units of size is pproximtely exponentil with instntneous rte ν p p until the ncestor s type chnge to. The probbility tht there is muttion nd colescence of ncestrl lineges in single genertion is O. So we do not see this event in limit s. The process ρt tht determines the frequency of individuls s we trce bck in time under rhe colescent rescling converges to time reversl of the Wright-Fisher diffusion. 8
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