Convergence of a Moran odel to Eigen s quasispecies odel Joseba Dalau Université Paris Sud and ENS Paris June 6, 2018 arxiv:1404.2133v1 [q-bio.pe] 8 Apr 2014 Abstract We prove that a Moran odel converges in probability to Eigen s quasispecies odel in the infinite population liit. 1 Introduction The concept of quasispecies was proposed by Manfred Eigen in order to explain how a population of acroolecules behaves when subject to an evolutionary process with selection and utation. In his celebrated paper [8], Eigen odels the evolution of a population of acroolecules via a syste of differential equations, which arises fro the laws of cheical kinetics. Selection is perfored according to a fitness landscape, and utations occur in the course of reproductions, independently at each locus with rate q. On the sharp peak landscape all but one sequence, the aster sequence, have the sae fitness and the aster sequence has higher fitness than the rest Eigen discovered that an error threshold phenoenon takes place: there exists a critical utation rate q such that if q > q then at equilibriu the population is totally rando, while if q < q then at equilibriu the population fors a quasispecies, i.e., it contains a positive fraction of the aster sequence along with a cloud of utants that closely reseble the aster sequence. The concepts of error threshold and quasispecies ight not only be relevant in olecular genetics, but also in several other areas of biology, naely population genetics or virology [7]. Nevertheless, in Eigen s odel the dynaics of the concentrations of the different genotypes is driven by a syste of differential equations, which is a ajor drawback for the viability of the odel in settings ore coplex than the olecular level [18]. A finite and stochastic version of Eigen s quasispecies odel would be uch ore suitable to expand the quasispecies theory to other areas [9, 16, 18]. 1
The issue of designing a finite population version of the quasispecies odel has been tackled by several authors. Different approaches have been considered in the literature: Alves and Fontanari [1] propose a finite population odel and they study the dependence of the error threshold on the population size, a siilar approach is taken by McCaskill [11], Park, Muñoz, Dee [14] and Saakian, Dee, Hu [15], who all suggest different kinds of finite population odels. In [13], Nowak and Schuster derive the error threshold for finite populations using a birth and death chain. More recently, in [2, 3], Cerf shows that the error threshold and quasispecies concepts arise for both the Moran odel and the classical Wright Fisher odel in the appropriate asyptotic regies. Soe other authors propose stochastic odels that converge to Eigen s odel in the infinite population liit, this is the approach taken by Deetrius, Schuster, Sigund [5], who use branching processes, Dixit, Srivastava, Vishnoi [6] or Musso [12]. Showing convergence of a finite population odel to Eigen s odel is in general a delicate atter; to our knowledge, all the works that have been done in this direction prove that soe stochastic process converges to Eigen s odel in expectation. As pointed out in [5], convergence in expectation can be isleading soeties, ainly due to the fact that variation ight increase as expectation converges, leading to a poor understanding of the asyptotic behaviour of the stochastic process. In the current work we consider the Moran odel studied in [2, 4] driving the evolution of a finite population subject to selection and utation effects. This Moran odel is shown to converge to Eigen s quasispecies odel in the infinite population liit, independently of the fitness landscape: on any finite tie interval, we prove convergence in probability for the supreu nor. The interest of our result not only lies on the type of convergence, but also on the choice of the odel: the Moran odel is possibly one of the siplest odels for which such a result can be expected. The result is proven by eans of a theore due to Kurtz [10], which gives sufficient conditions for the convergence of a sequence of Markov processes to a deterinistic trajectory, characterised by a syste of differential equations. The article is organised as follows: first we briefly introduce Eigen s quasispecies odel and the Moran odel. We state the ain result in section 3. In section 4 we adapt Kurtz s theore, which originally deals with continuous state space Markov chains, to the discrete state space setting. Finally, in section 5, we apply Kurtz s theore in order to prove the ain result. 2
2 The Eigen and Moran odels We present here Eigen s quasispecies odel and a discrete Moran odel. Consider a set of N different genotypes, labelled fro 1 to N. Both Eigen s odel and the Moran odel describe the evolution of a population of individuals having genotypes 1,...,N. In both odels the evolution of the population is driven by two ain forces: selection and utation. The selection and utation echaniss depend only on the genotypes, and are coon to both odels. Selection is perfored with a fitness landscape (f i ) 1 i N, f i being the reproduction rate of an individual having genotype i. The utation schee is encoded in a utation atrix(q ij ) 1 i,j N, Q ij being the probability that an individual having genotype i utates into an individual having genotype j. The utation atrix is assued to be stochastic, i.e., its entries are non negative and the rows add up to 1. Eigen s odel. Eigen originally forulated the quasispecies odel to explain the evolution of a population of acroolecules. The evolution of the concentration of the different genotypes is driven by a syste of differential equations, obtained fro the theory of cheical kinetics. Let us denote by S N the unit siplex, i.e., S N = {x R N : x i 0, 1 i N and x 1 + +x N = 1}. An eleent x S N represents a population in which the concentration of the individuals having the i th genotype is x i, for 1 i N. Let x 0 S N be the starting population and let us denote by x(t) the population at tie t > 0. Eigen s odel describes the dynaics of x(t) thorough the following syste of differential equations: ( ) x i(t) = N N f k Q ki x k (t) x i (t) f k x k (t), 1 i N, k=1 k=1 with initial condition x(0) = x 0. The first ter in the differential equation accounts for the replication rate and utations towards the i th genotype, while the second ter helps to keep the total concentration constant. A recent review on Eigen s quasispecies odel can be found in [17]. The Moran odel. Moran odels ai at describing the evolution of a finite population. The dynaics of the population is stochastic, the evolution is described by a Markov chain. Loosely speaking, the Moran odel evolves as follows: at each step of tie, an individual is selected fro the current 3
population according to its fitness, this individual then produces an offspring, which is subject to utations. Finally, an individual chosen uniforly at rando fro the population is replaced by the offspring. The state space of the Moran process will be the set PN of the ordered partitions of the integer in at ost N parts: P N = {z N N : z 1 + +z N = }. An eleent z P N represents a population in which z i individuals have the genotype i, for 1 i N. The only allowed changes at each tie step consist in replacing an individual fro the current population by a new one. If we denote by (e i ) 1 i N the canonical basis of R N, the only allowed changes in a population are of the for z z e i +e j 1 i,j N. Let λ be a constant such that λ ax{f i : 1 i N }. The Moran process is the Markov chain (Z n ) n 0 having state space PN and transition atrix p given by: for all z Pl+1 and i,j {1,...,N } such that i j, p(z,z e i +e j ) = z i 1 N f k Q kj z k. λ The other non diagonal coefficients of the transition atrix are null, the diagonal coefficients are arranged so that the atrix is stochastic, i.e., the entries are non negative and the rows add up to 1. k=1 3 Main result Our ai is to show that Eigen s quasispecies odel arises as the infinite population liit of the Moran odel. More precisely, we will prove the following result: Theore 3.1. Let (Z n ) n 0 be the Moran process described above. Suppose that we have the convergence of the initial conditions towards x 0 : 1 li Z 0 = x 0, and let x(t) be the solution of the syste of differential equations ( ) with initial condition x(0) = x 0. Then, for every δ,t > 0, we have ( li P sup 0 t T ) 1 Z λt x(t) > δ = 0. 4
This result is an iediate consequence of theore 4.7 in [10]. In order to prove the result, we proceed in two steps. We state first theore 4.7 in [10], and we show next that all the hypotheses needed to apply the theore are fulfilled in our particular setting. 4 Convergence of a faily of Markov chains Let d 1 and let E be a subset of R d. Let ( (X n ) n 0, 1 ) be a sequence of discrete tie Markov chains with state spaces E E and transition atrices (p (x,y)) x,y E. Let F : R d R d and consider the syste of differential equations x i (t) = F i(x(t)), 1 i N. Theore 4.7 in [10] gives a series of sufficient conditions under which the sequence of Markov chains (X ) 1 converges to a solution of the above syste of differential equations. The original stateent of theore 4.7 in [10] is written for the ore general setting of continuous state space Markov chains. We odify just the notation in [10] in order to state the result in a way which is ore suited to our particular setting. Theore 4.7 in [10] can be applied if the following set of conditions is satisfied. There exist sequences of positive nubers (α ) 1 and (ε ) 1 such that 1. li α = and li ε = 0. 2. sup sup α y x p (x,y) <. 1 x E y E 3. li sup α y x p (x,y) = 0. x E y E : y x >ε Define, for 1, F (x) = α (y x)p (x,y). y E 4. li sup x E F (x) F(x) = 0. 5. There exists a constant M such that x,y E, F(x) F(y) M x y. 5
Theore 4.1 (Kurtz). Suppose that conditions 1 5 are satisfied. Suppose further that we have the convergence of the initial conditions Then, for every δ,t > 0, we have ( li P sup 0 t T li X 0 = x0. X αt x(t) ) > δ = 0. 5 Proof of theore 3.1 Let (Z n ) n 0 be the Moran process defined in section 2. Our ai is to apply theore 4.1 to the sequence of Markov chains ( (Z n /) n 0, 1 ). We only need to find the appropriate sequences (α ) 1 and (ε ) 1 and verify that conditions 1 5 are satisfied in our setting. For 1, let α = λ and ε = 2/. The sequences (α ) 1 and(ε ) 1 obviously verify condition 1. As for condition 2, we have, for z P N z z p(z,z ) = z P N N e i + e j p(z,z e i +e j ) i,j=1 2. Thus, sup 1 sup z P N α z P N z z p(z,z ) λ 2 <, as required for condition 2. Since p(z,z ) > 0 if and only if z z 2, for all 1 and z PN, we have z P N : z z >ε z z p(z,z ) = 0, and condition 3 is also satisfied. Let F : R N R N be the function defined by i {1,...,N }, x R N, F i (x) = N N f j Q ji x j x i f j x j. j=1 j=1 6
Since all the partial derivatives of F are bounded on the siplex S N, F is a Lipschitz function on S N, i.e., condition 5 holds. Finally, let us copute, for 1 and x P N /, the value of F (x). By definition, F (x) = λ z P N ( z x ) p(x,z) = λ Thus, for i {1,...,N } we have N ( e i +e j )p(x,x e i +e j ). i,j=1 Fi (x) = λ p(x,x e k +e i ) λ i +e k ) k:k i k:k ip(x,x e = N x k f j Q ji x j N x i f j Q jk x j. k:k i j=1 k:k i j=1 Since x 1 + +x N = 1 and for all i {1,...,N }, Q i1 + +Q in = 1, F i (x) = (1 x i ) N N f j Q ji x j x i f j (1 Q ji )x j = j=1 j=1 N N f j Q ji x j x i f j x j. j=1 j=1 Thus, the function F coincides with the function F on the set P N /, which readily iplies condition 4. Since all five conditions are satisfied, we can apply theore 4.1 to the sequence of Markov chains ( (Z n /) n 0, 1 ) and we obtain the desired result. References [1] Doingos Alves and Jose Fernando Fontanari. Error threshold in finite populations. Phys. Rev. E, 57:7008 7013, 1998. [2] Raphaël Cerf. Critical population and error threshold on the sharp peak landscape for a Moran odel. preprint, 2012. [3] Raphaël Cerf. Critical population and error threshold on the sharp peak landscape for the Wright Fisher odel. preprint, 2012. [4] Raphaël Cerf and Joseba Dalau. The distribution of the quasispecies for a Moran odel on the sharp peak landscape. preprint, 2013. [5] Lloyd Deetrius, Peter Schuster, and Karl Sigund. Polynucleotide evolution and branching processes. Bulletin of Matheatical Biology, 47(2):239 262, 1985. 7
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