Universal tree structures in directed polymers and models of evolving populations

Transcription

1 PHYSICA REVIEW E 78, Universal tree structures in irecte polymers an moels of evolving populations Éric Brunet,* Bernar Derria, an Damien Simon aboratoire e Physique Statistique, École ormale Supérieure,, rue homon, 75 Paris Ceex 05, France Receive 0 June 008; publishe December 008 By measuring or calculating coalescence times for several moels of coalescence or evolution, with an without selection, we show that the ratios of these coalescence times become universal in the large size limit an we ientify a few universality classes. DOI: 0.0/PhysRevE PACS numbers: 05.0.a, k, Hc, 87..Kg Ranom trees appear in many contexts in biology, mathematics, an physics. In evolutionary biology, they represent the genealogies of reproucing populations. In physics, ranom trees appear in many systems such as iffusion limite aggregation DA, coarsening, river networks,, iagrams in perturbation theory, ultrametric structure of pure states in mean-fiel spin glasses,5, irecte polymers in a ranom meium 6,7, shocks in one-imensional turbulence 8 0, etc. From a mathematical point of view, one of the simplest examples of ranom trees is Kingman s coalescent,: it escribes the coalescence tree of particles, where each pair of particles has a probability t of coalescing into a single particle uring every infinitesimal time interval t. The ranom tree structures of Kingman s coalescent are ientical to the genealogies obtaine in simple mean fiel moels of neutral evolution such as the Wright-Fisher moel,. In such moels, each iniviual of a population of fixe size at a given generation gives birth to a ranom number of offspring an the population at the next generation is obtaine by choosing survivors at ranom among all these offspring. If one follows the evolution over a large enough number of generations for the initial conition to be forgotten, a steay state is reache where the statistics of the genealogical tree of a large population are ientical to those of Kingman s coalescent. Other ranom trees have been consiere in the mathematical literature, such as the coalescents 5 7, which generalize Kingman s coalescent an escribe a wier class of mean-fiel coalescence moels 8. Inthe coalescent, each subset of k particles among n particles has a probability n,k t of coalescing into a single particle uring an infinitesimal time t. As a set of n particles can be consiere as a subset of a larger set of n+ particles, the rates n,k have to satisfy some consistency relations: the coalescence of k particles in the subset of size n happens in two cases: either these k particles coalesce in the set of size n+ rate n+,k or they coalesce together with the n+-th particle rate n+,k+. Therefore n,k = n+,k + n+,k+. This recursion leas to the following general expression for the coalescence rates 5,6: n,k =0 x k x n k xx, where is some positive measure on the interval 0,. With these notations, Kingman s coalescent correspons to x = x. Another particular case, which has been stuie, in the context of spin glasses, is the Bolthausen-Sznitman coalescent 9 for which x=. Trees in the Kingman s coalescent an in the Bolthausen-Sznitman coalescent have ifferent statistical properties. In orer to compare ifferent moels of physical or biological systems which generate ranom trees an to try to ientify universality classes, we consier here simple quantities characteristic of these ranom tree structures. For a tree with a large number of en points, we efine T p as the istance one has to go up into the tree to fin the most recent common ancestor of p given points see Fig.. For moels of evolving populations, the istance T p is the age of the most recent common ancestor of p iniviuals chosen at ranom in the population. In general, it epens both on the generation at which these p iniviuals live, but also on the choice of the p iniviuals, even in the limit of very large trees. This ouble source of fluctuations for the T p is reminiscent of what happens in mean fiel spin glasses 5: as for the overlaps in Parisi s theory, the istribution of the T p remains broa even when the size of the population becomes very large 5,0. For a given moel, one can try to etermine averages T p or moments T p k of these times T p the averages are taken over all the branches of the tree, i.e., over all the population at a given generation, an over all the ranom trees, i.e., over all the generations in the language of moels of evolution. In recent works,, it was notice that for a large class of mean fiel moels of evolution with selection, the ratios of these average times T p take, for a large population, simple universal values inicating that the genealogical trees T T T *Eric.Brunet@lps.ens.fr Bernar.Derria@lps.ens.fr Damien.Simon@lps.ens.fr FIG.. The times T p are the ages of the most common ancestors of p iniviuals chosen at ranom /008/786/ The American Physical Society

2 BRUET, DERRIDA, AD SIMO are istribute accoring to the statistics of the Bolthausen- Sznitman coalescent. Therefore, at the mean-fiel level an for a large size of the population, two universality classes seem to emerge for moels of evolution: Kingman s trees in the case of neutral evolution for which T T =, T T =, T T =, T T = 6 9, an Bolthausen-Sznitman s trees in the case of selection T T = 5, T T = 5 8, T T =, T T =. The goal of the present work is to try to measure these coalescence ratios for other moels of evolution, in particular to analyze the effect of spatial fluctuations, an to argue that irecte polymers in a ranom meium are in the same universality classes as evolution moels in presence of selection. The paper is organize as follows. In Sec. I we consier, at the mean fiel level or in finite imension, coalescence moels which are equivalent, as we will see, to neutral moels of evolution. Above two imensions of space, the coalescence trees have the same statistics as in mean fiel with coalescence times given by Eq., whereas in one imension, they lea to a ifferent universality class for which we compute the ratios of coalescence times. In Sec. II, we consier the trees of optimal paths in the problem of irecte polymers in a ranom meium. Our numerical results will show that at the mean fiel level, the trees satisfy the Bolthausen-Sznitman statistics Eq., whereas the ratios of coalescence times vary with imension as expecte by the known universality classes of the problem. I. COAESCECE AD MODES OF EUTRA EVOUTIO A. Kingman s coalescent Kingman s coalescent, is a mean-fiel moel of coalescing particles: uring each infinitesimal time interval t every pair of particles has a probability of coalescing into a single particle. Therefore if one starts with p particles, there is a ranom waiting time p until a coalescence event occurs when these p particles become p particles. Then there is another ranom time p until a pair among these p particles coalesce an one is left with p particles, an so on. The times k are inepenent an istribute accoring to exponential istributions kk kk k k = exp 5 an the time T p for p particles chosen at ranom to coalesce is given by k T p = p + p This allows one to easily recover the values of Eq.. In fact the whole generating functions of the times T p can be calculate as PHYSICA REVIEW E 78, p e T kk p = k= kk. 7 In particular one can notice that the time T has an exponential istribution. B. Wright-Fisher moel The Wright-Fisher moel, is one of the simplest neutral moels of an evolving population. It escribes a population of constant size with nonoverlapping generations an asexual reprouction. At each generation, all the population is replace by new iniviuals with the following rule: each iniviual at a given generation has its parent ranomly chosen among the iniviuals at the previous generation. If one goes backwar in times, the lineage of an iniviual performs a ranom walk on a fully connecte graph of sites. Following the lineages of p iniviuals is the same as following p coalescing ranom walks on this fully connecte graph. Since the ranom walks are inepenent, the statistics of coalescence times can be easily calculate,: for p iniviuals chosen at ranom at generation g, the time T p is the age of their most recent common ancestor, i.e., T p is the number of time steps for the p ranom walkers on the fully connecte graph to coalesce. At each generation in the past, two istinct lineages have a probability / of merging, thus T scales as the size of the population. For fixe p, the probability that a pair of lineages coalesce is / whereas multiple coalescences occur with higher powers of / for large. One can then neglect these multiple coalescences an T p p + p + + +, 8 where the times k are istribute accoring to Eq. 5, implying that the statistics of the times T p are exactly the same as in Kingman s coalescent Eq.. C. Coalescing ranom walks in finite imension We are now going to look at coalescing ranom walks on an hypercube of = sites in imension with perioic bounary conitions. We consier the continuous time case, where uring infinitesimal time interval t, each walker on the hypercube has a probability t of hopping to each of its neighboring sites, an whenever two walkers occupy the same site, they instantaneously coalesce into a single walker. If T r is the coalescence time between two walkers at a istance r apart, its evolution is = T r t + T r with probability t, t + T r + e i with probability t, 9 where e i is one of the unit vectors on the hypercubic lattice. It is clear that the istance between the two walkers performs a ranom walk an that T is simply the first time that this istance vanishes. This is of course a very well known first passage problem,5 which can be solve easily it 060-

3 UIVERSA TREE STRUCTURES I DIRECTED reuces to the inversion of a aplacian: the generating function of T r satisfies for t an for r0 e T r = e t tet r +te T r+e i, i= 0 where enotes an average over all the ranom walks. At r=0, it satisfies the bounary conition e T 0 =. For r0, one can rewrite Eq. 0 as e T r +e T r+e i e T r =0 i= an this can be easily solve in Fourier space to give e T r = A n =0 n =0 in r exp + i= cos n i, where the constant A is fixe by the conition of Eq.. Starting with two particles at ranom positions on the lattice an averaging over these two positions leas to e T = + n0 This implies that T = n0 T = n0 n0 + + i= cos n i. i= cos n i, 5 i= i= cos n i cos n i. 6 For large, it is well known,5 that Eq. 5 gives for =, T ln for =, 7 for. On the other han, one can show that the secon term in the right-han sie of Eq. 6 grows as in imension an as in imension. Therefore for, the ratio T /T goes to when, as in the mean-fiel case Eq.. In fact it has been prove,6 that in an for large the whole genealogies of p iniviuals average over all their positions are given by the Kingman coalescent, up to the rescaling 7. This implies that the istribution of the time T is exponential an that the moments of all the coalescing times T p have their mean fie values above the upper critical imension = which is the well known upper critical imension of coalescing ranom walks 7. D. Coalescing ranom walks in one imension In imension, the two terms in the right-han sie of Eq. 6 are comparable, an the ratio T /T no longer converges to. In imension = the calculation of all the moments of the times T p is rather straightforwar. First one can easily solve Eq. for perioic bounary conitions with conition Eq. an one gets e T r = / r / r / /. 8 For large, this becomes a scaling function of an of r/ e T r cos r cos an, averaging over r, one gets PHYSICA REVIEW E 78, = e T tan, sin r + sin r sin 9 0 which shows that the istribution of T is no longer exponential. One can write own the equations satisfie by the generating functions of the times T p. For large an =O the solution is p sinrk / e T p r,...,r p, k= sin / where the r k are the istances between consecutive particles along the ring one has of course r + +r p =. In particular, for p=, r =r an r = r, one recovers Eq. 9. Averaging Eq. over all the positions of the p particles on the ring leas to e T ppp x sinx / x 0 sin p. / From Eq. one can then obtain all the moments of T p. For example, one has 060-

4 BRUET, DERRIDA, AD SIMO PHYSICA REVIEW E 78, p p + T p p +p + an one can show T T = 7 5, T T = 8 5, T T = 5, T T = 5, in contrast with Eqs. an. One coul repeat the calculations which lea to Eqs. 5 7 an Eq. for moels of coalescence on other lattices or with more general jumping rates. As long as the motion of the coalescing particles remains iffusive, one woul recover the same values Eq. or Eq. for the statistics of the trees. E. eutral evolution in finite imension One can try to generalize the Wright-Fisher moel to the finite-imensional case, for example, by consiering an hypercube with a finite population of fixe size m on each lattice site, an the case where each iniviual chooses its parent in the previous generation with a probability p on the same lattice site an with probability p on one of the neighboring sites. The stuy of the genealogies in this case is obviously the same problem as following the coalescences of the lineages which perform ranom walks on this lattice. Therefore in imension = an above, the trees are given by the statistics Eq. of Kingman s coalescent whereas in imension = they will be in the universality class Eq. of coalescing ranom walks in one imension. II. DIRECTED POYMERS I A RADOM MEDIUM Directe polymers in a ranom meium is one of the simplest examples of a strongly isorere system 7,8 0. It escribes irecte paths in a ranom energy lanscape. In its zero-temperature version, the problem reuces to fining the optimal path, i.e., the path of minimal energy in this ranom energy lanscape. The optimal paths starting at the same point but arriving at ifferent points give rise to a tree structure, that we try to characterize in this section by measuring the coalescence times T p. A irecte polymer in imension + is a line extening in one of the irections traitionnaly calle time, an which we represent as the vertical irection in Fig. an Fig. with some ranom excursions in the other transverse irections see Fig.. We consier here irecte polymers on a lattice which is infinite in the time irection but finite an perioic in the transverse irections. In each time section, there are = sites locate on a -imensional hypercube of linear size with perioic bounary conitions. Each site in a given time section is connecte to M = sites in the previous time section an it is also connecte to M other sites in the next time section. The way each site is connecte is shown for imension + in Fig.. In higher imensions, we generalize the lattice of Fig. in the following way: let x =x,x,...,x be the transverse coorinates of a given site; the x i are integers at even times an FIG.. eft a irecte polymer in imension +. The time irection is vertical. Right A irecte polymer arriving at A comes either from B or from C, whichever is more energetically favorable: In the example shown, the coalescence time of the irecte polymers arriving at B an C is. half-integers at o times, an the M = potential parent sites of x have coorinates x /,x /,...,x / in the previous time section. We consier also a mean-fiel version Fig., where there is no spatial structure in the transverse irections: A time section consists of a set of sites, an each of them is connecte to M sites chosen at ranom among the sites of the previous time section, where M might be any number between an. We assume that each link AB between two connecte sites A an B carries a ranom energy AB. The energy E of the polymer is then the sum of all the energies AB of the visite links. We choose an origin where the polymer starts, an for any given site A on the lattice, we call E A the minimal energy of the polymer over all the possible irecte paths connecting this origin to A. At zero temperature, the irecte polymer chooses the path which minimizes its energy an one has the simple recursion relation E A = mine B + AB,E C + AC,..., 5 where B,C,..., are the M potential parent sites of site A. For any pair of sites A an A in the same time section, we efine their coalescence time see Fig. as the number of up steps uring which the two optimal paths arriving at A an A iffer we suppose that the origin of the irecte polymers is at a remote enough time in the past for the paths to coalesce. In a similar way, we efine the coalescence times of any group of p ifferent sites as the maximal coalescence time of any pair within the p sites. All these quan- FIG.. Directe polymer in mean fiel. At a given time, each of the = sites is connecte to M = ranom sites at the previous time. B A C 060-

5 UIVERSA TREE STRUCTURES I DIRECTED T imensions + imensions + imensions (mean fiel) 0 0 FIG.. Average coalescence time T of two iniviuals for several moels of irecte polymers in imensions + an +, an in mean-fiel, as a function of the number of sites in each time section. The ata are compare to the preiction T / in otte lines for imensions + an +. ote that, by chance, two out of the four moels in imension + an two out of the three moels in imension + have nearly the same prefactor an their ata are unistinguishable. PHYSICA REVIEW E 78, tities epen on the chosen sites an on the realization of the isorer an, as in the previous section, we note by the average over the choice of sites an the isorer. In this section, we consier the average coalesence time T p an the average square of the coalesence time T p of p sites. We have simulate four moels in imension +; from top to bottom on Fig. : on the lattice of Fig. with a iscrete istribution of with values =0 or = with probabilities /, on the lattice of Fig. with a uniform istribution of in 0,, on the lattice of Fig., with negative values of, istribute accoring to =e, on a square lattice where each site is connecte to M = parents just above itself, on its right an on its left where takes positive values, with an exponentially ecreasing istribution: =e +. In imension +, we have simulate three moels all on the lattice with M = ancestors escribe above; from top to bottom on Fig. : with an exponentially increasing istribution =e +, with a uniform istribution of in 0, with an exponentially ecreasing istribution =e +. Finally, we have simulate two moels in mean-fiel with a uniform istribution of in 0, an either M = or M = ranom ancestors for each site M = is above M = in Fig.. Our ata for all these moels are plotte together with the same symbol for each imension to emphasize the universality of our results. To measure the T p s, the conceptualy simplest way is to upate a matrix containing for all pairs i, j of iniviuals the time T i, j of their most commun ancestor. Inee, for an arbitrary number p of iniviuals, one has T p i,...,i p =maxt i,i,t i,i,...,t i,i p, so that the matrix of the T s contains all the relevant information. Upating this matrix at each time step is easy: The T of two ifferent sites is one plus the T of their parents, an the T of a site with itself is zero. Because upating at each time step a matrix is time consuming, we use a more sophisticate metho, where we keep track of the genealogical tree of all the sites at a current time: there are of course sites at the current time, an at most noes, where a noe is a site from previous times which is the most recent common ancestor of two sites at the current time. At each time step, upating the whole tree takes a time linear in, an averaging the T p over all the choices of p iniviuals takes also a time linear in, as one simply has to recursively walk own the tree from its root an count for each noe the number of times it is the most recent commun ancestor of p sites in the current time. This algorithm is escribe in more etails in Ref.. For each ata point, we have run one long simulation an average our results over all the time steps once the steay state was reache. This is equivalent to averaging over many inepenent realizations if we run a simulation for a time much longer than the correlation time, which we estimate to be of the orer of magnitue of T. All of our simulations were at least times longer than T. In Fig., we plot the coalescence time T as a function of the system size. For irecte polymers on a lattice which is infinite both in the time irection an in the transverse irections, the transverse isplacement of the optimal path scales as t, where t is the length of the irecte polymer an is a universal exponent 7 equal to + =/ in imension + an in imension +. In our setup, with a finite lattice of linear size in the transverse irections, this scaling can only hol as long as tt corr with T corr == /. This time T corr is the correlation time on the scale of which the system forgets its initial conition. Moreover, if we consier several sites an the optimal paths arriving at these sites, these paths coalesce on a time scale of the orer of T corr, as can be seen in Fig.. In mean-fiel with a finite number M of potential ancestors per site, there is no notion of istance in the transverse irections, an the exponent is meaningless. We therefore expect a ifferent scaling. The problem of zero-temperature mean-fiel irecte polymers can be formulate as a noisy Fisher-Kolmogorov-Petrovsky-Pisconov Fisher-KPP like equation,. Recently, a phenomenological theory of coalescence trees in moels of Fisher-KPP fronts suggeste,,5 that the coalescence time in such moels shoul be of orer T corr ln. On Fig., one can see that the ata seem to have a slower growth than a power law, but the values of we simulate here are too small to check the ln preiction. Better simulations on a closely relate moel are presente in Ref., where the ln scaling appears clearly. We now turn to the ratios of coalescence times. Figure 5 shows the ratios T /T an T /T as a function of the system size for all the moels we stuy four moels in imension +, three in imension +, an two in meanfiel. umerically, all the symbols for a given imension overlap of large : these ratios seem to epen only on the imension, an not on the istribution of the bon energies, nor on the shape of the lattice. The results in meanfiel are compatible with the preiction that for an infinitely large system in the Fisher-KPP front equation class, the genealogical tree converges to a Bolthausen-Sznitman coalescent, with ratios given by Eq.. In imensions + an +, our numerical results inicate clearly that we have tree 060-5

6 BRUET, DERRIDA, AD SIMO T T statistics ifferent from the Bolthausen-Sznitman coalescent, an also ifferent from the Kingman coalescent for which T /T woul be / asineq.. On Fig. 6, we show the ratios T /T an T /T. Here, the situation is less clear: the symbols for the ifferent moels o not superpose an the ratios o not seem to have converge in particular, the mean-fiel ratios are rather far from the preiction. For some reason we o not unerstan, it seems that the T p /T nee much larger values of to converge to their final values than the T p /T.We alreay observe a similar phenomenon on an exactly solvable relate moel. We also measure the ratios T /T, where T is the age of the most recent common ancestor of the whole population an foun these ratios to be close to.9 in imensions + an +, while it iverges in meanfiel. A. ong tail istributions T T FIG. 5. Ratios of coalescence times for irecte polymers at zero temperature as a function of the size of the system in, from top to bottom, imension +, imension +, an meanfiel. The otte line represents the preiction Eq. for meanfiel in the limit of infinite size. eft ratios T /T. Right ratios T /T T.95.9 T T T FIG. 6. Ratios of moments of the coalescence times for irecte polymers at zero temperature as a function of the size of the system in, from top to bottom, imension +, imension +, an meanfiel. The otte line represents the preiction Eq. for meanfiel in the limit of infinite size. eft ratios T /T. Right ratios T /T. T T an T T T T In the irecte polymer problem, it is known that the scaling regime is moifie when the istribution of the energies of the bons ecays as a power law for large negative : when c with c 7, the irecte polymer in imension + has an anomalous scaling 7,6 an the exponent epens on. We have measure the coalescence times in imension + for a istribution of energies given by = size = 00 size = α A +, T T FIG. 7. Ratios T /T an T /T as a function of the exponent appearing in the noise Eq. 6, for two ifferent system sizes in imension +. 6 with an A such that is normalize, for sizes =00 an =00. The ratios T /T an T /T are presente in Fig. 7. We observe that, for large, these ratios converge towars the universal values shown on Fig. 5, while for +, they seem to converge close to, respectively,. an.5. As we expect T to scale like /, it is possible to obtain a rough estimate of the exponent from the only two atapoints at sizes =00 an =00. This estimate is shown in Fig. 8. ν PHYSICA REVIEW E 78, α FIG. 8. Estimate of the exponent of the irecte polymer as a function of the exponent appearing in the istribution of in Eq. 6. This exponent has been evaluate from the formula ln/lnt =00/T =00. The universal value + =/ for istributions ecaying fast enough is also shown

7 UIVERSA TREE STRUCTURES I DIRECTED T.8 T =6 = The exponent seems to converge towar the universal value + =/ for large, while it seems to be for. As with previous numerical stuies 7, our results are not precise enough to etermine accurately the critical c above which =/. B. Discrete istributions =6 =6 = =6 We are now going to iscuss the case where the energies of the bons take iscrete values. In this case, it may happen in Eq. 5 that there are several paths coming from ifferent potential parent sites in the previous time section with the same minimal energy, an the question is, of course, which path shoul be selecte as the parent site. The simplest iea is to choose ranomly at each time step with equal probabilities one of the parent site with the lowest energy. With this proceure, we have run numerical simulations in imension + for several sizes with a binary noise for the energies of the bons =0 with probability p, 7 with probability p, for several values of p. Our results for the ratio T /T as a function of p are shown in Fig. 9 as otte lines. As p varies, we observe a crossover between two values: for small p, T /T.6 as for irecte polymers in imension + when the istribution of energies is continuous see Fig. 5 an, for large p, T /T. which correspons to the coalescence of ranom walks in imension, as in Eq.. The crossover between the two regions becomes sharper as increases, which suggests a phase transition. The critical value of p is very consistent with the known threshol 0.67 for irecte percolation on the same lattice 7. Thus, the system behaves similar to the neutral moel when the =0 bons percolate. p FIG. 9. Ratios T /T as a function of p for the istribution of of Eq. 7 in imension +. The ashe lines correspon to the simplest proceure of choosing with equal probabilities one of the potential parent sites realizing the minimal energy, an the plain lines represent the results using the weights which correspons to the T 0 + limit of finite temperature irecte polymers. The vertical otte line inicates the irecte percolation threshol on the same lattice. PHYSICA REVIEW E 78, Instea of choosing with equal probabilities which bon the polymer follows when they are energetically equivalent, there is an alternative proceure which correspons to taking the limit T 0 + in the problem of irecte polymers at a finite temperature T. At finite temperature, we keep track for each site A of the partition function Z A of a polymer arriving on A. Assuming that the site A has M = potential parent sites B an C, we have the recursion Eq. 5 Z A = Z A B + Z A C, 8 where Z A B =Z B exp AB is the partition function of a irecte polymer arriving on A via the site B an where =/T. The probability that a polymer reaching A comes from B is given by probability the polymer comes from B = Z A B. 9 Z A At very low temperature, the partition function is ominate by the lowest energy paths Z e E, 0 where E is the minimal energy an the number of ways that this energy E can be obtaine, so that Eq. 8 reas, at low temperature, A e E A B e E A B + C e E A C, where E A B =E B + AB is the minimal energy of the path arriving at A through B. IfE A B E A C, then the first term in the right-han sie of Eq. ominates an we obtain E A =E A B an A = B. Furthermore, from Eq. 9, the chosen path comes from B. On the other han, if E A B =E A C, both terms in Eq. have the same orer of magnitue an we obtain E A =E A B =E A C an A = B + C. Then, from Eq. 9, the probability that the irecte polymer comes from B is B / A. In this way, we not only choose the optimal energy but we also keep track of entropy effects. We have run numerical simulations with the same parameters as above but with this new proceure. The ratios T /T are shown on Fig. 9 in plain lines. For small values of p, both proceures yiel the same results. For larger p, however, the ifference is striking, an the phase transition seems to have isappeare: on both sies of the percolation threshol the ata seem to be in the same universality class as they converge to.6. III. COCUSIO In this paper, we have presente analytical an numerical results showing the existence of universality classes in the tree structures which appear in several moels of evolution an in irecte polymers see Table I for a summary. Without selection, the genealogies of neutral moels like the Wright-Fisher moel or coalescing ranom walks are escribe above the critical imension c = by the Kingman coalescent. For = the universality class is ifferent: we have obtaine the istribution Eq. of the ages T p of the most recent common ancestor of p iniviuals. For irecte polymers in a ranom meium at zero temperature, the same coalescence times T p have been measure 060-7

8 BRUET, DERRIDA, AD SIMO PHYSICA REVIEW E 78, TABE I. Universal ratios an orer of magnitues of coalescence times for moels of evolution with an without selection, an irecte polymers in a ranom meium. We coul not reach large enough system sizes to give a reliable numerical preiction for the ratios T p /T p. T T T T T T T T T T T Kingman coalescent 6 Coalescing ranom walks in 9 eutral moels of evolution in eutral moel in = ln eutral moel in = Bolthausen-Sznitman s coalescent 5 5 Moels of evolution with selection ln 8 or irecte polymers at zero temperature in mean fiel Moels of evolution with selection in imension or irecte polymers at zero temperature in imension ?? / Moels of evolution with selection in imension or irecte polymers at zero temperature in imension ?? / numerically. In the mean fiel case, their values are compatible with Bolthausen-Sznitman s coalescent, which is alreay known to appear in spin glasses 9 an in branching ranom walks with a selection mechanism keeping the size constant,. In low imension at least = an =, the coalescence times belong to ifferent universality classes. It woul be interesting to preict analytically the values of T /T an T /T measure numerically in Fig. 5 for fast ecaying istributions of as well as the ones obtaine in Fig. 7 for power-law istributions of with exponent +. In the mean-fiel case, it woul also be interesting to know if the replica metho can be use in orer to etermine the coalescence times. The simulations presente in this paper eal only with irecte polymers at T = 0. Directe polymers exhibit a phase transition for as the temperature increases 8. We expect the tree statistics to change at T c from the universality class of irecte polymers at zero temperature to the universality class of coalescing ranom walks. The construction of the minimal energy path for irecte polymers can be relate to spatial moels in presence of selection. In population ynamics, selection can be taken into account through a parameter, calle the fitness or the aaptability, which characterizes the ability of an iniviual to survive an reprouce 9. Iniviuals with a higher fitness have a higher probability of having a escenance. This parameter is transmitte from parents to offspring up to fluctuations ue to mutations. An analogy can be rawn between the minimal energy of a irecte polymer arriving on a site, an minus the fitness of an iniviual living on a site. In presence of local selection, a spatial moel of population coul therefore be formulate as follows: on each site there woul be one or a finite number m of iniviuals; at each generation, each iniviual woul branch into k offspring with mutate fitnesses. These offspring iffuse an, uner the effect of selection, only the best or the m best iniviuals on each site woul be kept. Because of the similarity of such spatial moels of population ynamics in presence of selection with the irecte polymers, we expect these moels to belong to the same universality classes. We performe preliminary simulations on such a spatial moel of evolution with selection in imension + with m=5 iniviuals per site. Our results for the ratios T /T an T /T coincie with those of irecte polymers

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