Simulation of Biological Processes phần 3

44 KRAKAUER make simulation models partially phenomenological through simplifying approximations. We see therefore that the distinction between modelling approaches becomes somewhat arbitrary, as all models are phenomenological models. The di¡erences are not qualitative but quantitative, and relate to the number of variables and parameters we are happy to plug into our brains or into thecircuitryofacomputer.Asmallernumberofvariablesandparametersisalways preferable,butourwillingnesstomovetowardthephenomenological,dependson how reliable is the derivation of the macroscopic equations from the microscopic interactions.Aformalapproachtorescalingmany-bodyproblemsöamethodfor reducingthenumberofvariablesöistouserenormalizationgrouptheory(Wilson 1979). Here I am going to present an evolutionary perspective on this complex topic. RatherthandiscussMonteCarlomethods,agentbasedmodels,interactingparticle systems, and stochastic and deterministic models, and their uses at each scale. I restrict myself to a biological justi¢cation for phenomenological modelling. The argumentisasfollows.Naturalselectionworksthroughthedi¡erentialreplication ofindividuals.Individualsarecomplexaggregatesandyetthe¢tnessofindividuals is a scalar quantity, not a vector of component ¢tness contributions. This implies that the design of each component of an aggregate must be realized through the di¡erential replication of the aggregate as a whole. We are entitled therefore to characterize the aggregate with a single variable, ¢tness, rather than enumerate variablesforallof itscomponents. Thisamounts to stating thatidentifyinglevels of selection can be an e¡ective procedure for reducing the dimensionality of our statespace. Levelsofselection Here I shall brie£y summarize current thinking on the topic of units and levels of selection(forusefulreviewsseeKeller1999,Williams1995).Thelevelsofselection are those units of information (whether genes, genetic networks, genomes, individuals, families, populations, societies) able to be propagated with reasonable ¢delity across multiple generations, and in which these units, possess level-speci¢c ¢tness enhancing or ¢tness reducing properties. All of the listed levels are in principle capable of meeting this requirement (that is the total genetic information contained within these levels), and hence all can be levels of selection.Selectionoperatesatmultiplelevelsatonce.However,selectionismore e⁄cient in large populations, and drift dominates selection in small populations. As we move towards increasingly inclusive organizations, we also move towards smallerpopulationsizes.Thisimpliesthatselectionislikelytobemoree¡ectiveat the genetic level than, say, the family level. Furthermore, larger organizations are more likely to undergo ¢ssion, thereby reducing the ¢delity of replication. These LEVELS OF DESCRIPTION AND SELECTION 45 twofactorshaveledevolutionarybiologiststostressthegeneasaunitofselection. This is a quantitative approximation. In reality there are numerous higher-order ¢tnesstermsderivedfromselectionatmoreinclusivescalesoforganization. Fromtheforegoingexplanationitshouldbeapparentthattheeasewithwhicha component can be an independent replicator, helps determine the e⁄ciency of selection. In asexual haploid organisms individual genes are locked into permanent linkage groups. Thus individual genes do not replicate, rather whole genomes or organisms. The fact of having many more genes than genomes is not animportantconsiderationforselection.Thisisanextremeexamplehighlighting the important principle of linkage disequilibrium. Linkage disequilibrium describes a higher than random association among alleles in a population. In other words, picking an AB genome from an asexual population is more likely than ¢rst picking an A allele and subsequently a B allele. Whenever A and B are bothrequiredforsomefunctionweexpectthemtobefoundtogether,regardlessof whether the organism is sexual, asexual, or even if the alleles are in di¡erent individuals!(Considerobligatesymbioticrelationships.) ThisimpliesthattheAB aggregatecannowitselfbecomeaunitofselection.Thisprocesscanbeextendedto include potentially any number of alleles, spanningall levelsof organization. The important property of AB versus A and B independently is that we can now describe the system with one variable whereas before we had to use two. The challenge for evolutionary theory is to identify selective linkage groups, thereby exposing units of function, and allowing for a reduction in the dimension of the state space. These units of function can be genetic networks, signal transduction modules, major histocompatibility complexes, and even species. In the remainder of this paper I shall describe individual level models and their phenomenological approximations,motivatedbytheassumptionofhigherlevelsofselection. Levelsofdescriptioningenetics Population genetics is the study of the genetic composition of populations. The emphasis of population genetics has been placed on the changes in allele frequencies through time, and the forces preserving or eliminating genetic variability. Very approximately, mutation tends to diversify populations, whereas selection tends to homogenize populations. Population genetics is a canonical many-body discipline.Itwould appearthat weare requiredto track the abundance of every allele at each locus of all members in a randomly mating population. This would seem to be required assuming genes are the units of selection, and all replicate increasing their individual representation in the gene pool. However, even a cursory examination of the population genetics literature reveals this expectation to be unjusti¢ed. The standard assumption of population genetics modelling is that whole genotypes can be assigned individual ¢tness 46 KRAKAUER values. Consider a diploid population with two alleles. A1 and A2 and corresponding ¢tness values W11¼1, W12¼W21¼1ÿhs and W22¼1ÿs. The value s is the selection coe⁄cient and h the degree of dominance. Population genetics aims to capture microscopic interactions among gene products by varying the value of h.When h¼1then A1 is dominant. When05h51 then A1 is incompletely dominant. When h¼0, A1 is recessive. Denoting as p the frequency of A1 and 1ÿp the frequency of A2, the mean population ¢tness is givenby W ¼ 1 ÿ s þ 2s(1ÿ h)pÿ s(1ÿ 2h)p2 andtheequilibriumabundanceofA1, 1ÿ h 1ÿ 2h These are very general expressions conveying information about the ¢tness and composition of a genetic population at equilibrium. The system is reduced from twodimensions to one dimension by assuming that dominance relations among autosomal alleles can be captured through a single parameter (h). More signi¢cantly, the models assume that autosomal alleles are incapable of independent replication. The only way in which an allele can increase its ¢tness is through some form of cooperation (expressed through the dominance relation) withanotherallele. The situation is somewhat more complex in two-allele two-locus models (A1, A2, B1, B2). In this case we have 16 possible genotypes. The state space can be reduced by assuming that there is no e¡ect of position, such that the ¢tness of A1B1A2B2 is equal to that of A1B2A2B1. We therefore have 9 possible genotypes. We can keep the number of parameters in such amodel below 9 while preventing our system from becoming underdetermined, by assuming that genotype ¢tness is the result of the additive or multiplicative ¢tness contributions of individual alleles. This leaves us with us 6 free parameters. The assumption of additive allelic ¢tness means that individual alleles can be knocked outwithoutmortalityofthegenotype.Withmultiplicative¢tnessknockoutofany onealleleinagenomeislethal.Thesetwophenomenologicalassumptionsrelateto very di¡erent molecular or microscopic processes. Once again this modelling approach assumes that individual alleles cannot increase their ¢tness by going solo; alleles increase in frequency only as members of the complete genome and theycooperatetoincreasemean¢tness. When alleles or larger units of DNA (microsattelites, chromosomes) no longer cooperate,thatiswhentheybehavesel¢shly,thenthestandardpopulationgenetics approximations for the genetic composition of populations breaks down (Buss LEVELS OF DESCRIPTION AND SELECTION 47 1987).Thisrequiresthatindividualgeneticelementsratherthanwholegenotypes areassigned¢tnessvalues.Theconsequenceisalargeincreaseinthestatespaceof themodels. Levelsofdescriptioninecology Population genetics was described as the study of the genetic structure of populations. In a like fashion, ecology might be described as the study of the species composition of populations. More broadly, ecology seeks to study the interactions between organisms and their environments. This might lead one to expect that theory in ecology is largely microscopic, involving extensive simulation oflargepopulationsofdi¡erent individuals.Onceagainthis isnotthe case. The most common variable in ecological models is the species. In order to understand the species composition of populations, theoretical ecologists ascribe replication rates and birth rates to whole species, and focus on species level relations. We can see this by looking at typical competition equations in ecology. Assumethat wehave two speciesX and Y withdensitiesx and y. We assumethat these species proliferate at rates ax and dy. In isolation each species experiences density limited growth at rates bx2 and fy2. Finally, each species is able to interfere with the other such that y reduces the growth of x at a rate cyx and x reduces the growth of y at a rate exy.With these assumption we can write down a pairofcoupleddi¡erentialequationsdescribingthedynamicsofspecieschange, x_ ¼ x(aÿ bc ÿ cy) y_ ¼ y(d ÿ ex ÿ fy) Thissystemproducesoneoftwosolutions,stablecoexistenceorbistability.When theparametervaluessatisfytheinequalities, b a c e d f The system converges to an equilibrium in which both species coexist. When the parametervaluessatisfytheinequalities, c a b f d e thendependingontheinitialabundancesofthetwospeciesoneortheotherspecies is eliminated producing bistability. These equations describe in¢nitely large populations of identical individuals constituting two species. The justi¢cation for this approximation is derived from the perfectly reasonable assumption that 48 KRAKAUER evolution at the organismal level is far slower than competition among species. This separation of time scales is captured by Hutchinson’s epigram, ‘The ecological theatre and the evolutionary play’. In e¡ect these models have made thespeciesthevehicleforselection. An explicit application of the separation of time scales to facilitate dimension reduction lies at the heart of adaptive dynamics (Diekman & Law 1996). Here the assumption is made to allow individual species composition to be neglected in order to track changes in trait values. The canonical equation for adaptive dynamicsis, s_i ¼ ki(s) @s0 Wi(s0, s)js0¼si . The si with i¼1,..., Ndenote the values of an adaptive trait in a population of N species. The W(s0 , s) are the ¢tness values of individual species with trait values given by s2 when confronting the resident trait values s. The ki(s) values are the species-speci¢c growth rates. The derivative (@=@si)Wi(si, s)js0¼s points in the direction of the maximal increase in mutant ¢tness. The dynamics describes the outcome of mutation which introduces new trait values (si) and selection that determines their fateö¢xation or extinction. It is assumed that the rapid time scale of ecological interactions, combined with the principle of mutual exclusion, leads to a quasi-monomorphic resident population. In other words, populations for which the periods of trait coexistence are negligible in relation to the time scale of evolutionary ¢xation. These assumptions allow for a decoupling of population dynamics (changes in species composition) from adaptive dynamics (changesintraitcomposition). Whiletheselevelsofselectionapproximationshaveprovedveryuseful,thereare numerous phenomena for which we should like some feeling for the individual behaviours. This requires that we do not assume away individual contributions in order to build models, but model them explicitly, and derive aggregate approximations from the behaviour of the models. This can prove to be very important as the formal representation of individuals, can have a signi¢cant impact on the statistical properties of the population. Durret & Levin (1994) demonstrate this dependence by applying four di¡erent modelling strategies to a single problem: mean ¢eld approaches (macroscopic), patch models (macroscopic), reaction di¡usion equations (macroscopic) and interacting particle systems (microscopic). Thus the models move between deterministic mean ¢eld models, to deterministic spatial models, to discrete spatial models. Durret and Levin conclude that there can be signi¢cant di¡erences at the population level as a consequence of the choice of microscopic or macroscopic model. For example spatial and non-spatial models disagree when two species ... - tailieumienphi.vn