Non-Archimedean Replicator Dynamics and Eigen's Paradox
aa r X i v : . [ q - b i o . P E ] O c t NON-ARCHIMEDEAN REPLICATOR DYNAMICS AND EIGEN’SPARADOX
W. A. Z ´U ˜NIGA-GALINDO
Abstract.
We present a new non-Archimedean model of evolutionary dyna-mics, in which the genomes are represented by p -adic numbers. In this modelthe genomes have a variable length, not necessarily bounded, in contrast withthe classical models where the length is fixed. The time evolution of theconcentration of a given genome is controlled by a p -adic evolution equation.This equation depends on a fitness function f and on mutation measure Q .By choosing a mutation measure of Gibbs type, and by using a p -adic versionof the Maynard Smith Ansatz, we show the existence of threshold function M c ( f, Q ), such that the long term survival of a genome requires that its lengthgrows faster than M c ( f, Q ). This implies that Eigen’s paradox does not occurif the complexity of genomes grows at the right pace. About twenty years ago,Scheuring and Poole, Jeffares, Penny proposed a hypothesis to explain Eigen’sparadox. Our mathematical model shows that this biological hypothesis isfeasible, but it requires p -adic analysis instead of real analysis. More exactly,the Darwin-Eigen cycle proposed by Poole et al. takes place if the length ofthe genomes exceeds M c ( f, Q ). Introduction
In this article we present a new non-Archimedean model of evolutionary dyna-mics of replicating single-stranded RNA genomes, which constitutes a non-Archi-medean generalization of the classical Eigen quasispecies model. Mathematicallyspeaking, the new model is a class of p -adic pseudodifferential evolution equations,which depends on a fitness function f and on a mutation measure Qdx . In the newmodel a sequence (genome) is specified by a p -adic number:(1.1) x = x − k p − k + x − k +1 p − k +1 + . . . + x + x p + . . . , with x − k = 0,where p denotes a fixed prime number, and the x j s are p -adic digits, i.e. numbersin the set { , , . . . , p − } . Thus, in our model the sequences may have arbitrarylength. In the case p = 2, we obtain the set of binary sequences of arbitrary length.The set of all possible sequences constitutes the field of p -adic numbers Q p . Thereare natural field operations, sum and multiplication, on series of form (1.1), see e.g.[29]. There is also a natural norm in Q p defined as | x | p = p k , for a nonzero p -adicnumber x of the form (1.1). The field of p -adic numbers with the distance inducedby |·| p is a complete ultrametric space. The ultrametric property refers to the factthat | x − y | p ≤ max n | x − z | p , | z − y | p o for any x , y , z in Q p . Date : 25/01/2018.2000
Mathematics Subject Classification.
Primary 92D15, 92D25; Secondary 82B20, 32P05.
Key words and phrases.
Darwinian evolution, Eigen’s paradox, pseudodifferential evolutionequations, p -adic analysis.The author was partially supported by Conacyt Grant No. 250845. We denote by Z p the unit ball, which consists of the all the sequences withexpansions of the form (1.1) with k ≥
0, and by Z × p , the subset of Z p consisting ofthe p -adic numbers with norm 1. This last set is the disjoint union of sets of theform j + p Z p , for j ∈ { , . . . , p − } . Each set of the form j + p Z p is (in a naturalform) an infinite rooted tree. Then, all the sequences contained in the set j + p Z p are naturally organized in a ‘phylogenetic tree’, and the set Z × p is a forest formedby the disjoint union of p − Q p r { } is a countable disjoint union of scaled versions of the forest Z × p , more precisely, Q p r { } = F k =+ ∞ k = −∞ p k Z × p . The field of p -adic numbers has a fractal structure, seee.g. [1], [52]. For ‘pictures’ of p -adic spaces the reader may consult [17].Our model of evolutionary dynamics is a p -adic continuous model. The fitnesslandscape is given by a function f : Q p → R + . In this article, we assume that f is a test function, which means that f is a locally constant function with compactsupport. With respect to the mutation mechanism, we only assume the existence ofa mutation measure Q (cid:16) | x | p (cid:17) dx , where Q : R + → R + , dx is the normalized Haarmeasure of the group ( Q p , +), and R Q (cid:16) | x | p (cid:17) dx = 1, such that the probabilitythat a sequence x mutates into a sequence belonging to the set B is given by R B Q (cid:16) | x − y | p (cid:17) dy . In our model the concentration X ( x, t ) of the sequence x at thetime t is controlled by the following evolution equation:(1.2) ∂X ( x, t ) ∂t = Q (cid:16) | x | p (cid:17) ∗ n f (cid:16) | x | p (cid:17) X ( x, t ) o − Φ ( t ) X ( x, t ) , where Φ ( t ) = R Q p f (cid:16) | y | p (cid:17) X ( y, t ) dy .The term Q (cid:16) | x | p (cid:17) ∗ n f (cid:16) | x | p (cid:17) X ( x, t ) o represents the rate at which the se-quences are mutating into the sequence x . We now assume that the replicationreactions occur in a chemostat, see [51] and the references therein, which is a de-vice that allows the maintenance of a constant population size, this corresponds tothe term − Φ ( t ) X ( x, t ). From this discussion, we conclude that the instantaneouschange of the concentration X ( x, t ) is given by (1.2).The mathematical study of the equation (1.2), and in particular of the associatedCauchy problem, is an open problem. The functions f and Q may depend on time,but here we study only the Cauchy problem associated to (1.2) in a particular case,by using p -adic wavelets, see Section 8.A central problem in the origin of life is the reproduction of primitive organismswith sufficient fidelity to maintain the information coded in the primitive genomes.In the case of sequences (genomes) with constant length, and under the assump-tion of independent point mutations, that is, assuming that during the replicationprocess each digit (nucleotide) has a fixed probability of being replaced for anotherdigit, and that this probability is independent of all other digits, Eigen discoveredthat the mutation process places a limit on the number of digits that a genome mayhave, see e.g. [13], [14], [35], [46], [51]. This critical size is called the error thresh-old of replication. The genomes larger than this error threshold will be unable tocopy themselves with sufficiently fidelity, and the mutation process will destroy theinformation in subsequent generations of these genomes. This discussion naturallydrives to the following question: how is it possible the existence of large stable liv-ing organisms on earth? To create more complex organisms (that is to have more ON-ARCHIMEDIAN REPLICATOR DYNAMICS 3 genetic complexity), it is necessary to encode more information in larger genomesby using a replication mechanism with greater fidelity. But the information forcreating error-correcting mechanisms (enzymes) should be encoded in the genomes,which have a limited size. Hence, we arrive to the ‘Cath-22’ or Eigen’s paradox ofthe origin of life: “no large genome without enzymes, and no enzymes without alarge genome,” see [42, p. 317], [49]. Then main consequence of our p -adic model ofevolution is that it gives a completely ‘new mathematical’ perspective of the Eigenparadox.We show that there is a finite ultrametric space G M consisting of p -adic sequencesof finite length 2 M , which is a rooted tree with 2 M + 1 levels and p M branches atthe top level, and with a distance induced by the restriction of the p -adic norm to G M , such that the equation (1.2) admits a discretization (a finite approximation)of the form(1.3) ddt X ( J, t ) = 1 C X I ∈ G M Q (cid:16) | J − I | p (cid:17) f (cid:16) | I | p (cid:17) X ( I, t ) − Φ M ( t ) X ( J, t ) for J ∈ G M , which is exactly the Eigen model on G M , see Section 4.2. In the classical Eigenmodel the space of sequences is a finite metric space with a distance induced by theHamming weight, in our model this space is replaced by G M . The main restrictionof this last model is that the mutation matrix is a function of the p -adic distancebetween two sequences.Heuristically speaking, the limit when M tends to infinity of the Eigen system(1.3) is the evolution equation (1.2). This heuristics can be justified by using thetechniques from [53] and the references therein, more exactly, we have that thesolutions of the Cauchy problem associated to (1.2) can be very well approximatedby solutions of the Cauchy problem associated to (1.3), in a suitable function space,when M tends to infinity.In [42], Maynard Smith introduced a mathematical approximation (the MaynardSmith ansatz) that allows to study the error threshold problem without solving theoriginal Eigen system. This ansatz can be extended to the p -adic setting, see Section5. In this approximation the space of sequences is divided into two disjoint groups,each of them with a fix fitness, say a and b , with a > b . The ansatz provides aninequality which gives a necessary and sufficient condition for the long term survivalof the group of sequences with fitness a . It is remarkable that this inequality is theclassical one, see Section 5. We use this ansatz to study the Eigen paradox fortwo different families of mutation measures. The measures of the first family aresupported in the unit ball, the second is a family of Gibbs measures. In both caseswe show that the Eigen paradox does not occur if the length of the sequences growsto the right pace. We propose using a Gibbs type measure: C ( α,β ) e − β | x | αp dx , where α , β are positive constants, and C ( α, β ) is a normalization constant. The two mainreasons for this choice are: first, | x | αp is the simplest energy function which dependson log p | x | p , the ‘ p -adic length’ of the sequence, and second, the discretization ofthe p -adic replicator equation (1.2) attached to this mutation measure is connectedwith the matrices defining certain Markov chains, that typically are used in modelsof molecular evolution, see e.g. [33].The p -adic version of the Maynard Smith ansatz provides a necessary and suf-ficient condition for the long term survival of a group of sequences of the form W. A. Z´U˜NIGA-GALINDO I + p M Z p , where I is an infinite sequence which plays the role of the master se-quence, which are in competition with the group of sequences Q p r (cid:2) I + p M Z p (cid:3) . Weestablish the existence of an error threshold function M c ( s, α, β ), which depends onln s , with 1 − s = ba , and with f | I + p M Z p ≡ a > f | Q p r I + p M Z p ≡ b , such that the longterm survival of the sequences in the group I + p M Z p requires that M > M c ( s, α, β ).This means that under a ‘fierce competition’ between the two groups (i.e. when s → + ), the long term survival of the first group requires that all the sequences inthis group approaches to I , in such way that the logarithm of the p -adic norm of thedifference of any of these sequences and I is greater than M c ( s, α, β ). Notice thatthe set of sequences of finite length in I + p M Z p is a dense subset, the mentionedcondition implies that the long term survival of these sequences requires that thelength of them grow.On the other hand, if M is upper bounded, then M ≤ M c ( s, α, β ) for s suf-ficiently small, which is a version of the classical threshold condition. In conclu-sion, our p -adic model of evolution predicts that Eigen’s paradox does not occurif the complexity of the genomes grow at the right pace. About twenty yearsago, Scheuring [41] and Poole et al. [39] proposed a hypothesis to explain Eigen’sparadox. Our mathematical model gives life to this biological hypothesis, moreprecisely, the Darwin-Eigen cycle proposed in [39] takes place under the condition M > M c ( s, α, β ): larger genome size improves the replication fidelity, and this inturn increases the Eigen limit on the length of the genome, which allows the evo-lution of larger genome size. In turn, this allows the evolution of new function,which could further improve the replication fidelity, and so on. See Section 7.2 foran in-depth discussion about this matter.In Section 8, we study the Cauchy problem attached to the p -adic replicatorequation, in the case in which the initial concentration and the fitness are test func-tions. By using p -adic wavelets and the classical method of separation of variables,we show the existence of a solution X ( x, t ) for the mentioned Cauchy problem.Then we show that X ( x ) = lim t → + ∞ X ( x, t ) exists, and it is a probability densityconcentrated in the support of the fitness function. This steady state concentrationis the p -adic counterpart of the classical quasispecies. It is controlled by fitnessfunction and by the largest eigenvalue of the operator W ϕ ( x ) = Q ( | x | p ) ∗ n f (cid:16) | x | p (cid:17) ϕ ( x ) o for ϕ supported in a finite union of disjoint balls. The p -adic quasispecies behaveentirely different to the classical ones. An in-depth understanding of the p -adicquasispecies require developing of numerical methods for p -adic evolution equations.An ultrametric space ( M, d ) is a metric space M with a distance satisfying d ( A, B ) ≤ max { d ( A, C ) , d ( B, C ) } for any three points A , B , C in M . In the mid-dle of the 80s the idea of using ultrametric spaces to describe the states of complexbiological systems, which naturally possess a hierarchical structure, emerged in theworks of Frauenfelder, Parisi, Stein, among others, see e.g. [10], [15], [34], [40].Frauenfelder et al. proposed, based on experimental data, that the space of statesof certain proteins have an ultrametric structure, [15]. Mezard, Parisi, Sourlas andVirasoro discovered, in the context of the mean-field theory of spin glasses, thatthe space of states of such systems has an ultrametric structure, see e.g. [34], [40].A central paradigm in physics of complex systems (for instance proteins) assertsthat the dynamics of such systems can be modeled as a random walk in the energy ON-ARCHIMEDIAN REPLICATOR DYNAMICS 5 landscape of the system, see e.g. [15], [28], [30], and the references therein. Inthis framework, the energy landscape of a complex system is approximated by apair consisting of an ultrametric space and a function on this space describing thedistribution of the activation barriers, see e.g. [9]. The dynamics of a such systemcan is described by a system of equations of type(1.4) ∂u ( i, t ) ∂t = X j = i J ( j, i ) v ( j ) u ( j, t ) − X j = i J ( i, j ) v ( i ) u ( i, t ) , i = 1 , . . . , N, where the indices i , j number the states of the system (which correspond to localminima of energy), u ( i, t ) denotes the concentration of particles at the state i andat time t , J ( i, j ) ≥ i to j , and the v ( j ) > M, |·| ), wherethe distance comes from a norm |·| , and that J ( j, i ) = J ( | j − i | ), v ( j ) = v ( | j | ), P j J ( | j − i | ) = 1, then the master equations (1.4) take the form(1.5) ∂u ( i, t ) ∂t = X j = i J ( | j − i | ) v ( | j | ) u ( j, t ) − (1 − J (0)) v ( | i | ) ( i, t ) for i ∈ X .This system of equations is ‘similar’ but not equal to the ultrametric version ofEigen’s system of equations up to the term − Φ ( t ) u ( i, t ), see (1.3). The differenceis that in (1.5) there are no transitions from state i into i , and such transitions playan important role in the Eigen ultrametric system (1.3).It is relevant to mention that the kinetic models (1.4) are deeply connected withthe theory of spin glasses, see e.g. [6], [28], [37], [40], and the references therein, andthat techniques from this area have been successfully used in the study of Eigen’smodel, see e.g. [32], [36], [43]-[45].In [7], see also [8], Avetisov and Zhuravlev proposed using the one-dimensionalultrametric diffusion equation in the theory of biological evolution. This equation(also known as the p -adic heat equation) was introduced by Vladimirov, Volovichand Zelenov in [52]. A very general theory of such equations is now available see e.g.[19], [28], [54]. In [7], the authors proposed that an evolutionary model based on theultrametric diffusion equation corresponds to the hierarchical picture of the referentdescription of the biological world, and that the problem of error catastrophe hasa natural solution in the proposed framework. This approach does not allow toanalyze directly the error catastrophe in the usual sense.In the last thirty years there has been a strong interest in the developing ofultrametric models in biology, see for instance, [3]-[12], [15], [20]-[28], [30]-[31]. Theresults presented in this article are framed in this development and they confirm therelevance of the ultrametricity in modeling biological systems which have naturalhierarchical structures.We did our best to write a self-contained article addressed to a general audience(biologists, mathematicians, physicists, among others). Numerical simulations ofour model are a not a straightforward matter due to several new features, amongthem, the problem of visualization of p -adic objects.2. p -Adic Analysis: Essential Ideas The field of p -adic numbers. Along this article p will denote a prime num-ber. The field of p − adic numbers Q p is defined as the completion of the field of W. A. Z´U˜NIGA-GALINDO rational numbers Q with respect to the p − adic norm | · | p , which is defined as | x | p = x = 0 p − γ if x = p γ ab ,where a and b are integers coprime with p . The integer γ := ord ( x ), with ord (0) :=+ ∞ , is called the p − adic order of x .Any p − adic number x = 0 has a unique expansion of the form x = p ord ( x ) ∞ X j =0 x j p j , where x j ∈ { , . . . , p − } and x = 0. By using this expansion, we define thefractional part of x ∈ Q p , denoted { x } p , as the rational number { x } p = x = 0 or ord ( x ) ≥ p ord ( x ) P − ord p ( x ) − j =0 x j p j if ord ( x ) < . In addition, any non-zero p − adic number can be represented uniquely as x = p ord ( x ) ac ( x ) where ac ( x ) = P ∞ j =0 x j p j , x = 0, is called the angular component of x . Notice that | ac ( x ) | p = 1.For r ∈ Z , denote by B r ( a ) = { x ∈ Q p ; | x − a | p ≤ p r } the ball of radius p r withcenter at a ∈ Q p , and take B r (0) := B r . The ball B equals Z p , the ring of p − adicintegers of Q p . We also denote by S r ( a ) = { x ∈ Q p ; | x − a | p = p r } the sphere ofradius p r with center at a ∈ Q p , and take S r (0) := S r . We notice that S = Z × p (the group of units of Z p ). The balls and spheres are both open and closed subsetsin Q p . In addition, two balls in Q p are either disjoint or one is contained in theother.The metric space (cid:16) Q p , |·| p (cid:17) is a complete ultrametric space. As a topologicalspace ( Q p , | · | p ) is totally disconnected, i.e. the only connected subsets of Q p arethe empty set and the points. In addition, Q p is homeomorphic to a Cantor-likesubset of the real line, see e.g. [1], [52]. A subset of Q p is compact if and only if itis closed and bounded in Q p , see e.g. [52, Section 1.3], or [1, Section 1.8]. The ballsand spheres are compact subsets. Thus ( Q p , | · | p ) is a locally compact topologicalspace. Notation 1.
We will use
Ω ( p − r | x − a | p ) to denote the characteristic function ofthe ball B r ( a ) . We will use the notation A for the characteristic function of a set A . Some function spaces.
A complex-valued function ϕ defined on Q p is calledlocally constant if for any x ∈ Q p there exist an integer l ( x ) ∈ Z such that(2.1) ϕ ( x + x ′ ) = ϕ ( x ) for x ′ ∈ B l ( x ) . A function ϕ : Q p → C is called a Bruhat-Schwartz function (or a test function) if it is locally constant with compact support. In this case, we can take l = l ( ϕ )in (2.1) independent of x , the largest of such integers is called the parameter oflocal constancy of ϕ . The C -vector space of Bruhat-Schwartz functions is denotedby D := D ( Q p , C ). We will denote by D R := D ( Q np , R ), the R -vector space of testfunctions. ON-ARCHIMEDIAN REPLICATOR DYNAMICS 7
Given ρ ∈ [0 , ∞ ), we denote by L ρ := L ρ ( Q p ) := L ρ ( Q p , dx ) , the C − vectorspace of all the complex valued functions g satisfying R Q p | g ( x ) | ρ dx < ∞ , and L ∞ := L ∞ ( Q p ) = L ∞ ( Q p , dx ) denotes the C − vector space of all the complex valuedfunctions g such that the essential supremum of | g | is bounded. The corresponding R -vector spaces are denoted as L ρ R := L ρ R ( Q p ) = L ρ R ( Q p , dx ), 1 ≤ ρ ≤ ∞ .2.3. Integration on Q p . Since ( Q p , +) is a locally compact topological group,there exists a Borel measure dx , called the Haar measure of ( Q p , +), unique upto multiplication by a positive constant, such that R U dx > U ⊂ Q p , and satisfying R E + z dx = R E dx for every Borel set E ⊂ Q p ,see e.g. [16, Chapter XI]. If we normalize this measure by the condition R Z p dx = 1,then dx is unique. From now on we denote by dx the normalized Haar measure of( Q p , +).A test function ϕ : Q p → C can be expressed as a linear combination of char-acteristic functions of the form ϕ ( x ) = P li =1 c i Ω ( p − r i | x − a i | p ), where c i ∈ C andΩ ( p − r i | x − a i | p ) is the characteristic function of the ball a i + p − r i Z p , for every i .In this case Z Q p ϕ ( x ) dx = l X i =1 c i Z a i + p − ri Z p dx = l X i =1 c i Z p − ri Z p dx = l X i =1 c i p r i , where we use the facts that dx is invariant under translations and that R p − ri Z p dx = p r i . By using the fact that D ( Q p ) is a dense subspace of C ( Q p ), the space ofcontinuous functions with compact support, the functional ϕ → R Q p ϕ ( x ) dx , ϕ ∈D ( Q p ) has a unique extension to C ( Q p ). For integrating more general functions,say locally integrable functions, the following notion of improper integral is used. Definition 1.
A function ϕ ∈ L loc is said to be integrable in Q p if lim m → + ∞ Z B m (0) ϕ ( x ) dx = lim m → + ∞ m X j = −∞ Z S j (0) ϕ ( x ) dx exists. If the limit exists, it is denoted as R Q p ϕ ( x ) dx , and we say that the (im-proper) integral exists. Analytic change of variables.
A function h : U → Q p is said to be analytic on an open subset U ⊂ Q p , if for every b ∈ U there exists an open subset e U ⊂ U ,with b ∈ e U , and a convergent power series P i a i ( x − b ) i for x ∈ e U , such that h ( x ) = P i ∈ N a i ( x − b ) i for x ∈ e U . In this case, ddx h ( x ) = P i ∈ N a i ddx ( x − b ) i is aconvergent power series.Let U , V be open subsets of Q p . Let ϕ : V → C be a continuous function withcompact support, and let h : U → V be an analytic mapping. Then R V ϕ ( y ) dy = R U ϕ ( h ( x )) (cid:12)(cid:12)(cid:12)(cid:12) ddx h ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p dx ,see e.g. [52]. W. A. Z´U˜NIGA-GALINDO
Fourier transform.
Set χ p ( y ) = exp(2 πi { y } p ) for y ∈ Q p . The map χ p ( · ) isan additive character on Q p , i.e. a continuous map from ( Q p , +) into S (the unitcircle considered as multiplicative group) satisfying χ p ( x + x ) = χ p ( x ) χ p ( x ), x , x ∈ Q p . The additive characters of Q p form an Abelian group which is iso-morphic to ( Q p , +), the isomorphism is given by ξ → χ p ( ξx ), see e.g. [1, Section2.3].If f ∈ L its Fourier transform is defined by( F f )( ξ ) = Z Q p χ p ( ξx ) f ( x ) dx, for ξ ∈ Q p . We will also use the notation F x → ξ f and b f for the Fourier transform of f . TheFourier transform is a linear isomorphism from D onto itself satisfying(2.2) ( F ( F f ))( ξ ) = f ( − ξ ) , for every f ∈ D , see e.g. [1, Section 4.8]. If f ∈ L , its Fourier transform is definedas ( F f )( ξ ) = lim k →∞ Z | x | p ≤ p k χ p ( ξ · x ) f ( x ) d n x, for ξ ∈ Q p ,where the limit is taken in L . We recall that the Fourier transform is unitary on L , i.e. || f || L = ||F f || L for f ∈ L and that (2.2) is also valid in L , see e.g. [50,Chapter III , Section 2]. 3.
The model
A replicator is a model of an entity with the template property, which meansthat it serves as a pattern for the generation of another replicator. This copyingprocess is subject to errors (mutations). Along this article we will use replicators,genomes, macromolecules and sequences as synonyms. The assumption of the exis-tence of replicators implies that the information stored in the replicators is modifiedrandomly, and that part of it is fixed due to the selection pressure, which in turnis related with the self-replicate capacity of the replicators (their fitness).In this section we introduce a p -adic version of Eigen’s equation, see e.g.[13],[14], [35], [48], [51], [46], which describes mutation-selection process of replicatingsequences.3.1. The space of sequences.
In our model each sequence corresponds to a p -adicnumber: x = x − m p − m + x − m +1 p − m +1 + . . . + x + x p + . . . where the digits x i s run through the set { , , . . . , p − } . In the case p = 2, thesequences are binary words. Consequently, in our model the sequences are words ofarbitrary length written in the alphabet 0, 1, . . . , p −
1, and the space of sequencesis (cid:16) Q p , |·| p (cid:17) , which is an infinite set. A key feature of our model is the use ofsequences of variable length. It is relevant to mention that Poole et al. [39] andScheuring [41] already proposed solutions for Eigen’s paradox which require theusage of sequences of variable length.On the other hand, p -adic numbers naturally appear in models of the geneticcode. For an in-depth discussion the reader may consult [11], [27], [28, Chapter 4]and the references therein. ON-ARCHIMEDIAN REPLICATOR DYNAMICS 9
Concentrations.
The concentration of sequence x ∈ Q p at the time t ≥ X ( x, t ), this is a real number between zero and one. In addition, weassume that(3.1) Z Q p X ( y, t ) dy = 1 for t > t > The mutation measure.
We fix a function Q : R + → [0 , ≤ Q ( | x | p ) ≤ Z Q p Q (cid:16) | y | p (cid:17) dy = 1 . We call Q ( | x | p ) dx the mutation measure . Given a Borel subset E ⊆ Q p and x ∈ Q p ,the integral Z E Q (cid:16) | x − y | p (cid:17) dy gives the probability that sequence x will mutate into a sequence belonging to E .In addition to this hypothesis we do not need additional assumptions about themutation mechanism. The results presented in this article can be easily extendedto case in which the mutation measure depends on time, more precisely, when Q ( | x | p , t ) dx is the transition function of a Markov process.3.4. The fitness function.
The fitness function f is a non-negative real-valuedtest function. Notice that f is a radial function, and by abuse of notation, we willuse the notation f (cid:16) | x | p (cid:17) . The assumption that function f has compact supportsays that the evolution process is limited to a certain region of the space of se-quences, which is infinite. This assumption allows very general fitness landscapes,for instance, c if | x | p ≤ p L g (cid:16) | x | p (cid:17) if p L < | x | p ≤ p L | x | p > p L , where c is a nonnegative constant, L , L ∈ Z , with L < L , and g : R + → R + is a function.3.5. The non-Archimedean replicator equation.
We set( W ϕ ) ( x ) = Q (cid:16) | x | p (cid:17) ∗ n f (cid:16) | x | p (cid:17) ϕ ( x ) o = Z Q p Q (cid:16) | x − y | p (cid:17) f (cid:16) | y | p (cid:17) ϕ ( y ) dy, where Q and f are as before. Notice that for 1 ≤ ρ ≤ ∞ , L ρ ( Q p , C ) → L ρ ( Q p , C ) ϕ → W ϕ is a well-defined continuous operator.Now our non-Archimedean mutation-selection equation has the form(3.2) ∂X ( x, t ) ∂t = W X ( x, t ) − Φ ( t ) X ( x, t ) , x ∈ Q p , t ∈ R + , where Φ ( t, X ) := Φ ( t ) = Z Q p f (cid:16) | y | p (cid:17) X ( y, t ) dy for t ≥ t ) is used to maintain constant the total concentration in thechemostat. Later on, we will consider the Cauchy problem associated to (3.2) withinitial datum(3.3) X ( x,
0) = X ∈ D R By changing variables as(3.4) X ( x, t ) = Y ( x, t ) exp (cid:18) − Z t Φ ( τ ) dτ (cid:19) , like in the classical case, (3.2)-(3.3) becomes(3.5) ∂Y ( x,t ) ∂t = W Y ( x, t ) , x ∈ Q p , t ∈ R + Y ( x,
0) = X ∈ D R . Comments. (i) Archimedean evolution equations appear in several models ofevolution, see e.g. [38] and [2]. In particular the proposed equation (3.2) makessense in R , however, the interpretation of the space variable x as a sequence ofvariable length is only natural in the p -adic setting.(ii) The Equation (3.2) is really a family of models depending on f , Q . Weformulated these models in dimension one, that is, just for ‘one type of sequences’.However, the equations introduced here can be extended to dimension n , that is,to the case of n ‘types of sequences’. In principle, the models introduced here maybe extended to include recombination and sexual reproduction.4. An ultrametric Version of the classical replicator equation
Eigen’s evolution model consists of a systems of non-linear ordinary differentialequations describing the evolution of the concentrations of sequences, which belongto a finite metric space, where the distance is induced by the Hamming weight.If we use binary sequences, the space of sequences is Z / M Z , for some positiveinteger M .In this section by using ideas from [53], we show a discretization of the equation(3.2) that agrees with the Eigen model in an ultrametric space formed by finite p -adic sequences, where the mutation probability is a function of the ultrametricdistance between sequences. By a suitable normalization the space of sequences is Z p /p M Z p , and the ultrametric distance is induced by the restriction of the p -adicnorm to Z p /p M Z p . This fact shows that our equation (3.2) is a non-Archimedeangeneralization of the Eigen model. Notation 2.
The set of non-negative integers is denoted as N . The space of sequences G M . We fix M ∈ Nr { } and set G M := p − M Z p /p M Z p . We consider G M as an additive group and fix the following systems of representa-tives: any I ∈ G M is represented as(4.1) I = I − M p − M + I − M +1 p − M +1 + · · · + I + · · · + I M − p M − , ON-ARCHIMEDIAN REPLICATOR DYNAMICS 11 where the I j s belong to { , , . . . , p − } . Furthermore, the restriction of |·| p to G M induces an absolute value such that | G M | p = (cid:8) , p − ( M +1) , · · · , p − , , · · · , p M (cid:9) .We endow G M with the metric induced by |·| p , and thus G M becomes a finiteultrametric space. In addition, G M can be identified with the set of branches(vertices at the top level) of a rooted tree with 2 M + 1 levels and p M branches.Any element I ∈ G M can be uniquely written as p M e I , where e I = e I + e I p + · · · + e I M − p M − ∈ Z p /p M Z p , with the e I j s belonging to { , , . . . , p − } . The elements of the Z p /p M Z p are inbijection with the vertices at the top level of the above mentioned rooted tree. Bydefinition the root of the tree is the only vertex at level 0. There are exactly p vertices at level 1, which correspond with the possible values of the digit e I in the p -adic expansion of e I . Each of these vertices is connected to the root by a non-directed edge. At level ℓ , with 1 ≤ ℓ ≤ M , there are exactly p ℓ vertices, each vertexcorresponds to a truncated expansion of e I of the form e I + · · · + e I ℓ − p ℓ − . The vertexcorresponding to e I + · · · + e I ℓ − p ℓ − is connected to a vertex e I ′ + · · · + e I ′ ℓ − p ℓ − at the level ℓ − (cid:16)e I + · · · + e I ℓ − p ℓ − (cid:17) − (cid:16)e I ′ + · · · + e I ′ ℓ − p ℓ − (cid:17) isdivisible by p ℓ − . Notice that there are other geometric realizations of G M . Forinstance, we can chose a forest formed by p rooted trees, each of them with 2 M levels and p M − branches.4.2. A discretization of equation (3.2).
We denote by D − MM the R -vector sub-space of D R spanned by the functionsΩ (cid:16) p M | x − I | p (cid:17) , I ∈ G M .Notice that Ω (cid:0) p M | x − I | (cid:1) Ω (cid:0) p M | x − J | (cid:1) = 0 for any x , if I = J . Thus, anyfunction ϕ ∈ D − MM has the form ϕ ( x ) = X I ∈ G M ϕ ( I ) Ω (cid:16) p M | x − I | p (cid:17) , where the ϕ ( I )s are real numbers. Notice that the dimension of D − MM ( Q p ) is G M = p M .In order to explain the connection between the non-Archimedean replicator equa-tion (3.2) and the classical one, we assume that Q (cid:16) | x | p (cid:17) , f (cid:16) | x | p (cid:17) and X ( x, t )belong to D − MM , for any t . Then Q (cid:16) | x | p (cid:17) = 1 C M X I ∈ G M Q (cid:16) | I | p (cid:17) Ω (cid:16) p M | x − I | p (cid:17) , with C M = p − M P I ∈ G M Q (cid:16) | I | p (cid:17) , f (cid:16) | x | p (cid:17) = X I ∈ G M f (cid:16) | I | p (cid:17) Ω (cid:16) p M | x − I | p (cid:17) , and X ( x, t ) = X I ∈ G M X ( I, t )Ω (cid:16) p M | x − I | p (cid:17) for any t ≥ where each X ( I, t ) is a real-valued function of class C in t . Now Z Q p Q (cid:16) | x − y | p (cid:17) f (cid:16) | y | p (cid:17) X ( y, t ) dy =(4.2) ( C M X K ∈ G M X I ∈ G M Q (cid:16) | K | p (cid:17) f (cid:16) | I | p (cid:17) X ( I, t ) ) Ω (cid:16) p M | x − K | p (cid:17) ∗ Ω (cid:16) p M | x − I | p (cid:17) , and by using that Ω (cid:16) p M | x − K | p (cid:17) ∗ Ω (cid:16) p M | x − I | p (cid:17) = p M Ω (cid:16) p M | x − ( I + K ) | p (cid:17) and the fact that G M is an additive group, the right-hand side of (4.2) can berewritten as ( C X I ∈ G M Q (cid:16) | J − I | p (cid:17) f (cid:16) | I | p (cid:17) X ( I, t ) ) Ω (cid:16) p M | x − J | p (cid:17) , with C = P I ∈ G M Q (cid:16) | I | p (cid:17) . Finally, using the fact that the Ω (cid:16) p M | x − I | p (cid:17) , I ∈ G M are R -linearly independent, we get(4.3) ddt X ( J, t ) = 1 C X I ∈ G M Q (cid:16) | J − I | p (cid:17) f (cid:16) | I | p (cid:17) X ( I, t ) − Φ M ( t ) X ( J, t ) for J ∈ G M , where Φ M ( t ) = p − M X I ∈ G M f (cid:16) | I | p (cid:17) X ( I, t ) , which is exactly the Eigen model on G M , but G M is an ultrametric space, wherethe distance comes from the p -adic norm.4.3. The limit M tends to infinity. Heuristically speaking, the limit of the sys-tem of equations (4.3) when M tends to infinity is the evolution equation (3.2). In[6], Avetisov, Kozyrev et al. established (using physical arguments) that certainnon-Archimedean kinetic models, similar to (4.3), have ‘continuous p -adic limits’ asreaction-ultradiffusion equations. A mathematical explanation of these construc-tions is given in our article [53]. From a mathematical perspective, the Cauchyproblem associated with (3.2) can be very well approximated by an initial valueproblem associated with the system (4.3), in the sense that any solution of (3.2) isarbitrarily close to a suitable solution of (4.3) in a certain function space, when M tends to infinity, see introduction of [53]. This result follows by applying techniquesused in [53] to the system (3.5) with X a test function, see (3.5).Now, if consider the system (4.3) as a system of ODEs in R p M , then it seemsnot plausible that the ‘limit N → ∞ ’ can be defined.5. The Maynard Smith Ansatz
In this section we present a p -adic version of Maynard Smith approach to theerror threshold problem, see [42], [48]. We divide the space of sequences into twodisjoint sets:(5.1) Q p = (cid:2) I + p M Z p (cid:3) G (cid:2) Q p r I + p M Z p (cid:3) , where I is a fixed sequence as in (4.1), with M ≥
1, and assume that(5.2) f | I + p M Z p ≡ a , f | Q p r I + p M Z p ≡ b , with a > b , ON-ARCHIMEDIAN REPLICATOR DYNAMICS 13 here “ ≡ ” means identically equal. This means that the group I + p M Z p contains thefittest sequences and that all these sequences coincide with I up to digit I M − . Wedenote by X ( x, t ) the concentration of I + p M Z p and by Y ( x, t ) the concentrationof Q p r (cid:2) I + p M Z p (cid:3) . Notice that the supports of X ( x, t ) and Y ( x, t ) are disjoint.We denote by q := q ( M, Q ) the probability that a sequence in I + p M Z p mutatesinto a sequence belonging to Q p r (cid:2) I + p M Z p (cid:3) , and we denote by r := r ( M, Q )the probability of mutation of a sequence from Q p r (cid:2) I + p M Z p (cid:3) into a sequence in I + p M Z p . The system of equations governing the development of these populationsis(5.3) ∂X ( x,t ) ∂t = a (1 − q ) X ( x, t ) + brY ( x, t ) − Φ ( t ) X ( x, t ) ∂Y ( x,t ) ∂t = aqX ( x, t ) + b (1 − r ) Y ( x, t ) − Φ ( t ) Y ( x, t ) , where Z Q p X ( x, t ) dx + Z Q p Y ( x, t ) dx = 1 , and Φ ( t ) = Z I + p M Z p f (cid:16) | x | p (cid:17) X ( x, t ) dx + Z Q p r I + p M Z p f (cid:16) | x | p (cid:17) Y ( x, t ) dx = a Z I + p M Z p X ( x, t ) dx + b Z Q p r I + p M Z p Y ( x, t ) dx. We assume that for M sufficiently large, r ( M, Q ) is very small (this conditionshould be verified for each particular choice of Q ), so we can assume that system(5.3) has the form(5.4) ∂X ( x,t ) ∂t = a (1 − q ) X ( x, t ) − Φ ( t ) X ( x, t ) ∂Y ( x,t ) ∂t = aqX ( x, t ) + bY ( x, t ) − Φ ( t ) Y ( x, t ) . By taking Z ( x, t ) = X ( x,t ) Y ( x,t ) , system (5.4) becomes ∂Z ( x, t ) ∂t = Z ( x, t ) { a (1 − q ) − aqZ ( x, t ) − b } . Assuming that concentration Z ( x, t ) achieves a steady concentration Z ( x ) over thetime, we get Z ( x ) = a (1 − q ) − baq . The original population persists, i.e. the sequences in I + p M Z p survive in a longterm, if and only if Z ( x ) >
0, i.e. if and only if1 − q > ba . By writing ba = 1 − s , with s ∈ (0 , q < s. This is exactly the classical condition determining the error threshold, see e.g. [42],[48].
Comments.
Notice that by using the ultrametric property | x − y | p = | y | p for x ∈ p M Z p and y ∈ Q p r p M Z p , we have q = q ( M, α ) = Z I + p M Z p Z Q p r [ I + p M Z p ] Q (cid:16) | x − y | p (cid:17) dydx = Z p M Z p Z Q p r p M Z p Q (cid:16) | x − y | p (cid:17) dydx = Z p M Z p Z Q p r p M Z p Q (cid:16) | y | p (cid:17) dydx = p − M Z Q p r p M Z p Q (cid:16) | y | p (cid:17) dy = p − M Z supp Q ∩ [ Q p r p M Z p ] Q (cid:16) | y | p (cid:17) dy, which implies that q is independent of I , r = r ( M, Q ) = q , and that conditions(5.1)-(5.2) can be replaced bysupp Q = (cid:2) I + p M Z p (cid:3) G (cid:2) supp Q r (cid:2) I + p M Z p (cid:3)(cid:3) , where I is a fixed sequence as in (4.1), with M ≥
1, and by f | I + p M Z p ≡ a , f | supp Q r [ I + p M Z p ] ≡ b , with a > b .6. The Error Threshold Problem Using a mutation measuresupported in unit ball
In this section we study the error threshold problem using the Maynard Smithansatz with a mutation measure supported in the unit ball.6.1.
A class of mutation measures supported in the unit ball.
Take α ≥ Q (cid:16) | x | p ; α (cid:17) = | x | αp Ω (cid:16) | x | p (cid:17) Z ( α ) , where for γ ∈ R ,(6.2) Z ( γ ) = Z Z p | x | γp dx = 1 − p − − p − − γ for γ > − . Then, Q (cid:16) | x | p ; α (cid:17) dx gives rise to a family of mutation measures, which includethe uniform distribution for α = 0.We now fix a sequence I ∈ Z p , which plays the role of the master sequence, anddivide the space of sequences Z p into two subsets: I + p M Z p and Z p r (cid:2) I + p M Z p (cid:3) for some positive integer M . The set I + p M Z p consists of the sequences in the unitball that coincide with the sequence I up to the digit M −
1. We denote by H M afixed set of representatives of Z p /p M Z p . We also assume that f | I + p M Z p ≡ a , f | Z p r I + p M Z p ≡ b , with a > b .Notice that by the remarks made at the end of Section 5, we can apply the MaynardSmith ansatz to establish (5.5) under the above mentioned conditions. We nowcompute the probability that a sequence in the set I + p M Z p mutates into a sequence ON-ARCHIMEDIAN REPLICATOR DYNAMICS 15 belonging to the set Z p r (cid:2) I + p M Z p (cid:3) (notice that this probability does not dependedon I ): q ( α ) := 1 Z ( α ) Z I + p M Z p Z Z p r I + p M Z p | x − y | αp dydx = 1 Z ( α ) Z | x − I | p ≤ p − M Z | y − I | p >p − M | x − y | αp dydx = 1 Z ( α ) Z p M Z p Z Z p r p M Z p | x − y | αp dydx = 1 Z ( α ) Z p M Z p Z Z p r p M Z p | y | αp dydx = p − M Z ( α ) Z Z p r p M Z p | y | αp dy = p − M Z ( α ) X J ∈ H M , J =0 | J | αp , where we use the ultrametric property: | x − y | p = | y | p for | x | p ≤ p − M and | y | p >p − M .Notice that q ( α ) > p − M Z ( α ) (cid:12)(cid:12) p M − (cid:12)(cid:12) αp = p − M − ( M − α Z ( α ) > p − M − Mα Z ( α ) > p − M − Mα , since Z ( α ) ∈ h , − p − (cid:17) for α ∈ [0 , + ∞ ).6.1.1. Classical error threshold: M and α fixed. We analyze now weather or notthe condition (5.5) is satisfied, when M and α ∈ [0 , + ∞ ) are fixed. The condition M fixed can be relaxed to ‘ M is upper bounded.’ Taking into account that s > M c such that q ( α ) > p − M c − M c α ≥ s, which implies the existence of a classical error threshold:(6.3) M c ≤ − ln s (2 + α ) ln p for s ∈ (0 ,
1) and α fixed.6.1.2. Overcoming Eigen’s paradox: M variable and α fixed. If M can grow and α is fixed, the condition (5.5) is satisfied if p − M − Mα < q ( α ) < s , which impliesthat M > − ln s (2 + α ) ln p for s ∈ (0 ,
1) .Under a ‘fierce competition’ between the groups I + p M Z p , Z p r (cid:2) I + p M Z p (cid:3) ,i.e. when rate b approaches from the left to rate a (i.e. s → + ), M must grow,which means that the survival of the sequences in the group I + p M Z p demandsthat they get closer to master sequence I , which means, that they must increasetheir lenghts. Then in this model the ‘classical Eigen’s paradox does not occur’because the length of the genomes can grow during the evolution process.7. The Error Threshold Problem Using a mutation measure of Gibbstype
In this section we study the error threshold problem using the Maynard Smithansatz with a mutation measure of Gibbs type depending only on |·| p . A mutation measure of Gibbs type.
We propose a mutation measure oftype(7.1) e − βE ( | x | p ) Z ( β, E ) , where β > E : R + → R + , and Z ( β, E ) = R Q p e − βE ( | x | p ) dx < ∞ . A Gibbsmeasure is a natural choice when dealing with infinite systems. On the other hand,mutations matrices of type Q ( I, J ) = e − βE ( I,J ) C appear naturally in the models ofevolution using spin glasses technique, see e.g. [32, Equations (6) and (8)]. We pick E (cid:16) | x | p (cid:17) = | x | αp , with α >
0, which corresponds to the simplest energy function. Byassuming that β > − β · ), our hypothesis on the mutation measure implies that the mostprobable mutations of a given sequence I happen to sequences which are very closeto I in the p -adic norm, which are sequences belonging to a ball of type I + p M Z p ,with M sufficiently large. In practical terms, this means that the replicators arenot too dispersed on Q p . It is interesting to quote here that according to [47]:“in silico simulations reveal that replicators with limited dispersal evolve towardshigher efficiency and fidelity.”We assume that the probability that a sequence x ∈ Q p mutates into a sequencebelonging to set B (a Borel subset of Q p ) is given by P ( x, B ; α, β ) = 1 C Z B e − β | x − y | αp dy, where C ( α, β ) := C , and α , β are positive constants such that C R Q p e − β | x − y | αp dy =1. Notice that P ( x, B ; α, β ) is a space homogeneous Markov transition function,the parameter β (which is typically interpreted as proportional to the inverse of thetemperature) plays the role of time. We fix a sequence I = ∞ X l = − k I l p l ,and assume that I has ‘infinite length’, and consider the ball I + p M Z p , M ∈ N ,which contains all the sequences that coincide with I up to the digit I M − . We canconsider I as the master sequence, and M is the minimum number of nucleotidesthat a sequence in the set I + p M Z p shares with I .The probability that a sequence x mutates into a sequence belonging to Q p r (cid:2) I + p M Z p (cid:3) is P ( x, Q p r (cid:2) I + p M Z p (cid:3) ; α, β ), and the probability that any sequencefrom the ball I + p M Z p mutates into a sequence in Q p r (cid:2) I + p M Z p (cid:3) is given by q ( M, α, β ) := 1 C Z I + p M Z p Z Q p r [ I + p M Z p ] e − β | x − y | αp dydx. We assert that(7.2) q ( M, α, β ) > ( p − p − M C e − βp α . ON-ARCHIMEDIAN REPLICATOR DYNAMICS 17
Indeed, by using the fact that the measure dydx is invariant under translations,and the ultrametric property of |·| p , we have q ( M, α, β ) = 1 C Z I + p M Z p Z Q p r [ I + p M Z p ] e − β | ( x − I ) − ( y − I ) | αp dydx = 1 C Z p M Z p Z Q p r p M Z p e − β | x − y | αp dydx = 1 C Z | x | p ≤ p − M Z | y | p >p − M e − β | x − y | αp dydx = 1 C Z | x | p ≤ p − M Z | y | p >p − M e − β | y | αp dydx = p − M C Z | y | p >p − M e − β | y | αp dy = p − M C ∞ X j = − M +1 Z | y | p = p j e − β | y | αp dy> p − M C Z | y | p = p − M +1 e − β | y | αp dy = (cid:0) − p − (cid:1) p − M +1 C e − βp ( − M +1) α = ( p − p − M C e − βp α p − Mα > ( p − p − M C inf M ∈ N e − βp α p − Mα = ( p − p − M C e − βp α . We also notice that(7.3) q ( M, α, β ) = p − M C Z | y | p >p − M e − β | y | αp dy < p − M C Z Q p e − β | y | αp dy = C p − M , where C = C ( α, β ) is a positive constant independent of M , since e − β | y | αp is anintegrable function, see e.g. the proof of Lemma 4.1 in [19].On the other hand, we denote by r := r ( M, α, β ) the probability of mutation ofa sequence from Q p r (cid:2) I + p M Z p (cid:3) into I + p M Z p . Then(7.4) r ( M, α, β ) = q ( M, α, β ) . The Eigen paradox.
Notice that by (7.3), r ( M, α, β ) = q ( M, α, β ) decayswith M , so we can use the Maynard Smith ansatz to estimate the error threshold,see Section 5. From (5.5), by using that q ( M, α, β ) > ( p − p − M C e − βp α , with p , α , β , s fixed, we have ( p − p − M C e − βp α < s, which implies that(7.5) M > − ( βp α + ln s )2 ln p + ln p − C p . Under a ‘fierce competition’ between the groups I + p M Z p , Q p r (cid:2) I + p M Z p (cid:3) , i.e.when rate b approaches from the left to rate a (i.e. s → + ), M must grow, whichmeans that the survival of the sequences in the group I + p M Z p demands thatthey get closer to master sequence I , which means, that they must increase theirlenghts. Then in our model the ‘classical Eigen’s paradox does not occur’ becausethe length of the genomes can grow during the evolution process. Notice that if M does not satisfy (7.5), then the sequences in the set I + p M Z p will not survive inthe long term.For arbitrary f and Q , we propose the existence of threshold function for thelength of the genomes M c ( f, Q ) such that M > M c ( f, Q ) is a necessary and suffi-cient condition for the long term survival of the genomes. Formula (7.5) gives anestimate for the function M c ( f, Q ) for the particular case in which Q has the form(7.1). We propose that under the condition M > M c ( f, Q ), the Darwin-Eigen cycle proposed by A. Poole, D. Jeffares and D. Penny takes place, see [39]: the Darwin-Eigen cycle is a positive feedback mechanism. Larger genome size improves thefidelity replication ( q ( M, α, β ) decays when M grows cf. (7.3)), and this increasesthe Eigen limit (see (7.5)) on the length of the genome, and this allows the evolu-tion of larger genome size. In turn, this allows the evolution of new function, whichcould further improve the replication fidelity, and so on. See [39], [41], and [18], fora detailed biological discussion. On the other hand, if M ≤ M c ( f, Q ) then genomesdo not survive in the long term. This is a type of classical error threshold, ‘similar’to the one provided by the Eigen evolution model with point mutation matrices.8. The Cauchy Problem for the p -adic replicator equation In this section we show the existence of a solution for the Cauchy problem (3.5),which in turn implies the existence of quasispecies for the p -adic replicator equationintroduced here. This goal is achieve by using the classical method of separationof variables and p -adic wavelets, see [1] and [28]. We assume that the function Q satisfies only the hypotheses given in Section 3.3.8.1. Some remarks on p -adic wavelets. We take K = C , R . We denote by C ( Q p , K ) the K -vector space of continuous K -valued functions defined on Q p .We fix a function a : R + → R + and define the pseudodifferential operator D → C ( Q p , C ) ∩ L ϕ → A ϕ, where ( A ϕ ) ( x ) = F − ξ → x n a (cid:16) | ξ | p (cid:17) F x → ξ ϕ o .The set of functions { Ψ rnj } defined as(8.1) Ψ rnj ( x ) = p − r χ p (cid:0) p r − jx (cid:1) Ω (cid:16) | p r x − n | p (cid:17) , where r ∈ Z , j ∈ { , · · · , p − } , and n runs through a fixed set of representatives of Q p / Z p , is an orthonormal basis of L ( Q p , C ) consisting of eigenvectors of operator A :(8.2) A Ψ rnj = a ( p − r )Ψ rnj for any r , n , j .This result is due to S. Kozyrev see e.g. [28, Theorem 3.29], [1, Theorem 9.4.2].Notice that b Ψ rnj ( ξ ) = p r χ p (cid:0) p − r n (cid:0) ξ + p r − j (cid:1)(cid:1) Ω (cid:16)(cid:12)(cid:12) p − r ξ + p − j (cid:12)(cid:12) p (cid:17) , and then a (cid:16) | ξ | p (cid:17) b Ψ rnj ( ξ ) = a ( p − r ) b Ψ rnj ( ξ ) . ON-ARCHIMEDIAN REPLICATOR DYNAMICS 19
The Fourier transform b Q of Q is a real-valued function, which is radial in Q p r { } .For this reason, we use the notation b Q (cid:16) | ξ | p (cid:17) .We set W ϕ = Q ∗ ϕ for ϕ ∈ D , as before. Then(8.3) W Ψ rnj ( x ) = b Q (cid:0) p − r (cid:1) Ψ rnj ( x ) , where b Q (cid:0) p − r (cid:1) is a real number satisfying (cid:12)(cid:12)(cid:12) b Q (cid:0) p − r (cid:1)(cid:12)(cid:12)(cid:12) ≤ The Cauchy problem for operator W . We now consider the followinginitial value problem:(8.4) Y : Q p × R + → R , Y ( · , t ) ∈ L R , Y ( x, · ) ∈ C ( R + , R ) ∂Y ( x,t ) ∂t = W Y ( x, t ) , x ∈ Q p , t > Y ( x,
0) = Y ( x ) ∈ D R . Notice that the conditions Y ( x ) ≥ R Q p Y ( x ) dx = 1 constitute naturalphysical restrictions for the function Y , however, here we do not use them.We solve (8.4) by using the separation of variables method. We first look for acomplex-valued solution of (8.4) of the form e Y ( x, t ) = X rjn C rjn ( t ) Ψ rnj ( x )(8.5) = X rjn C rjn ( t ) p − r χ p (cid:0) p r − jx (cid:1) Ω (cid:16) | p r x − n | p (cid:17) , where C rjn ( t ) are complex-valued functions, which admit continuous temporalderivatives. We fix a countable disjoint covering of Q p by balls of the form: B r i (cid:0) p − r i n i (cid:1) = p − r i n i + p − r i Z p ,where r i ∈ Z , and n i = a − k i p − k i + · · · + a − p − ∈ Q p / Z p , and the digits a j s runsthrough the set { , · · · , p − } , such that(8.6) f | B ri ( p − ri n i )= f ( (cid:12)(cid:12) p − r i n i (cid:12)(cid:12) p ) . Consequently(8.7) f (cid:16) | x | p (cid:17) = l ( f ) X i =0 f ( (cid:12)(cid:12) p − r i n i (cid:12)(cid:12) p )Ω (cid:16) | p r i x − n i | p (cid:17) , l ( f ) ∈ N ,due to the fact that the fitness function f is a locally constant function with compactsupport.We now describe the balls contained in B r i ( p − r i n i ). Any such ball has the form: a − k i p − r i − k i + · · · + a − p − r i − + b p − r i + · · · + b r − p r − + p r Z p = p − r i n i + p r (cid:0) b p − r − r i + · · · + b r − p − (cid:1) + p r Z p =: p − r i n i + p r n r + p r Z p = p r (cid:0) p − r − r i n i + n r (cid:1) + p r Z p , (8.8) for some integer r ≥ − r i and some n r ∈ Q p / Z p . The amount of such balls isexactly p r − . Now, for a fixed Ω (cid:16) | p r i x − n i | p (cid:17) and a variable Ω (cid:16) | p r x − n | p (cid:17) , wehave(8.9)Ω (cid:16) | p r x − n | p (cid:17) Ω (cid:16) | p r i x − n i | p (cid:17) = Ω (cid:16) | p r x − n | p (cid:17) if B r ( p − r n ) ⊂ B r i ( p − r i n i )0 if B r ( p − r n ) " B r i ( p − r i n i ) . By using (8.8)-(8.9), we have f (cid:16) | x | p (cid:17) e Y ( x, t ) = l ( f ) X i =0 f ( (cid:12)(cid:12) p − r i n i (cid:12)(cid:12) p ) p − X j =1 X r ≥− r i X p − r − ri n i + n r C ( − r ) j ( p − r − ri n i + n r ) ( t ) × p r χ p (cid:0) p − r − jx (cid:1) Ω (cid:16)(cid:12)(cid:12) p − r x − (cid:0) p − r − r i n i + n r (cid:1)(cid:12)(cid:12) p (cid:17) , and by using (8.3), W (cid:16) f (cid:16) | x | p (cid:17) e Y ( x, t ) (cid:17) =(8.10) l ( f ) X i =0 f ( (cid:12)(cid:12) p − r i n i (cid:12)(cid:12) p ) p − X j =1 X r ≥− r i X p − r − ri n i + n r C ( − r ) j ( p − r − ri n i + n r ) ( t ) × b Q ( p r ) p r χ p (cid:0) p − r − jx (cid:1) Ω (cid:16)(cid:12)(cid:12) p − r x − (cid:0) p − r − r i n i + n r (cid:1)(cid:12)(cid:12) p (cid:17) . By replacing (8.5) and (8.10) in (8.4), we obtain e Y ( x, t ) = l ( f ) X i =0 f ( (cid:12)(cid:12) p − r i n i (cid:12)(cid:12) p ) p − X j =1 X r ≥− r i X p − r − ri n i + n r C ( − r ) j ( p − r − ri n i + n r ) (0) × (8.11)exp (cid:16) f ( (cid:12)(cid:12) p − r i n i (cid:12)(cid:12) p ) b Q ( p r ) t (cid:17) p r χ p (cid:0) p − r − jx (cid:1) Ω (cid:16)(cid:12)(cid:12) p − r x − (cid:0) p − r − r i n i + n r (cid:1)(cid:12)(cid:12) p (cid:17) , where Y ( x ) = l ( f ) X i =0 f ( (cid:12)(cid:12) p − r i n i (cid:12)(cid:12) p ) p − X j =1 X r ≥− r i X p − r − ri n i + n r C ( − r ) j ( p − r − ri n i + n r ) (0) × p r χ p (cid:0) p − r − jx (cid:1) Ω (cid:16)(cid:12)(cid:12) p − r x − (cid:0) p − r − r i n i + n r (cid:1)(cid:12)(cid:12) p (cid:17) . Now, we set α rr i n i := f ( (cid:12)(cid:12) p − r i n i (cid:12)(cid:12) p ) b Q ( p r ) , then exp h f ( | p r i n i | p ) b Q ( p r ) t + 2 πi (cid:8) p − r − jx (cid:9) p i = e tα rrini h cos (cid:16) π (cid:8) p − r − jx (cid:9) p (cid:17) + √− (cid:16) π (cid:8) p − r − jx (cid:9) p (cid:17)i , and by setting A rjr i n i n r := Re C ( − r ) j ( p − r − ri n i + n r ) (0) , B rjr i n i n r := Im C ( − r ) j ( p − r − ri n i + n r ) (0) , ON-ARCHIMEDIAN REPLICATOR DYNAMICS 21 the solution Y ( x, t ) is the real part of e Y ( x, t ): Y ( x, t ) = l ( f ) X i =0 p − X j =1 X r ≥− r i X p − r − ri n i + n r p r e tα rrini Ω (cid:16)(cid:12)(cid:12) p − r x − (cid:0) p − r − r i n i + n r (cid:1)(cid:12)(cid:12) p (cid:17) × (8.12) h A rjr i n i n r cos (cid:16) π (cid:8) p − r − jx (cid:9) p (cid:17) − B rjr i n i n r sin (cid:16) π (cid:8) p − r − jx (cid:9) p (cid:17)i . In the next section we show that Y ( x, t ) is really a finite sum, if Y ∈ D R .8.2.1. Some remarks about the Fourier coefficients. If Y ∈ D R , then C ( − r ) j ( p − r − ri n i + n r ) (0) = 0 ⇔ − r i ≤ r ≤ l − , where l is the index of local constancy of Y . This implies that almost all the C ( − r ) j ( p − r − ri n i + n r ) (0)s are zero.Indeed, f ( (cid:12)(cid:12) p − r i n i (cid:12)(cid:12) p ) C ( − r ) j ( p − r − ri n i + n r ) (0) = p r Z p − ri n i + p r n r + p r Z p Y ( x ) χ p (cid:0) p − r − jx (cid:1) dx .We now use the subdivision p − r i n i + p r n r + p r Z p = G n l p − r i n i + p r n r + p l n l + p l Z p . Then f ( | p − r i n i | p ) C ( − r ) j ( p − r − ri n i + n r ) (0) becomes a finite sum of terms of the form Y (cid:0) p − r i n i + p r n r + p l n l (cid:1) Z p − ri n i + p r n r + p l n l + p l Z p χ p (cid:0) p − r − jx (cid:1) dx = p − l Y (cid:0) p − r i n i + p r n r + p l n l (cid:1) χ p (cid:0) p − r − j (cid:8) p − r i n i + p r n r + p l n l (cid:9)(cid:1) × Z Z p χ p (cid:0) p l − r − jy (cid:1) dy = p − l Y (cid:0) p − r i n i + p r n r + p l n l (cid:1) χ p (cid:0) p − r − j (cid:8) p − r i n i + p r n r + p l n l (cid:9)(cid:1) × r ≤ l −
10 if r > l − . For details about the calculation of the integral involving the additive character,the reader may consult for instance [52, p. 42-43].8.3.
The Cauchy Problem for the p -adic replicator equation. We now con-sider the following Cauchy problem:(8.13) X : Q p × R + → R , X ( · , t ) ∈ L R ∩ L R , X ( x, · ) ∈ C ( R + , R ) ∂X ( x,t ) ∂t = W X ( x, t ) − Φ ( t ) X ( x, t ) , x ∈ Q p , t > X ( x,
0) = Y ∈ D R . By using (3.1), (3.4) and (8.12), we haveexp (cid:18)Z t Φ ( τ ) dτ (cid:19) = Z Q p Y ( x, t ) dx = l ( f ) X i =0 p − X j =1 X r ≥− r i X p − r − ri n i + n r p r e tα rrini D rjr i n i n r =: Y ( t ) , where the D rjr i n i n r s are real constants, which are obtained by integrating (8.12)termwise with respect to x .Therefore the solution of initial value problem (8.13) is given by X ( x, t ) = 1 Y ( t ) l ( f ) X i =0 p − X j =1 X r ≥− r i X p − r − ri n i + n r p r e tα rrini Ω (cid:16)(cid:12)(cid:12) p − r x − (cid:0) p − r − r i n i + n r (cid:1)(cid:12)(cid:12) p (cid:17) (8.14) × h A rjr i n i n r cos (cid:16) π (cid:8) p − r − jx (cid:9) p (cid:17) − B rjr i n i n r sin (cid:16) π (cid:8) p − r − jx (cid:9) p (cid:17)i . Notice that R Q p X ( x, t ) dx = 1 for t >
0, and thus the hypothesis (3.1) holds, andconsequently X ( · , t ) ∈ L R for t > The p -adic quasispecies. In this section we consider the steady state con-centration:(8.15) X ( x ) := lim t → + ∞ X ( x, t ) , which corresponds, in the classical terminology, to the p -adic quasispecies . Wedefine λ max = max ≤ i ≤ l ( f ) r ≥− r i n f ( (cid:12)(cid:12) p − r i n i (cid:12)(cid:12) p ) b Q ( p r ) o . We recall that the condition r ≥ − r i involves only a finite number of r s, since allthe sums in (8.14) run through finite sets, this is a consequence of the fact that theFourier expansion of e Y ( x, t ) is finite. We now define T = n rr i n i ; f ( (cid:12)(cid:12) p − r i n i (cid:12)(cid:12) p ) b Q ( p r ) = λ max o . Then, we have X ( x ) = 1 C X rr i n i ∈ T p − X j =1 p r Ω (cid:16)(cid:12)(cid:12) p − r x − (cid:0) p − r − r i n i + n r (cid:1)(cid:12)(cid:12) p (cid:17) × (8.16) h A rjr i n i n r cos (cid:16) π (cid:8) p − r − jx (cid:9) p (cid:17) − B rjr i n i n r sin (cid:16) π (cid:8) p − r − jx (cid:9) p (cid:17)i , where C = X rr i n i ∈ T p − X j =1 p r D rjr i n i n r . Notation 3. If U is an open and compact subset of Q p , for instance a finite unionof balls, we denote by D R ( U ) , the R -vector space of test functions with supports in U . ON-ARCHIMEDIAN REPLICATOR DYNAMICS 23
Consider the following eigenvalue problem:(8.17) W ϕ = λϕ , for λ ∈ R , ϕ ∈ D R ( l ( f ) a i =0 B r i (cid:0) p − r i n i (cid:1) ) , where ` l ( f ) i =0 B r i ( p − r i n i ) is the support of f , see (8.7). Then λ max is the largesteigenvalue associated with (8.17).On the other hand, from (8.16) we obtain that X ( x ) is a continuous function withcompact support, consequently, this function is integrable. By using the dominatedconvergence theorem and condition (3.1), we have1 = lim t → + ∞ Z Q p X ( x, t ) dx = Z Q p lim t → + ∞ X ( x, t ) dx = Z Q p X ( x ) dx. We now use the fact that X ( x, t ) ≥ X ( x ) ≥
0, in this way we reach the conclusion that X ( x ) is a probability density supported in supp f .This behavior is completely different from the one presented in the Eigen model.In the classical case, under the hypothesis that the mutation matrix has one largesteigenvalue, the steady state concentration is a constant vector having exactly onenon-zero entry. Acknowledgement 1.
The author wishes to thank the referees for their carefulreading of the original manuscript and for their helpful suggestions.
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Centro de Investigaci´on y de Estudios Avanzados del Instituto Polit´ecnico Na-cional, Departamento de Matem´aticas, Unidad Quer´etaro, Libramiento Norponiente
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