Modeling Evolution Using the Probability of Fixation: History and Implications

Modeling Evolution Using the Probability of Fixation: History and ImplicationsAuthor(s): David M. McCandlish, Arlin StoltzfusSource: The Quarterly Review of Biology, Vol. 89, No. 3 (September 2014), pp. 225-252Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/10.1086/677571 .

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MODELING EVOLUTION USING THE PROBABILITY OF FIXATION:HISTORY AND IMPLICATIONS

David M. McCandlishBiology Department, Duke University, Durham, North Carolina 27708 and Department of Biology,

University of Pennsylvania, Philadelphia, Pennsylvania 19104 USA

e-mail: [email protected]

Arlin StoltzfusBiosystems and Biomaterials Division, National Institute of Standards and Technology, and Institute for

Bioscience and Biotechnology ResearchRockville, Maryland 20850 USA

e-mail: [email protected]

keywordspopulation genetics, modern synthesis, weak mutation, SSWM,

mutation-selection-drift, mutationism

abstractMany models of evolution calculate the rate of evolution by multiplying the rate at which new mutations

originate within a population by a probability of fixation. Here we review the historical origins, contem-porary applications, and evolutionary implications of these “origin-fixation” models, which are widely usedin evolutionary genetics, molecular evolution, and phylogenetics. Origin-fixation models were first intro-duced in 1969, in association with an emerging view of “molecular” evolution. Early origin-fixation modelswere used to calculate an instantaneous rate of evolution across a large number of independently evolving loci;in the 1980s and 1990s, a second wave of origin-fixation models emerged to address a sequence of fixation eventsat a single locus. Although origin-fixation models have been applied to a broad array of problems in contemporaryevolutionary research, their rise in popularity has not been accompanied by an increased appreciation of theirrestrictive assumptions or their distinctive implications. We argue that origin-fixation models constitute a coherenttheory of mutation-limited evolution that contrasts sharply with theories of evolution that rely on the presence ofstanding genetic variation. A major unsolved question in evolutionary biology is the degree to which these modelsprovide an accurate approximation of evolution in natural populations.

Introduction

ALTHOUGH many of the basic themesof theoretical population genetics had

already emerged by 1932, the body of formal

theory applied by evolutionary researchershas continued to expand in new directions.Examples of theoretical developments nota-ble for their wide use include the theory of

The Quarterly Review of Biology, September 2014, Vol. 89, No. 3

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Volume 89, No. 3 September 2014THE QUARTERLY REVIEW OF BIOLOGY

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kin selection (Hamilton 1964), evolutionarygame theory (Maynard Smith and Price 1973),the coalescent (Kingman 1982), and the mul-tivariate generalization of quantitative genetics(Lande and Arnold 1983).

Here we draw attention to a theoreticalinnovation that has achieved wide use cryp-tically, without having been recognized as aninnovation: specifying the rate of evolutionas the product of two factors, the rate atwhich new alleles originate within the popu-lation, and the probability that they reachfixation. By rendering evolutionary change asa two-step procedure of (1) mutation followedby (2) fixation or loss, such models skip lightlyover the complex within-population processesthat are the predominant focus of theoreticalpopulation genetics, and which typically pre-vent analytical solutions to the dynamics oflong-term evolutionary change.

Models of this form are in common usetoday, appearing in separate patches ofthe literature, and sometimes associatedwith labels such as “weak mutation” (Gil-lespie 1983a), “selection-mutation-drift”(Bulmer 1991), “mutation-selection” (Yangand Nielsen 2008), “mutation-limited”(Kopp and Hermisson 2009b), “new muta-tion” (Hill 1982), “mutation-driven” (Li 1997)and, in the context of molecular phylogenetics,“mechanistic” (as opposed to empirical) mod-els (Ren et al. 2005). Because none of thesenames uniquely specify the class of models wehave in mind, we call models that determinethe rate of evolution by multiplying the rate ofmutational origination by the probability of fix-ation “origin-fixation” models, in reference totheir mathematical form (readers should notbe confused by the fact that Gillespie, who con-tributed importantly to the development ofthese models, uses the terms “origination pro-cess” and “fixation process” to refer to some-thing related but different, e.g., Gillespie 1991,1994a).

Below, we begin with a brief mathematicaldescription of these models, including widelyknown special cases such as K�4Nus and themove-rule used in the mutational landscapemodel. Two general classes may be dis-cerned: aggregate-rate models that considerthe total rate of evolution over a large num-ber of independently evolving loci, and se-

quential fixation models that treat a series ofsubstitutions at a single locus.

We then review the diverse ways in whichthese models have been developed and ap-plied. Whereas the probability of fixation it-self is a classical result, true origin-fixationmodels did not appear until 1969, in thecontext of the neutral theory, taking theform of aggregate-rate models of neutral orbeneficial changes (King and Jukes 1969; seealso Kimura and Maruyama 1969). Overtime, such models became fundamental to theanalysis of molecular evolution. The first influ-ential sequential fixations model emerged in1983 as a model of genetic adaptation (Gil-lespie 1983b), while later models of sequentialfixations allowing deleterious and neutral mu-tations were proposed independently in sev-eral different fields. As shown below, suchmodels have been applied to a range of issues,including heterogeneity in evolutionary rates,codon usage bias, and phylogenetic inference.

Finally, we draw attention to origin-fixationmodels as a distinctive theory of evolutionarychange. Today, origin-fixation models are fre-quently invoked without regard to either theirunderlying assumptions or their implications,as if they represented a generic frameworkfor modeling the evolutionary process. Rather,origin-fixation models depict a specific re-gime of mutation-limited evolution in whichthe power of mutation to determine the rateand direction of evolution is particularlystrong. Because this regime does not neces-sarily occur widely in nature, we argue thatorigin-fixation models should be treated ashypotheses to be tested rather than merely amethod to be applied. Determining the de-gree to which evolution in nature occurs asdepicted by origin-fixation models providesan important direction for future research.

The Origin-Fixation FormalismTo understand the broad appeal of origin-

fixation models, it is helpful to recognizethat the long-term behavior of a reproducingpopulation of interacting individuals subjectto mutation, selection, drift, and recombina-tion is typically mathematically intractable.Even short-term models of population genet-ics often depend on radical simplifying as-sumptions such as neutrality, complete linkage

226 Volume 89THE QUARTERLY REVIEW OF BIOLOGY



(or full recombination), the absence of epista-sis (fitness interactions), and so on.

In origin-fixation models, the key simplify-ing assumption is that any time a mutationenters the population, it enters the popula-tion at a locus where only one other allele ispresent in the population. The great benefitof making this assumption is that there is abody of classical theory that describes thedynamics of a locus where at most two allelesare segregating and, in particular, there aresimple mathematical formulas for the prob-ability that a new mutation will reach fre-quency one in the population, i.e., achievefixation. Using such expressions, one canthen write down a model in which mutationsenter the population one at a time, and eachmutation either goes to fixation or is lostdepending on its probability of fixation. Suchmodels typically fall into one of two classes.Below we first describe, and then compare,these two classes of models.

aggregate-rate modelsIn the first of class of models, we consider

an infinite collection of independently evolv-ing loci where the total mutation rate acrossthis infinite collection is finite. Because themutation rate at any individual locus is infin-itesimally small, each new mutation will oc-cur at a different, previously monomorphic,locus. Since the typical goal of such a modelis to calculate a total rate of evolution acrossthis infinite collection of loci, we will callthese models “aggregate-rate” models.

Let us construct a simple version of such amodel and use it to derive some familiarresults. Suppose in particular we have a dip-loid population of size N and we assume thatmutations appear with a total rate u acrossour infinite collection of loci (this rate canbe arbitrarily large, yet the assumption ofinfinite loci ensures that each mutation oc-curs at a new locus). Then the total rate thatsuch mutations appear in the population as awhole is simply 2Nu. Of course, only a frac-tion of these mutations will eventually go tofixation. Thus, to calculate the rate of evolu-tion, K, we need to multiply this originationrate with the probability of fixation of a newmutation. Let us assume that dominance isabsent, and that each of these new mutations

has the same selection coefficient s. Writingthe probability of fixation of a new mutationwith selection coefficient s as �(s,N), we con-clude that the rate of evolution is given by

K � 2Nu � ��s,N�. (1)

Several classical results then follow immedi-ately by making particular choices for s and�(s,N). For instance, if we assume that mu-tations with a positive selection coefficient sare occurring with rate u, we can use Ha-ldane’s (1927) approximation �(s,N)�2s torecover the famous formula,

K � 2Nu � 2s � 4Nus. (2)

Similarly, if neutral rather than beneficialmutations are arising at rate u, we can usethe neutral probability of fixation, �(s,N)�1/(2N), to derive

K � 2Nu �1

2N� u. (3)

Although the above models are extremelysimple, more complex aggregate-rate mod-els can be constructed, e.g., by allowing theselection coefficients of new mutations to bedrawn from a probability distribution (e.g.,Ohta 1977; Kimura 1979) or by using thesesimple formulas as one component of amore complex model (e.g., Hill 1982; Sawyerand Hartl 1992).

sequential fixation modelsWhile the aggregate-rate models assume

that each mutation occurs at a new locus,models in our second major class insteadassume that mutations at any one locus oc-cur sufficiently infrequently that each newmutation is lost or goes to fixation prior tothe appearance of the next mutation. Thus,it is still the case that each new mutationoccurs at a locus that is otherwise monomor-phic. Such models typically are presented asone-locus, multiallelic models where thequestions of interest concern the trajectoryof a population through some kind of ex-plicit genotypic space (e.g., sequence space).We will call these models “sequential fixa-tion” models. This class of models, althoughvery important (see below), is perhaps lesswell known than the aggregate-rate models.

September 2014 227THE ORIGIN-FIXATION FORMALISM



Because sequential fixation models as-sume that the time during which any partic-ular mutation segregates in the population ismuch shorter than the waiting time fromone mutation to the next, the populationmust be monomorphic the vast majority ofthe time. Thus, in sequential fixation mod-els, one conventionally ignores both the timeit takes for a successful mutation to fix andthe periods during which the population ispolymorphic due to mutations destined forloss. This leads to a model in which a popu-lation is viewed as jumping directly from onefixed state to another at the introduction ofeach new mutation that is destined for fixa-tion (Figure 1).

More formally, sequential fixations mod-els treat evolution as a Markov chain wherethe states are the possible alleles at a singlelocus. If uij is the gametic mutation rate fromallele i to allele j, then the total rate at whichmutants of allele j originate within a diploidpopulation is given by 2Nuij. Furthermore,each such mutant is destined for fixationwith probability �(sij,N), where sij is the selec-tion coefficient of alternative allele j relativeto prevailing allele i. Thus, fixations from i toj occur at rate:

Q�i, j� � 2Nuij � ��sij,N�. (4)

That is, the fixation rate is again equal to theorigination rate times the probability of fix-ation.

The basic behavior of such a Markov chainis easy to understand. A population currently

fixed for some genotype i stays at genotype ifor an exponentially distributed amount oftime, where the mean of this exponentialdistribution is equal to the inverse of thesubstitution rate at genotype i, and the sub-stitution rate at i is equal to the sum of therates to its neighbors, �k � iQ�i,k�. Further-more, if we want to know the probability thatthe next fixation will be to some genotype j,this probability is given by

P�i, j� �Q�i, j�

�k�iQ�i,k�. (5)

which is simply the ratio between the rate offixations from i to j and the total substitutionrate at i. The great power of these relativelysimple dynamics is that it becomes possible tomodel the evolution of a population through alarge space of possible genotypes.

Equation 5 also provides an alternative wayto specify an origin-fixation model. Ratherthan keeping track of absolute time, we canconsider a Markov chain where each timestep consists of a single substitution. In math-ematics, this kind of discrete-time version ofa continuous-time process is called the “em-bedded Markov chain,” and P is the transi-tion matrix for this embedded Markov chain.We refer to Equation 5 as the “move-rule”form of origin-fixation dynamics, as it is con-veniently used as a rule to choose the nextstep in an adaptive walk. For instance, themove-rule in Orr’s (2002) mutational land-

Figure 1. A Comparison Between a Sequential Fixations Model and the Full EvolutionaryDynamics

In a sequential fixations model, the population jumps from one state to another (upper figure) by a processthat, in effect, is an instantaneous event of mutation-and-fixation. In an actual population, this process musttake some period of time as an allele frequency traverses from a starting value to fixation (lower figure).However, if the gaps between these polymorphic periods are sufficiently long, an origin-fixation model canprovide a good approximation to the full evolutionary dynamics.




scape model is a simplified version of Equa-tion 5.

comparing classes of modelsAn important similarity between the

aggregate-rate and sequential fixation mod-els is that both classes of models rely on thesame set of well-known expressions for theprobability of fixation. Roughly speaking,these are Haldane’s (1927) formula for theprobability of fixation of an advantageousallele, Wright and Fisher’s approximate for-mulas (Fisher 1930a,b; Wright 1931), Kimu-ra’s (1957, 1962) general formula for thediffusion approximation, and Moran’s (1959)exact expression for the Moran process.

Despite being based on the same buildingblocks, aggregate-rate and sequential fixa-tion models rely on different assumptions. Inan aggregate-rate model, one assumes thatthe loci in question evolve independentlyfrom each other. This requires that the locibe unlinked (i.e., there is free recombina-tion between them), that the fitness effects ofa mutation at one locus do not depend onsegregating variants at another locus (i.e., anabsence of epistasis), and that selection isweak in the sense that the variance in fitnesswithin the population is small (e.g., Barton1995; this requirement can be relaxedthrough the use of an effective popula-tion size).

In addition to the assumption that dif-ferent loci evolve independently, one mustalso assume that each new mutation happensat a location in the genome where no varia-tion is currently segregating. This second as-sumption is known in the literature as the“infinite-sites” assumption (Kimura 1969).Thus, aggregate-rate models are a subset ofinfinite-sites, free-recombination models (adifferent, but still common, class of modelsmakes the infinite-sites assumption but alsoassumes complete linkage between sites, e.g.,in Watterson 1975).

By contrast, the key assumption underly-ing sequential fixation models concerns therate of mutational origination. In sequentialfixation models, mutations must occur suffi-ciently infrequently that each new mutationis lost or becomes fixed in the populationbefore the next new mutation enters the

population (whereas in aggregate-rate mod-els, the total rate at which mutations at allloci enter the population, 2Nu, can be arbi-trarily large). For a diploid locus undergoingfrequency-independent genic selection, themutations that segregate for the longest timein a population are neutral mutations thatare destined for fixation, which segregate onthe order of 4N generations (Kimura andOhta 1969). The expected number of muta-tions to enter the population during this pe-riod is then 4N�2Nu�8N 2u, so that acondition for the strict applicability of amodel of sequential fixations is 8N 2u �� 1(e.g., see Ishii et al. 1982; Lynch and Abegg2010). This condition is, however, just a ruleof thumb, and the assumptions necessary foran origin-fixation model to faithfully capturethe true evolutionary dynamics will in gen-eral depend on other factors in addition tothe population size and mutation rate. Wewill return to this point in the Discussion.

It is important to realize that in practicemany models of sequential fixations assumefree recombination and the absence of epis-tasis in addition to assuming that mutationsenter the population sufficiently infrequently.Such models are typically constructed by con-sidering a collection of independently evolvingsequential-fixations Markov chains. For exam-ple, Halpern and Bruno (1998) construct amodel for the evolution of coding sequenceswhere each codon is modeled as an indepen-dent sequential-fixations Markov chain. Forsuch models, the mutation rate u in the condi-tion 8N 2u �� 1 then refers to the mutationrate for each independent Markov chain,rather than the total mutation rate over thecollection as a whole.

Origin-fixation models may be distinguishedfrom models that share some, but not all, oftheir features, as follows. The assumptionthat mutations at any given locus enter apopulation extremely rarely is sometimesknown as the “weak mutation” assumption.Whereas origin-fixation models are alwaysweak-mutation models, not all weak-mutationmodels are origin-fixation models. For in-stance, many weak-mutation models work byintegrating over a distribution of gene fre-quencies in the limit of weak mutation ratherthan multiplying a rate of mutational origin




by the probability of fixation (e.g., Wright1931; Li 1987; Zeng et al. 1989; Kondrashov1995). It is also important to note that thereare many classes of models that contain dis-crete origination events that are not origin-fixation models, such as coalescent models,simulation models that treat the dynamics ofsegregating alleles in full detail, or recent mod-els of polymorphic asexual populations that, insome ways, extend origin-fixation thinking be-yond the weak-mutation regime (e.g., Weiss-man et al. 2009).

Finally, it may also be worth noting thatthe origin-fixation formalism might be ap-plied, not only to new mutations in the con-ventional sense, but also wherever discreteevents introduce new heritable types into apopulation, e.g., events of lateral transfer,migration, infection, and endosymbiogenesis,among others. In principle, the formalismalso might be applied to premeiotic clustersof mutations (Woodruff et al. 1996), withsuitable adjustments for mutation rate andprobability of fixation.

Early History of Origin-FixationModels

classical periodMathematical formulas for the probability

of fixation were originally proposed by Hal-dane, Fisher, and Wright (Haldane 1927;Fisher 1930a; Wright 1931; see also Fisher1922, 1930b; Haldane 1932). Given a formulafor the probability of fixation, the constructionof an origin-fixation model is trivial—ignoringpossible conceptual barriers (e.g., conceptual-izing genetic state-space)—yet none of theseearly works contains an origin-fixation model.For instance, Fisher (1930a:93) derives theprobability of fixation by assuming that onenew mutation enters the population per gen-eration and then noting that the equilibriumrate at which such mutations reach fixationmust be equal to the probability of fixation.But Fisher never takes the next step of writingthe rate of evolution, K, as the product of 2Nuand the probability of fixation.

To our knowledge, the first paper to de-scribe the general relationship between K,2Nu, and the probability of fixation isWright’s well-known paper, “The distri-

bution of gene frequencies under irre-versible mutation” (Wright 1938). In thispaper, Wright notes that, in the specialcase of his model where dominance is absent,

“K �4Nvs

1 � e� 4Ns approaching v�1 � 2Ns� as

4Ns decreases” (Wright 1938:258). Here, K isthe instantaneous equilibrium rate of fix-ations at a single site and v is analogousto u, the mutation rate, in our notation.Clearly, this formula reduces to K�u(Equation 3 above) as Ns approaches 0and K�4Nus (Equation 2 above) forlarge Ns (the formula K�u for the neu-tral case is also stated explicitly as Wright’sEquation 15). However, these results werenot derived by assuming the origin-fixationformalism, but by considering the full prob-ability distribution for the frequency of thenew allele during the period that it segre-gates in the population.

Surprisingly, Wright’s 1938 paper does notappear to be the source of later origin-fixationmodels. Looking at all citations to this paperfrom 1938 to 1985, we do not find a singlearticle that cites Wright for an origin-fixationformalism or for the formula K�4Nus (51 re-cords, Web of Science search, accessed 4March 2014). This is particularly odd becauseKimura, who did much to popularize both for-mulas, habitually cited Wright (1938) for thegeneral form of the distribution of allele fre-quencies under reversible mutation that oc-curs earlier in the paper.

Indeed, we are not aware of any origin-fixation model proposed prior to the molec-ular era (see next section). Before this time,the typical uses for the probability of fixationwere to inquire as to the number of times aparticular advantageous mutant must enterthe population before it is relatively assurredof reaching fixation (Haldane 1927:839, 1932,1990:114; Fisher 1930a:77; Wright 1949, inthe section on nonrecurrent mutation) or toremark on the extremely small probability offixation for disadvantageous mutations (Fisher1930a:93-94; Wright 1938).

In addition to being absent from themathematical theories of Wright, Fisher,and Haldane, origin-fixation models arebroadly incompatible with their stated views of




how evolution occurs in nature. In gen-eral, Wright, Fisher, and Haldane believedthat evolutionary change is due largely to nat-ural selection acting on segregating (i.e.,standing) variation (Wright 1960, 1980:829),and were impressed both with the speed of thisprocess, and with the ability of the joint actionof natural selection and recombinationto make genotypes common that otherwisewould have been vanishingly rare (e.g., Fisher1930a:96; Haldane 1932:94-95). Fisher andWright also placed a strong emphasis on fit-ness interactions (epistasis) among segregat-ing alleles, which precluded the possibilitythat an allele would have a constant selectioncoefficient during its sojourn in a population(as per the end of the section on nonrecur-rent mutation in Wright 1949; or Fisher1930a:95). Thus, for Fisher and Wright, for-mulas assuming an independent fixationprobability for each new mutation had littlerelevance to evolution in actual populations.Finally, whereas Fisher and Haldane rejecteddrift as being too slow to substantially affectevolution in natural populations (Haldane1932:117), origin-fixation models rely onthe waiting time between mutations beingeven longer than the time-scale of drift.

As the mid-20th-century evolutionary syn-thesis progressed, it came to include a ver-bal theory of evolutionary genetics even lesscompatible with origin-fixation thinking. Al-though today evolutionary geneticists considerit an open question whether adaptation occursprimarily through “new mutations, fixation ofstanding variation or modest changes at manyloci” (Olson-Manning et al. 2012:869), the ar-chitects of the mid-century consensus arguedthat the abundance of variation in the “genepool” and the ability of recombination to gen-erate further multilocus variation ensured thatselection would never need to wait for a newmutation. For these authors, evolution was nota process wherein individual mutations fixedone at a time; rather, evolution became synon-ymous with a shift from one multilocus equi-librium of segregating variation to another(e.g., see Dobzhansky 1955:282; Mayr 1963:613; Stebbins 1966:29-31, 1982:160; Dobzhan-sky et al. 1977:6, 72).

the molecular revolutionAt the same time that mainstream evolu-

tionary biology was emphasizing the role ofsegregating variation in the evolutionary pro-cess, a very different vision of evolution wasemerging, starting in the late 1950s, basedon comparing protein sequences from dif-ferent species. A key aspect of this view was apowerful new mode of inference, by which amacromolecular subsequence (part of a gene,protein, or RNA) is inferred to be not “crit-ical” or not “essential” in determining someproperty of the macromolecule if the subse-quence differs among species while the prop-erty does not (e.g., Anfinsen 1959:149-155).Evolutionary change was understood as a se-quence of distinct amino acid substitutions,each determined by a mutation (e.g., Zuck-erkandl and Pauling 1962, 1965; Margoliash1963). Natural selection was said to “accept”or “reject” mutations, and was invoked mainlyin the role of a filter preventing harmfulchanges (e.g., Eck and Dayhoff 1966:161,200).

It was in this more favorable climate thatan explicit origin-fixation formalism rose toprominence in association with the neutraltheory of molecular evolution, which wasproposed in parallel by Kimura (1968) andKing and Jukes (1969). Indeed, origin-fixationmodels played an integral role in both theproposal and development of the neutraltheory (see Dietrich 1994 for a history of thisera).

In his original proposal of the neutral the-ory, Kimura (1968) argued that the substitu-tional load (i.e., excess mortality) necessaryto account for the observed genome-widerate of substitution would be unacceptablylarge unless most substitutions are neutral,noting that for neutral mutations the rate ofevolution is equal to the rate of mutation(K�u). Whereas some sources (e.g., Charles-worth and Charlesworth 2010) cite Kimura’s1968 paper as the source of an origin-fixationderivation of K�u, the origin-fixation formal-ism is not presented explicitly, and only (ar-guably) arises if one tries to reconstruct themissing steps in Kimura’s argument. Themere presence of the relation K�u does notindicate origin-fixation thinking because K�u




holds for neutral evolution under broadconditions, even if one does not assume amutation-limited regime. Among Kimura’sworks, the equation K�4Nus first appears ina subsequent technical paper (Kimura andMaruyama 1969) where it is derived usingthe origin-fixation formalism.

In contrast to the ambiguity in Kimura’spresentation, King and Jukes (1969) deriveboth K�u and K�4Nus using a fully explicit,and clearly explained, origin-fixation argu-ment as part of their parallel proposal of theneutral theory. Furthermore, while Kimura’sproposal of the neutral theory emerged pri-marily from a theoretical argument, the pro-posal by King and Jukes (1969) invoked avariety of empirical arguments based on pat-terns of molecular evolution, e.g., the obser-vation that the least important parts ofmolecules seemed to be changing the mostrapidly. One such argument was that, for anygiven protein, the rate of evolution appearedto be remarkably constant (Zuckerkandl andPauling 1962, 1965), a result that seemedmore consistent with the neutral formulaK�u than with K�4Nus. Indeed, by 1971,the contrast between a constant neutral rateand a variable selective one had become aleading argument for the neutralists, withthis argument often expressed in an explicitorigin-fixation framework (Kimura and Ohta1971a,c; see also Kimura and Ohta 1971b;Kimura 1971).

The introduction of Ohta’s theory ofslightly deleterious fixations (see Ohta 1992;Ohta and Gillespie 1996 for a review) led toan additional class of origin-fixation modelsthat focused on deleterious, rather than ad-vantageous, fixations. These models wereproposed with the positive goal of explainingthe observed data better than the simplermodel K�u (e.g., Ohta 1973). Note that thiswas in contrast to the case of K�4Nus, whichwas used by neutralists as a straw man toargue against, at a time when selectioniststhemselves had not embraced the concept ofevolution as a stochastic proposal-acceptanceprocess (e.g., Maynard Smith 1975:106).

Another notable advance during this erawas the introduction of the so-called “shift”models, which extended the calculation ofthe substitution rate to the situation where

the selection coefficients of new mutationsare drawn from a distribution rather thantaking a constant value (Ohta 1977; Kimura1979; see also Ohta and Kimura 1971;Kimura and Ohta 1972). The name comesfrom the fact that, after each non-neutralsubstitution, the old distribution of fitnessesmust shift in order for the new distributionof selection coefficients to remain the same.

Finally, it is important to note that theorigin-fixation models used during this pe-riod are best understood as aggregate-ratemodels, even though the assumptions of in-finite sites and free recombination underly-ing such models are not always stated explicitly.For instance, Kimura and Ohta (1971a)state both assumptions, while Ohta andKimura (1971) state neither, even though thetwo articles appear back-to-back in theinaugural issue of the Journal of MolecularEvolution; of the original papers that de-rive K�4Nus, Kimura and Maruyama(1969) state only the infinite sites assump-tion, while King and Jukes (1969) state nei-ther assumption.

Subsequent Development andCurrent Applications

After featuring prominently in the adventof molecular evolution, origin-fixation mod-els of the aggregate-rate form became astaple of textbook treatments of molecu-lar evolution (Nei 1987; Li 1997; Gillespie1998; Graur and Li 2000; Charlesworth andCharlesworth 2010). Such models remain inwide use today, with the major innovationsbeing the use of such models to track thechange in phenotype or fitness with time(e.g., Hill 1982; Lande 1994), or the incorpo-ration of aggregate-rate components withinmore complex population-genetic models thatalso make predictions about polymorphism(e.g., Sawyer and Hartl 1992; Piganeau andEyre-Walker 2003).

More recent progress in the extensionof the origin-fixation framework has cen-tered on the introduction and developmentof models of sequential fixation. Like theaggregate-rate models, the intellectual rootsof models of sequential fixation go back tothe molecular revolution. Whereas in thecase of aggregate-rate models, the main in-




spiration was observations about the tempo(i.e., rate) of molecular evolution, models ofsequential fixation instead draw on an obser-vation about its mode: molecular evolutionlargely occurs by a succession of single aminoacid (or later nucleotide) substitutions (Zuck-erkandl and Pauling 1962, 1965), so that apopulation can be viewed as having traced anevolutionary path through “sequence space,” aconcept first proposed by Maynard Smith(1962, 1970).

Unlike the aggregate-rate models, whichcan be associated historically and conceptu-ally with the advent of the neutral theory, thehistory of models of sequential fixation ismore complex (see Figure 2). Whereasorigin-fixation models using Haldane’s2s as the fixation probability clearly orig-inate with Gillespie’s pioneering work inthe 1980s (Gillespie 1983b, 1984a), mod-els including deleterious or neutral fixa-tions were described independently at least

Figure 2. Historical Timeline of Origin-Fixation ModelsLight, curved arrows indicate that one publication draws methods from another; dark, straight arrows

indicate that one origin-fixation model builds on another. The shaded triangles at the right indicate segmentsof the contemporary literature that could be represented by multiple publications not listed. Whereas we depictthe initial proposal of aggregate-rate models in the 1969 work of Kimura, Maruyama, King, and Jukes, we donot attempt to follow their subsequent influence, as such models are used widely and recognized in textbooksof molecular evolution (see text). Independent proposals of models with sequential fixations are marked withan asterisk (*).




seven times in the late 1980s and early 1990s(Iwasa 1988; Golding and Felsenstein 1990;Shields 1990; Rousset et al. 1991; Tachida1991; Eyre-Walker 1992; Gabriel et al. 1993)and arguably several more times subsequently(Hartl and Taubes 1998; van Nimwegen etal. 1999; Nielsen and Yang 2003; Berg et al.2004; see also the introduction to Ishii et al.1982; and an early, flawed attempt in Vogeland Zuckerkandl 1971). Thus, models of se-quential fixation have been proposed inde-pendently or semi-independently so manytimes between the mid-1980s and today thatno single suggestion for a canonical sourceseems justified. In what follows, we attemptto summarize the complex history of modelsof sequential fixation by discussing five dif-ferent clusters of models whose literatureand methods have developed relatively inde-pendently.

models of pure adaptationThe first influential models of sequential

fixation, proposed by Gillespie (1983b, 1984a)had two distinctive features. First, they wererestricted to beneficial changes, using Ha-ldane’s probability of fixation, 2s. Becausethe use of this formula assumes that Ns islarge, and the use of the origin-fixation for-malism implies that Nu is small, they becameknown as strong-selection weak-mutation(SSWM) models. Second, the fitnesses ofgenotypes were assumed to be random drawsfrom the tail of a single distribution (imply-ing that the fitness landscape is uncorre-lated). This justified the use of extreme valuetheory, which allowed predictions that werelargely independent of the exact form of thisdistribution (Gillespie 1983b; reviewed in Orr2005a).

Whereas Gillespie (1983b) was certainlyaware of the origin-fixation models of Kimuraand others, his SSWM formalism was con-structed to cover a broader set of cases, includ-ing cases in which the population is typicallypolymorphic, e.g., models of symmetric over-dominance (Gillespie 1983a, 1991). In otherwords, Gillespie’s models (like Wright’s 1938model) converged on origin-fixation dy-namics without being derived from earlierorigin-fixation models, and without taking theorigin-fixation formalism as a starting point.

Gillespie (1984a) introduced the “muta-tional landscape model,” a SSWM model inwhich each allele has a fixed number ofneighbors and each substitution results in anew set of neighbors (studies in this frame-work typically use a move-rule formalism basedon Equation 5). In the mutational landscapemodel, evolution continues until the popu-lation becomes fixed at a genotype whoseneighbors are all less fit. In recent years, thismodel has gained attention due to exten-sions that address the fitness effects and rankof fixed mutations (Orr 2002, 2005b; Rokytaet al. 2006; Unckless and Orr 2009) or applyto broader classes of fitness landscapes andstarting conditions (Orr 2006; Joyce et al.2008; Jain and Seetharaman 2011).

Another recent body of work retains theSSWM assumptions but dispenses with theassumption that fitnesses are drawn from anunchanged distribution. Often these studiesfocus on the set of possible adaptive paths (andtheir relative probabilities) from a given start-ing point (Weinreich 2005; Weinreich et al.2005, 2006; Stoltzfus 2006; Unckless and Orr2009; Draghi and Plotkin 2013). The sameapproach may be applied to analyze evolu-tionary dynamics on empirical fitness land-scapes, when the fitness is known for allpossible subsets of a fixed set of mutations(Weinreich et al. 2006; DePristo et al. 2007;de Visser et al. 2009; Lozovsky et al. 2009;Brown et al. 2010).

A similar set of approaches have beenused to analyze an influential model ofstepwise evolution known as the “NK” land-scape (Kauffman and Levin 1987; Kauffman1993). Although analyses of this model inthe physics literature were originally con-ducted without a biologically inspired move-rule (Macken and Perelson 1989; Weinberger1991; Flyvbjerg and Lautrup 1992; Kauffman1993; Perelson and Macken 1995), theseanalyses were later adapted to incorpo-rate origin-fixation dynamics within a SSWMframework (e.g., Welch and Waxman 2005;Orr 2006; Stoltzfus 2006; Kryazhimskiy et al.2009; Jain and Seetharaman 2011). Finally,while the above models all consider a fixedfitness landscape, SSWM models have occa-sionally also been used to investigate evolution




in a changing environment (e.g., Kopp andHermisson 2009b).

It is important to note that a SSWM origin-fixation formalism sometimes also appearsin the “adaptive dynamics” literature as a stepin deriving the canonical equation of adap-tive dynamics (Dieckmann and Law 1996;Champagnat et al. 2001). However, the as-sumption of vanishingly small mutationsizes in adaptive dynamics gives these mod-els a deterministic character that is qualita-tively different from other origin-fixationmodels, so we do not review this literaturehere.

various models not restricted toadvantageous fixations

As was mentioned earlier, models of se-quential fixations that include deleteriousand neutral fixations arose independentlymany times, starting in the late 1980s andearly 1990s. These scattered introductionslead to a number of distinct literatures; wereview a few of the smaller literatures here,while tackling the larger literatures in thenext several sections.

One set of models of sequential fixationwas focused on understanding the statisticalproperties of sequence evolution. The first ofthese models emerged out of the analysis ofthe so-called “fixed” model (Ohta and Tachida1990), where the fitness of each new mutantis drawn at random from a fixed distribution(this model is also known as the “house ofcards model”; Kingman 1977, 1978) in con-trast to the “shift” models discussed earlier(Ohta 1977; Kimura 1979). The introduc-tion of this new class of models paralleled achange in Ohta’s emphasis from deleteriousfixations (Ohta 1973) to compensatory ornearly neutral evolution involving a mix ofslightly deleterious and slightly advantageousfixations (Ohta 1992). Subsequently, origin-fixation models with sequential fixations wereintroduced to analyze the fixed model in theweak-mutation limit (Tachida 1991, 1996; Gil-lespie 1994b). Notable later developmentsinclude the use of an origin-fixation modelto explore the relationship between the NKmodel and the fixed model (Welch andWaxman 2005), and the use of an origin-fixation model to investigate a broader

class of models that subsumes both theshift and fixed models (Kryazhimskiy etal. 2009).

A related set of origin-fixation models wasfocused on explaining the index of disper-sion of the molecular clock (e.g., Gillespie1984b, 1986), which is a measure of howirregularly substitutions occur in time. Themain early contribution to this question us-ing a model of sequential fixations was madeby Iwasa (1993; drawing on Iwasa 1988), whosuggested that the substitution rate tends toincrease after deleterious fixations. A differ-ent, neutralist, explanation was given byTakahata’s (1987) fluctuating neutral spacemodel, which proposed to explain the over-dispersion of the molecular clock by in-voking a change in the space of neutralpossibilities (and, thus, the neutral mu-tation rate) after each neutral substitu-tion. A clear mechanistic justification forthis second type of model was providedwhen Bastolla and colleagues (Bastolla etal. 1999, 2002, 2003a,b) argued, on bio-physical grounds, that different protein se-quences will have different numbers ofmutational neighbors that share the samefold and, hence, will have different neutralsubstitution rates. However, neither Taka-hata nor Bastolla and colleagues invoked anorigin-fixation framework. Meanwhile, vanNimwegen et al. (1999) proposed a model ofsequential fixations (for a population evolv-ing on a network of selectively neutral allelesembedded within a larger space of highlyunfit alleles) that could be interpreted asbeing isomorphic with the earlier phenome-nological models by Takahata and by Bas-tolla et al., although this connection was notnoted. Finally, Wilke (2004) put the piecestogether and pointed out that the earlierefforts by Bastolla and Takahata could beconstrued in origin-fixation terms using theframework of van Nimwegen et al. (1999).Subsequently, studies in the same vein(Bloom et al. 2007; Raval 2007), including byBastolla and colleagues (Bastolla et al. 2007;Bastolla 2014), have invoked origin-fixationdynamics explicitly.

Another set of origin-fixation models aris-ing from the literature of the early 1990swere models of sequential fixation meant to




address the evolution of codon bias, particu-larly the simple two-allele case with preferredand nonpreferred codon types (Shields 1990;Bulmer 1991). Although the simplicity ofthis framework led to its re-use (e.g., in Eyre-Walker and Bulmer 1995; Takano-Shimizu1999; Charlesworth and Eyre-Walker 2007;Kondrashov et al. 2010), the two-allele case isso simple that the origin-fixation formalismis unnecessary: a full diffusion-based analysisof population dynamics is possible and pro-vides greater insight (McVean and Charles-worth 1999). Similarly, Rousset et al. (1991)proposed a simple origin-fixation model forthe coevolution of paired bases in non-coding RNA molecules, but subsequentwork on this problem from a population-genetic perspective has tended to be basedon explicit two-locus models that include re-combination (following Kimura 1985; Higgs1998).

Several other models of sequential fixa-tion were also proposed in the early 1990s.Eyre-Walker (1992) proposed an early neu-tral origin-fixation model in the process ofarguing that increasing constraint can some-times increase the rate of evolution. Gabrielet al. (1993) discussed a model of Muller’sRatchet in which deleterious fixations ac-cumulate by sequential fixations. Forceet al. (1999) and Lynch and Force (2000)later went on to use a neutral model ofsequential fixations in their analysis of theduplication-degeneration-complementation(DDC) model for the preservation of du-plicate genes in the limit of low mutationrates.

We also note in passing that, just as onecan derive adaptive dynamics models by as-suming that the probability of fixation is pro-portional to the selective advantage and allmutations have infinitesimally small effects,one also can modify such models to includedeleterious fixations, which yields a diffusionprocess rather than the deterministic dynam-ics that appear when only advantageous fix-ations are considered. Oddly, this type ofmodel has been proposed only rarely (al-though see Proulx and Day 2001; Day andOtto 2007:655, 667).

applications to fisher’s geometricmodel

In a famous argument for Darwinian in-finitesimalism, Fisher (1930a) imagined amultidimensional phenotypic space in whichfitness is a decreasing function of the dis-tance from an arbitrary optimum, conclud-ing that the smallest mutations were themost likely basis of evolutionary change onthe grounds that, starting from a suboptimalphenotype, a smaller change in a randomdirection is more likely to be beneficial, par-ticularly when the phenotypic space is high-dimensional. This framework was revived inthe modern literature by Kimura (1983),who noted that if one assumes that selectioncoefficients for new mutations are propor-tional to the size of a mutation’s effect onphenotype, then in fact mutations with inter-mediate effect sizes have the highest jointprobability of being beneficial and escapingrandom loss.

Many subsequent authors have addressedFisher’s geometric model in Kimura’s origin-fixation framework, where only a single sub-stitution is considered (Hartl and Taubes1996; Welch and Waxman 2005; Martin andLenormand 2008; Chevin et al. 2010). As afurther development, models of sequentialfixation that consider the longer-term dy-namics of evolution under Fisher’s geomet-ric model were introduced by Hartl andTaubes (1998) and Orr (1998). Subsequentmodels of sequential fixations have beenposed both in terms of models of adaptationtoward the optimum using Haldane’s 2s (asin Orr 1998, 2000; Welch and Waxman 2003),and as models of nearly neutral evolutionthat include deleterious fixations (Hartl andTaubes 1998; Poon and Otto 2000; Sella andHirsh 2005; Gros and Tenaillon 2009; Groset al. 2009; Sella 2009; Lourenco et al. 2011;Razeto-Barry et al. 2011). Many of the latterclass of studies have discussed the differencebetween the fitness at the optimum and theexpected fitness of a population at equilib-rium (the drift load) as a function of popu-lation size and the number of dimensions inthe phenotypic space; Tenaillon et al. (2007)were able to use these results to estimateempirically the dimensionality of the pheno-




typic space for two viruses. Barton (2001)and Gu (2007a,b) also treat Fisher’s geomet-ric model, but in something closer to anaggregate-rate framework.

models from phylogeneticsThe models described in this section and

the next rely on the fact that models of se-quential fixation can be easily configured tosatisfy a technical characteristic known inmathematics as reversibility, which refers to aproperty of a Markov chain at equilibriumwhen time is reversed. Allowing each type oftransition to be reversed at some rate is anecessary (though not sufficient) conditionfor reversibility. Therefore, a model cannotbe reversible if it includes only categoricallydeleterious changes, or only categoricallybeneficial ones.

In molecular phylogenetics, it is conve-nient to have a model of sequence evolutionthat takes the form of a reversible Markovchain because the likelihood of any particu-lar phylogeny under such a model can befound efficiently using Felsenstein’s (1981)pruning algorithm. This made origin-fixationmodels an obvious choice to be incorporatedinto models of phylogenetic reconstruction.

The first phylogenetic model to includean explicit origin-fixation formalism was byGolding and Felsenstein (1990). This in-spired Halpern and Bruno (1998; cf. Bruno1996) to construct an influential model thatassigns—to each amino acid in a particularposition—a scaled selection coefficient (Ns),with the transition rates between codons de-termined by multiplying the mutation rateby the probability of fixation (see also Ro-drigue et al. 2010; Tamuri et al. 2012, 2014;Rodrigue 2013). Similar methods have alsobeen applied to investigate selection for co-don usage by assigning a fitness to each co-don regardless of its place in the protein(McVean and Vieira 2001; Nielsen et al.2007; Yang and Nielsen 2008) and adaptedto study the evolution of transcription factor-binding sites (Moses et al. 2003; Doniger andFay 2007; Kim et al. 2009).

We note that an explicit origin-fixation ar-gument is not always presented in papersinspired by the Halpern and Bruno (1998)model. This is because their Equation 10

allows the construction of a rate matrix in-corporating natural selection given just amutational rate matrix and an observed vec-tor of equilibrium frequencies under selec-tion. Some models derived from the Halpernand Bruno (1998) model (e.g., Moses et al.2004; Wong and Nielsen 2007) simply usethis equation rather than recapitulating theentire model. A special case of this relationalso was rediscovered, in an origin-fixationframework, by Knudsen and Miyamoto (2005),who related it to generalized weighted fre-quencies (“�gwF”) models (Goldman andWhelan 2002).

Another approach influenced by Halpernand Bruno (1998) attempts to incorporateknowledge about the structure of evolvingmacromolecules to make phylogenetic infer-ences as well as inferences about the strengthof selection. In this approach, each possiblenucleic acid sequence is assigned a scaledselection coefficient based on its compat-ibility with a specified, and presumablyselectively favored, protein structure or RNAsecondary structure (Robinson et al. 2003; Yuand Thorne 2006; Choi et al. 2007, 2008;Thorne et al. 2007; Rodrigue et al. 2009; Pol-lock et al. 2012; reviewed by Thorne et al.2012), so that the probability of fixation ofeach mutant depends on its predicted effectson the structure of the relevant macromole-cules.

Origin-fixation models have also beenused to interpret the parameter known as �(omega), which can be viewed as the ratio ofthe fixation probabilities of nonsynonymousversus synonymous substitutions (Nielsen andYang 2003; Subramanian 2013; see also Krya-zhimskiy and Plotkin 2008), and have beenused to understand the impact of long-termstabilizing selection and biased gene conver-sion on phylogenetic inference (Lawrie et al.2011). Speaking more broadly, the rise oforigin-fixation models in phylogenetics isbest seen as part of the increasing popu-larity of “mechanistic” models that attempt todirectly model certain causal aspects of theevolutionary process, rather than treatingevolutionary rates as purely phenomenolog-ical (see the recent review by Rodrigue andPhilippe 2010).




models based on statisticalmechanics

In addition to being useful in a phy-logenetic context, reversible origin-fixationMarkov chains are easy to analyze using asuite of techniques from statistical physics, anobservation that has been noted indepen-dently on at least three occasions (Iwasa1988; Berg and Lässig 2003; Berg et al. 2004;Sella and Hirsh 2005; see also Vogel andZuckerkandl 1971). In particular, in the spe-cial case in which a mutational model thatwould be reversible in the absence of naturalselection is used and each allele is assigned afixed fitness, the choice of any standard formof the fixation probability (as long as it allowsboth advantageous and deleterious fixa-tions) results in a reversible origin-fixationMarkov chain. One may then view fitness asbeing analogous to the negative of the en-ergy of a state, and population size as beinganalogous to inverse temperature, so thatthe stationary distribution for the evolutionaryMarkov chain corresponds to a Boltzmann dis-tribution, and a quantity, analogous to free en-ergy, can be defined whose expectation isnondecreasing over time.

Whereas these properties are really prop-erties of reversible Markov chains in generaland not origin-fixation models per se (Iwasa1988; Barton and Coe 2009), the simple andtractable description of the origin-fixationdynamics and equilibrium distribution pre-sented by Berg et al. (2004) and Sella andHirsh (2005) have proved useful in a varietyof contexts. These include studying the evo-lution of transcription factor-binding sites(Mustonen and Lässig 2005; Mustonen et al.2008; Nourmohammad and Lässig 2011),codon usage bias (Wilke and Drummond2006; Gilchrist 2007; Gilchrist et al. 2009;Shah and Gilchrist 2011), protein evolution(Mendez et al. 2010; Ramsey et al. 2011; Pol-lock et al. 2012), deleterious mutation accu-mulation (Tenaillon et al. 2007), the evolutionof epistasis (Gros and Tenaillon 2009; Gros etal. 2009), gene network evolution (Le Nagardet al. 2011), Fisher’s Geometric Model (Sella2009), and evolutionary dynamics on high-dimensional fitness landscapes (McCandlish2011, 2013).

DiscussionIn origin-fixation models, evolutionary

change consists of events in which a variant isintroduced into a population by mutation,and then is accepted or rejected based on itsindividual effects. Such models seem obvioustoday, yet they remained undiscovered forover 40 years after initial work on the prob-ability of fixation. Through a complex anduneven process that began with the proposalof the neutral theory, origin-fixation modelshave risen to prominence and now representa widely used branch of theoretical evolu-tionary genetics.

Above, we reviewed the history, develop-ment, and technical uses of these modelswithout addressing their validity or realism.In fact, the rise in popularity of origin-fixation models (with a few exceptions) hasnot been accompanied by increased atten-tion to their problematic assumptions anddistinctive implications. Instead of beingtreated as a conjecture subject to evaluation,they are invoked instrumentally, as thoughthey were a generic tool that may be in-serted into a method of inference withoutintroducing any dangers.

However, the true dynamics of evolution-ary change are shaped to an unknown degreeby phenomena that origin-fixation modelssimply ignore. Various phenomena thatdepend intrinsically on polymorphism maybecome relevant when the weak-mutation as-sumption fails. Examples include the evolu-tion of quantitative traits based on abundantstanding variation, overdominance (whichcan result in prolonged polymorphism),stochastic tunneling (where mutations thatarise on a deleterious background sweep tofixation, e.g., Iwasa et al. 2004; Weinreichand Chao 2005), selection for mutational ro-bustness (van Nimwegen et al. 1999), andinterference between multiple beneficial mu-tations at either a single locus (Gerrish andLenski 1998; Desai and Fisher 2007; Rouzine etal. 2008) or several partially linked loci (Hilland Robertson 1966).

Even when the weak-mutation assumptionholds, there are other reasons to questionthe validity of origin-fixation models. Gillespie(2004) argues that selection coefficients do




not remain sufficiently constant over time tojustify a model of fixation based on a singleselection coefficient. Origin-fixation modelsneglect the time that it takes for a new mu-tation to reach fixation (see Figure 1), andthus may miscalculate the rate of evolution atshort time scales (Stephan and Kirby 1993).The same approximation means that incom-plete lineage sorting (i.e., the persistence ofa polymorphism through a speciation event),relevant in the context of phylogenetic analysis,is overlooked.

By ignoring the complications of within-population processes, origin-fixation modelsprovide a formal view of the evolutionarytraversal of a genotypic or phenotypic spacevia multiple changes. Yet, the same assump-tion that makes the origin-fixation formalismuseful also guarantees that it is not a genericmodel of the evolutionary process, but a spe-cialized model—specialized for a particularregime of “mutation-limited” evolution thatside-steps various phenomena consideredimportant in theoretical population genet-ics. In strict terms, this regime is narrow andvery difficult to justify, yet the same might besaid for the mid-20th-century orthodoxy ofpopulation genetics (noted above, and de-scribed further below). Because of theprevalence of origin-fixation models in thecontemporary literature, it is important todetermine the extent to which this origin-fixation regime actually occurs in nature.

Thus, in the following two sections, wetreat origin-fixation models, not merely as auseful technical innovation, but as a theoryof evolution whose importance is subject toinvestigation. To invoke an origin-fixationmodel is to invoke a particular mode of evolu-tionary change with distinctive implications;these implications may be used to evaluate theextent to which the true evolutionary dynamicsfall within the origin-fixation regime.

distinctive implications of origin-fixation models

By merit of both their assumptions andtheir mathematical form, accepting origin-fixation models as an accurate representationof evolution in nature would have definite im-plications for our understanding of the evo-lutionary process. Here, we focus on three

implications regarding the rate of evolution,the direction of evolution, and the roleof selection.

The first implication is that, under origin-fixation models, the rate of evolution is di-rectly proportional to the mutation rate. In asimple aggregate-rate model, doubling therate of mutation doubles the rate of evolu-tion; doubling the mutation rate in a simplemodel of sequential fixations generally re-sults in behavior equivalent to the originalmodel run at twice the speed. However, byembedding an origin-fixation model withina larger model that includes environmentalchange, one may discover conditions underwhich most fixations occur after an environ-mental change, so that the long-term rate ofevolution is decoupled from the mutationrate and is instead controlled by the rate ofenvironmental change (e.g., Gillespie 1984b,1991, 2004; Razeto-Barry et al. 2012). None-theless, in such models the short-term rate ofadaptation during the adaptive transient re-mains proportional to the mutation rate.

The second major implication is that bi-ases in mutation rates can bias the course ofevolution, including adaptive evolution. Forexample, consider a model of sequential fix-ations in which our population is currentlyfixed at allele i and can mutate to a variety ofother alleles. The odds that the populationwill next become fixed at allele j rather thanallele k (based on the move-rule in Equation5) are given by

P�i, j�P�i,k�

�2Nuij��sij,N �

2Nuik��sik,N ��

uij

uik�

��sij,N �

��sik,N �.

(6)

This can be described as the product of twofactors: a ratio of mutation rates and a ratioof fixation probabilities. For the case of ben-eficial fixations, this can be reduced to

P�i, j�P�i,k�

�uij

uik�

sij

sik, (7)

by assuming that the probability of fixation iswell-approximated by 2s (see Yampolsky andStoltzfus 2001). Thus, doubling the mutationrate to an allele or doubling its probability offixation both have the same effect on theodds that it is the next allele to be fixed. This




means that a sustained bias in mutation cancause a long-term trend, even when all fixa-tions are due to selection (Stoltzfus 2006).Note that one could construct analogous for-mulas for aggregate-rate models that wouldaddress the ratio between the rates of evolu-tion for two different classes of mutations,e.g., transitions versus transversions or inser-tions versus deletions.

Equations of this general form provide auseful framework for assessing the relativeimportance of mutation and selection and,more specifically, for comparing the impactof biases in the origination process to theimpact of biases in the fixation process (Yam-polsky and Stoltzfus 2001; Stoltzfus and Yam-polsky 2009; Farlow et al. 2011; Streisfeld etal. 2011). For instance, Streisfeld et al. (2011)use Equation 7 as the framework for a test ofmutational and selective causes of a bias inthe observed pattern of adaptive changes infloral evolution. A related result, discussedfurther below, is the finding of Rokyta et al.(2005) that both biases in mutation and dif-ferences in fixation probability affect thechances of parallel evolution in a laboratoryphage population.

The final major implication of origin-fixation models concerns the role of naturalselection in evolution. Clearly, in these mod-els, selection acts as a stochastic sieve or filterthat accepts or rejects new mutations one ata time depending on their fitness effects(and the population size).

These implications of origin-fixation mod-els are not shared by all theories of evolution.In particular, the mid-20th-century ortho-doxy of population genetics rejected each ofthe above implications on the grounds thatevolution is intrinsically polygenic and epi-static (reliant on gene interactions): “Naturalselection directs evolution not by acceptingor rejecting mutations as they occur, but bysorting new adaptive combinations out of agene pool of variability which has been builtup through the combined action of muta-tion, gene recombination, and selection overmany generations” (Stebbins 1966:31; seealso Dobzhansky et al. 1977:6; Mayr 1963:613; Simpson 1964:1536).

First, in this view, the buffering effect ofthe “gene pool” means that change is initi-

ated by the environment (not by a new mu-tation), and the rate of evolution is unlinkedfrom the mutation rate (e.g., Dobzhansky etal. 1977:77; Stebbins 1966:29; Mayr 1994:38),a view that led some to oppose the idea of amolecular clock (see Dietrich 1998). Sec-ond, in regard to the causes of orientation ordirection in evolution, the mid-20th-centuryconsensus holds that “selection is the onlydirection-giving factor in evolution” (Mayr1980:3), i.e., the course of evolution is deter-mined externally, by the environment, whereasthe contrary notion of any intrinsic sourcesof direction is dismissed (e.g., Simpson 1967:159). Various authors invoked theoreticalpopulation genetics to support this position(reviewed in Yampolsky and Stoltzfus 2001).Based on the equation for the equilibriumgene frequency in an infinite population atmutation-selection balance, Fisher arguedthat mutation cannot control the directionof evolution, on the grounds that mutationrates are small compared to selection coeffi-cients (Fisher 1930a,b; see also Wright 1931;Haldane 1933). This argument, invoking therelative weakness of the “force” of mutation,was repeated later by others (Huxley 1942:56; Mayr 1960:355; Simpson 1967:159; Gould2002:509).

Importantly, the logic of mutation and se-lection as opposing pressures is repeated intextbooks and used by researchers, leadingto the common assumption that biases invariation are ineffectual except when the op-posing pressure of selection is effectively ab-sent, i.e., the special case of neutral evolution(e.g., as in the rationale given in MaynardSmith et al. 1985:282; for other examples,see Stoltzfus 2006). Clearly, the origin-fixationview, via Equation 6, has a completely differentimplication: a bias toward a particular type ofevolutionary change may reflect a bias in mu-tational origin, whether fixations occur by se-lection or drift.

Finally, in regard to the role of selection,the mid-20th-century consensus rejects theidea that selection acts as a sieve or filter onseparate mutations, a notion that is con-demned as “downright misleading” on thegrounds that, due to epistasis in a polymor-phic context, an allele has no fixed meaning,independent of the rest of the gene pool




(Dobzhansky 1974). Instead, natural selec-tion is understood as an artist or creativeagent (see Chapter 4 of Gould 1977), withthe gene pool supplying the abundant rawmaterials for selection to craft an adaptiveresponse to any change in conditions.

The contrast between these two positions,only rarely treated as a multifaceted clash ofworldviews (e.g., Nei 2013), is suggestedmore subtly in the literature on the geneticsof adaptation which frequently references adistinction between new mutations and stand-ing variation (Orr and Betancourt 2001; Her-misson and Pennings 2005; Barrett andSchluter 2008; Pritchard and Di Rienzo2010; Jerome et al. 2011; Nuismer et al. 2012;Olson-Manning et al. 2012). Some authorswho emphasize the role of standing variationhave suggested that “[m]ost of the currenttheory on the genetics of adaptation assumesthat adaptation occurs exclusively from newmutations rather than from standing varia-tion” (Barrett and Schluter 2008:39; see alsoHermisson and Pennings 2005). Indeed,such an assumption is evident in Orr’s (2002)characterization of three possible move-rulesfor evolutionary walks. After describing threesuch rules—choose the fittest neighbor (theperfect or greedy rule), choose any fitterneighbor at random (the random rule), orchoose a fitter neighbor with a probabilityproportional to the fixation probability for anew mutation—Orr refers to the first two as“idealized” rules, and to the third as “adap-tation by natural selection,” as though thiswere the only possible implication of selec-tion in evolutionary theory. Instead, the mid-20th-century orthodoxy definitively rejectsany single-locus move-rule, and suggests in-stead that evolution in nature works by some-thing like a multilocus greedy rule (craftingan optimal combination of allele frequenciesfrom the diversity of the gene pool).

It is also useful to compare evolution inthe origin-fixation regime to neutral evolu-tion. Of course, there is a precise overlap forthe case of origin-fixation models in whichfixation takes place by drift, but many simpleneutral models that allow polymorphismshare the implications described above. Insuch models, the rate of evolution is propor-tional to the (neutral) mutation rate, the direc-

tion of evolution is determined by mutationalbiases in mutation, and natural selection acts asa sieve that removes deleterious mutations.

More generally, however, neutral evolu-tion is a different kind of concept than theorigin-fixation regime. Whether neutral evo-lution will happen is mainly an issue of thesizes of fitness differences in relation to pop-ulation size, while the relevance of the origin-fixation regime is an issue of the relationshipof mutation rates to population size and, foraggregate-rate models, independence betweenloci. Just as there are origin-fixation modelsthat are not neutral models (e.g., the muta-tional landscape model), one may constructmodels of neutral evolution that dependspecifically on phenomena of polymorphismexcluded from origin-fixation models. For in-stance, Kimura’s model of compensatory neu-tral evolution invokes a new mutation thatoccurs in the background of a segregating del-eterious allele, reversing its deleterious effect(Kimura 1985). This model was introducedspecifically because the rate of paired changes(e.g., in structural RNAs) was too high to re-flect the rate of an origin-fixation processbased on double mutations.

The general implications of origin-fixationmodels overlap most easily with the Mendelian-mutationist view of evolution developed a cen-tury ago by early geneticists such as Punnett,Johannsen, Bateson, and Morgan. Their viewallowed smooth Darwinian change, but alsoemphasized the creative and dispositional roleof mutation (via discrete effects, mutationalbiases, and the uniqueness of individualvariants), with selection often cast in therole of a filter that accepts or rejects mu-tations (Stoltzfus and Cable 2014). Thearchitects of the 20th-century orthodoxybelieved that they could dispense with theimportance of individual mutations for de-termining the direction of evolution—whatFisher (1930a) called “evolution workedby mutation” (Chapter 1)—on the groundsthat, via recombination among segregatingalleles, selection can move a population wellbeyond its initial phenotypic range, withoutany new mutations (Stoltzfus and Cable2014; see also Provine 1971:108-129). This iswhy mid-20th-century writings are rich instatements (some quoted above) that reject




the origin-fixation view, even before it wasformalized in 1969: when the architects ofthis orthodoxy were arguing for their dis-tinctive “shifting gene frequencies” viewthroughout the 1960s and 1970s, they werearguing against both the earlier Mendelian-mutationist view (Stoltzfus and Cable2014) and the emerging “molecular evo-lution” view. However, the degree to whichevolution can be viewed as an origin-fixationprocess cannot be established by either verbalplausibility arguments or mathematics, butmust be determined empirically.

testing the origin-fixationhypothesis

As explained in the previous section,origin-fixation models have distinctive impli-cations because they reflect a distinctive setof assumptions about how evolution actuallyworks. Thus, an important research programfor the future is to assess the extent to whichevolution in nature may be characterized asan origin-fixation process.

Unfortunately, the set of systems thatorigin-fixation models have been applied toin the literature provides little insight intothis problem. For instance, in the case ofmodeling sequence evolution in Drosophilaspecies, both theory (Durrett and Schmidt2007) and empirical observations (e.g., theappearance of independent, single nucleo-tide insecticide residence mutations in Kara-sov et al. 2010) indicate that evolution is notunder weak mutation even for regions of thegenome that are only a few nucleotides long.Yet, origin-fixation models are often appliedto make inferences, not only about Drosophilabut also unicellular microorganisms witheven larger population sizes (e.g., yeast andE. coli) for which the weak-mutation assump-tion is even less likely to apply. How, then,can one test whether a particular system isevolving according to an origin-fixation pro-cess?

One approach is to begin with a theoreti-cal understanding of conditions that wouldjustify an origin-fixation model and then as-sess whether those conditions hold for a par-ticular evolving system. This is problematiceven when the relevant population parame-ters can be estimated because origin-fixation

models often are employed in situationswhere they cannot be subsumed within amore general population-genetic model thatis sufficiently tractable to derive the relevantcondition for their validity.

Furthermore, in the few simple modelswhere we do understand the boundaries ofthe origin-fixation regime, these boundariesdo not depend solely on genome-wide char-acteristics like population size and mutationrate. For instance, while the strict condition8N2u �� 1 mentioned earlier for the validityof sequential fixations is meant to precludethe possibility of complications of polymor-phism, the degree to which the origin-fixationregime extends to situations that violate thiscondition depends on the structure of the fit-ness landscape (Kim and Orr 2005; Desai andFisher 2007; Kopp and Hermisson 2007,2009a; Rouzine et al. 2008; Weissman et al.2009; Park et al., 2010). At the same time, thecondition 8N2u �� 1 is based on the assump-tion that there is no heterozygote advantage(which could result in prolonged polymor-phism); the validity of this assumption againdepends on the structure of the genotype-phenotype-fitness map (Sellis et al. 2011). Asimilar situation arises for aggregate rate mod-els, where deviations from the aggregate ratepredictions due to linkage and epistasis willalso depend on the distribution of selectioncoefficients introduced by mutation (Bar-ton 1994, 1995; Stephan et al. 1999; Ne-her et al. 2010; Weissman and Barton 2012;see also Neher 2013). Finally, there re-mains the aforementioned concern, rele-vant to both classes of models, that selectioncoefficients remain sufficiently constantover the course of a substitution (see Gil-lespie 2004).

Fortunately, there are other ways to assessthe validity of an origin-fixation model, includ-ing (1) using simulations to clarify the limits ofthe origin-fixation regime, (2) comparing thelikelihood of an origin-fixation model to thatof another model, when applied to data fromnatural populations, (3) evaluating the inter-nal consistency of an origin-fixation modelwhen applied to data from natural popula-tions, and (4) rare cases in which experimentaldata are of sufficient depth to allow a test.

Simulations are likely to be extremely use-




ful in determining the applicability of origin-fixation models because they overcome manyof the difficulties with the strategy based onanalytically deriving the limits of the origin-fixation regime. Even if a population-geneticmodel is too complex to derive some simple,analytical condition for the boundariesof the origin-fixation regime, violationsof origin-fixation behavior should still beeasy to detect by simulation.

In some cases, it may be possible to com-pare an origin-fixation model with a morecomplex population-genetic model to deter-mine which model provides a better fit todata on natural divergence. Such an ap-proach has become possible only recently,due to the advent of phylogenetic modelsthat can accommodate both polymorphismand mutation (e.g., Bryant et al. 2012, whichis limited to biallelic sites; or De Maio et al.2013, which also includes selection). As thesemethods mature, a broader range of com-parisons will become possible.

A different approach is provided by testsof internal consistency, which provide a wayto evaluate origin-fixation models without re-quiring a fully specified alternative model.An example of such a test would be to assessthe effect of transition-transversion bias inthe context of phylogenetic inference meth-ods. Such methods often implement a fittedparameter for transition-transversion bias,which in the case of origin-fixation models(e.g., Yang and Nielsen 2008) is understoodas a bias in the origination process. Whenstrict origin-fixation dynamics apply to allevolutionary changes, it will not matter if(for instance) the synonymous changes aremostly neutral and the nonsynonymousones are mostly beneficial: the dependenceof rates on the mutational bias factor (as inEquation 6) is independent of the mode offixation. Thus, if the transition-transversionbias parameter is inferred separately for twodifferent categories of changes (e.g., synon-ymous versus nonsynonymous) the inferredvalues should be the same.

Finally, in some cases, an origin-fixationmodel may be evaluated experimentally. Forinstance, Rokyta et al. (2005) repeatedlycarried out one-step adaptation with abacteriophage, comparing the results to

the predictions of Orr’s (2002) muta-tional landscape model. Orr’s model pro-vided a good fit to the data, but only afterbiases in mutation rates (absent in theoriginal model) were taken into account.In fact, the best-fitting model did notemploy the distinctive features of the mu-tational landscape model (e.g., extremevalue theory), but was simply the origin-fixation move-rule (Equation 5) with em-pirical parameter values (mutation ratesand selection coefficients) and an im-proved formula for the probability of fix-ation. A number of studies conclude againstorigin-fixation dynamics based on deviationsfrom origin-fixation predictions (e.g., de Vis-ser et al. 1999; Miralles et al. 1999), however,for studies where multiple fixation events haveoccurred, care must be taken to avoid rejectingtoo simplistic an origin-fixation model, e.g.,one that does not include epistasis. This prob-lem can be circumvented by directly observingcompetition between multiple segregat-ing mutations (Kao and Sherlock 2008;Lang et al. 2013). This second approachis probably preferable and should be-come increasingly practical as sequenc-ing costs continue to fall.

The application of the approaches de-scribed above, we suspect, will show thatmany systems previously analyzed using origin-fixation models are not evolving in a pureorigin-fixation regime. However, inferencesmade from these models are not necessar-ily incorrect: there must be conditionsunder which origin-fixation models provide arough approximation to the true evolu-tionary dynamics, and this regime of approx-imate applicability may be quite large. Thus,another important area for future re-search is to assess the strengths and weak-nesses of origin-fixation models in those theregions of population-genetic parameter spacein which their behavior is approximate, with aview to identifying their suitability for specificpurposes.

In general, studies to address the robust-ness of origin-fixation models to violations inthe underlying assumptions will have to betailored to specific inferences. This is becausemany different behaviors are possible outsidethe origin-fixation regime and these different




behaviors will have different effects on differ-ent inferences (e.g., incomplete lineage sort-ing may be relevant to inferring phylogeniesbut irrelevant to studies of adaptation).

Nonetheless, developing a qualitative un-derstanding of how certain simple phenom-ena change as one leaves the weak-mutationregime may still be worthwhile. For instance,Yampolsky and Stoltzfus (2001) showed that,as the weak-mutation assumption for advan-tageous mutations breaks down, the influ-ence of mutational biases on the directionof evolution is reduced relative to origin-fixation expectations (Equation 7) becausenatural selection is very effective at increas-ing the frequency of the more fit of a pair ofadvantageous mutants that are simultane-ously segregating in a single population.However, it is currently unknown whatwould occur in the more general context ofEquation 6, e.g., for the case of deleteriousmutations or a mixture of deleterious andadvantageous mutations. This type of basicunderstanding could be broadly useful inunderstanding the implications of violatingthe weak-mutation assumption.

A final issue relevant to hypothesis testingis whether the predictions of origin-fixation

models are unique. Many of the distinctiveimplications of origin-fixation models wouldstill apply if the rate of evolution was merelyproportional to (rather than equal to) theproduct of (1) the rate of mutational originand (2) some function of the selection coef-ficient. As discussed above, these conditionshold for some neutral models, which explainscertain similarities between the implications ofneutral evolution and the implications of evo-lution in the origin-fixation regime. It is there-fore important to determine the broader setof population-genetic conditions outside theorigin-fixation regime where some subset ofthe basic implications of origin-fixation modelscontinue to hold.

acknowledgments

We thank Brian and Deborah Charlesworth, AdamEyre-Walker, Joe Felsenstein, Joanna Masel, K. RyoTakahasi, and Michael Dietrich for discussion, andDan Hartl, Etienne Rajon, Premal Shah, Jeff Thorne,Pablo Razeto-Barry, Josh Plotkin, and three anony-mous reviewers for comments on the manuscript. Theidentification of any specific commercial product isfor the purpose of specifying a protocol, and does notimply a recommendation or endorsement by the Na-tional Institute of Standards and Technology.

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Modeling Evolution Using the Probability of Fixation: History and Implications

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