When phenotypic traits are characterized at the molecular level, many have a complex GENETIC ARCHITECTURE1,2 (see Glossary). By genetic architecture, we mean the genes as well as the INTERACTIONS among them (epistasis), and between genes and environments that affect trait expression. Epistasis is particularly important in several areas of current evolutionary research, including speciation, CANALIZATION, inbreeding depression, the evolution of sex and interdemic selection in Wright's shifting balance theory3. In speciation, epistasis plays a key role in reproductive isolation – genes that function well in conspecific genetic backgrounds function poorly when combined in interspecific hybrids. Diminished effects of one locus as a result of interactions with another are fundamental to concepts of developmental homeostasis and canalization4. With inbreeding and other systems of nonrandom mating, negative synergism between deleterious alleles can contribute to the decline and extinction of small, endangered populations5 and so therefore is a concern for conservation biologists. Similarly, epistasis for fitness (FITNESS EPISTASIS) might be essential to the evolution of sex and recombination6–8. The shifting balance theory was proposed specifically as a mechanism for the evolution of 'co-adapted gene complexes'9–11. Definitions of epistasis Although there are several ways of defining epistasis12–15, most definitions in evolutionary genetics are concordant with the concept that epistasis occurs when the differences in the phenotypic value of an allele at one locus are dependent on differences in specific alleles at one or more other loci. It is illustrative to consider epistasis, or genetic context, in the light of the somewhat better understood phenomenon of genotype-by-environment interaction (G × E), or environmental context. Just as variation in the environment can result in changes in the magnitude or order of ALLELIC VALUES with G × E (Fig. 1a), with epistasis, changes in the magnitude or order of allelic values occur with variations in genetic background (Fig. 1b)14–18. Whenever effects at one locus depend in this way on the genetic background at other loci, genetically subdividing a population into DEMES creates an opportunity for different genetic backgrounds to arise via random genetic drift, selection, or mutation. It is then possible for the effect on viability fitness of allele A1 to be greater than that of alternative allele A2 in one deme, but lower than it in another deme (Fig. 1c). This kind of change in the order of allelic effects among demes means that migration between demes will mix positive and negative gene frequency changes and retard local adaptation. In this sense, epistasis represents a kind of genetic constraint on evolution. Population genetics and independence The concept of allelic INDEPENDENCE is used in several different ways, leading to different definitions and different families of models19. In the most commonly used POPULATION GENETIC20–22 model of epistasis, the MULTIPLICATIVE MODEL, the probablistic definition of independence of events is used. Thus, the joint effects on the viability of alleles at two different loci are considered to be independent (i.e. no epistasis) when the total viability of a genotype equals the product of the viability effects of the component loci. For example, if the viability fitness of genotype A2A2 is (1 − s)2 and that of genotype B2B2 is (1 − t)2, the viability of the A2A2B2B2 double homozygote would be (1 − t)2(1 − s)2 under this model (where t, s>0; t and s are the selection coefficients). Epistasis is modeled by introducing additional selection terms to the fitness of one or another of the multi-locus genotypes. Often, to reduce dimensionality and simplify mathematical analysis, the fitness of one genotype is assigned a value of one and other genotypes are assigned fitnesses relative to this standard, as one minus a selection coefficient (cf. Table I in Box 1). Empirically, TRENDS in Ecology & Evolution Vol.16 No.9 September 2001 http://tree.trends.com 0169-5347/01/$ – see front matter © 2001 Elsevier Science Ltd. All rights reserved. PII: S0169-5347(01)02213-3 498 Opinion Alternative definitions of epistasis: dependence and interaction Michael J. Wade, Rasmus G. Winther, Aneil F. Agrawal and Charles J. Goodnight Although epistasis is at the center of the Fisher-Wright debate, biologists not involved in the controversy are often unaware that there are actually two different formal definitions of epistasis. We compare concepts of genetic independence in the two theoretical traditions of evolutionary genetics, population genetics and quantitative genetics, and show how independence of gene action (represented by the multiplicative model of population genetics) can be different from the absence of gene interaction (represented by the linear additive model of quantitative genetics). The two formulations converge with weak selection but not with strong selection or, for multiple loci, when the aggregated interaction terms are not negligible. As a result of the different formulations of gene interaction, the presence or absence of linkage disequilibrium, D, does not necessarily indicate the presence or absence of fitness epistasis. Indeed, linkage disequilibrium is generated in 'additive' models in quantitative genetics whenever two (or more) loci experience simultaneous selection. As a research strategy, it is often practical, for theoretical or experimental reasons, to minimize gene interaction by assuming independence of gene action in regard to fitness, or by assuming linear additive effects of multiple loci on a phenotype. However, minimizing the role of epistasis in theoretical investigations hinders our understanding of the origins of diversity and the evolution of complex phenotypes. Michael J. Wade* Rasmus G. Winther Dept of Biology and Dept of History and Philosophy of Science, Indiana University, Bloomington, IN 47405, USA. *e-mail: mjwade@bio.indiana.edu Aneil F. Agrawal Dept of Biology, Indiana University, Bloomington, IN 47405, USA. Charles J. Goodnight Dept of Biology, University of Vermont, 1155 Marsh Life Science Building, Burlington, VT 05405-0086, USA. this procedure is problematic because: (1) it is rarely clear which genotype should be designated as the standard; (2) the average viability of the standard cannot usually be measured without error; (3) general environmental influences on viability negatively affect the statistical properties of viability ratios; and (4) the expected frequency of the standard genotype becomes vanishingly small as the number of loci increases. At present, multi-locus population genetic theory resembles a branch of applied mathematics whose findings inform evolutionary discussion, but whose many simplifying assumptions hinder empirical testing. Quantitative genetics and interaction In QUANTITATIVE GENETICS, there is only one family of models, based on linear regression23. In this regression model, alleles at loci, A and B (the independent variables), affect the phenotype (the dependent variable, P) in a manner described by Eqn 1: P = b1A + b2B + b12 AB + e [1] where b1 and b2 are the average effects of alleles at the A and B loci, respectively, on the phenotype P, whereas b12 describes the effect of interactions between alleles at the two loci (e reflects stochastic variations arising from the environment or from loci not being considered). The absence of epistasis means that the interaction coefficient b12 is zero; that is, the loci act additively (independently). This model is called the LINEAR ADDITIVE MODEL. For example, with fitness as the phenotype, if the deleterious effects on viability of allele A2 are (−s), and B2 are (−t), then the viability of the A2A2B2B2 double homozygote is (1 − t − s) in the absence of epistasis (where mean viability is one; Box 1). For two bi-allelic loci, there are four ways in which epistasis can be added to the regression24,25: (1 and 2) interactions between homozygotes at locus A and heterozygotes at locus B or vice versa (additive-bydominance epistasis); (3) interactions between homozygotes at both loci (additive-by-additive epistasis; Fig. 1b); and (4) interactions between heterozygotes at both loci (dominance-by-dominance epistasis). The quantitative genetic approach lends itself to the experimental study of individual loci and their interactions when controlled breeding designs are coupled with molecular genetic markers. However, because the number of possible interactions increases much faster than do the numbers of loci, the application of the approach has its statistical limitations. The two definitions are not equivalent Terms or coefficients considered nonepistatic in the multiplicative model of population genetics can be epistatic in the linear additive model of quantitative genetics, and vice versa. This incongruence has implications both for how epistasis is measured in natural and laboratory populations, and for how inferences about its relevance to the evolutionary process are drawn. Simply put, observation is always theory laden26–28; with epistasis, there are two different theories, which are not concordant in the way they guide empirical observation. For example, although random genetic drift, mutation, migration or selection create associations between loci (linkage disequilibria), persistent associations are often considered to be evidence of selection20. However, whereas selection alone in the absence of epistasis, sensu the multiplicative model, does not generate LINKAGE DISEQUILIBRIUM, selection will generate it in the absence of epistasis, sensu the linear additive model20. Thus, depending on the model, the observation of linkage disequilibria might or might not signify epistasis for fitness, even in the absence of other evolutionary forces. To emphasize the difference between the approaches, we provide an illustration that the absence of epistasis in one model is not equivalent to the absence of epistasis in the other. There is no doubt that interactions complicate theoretical analysis and that a variety of methods (such as transformation) or assumptions (such as independence), or both, have been made to eliminate the complicating features of epistasis from evolutionary theory. To paraphrase an earlier commentator29, with respect to epistasis, realism has been traded for generality in both theories. Realism would require that epistasis be more explicitly incorporated in evolutionary theory. TRENDS in Ecology & Evolution Vol.16 No.9 September 2001 http://tree.trends.com 499Opinion 1.00 0.75 0.50 1.00 0.75 0.50 A1A1 A1A2 A2A2 24 C 29 C 34 C Genotype x environment interaction A1A1 B2B2 B1B2 B1B1 A1A2 A2A2 Gene x gene interaction or epistasis 1 5 p B1 >0.5 ∆p A1 >0 p B1 <0.5 ∆p A1 <0 p B1 <0.5 ∆p A1 <0 p B1 >0.5 ∆p A1 >0 p B1 = 0.5 ∆p A1 = 0 2 3 4 6 (a) (c) (b) TRENDS in Ecology & Evolution p B1 <0.5 ∆p A1 <0 P he no ty pi c va lu e P he no ty pi c va lu e Fig. 1. A graphical view of interactions. (a) Genotype-by-environment interaction illustrating the change in the phenotypic values at the A-locus as a function of temperature. (b) Additive-by-additive epistasis illustrating the change in the phenotypic values at the A-locus as a function of genotype at the B-locus. (c) As a consequence of variations among demes in the frequency of the B1 allele, pB1, the allelic value of the A1 allele changes from deme to deme. With natural selection favoring larger phenotypic values in every deme, this causes among-deme variations in the direction of selection experienced by the A1 allele, ∆pA1. If the positive and negative values of ∆pA1 are combined by migration, then total selection on the A1 allele is reduced. The relationship between the models in the absence of epistasis Independence, sensu the multiplicative model, is not equivalent to the absence of gene interaction, sensu the linear additive model. We illustrate this with two bi-allelic loci, A and B. The A2 allele reduces fitness by a factor s relative to the A1 allele. The B2 allele reduces fitness by a factor t relative to the B1 allele. Under the definition of independence from the multiplicative model, the absence of epistasis produces the effects shown in Table I in Box 1. However, when converted to the linear additive model, epistasis is clearly present (Table II in Box 1). Reciprocally, the absence of gene interaction in the linear additive model produces the effects shown in Table III in Box 1. When we translate this into the terms of the multiplicative model, again, epistatic terms are manifest (Table IV in Box 1). Note, however, that all of the epistatic terms in Box 1 are second or higher order functions of s and t. Terms of this magnitude are often (but not always) assumed to be zero in analyses of population genetic models. In a two locus bi-allelic model, these terms are not only small with weak selection (i.e. 'nearly neutral'alleles) but also few in number. When more than two loci are considered, the number of higher order terms can grow so large that even extremely small pairwise effects might be large when summed together18. We now turn to two phenomena, linkage disequilibrium and episodic selection, in which differences between the models generate epistasis. In the light of this, clearer definitions of, and empirical measurement criteria for, epistasis are required. The generation of linkage disequilibrium with and without epistasis Although several evolutionary forces, such as migration, random genetic drift, mutation and selection, can cause linkage disequilibria, its persistence is often regarded as the signature of multi-locus epistatic selection20,22,30,31. For single large panmictic populations without mutation, gametic phase disequilibria between recombining loci eventually disappear without epistatic selection. One of the mathematical advantages of the multiplicative model is that selection does not TRENDS in Ecology & Evolution Vol.16 No.9 September 2001 http://tree.trends.com 500 Opinion Converting the multiplicative model to the linear additive model The multiplicative model (Table I) with independence of gene action (i.e. no epistasis), when converted to the linear additive model (Table II), creates epistasis as indicated. The terms t and s are the selection coefficients. The first row and column of both tables contain the single-locus genotypes and the associated allelic effects appropriate to the respective model. Note that the epistatic terms in the linear additive model are all equal to or higher than second order (i.e. t 2, s2, or ts). Hence, when selection is weak, the models converge. Converting the linear additive model to the multiplicative model The linear additive model (Table III) with absence of gene interaction (i.e. no epistasis), when converted to the multiplicative model (Table IV), creates epistasis as indicated. The first row and column of both tables contain the single-locus genotypes and the associated allelic effects appropriate to the respective model. Note that the epistatic terms in the multiplicative model are all equal to or higher than second order (i.e. t 2, s2, or ts). Hence, when selection is weak, the models converge. Box 1. Converting models Table I. The multiplicative model: no epistasis A 1 A 1 A 1 A 2 A 2 A 2 1 1 – s (1 – s)2 B 1 B 1 1 1 – s (1 – s)2 1 B 1 B 2 1 – t (1 – t)(1 – s) (1 – t)(1 – s)2 1 – t B 2 B 2 (1 – t)2 (1 – t)2(1 – s) (1 – t)2(1 – s)2 (1 – t)2 Table II. The linear additive model: epistasis A 1 A 1 A 1 A 2 A 2 A 2 1 + s 1 1 – s B 1 B 1 1 + s + t 1 + t 1 + t – s + s2 1 + t B 1 B 2 1 + s 1 + ts 1 – s + s2 + 2ts – ts2 1 B 2 B 2 1 – t + s + t2 1 – t + t2 + 2ts – t2s 1 – t – s + t2 + 4ts + s2 – 2t2s – 2ts2 + t 2s 1 – t Table IV. The multiplicative model: epistasis A 1 A 1 A 1 A 2 A 2 A 2 1 1 – s (1 – s)2 B 1 B 1 1 1 – s (1 – s)2 – s2 1 B 1 B 2 1 – t (1 – t)(1 – s) – ts (1 – t)(1 – s)2 – s2 – 2ts + ts2 1 – t B 2 B 2 (1 – t)2 – t2 (1 – t)2(1 – s) – t2 – 2ts + t2s (1 – t)2(1 – s)2 – t 2 – 4ts – s2 + 2t 2s + 2ts2 – t2s2 (1 – t)2 Table III. The linear additive model: no epistasis A 1 A 1 A 1 A 2 A 2 A 2 1 + s 1 1 – s B 1 B 1 1 + t + s 1 + t 1 + t – s 1 + t B 1 B 2 1 + s 1 1 – s 1 B 2 B 2 1 – t + s 1 – t 1 – t – s 1 – t generate linkage disequilibria. That is, if D is zero before selection, it is also zero after a single generation of selection (D′ = 0). This conclusion is clear when linkage disequilibrium after selection, D′, is expressed as a multiplicative function of the linkage disequilibrium before selection (Box 2). When this multiplicative model is analyzed from the perspective of the linear additive model, there is epistasis (Box 1). This means that there is at least one form of epistatic selection under the linear additive model that does not generate linkage disequilibrium. We can also show the opposite. With the linear additive model and no gene interaction (Box 2), if D is zero before selection, it is not zero after selection (D′ ≠ 0). In this case, we cannot express D′ as a multiplicative function of D. Instead, selection without gene interaction for fitness produces an amount of linkage disequilibrium, D′, proportional to the product, ts. Consistent with our earlier remarks, when s and t are small, the linkage disequilibrium created by selection, D′, will also be small. This linkage disequilibrium, created by selection in the absence of gene interaction, sensu the linear additive model, contributes to the phenomenon known as INTERFERENCE32,33, whereby the change in gene frequency at one locus is slowed by simultaneous selection at another locus. Note that, because the term ts is always positive (Box 2), selection always generates a negative genetic correlation between advantageous alleles at different loci or between disadvantageous alleles (i.e. D<0). Thus, alleles at different loci simultaneously selected in the same direction always interfere with one another to reduce the rate of single gene evolution. Conversely, when an allele is positively selected at one locus and an allele at another locus is negatively selected, then the term ts is always negative (Box 2) and selection always generates a positive genetic correlation between them. This is also interference. The generation of linkage disequilibrium by selection in the absence of gene interaction for fitness is counterintuitive for many who associate the existence of nonzero D with epistatic selection. However, under the linear additive model, the fitness of the least-fit genotype, A2A2B2B2, for example, is lower than it would be under multiplicative independence (compare entries in Tables III and IV of Box 1). When genotypic fitnesses are not multiplicatively independent, linkage disequilibrium is generated by selection and there is interference. In summary, considering both cases of no epistasis, it is clear that there is no necessary connection between epistatic selection and the existence of nonzero values of D, as is often expected (Table I in Box 2). TRENDS in Ecology & Evolution Vol.16 No.9 September 2001 http://tree.trends.com 501Opinion Consider the haploid version of the linear additive model (Box 1), in which the fitness of an individual of genotype A1B1 = 1, A1B2 = (1 − t), A2B1 = (1 − s), and A2B2 = (1 − t − s). Let the frequency of the A1 allele be pA and that of the B1 allele be pB. If we assume that a population is initially in linkage equilibrium (i.e. D = 0), then the frequencies of the four haploid genotypes before selection conform to the Hardy-Weinberg expectations. The frequency of each genotype after selection equals the product of its frequency before selection times its relative fitness, which equals its genotypic fitness divided by the mean population fitness, W, which is given by Eqn I: W = 1 − (1 − pB)(t) − (1 − pA)(s) [I] The linkage disequilibrium generated by a single generation of selection, is given by Eqn II: D ′ = −(1 / W)2(pB)(pA)(1 − pB)(1 − pA)(ts) [II] Clearly, the linkage disequilibrium generated in the linear additive model is proportional to the product ts. Here, linkage disequilibrium is generated despite the absence of epistasis for fitness. By contrast, with the haploid version of the multiplicative model of fitness (Table I in Box 1), the fitness of the A2B2 genotype is (1 − t)(1 − s). If we assume that a population is initially in linkage equilibrium (i.e. D = 0), then it will remain so despite selection. This happens because the initial value of D appears as a factor of D′, given in Eqn III: D ′ = (1 / W)2(1 − s)(1 − t)(D) = 0 [III] where the average fitness in the population, W, equals that above with the added term of (+ts)(1 − pA)(1 − pB). Here, linkage disequilibrium is not generated in the absence of epistasis in the multiplicative model, and despite the presence of epistasis in the linear additive model. Table I summarizes the differences between the two models with respect to whether linkage disequilibrium is a signature of epistatic selection. Box 2. Why linkage disequilibrium is not always a signature of epistatic selection Table I. The generation of linkage disequilibrium by epistatic selection Epistasis? Multiplicative model Linear additive model No D′ = 0 D′ ≠ 0 Yes D′ ≠ 0 D′ ≠ 0 or D′ = 0 Additive traits and multiplicative fitnesses: a recipe for epistatic selection In studies of phenotypic selection in nature, selection is partitioned into sequential episodes, such as germination or birth, early viability, late viability and fecundity20,34–36. In these studies, total lifetime fitness is equal to the product of the fitnesses at each stage in the life history. The logarithmic transformation of lifetime fitness defined as a product of components then becomes the sum of the logs of individual components. Logarithmic transformation is also recommended as a statistical means of removing epistasis (as could be the case with lifetime fitness) when '...the effects of different loci [are] combined by multiplication rather than by addition'23. This is an implicit reference to the multiplicative model as the basis of the underlying biology: we could hypothesize that the A locus affects early viability whereas the B locus affects later viability, for example. However, removing epistasis by switching between absolute and logarithmic scales can itself be problematic23 because epistasis cannot be removed by such a transformation whenever a difference in rank order of fitnesses exists among genotypes at one locus when measured across different genetic backgrounds. (This is always the case with the kind of additive-byadditive epistasis shown in Fig. 1b.) Furthermore, with linear additive traits contributing to multiplicative components of total fitness, epistasis cannot be removed by switching between scales as we show here. If allelic effects are additive for any of the fitness components (as is typically assumed of fecundity and other size-correlated fitness attributes), total fitness will exhibit epistasis under both models, and this epistasis cannot be eliminated by logarithmic transformation. For example, let two additively acting loci, A and B (e.g. Table III in Box 1), affect eggto-adult viability fitness, WV, and two other similarly additively acting loci, C and D with effects u and v, affect fecundity fitness, WF. If viability and fecundity affect fitness independently, the total lifetime fitness, WT, of a genotype equals the product, (WV)(WF). An individual of genotype, A2A2B2B2 C2C2D2D2, would have WV of (1 − 2t − 2s), WF of (1 − 2u − 2v) and lifetime fitness of (1 − 2t − 2s)(1 − 2u − 2v). Taking the logarithm of lifetime fitness would serve to make the fitness components 'additive' on a log scale (Eqn 2): Log[WT] = Log[(WV)(WF)] = Log[WV] + Log[WF] [2] However, the ADDITIVITY of the underlying loci of each fitness component so transformed is not preserved, because (Eqn 3): Log[WV] = Log[(1 − 2t − 2s)] ~ Log[(1 − 2t)] + Log[(1 − 2s)] [3] to order s2, t2 and st, but only when s and t are both small (<0.01). When s and t are equal to or greater than 0.05, then Eqn 2 is accurate only to order s or t. (A similar argument applies to WF.) Genetic inviability or infertility for some genotypes at any stage in the life history not only represents strong selection but also presents the problem of taking the logarithm of zero37. Thus, with additive traits and multiplicative fitness components, that is, the case deemed most representative of natural populations, there will always be epistasis for fitness with strong selection. Conclusion: independence and reductionism The definitions of independence and interaction are not congruent. The definition employed determines how we observe epistasis and also the evolutionary consequences attributed to it (e.g. the generation of linkage disequilibrium). There is no doubt that theoretical analyses in both population genetics and quantitative genetics are facilitated by diminishing or removing epistasis. However, attempts to remove it under one model produce it for the other (Box 1) and the models only converge when all higher order selection terms are relatively small in aggregate. What difference does it make for our perception of the evolutionary process if epistasis is diminished or absent in much of the mathematical theory? How do the simplifying assumptions, which stem in large part from mathematical convenience and might not be representative of living systems, contribute to TRENDS in Ecology & Evolution Vol.16 No.9 September 2001 http://tree.trends.com 502 Opinion Additivity: characterizes the mapping of phenotype onto genotype when the population mean genotypic value equals the sum of the genotypic values for each locus considered separately. Allelic value: the regression of genotypic value on allele number. In the absence of dominance (allelic interactions within a locus) and epistasis, genotypic value is the sum of allelic values. Canalization: the internal process of regulation during development that constrains phenotypic variation. Deme: a group of individuals among which mating is random. Genetic architecture: the characterization of a phenotype in terms of the direct effects of genes and environment as well as the genetic and environmental interactions affecting trait expression. Fitness epistasis: occurs when the fitness value of an allele at one locus depends upon alleles at one or more other loci. Independence: refers to a pair of events, x and y, such that the probability of x conditional on y is the same as the probability of x. Interaction: refers to cases when the phenotype cannot be predicted from knowledge of the average effects of either genes or environment alone. Interference: the reduction in the rate of evolution of one allele that occurs whenever there is simultaneous selection at another locus. Linear additive model: the family of regression models in quantitative genetics. Under this model, the phenotype of a multi-locus genotype does not include any terms for interactions among loci in the absence of epistasis. Linkage disequilibrium (D): the statistical association, or covariance, of alleles at two loci. For alleles A1 and B1 occurring at frequencies pA1 and pB1 at loci A and B, respectively, linkage disequilibrium equals (pA1B1 − pA1pB1), where pA1B1 is the frequency of A1 − B1 haplotypes. Multiplicative model: one family of fitness models in population genetics. Under this model, the fitness of a multi-locus genotype equals the product of the fitnesses of the single-locus genotypes. Population genetics: the study of specific genes in populations and the micro-evolutionary forces that affect their frequencies. Experimental population genetics tends to be concerned with change in the frequency of specific genes or molecular markers. Quantitative genetics: the study of polygenic traits and the statistical genetic properties of such traits when subjected to artificial selection and micro-evolutionary forces. Although founded on explicit genetic models, experimental quantitative genetics, in practice, tends to be concerned with measurable properties of individuals and populations. Glossary shaping what has been called the 'simple reductionistic'38 view of allelic effects and the debate over the importance of epistasis in evolution? The simplifying assumptions can be summarized as two heuristics that are common in evolutionary discussion: the ceteris paribus ('other things being equal') clause and reductionism, which conflates the structural and functional identification of loci. Scientific models are known to be sensitive to the ceteris paribus clause, which refers to background factual conditions and conceptual assumptions. For example, in physics, it has been shown that the apparently context-independent laws of physics are really 'ceteris paribus laws'which are not only sensitive to context but also are false (i.e. they do not correspond to nature) whenever the clause is omitted (i.e. when all pertinent factual conditions are not specified)39. We believe that an analogous situation exists in evolutionary biology regarding assumptions about genetic backgrounds and gene interactions. Here, the ceteris paribus clause includes the assumption of an absence of epistasis because of the related assumption of the averaging out of genetic context in sufficiently large natural populations40,41. Presupposing the absence of epistasis justifies both the simplifying assumptions mitigating interaction in models, and the employment of transformation to statistically remove epistasis in observations. Because the clause is often implicit or hidden, the absence of epistasis in data analyses or theoretical models can be falsely interpreted as its true absence in nature, rather than as a consequence of employing the research heuristic of ceteris paribus as an unstated background assumption regarding the absence of epistasis. Reductionism is characteristic of any genetic analysis, in which properties of the genetic system as a whole are derived solely from the independent properties of the loci from which the system is constructed. Relationships between gene loci are often ignored because loci and their effects are assumed to be not only structurally decomposable (genes can be linearly mapped) but also functionally additive (no interaction effects exist). However, as we have demonstrated, the meaning of functional additivity (independence or absence of interaction) depends on the type of model employed. With epistasis, gene function cannot be decomposed as easily as gene structure27,41–45. In some discussions, an individual organism is considered to be embedded in a biological hierarchy composed of nested levels of interacting components46–49, ranging from chromosomes, to cells, to populations, and even to ecological communities. In this formulation, interaction can occur at any level of organization above the locus. This means that the functional properties of structurally decomposable single genes, in general, cannot be understood as invariant, monadic properties of gene loci and epistasis must be considered in evolutionary analysis. Epistasis is not consistently defined in the different theories, making it difficult to empirically assess its role in evolutionary genetics and adaptive evolution. Some of the most favored single-gene examples of this field, such as sickle cell anemia, reveal significant epistasis when examined more thoroughly17. We believe that a comprehensive reconsideration of the importance of epistasis in the evolutionary process is warranted3. TRENDS in Ecology & Evolution Vol.16 No.9 September 2001 http://tree.trends.com 503Opinion Acknowledgements We thank Jeff Demuth, Elihu Gerson, James Griesemer, Norman Johnson, Doug KarpaWilson, Sandhya Kilaru, Jake Moorad and Meaghan Saur for useful comments and discussion. References 1 Clark, A.G. and Wang, L. (1998) Epistasis in measured genotypes: Drosophila P element insertions. Genetics 147, 157–163 2 Mackay, T. and Fry, J. (1996) Polygenic mutation in Drosophila melanogaster: genetic interactions between selection lines and candidate quantitative trait loci. Genetics 144, 671–688 3 Wolf, J.B. et al., eds (2000) Epistasis and the Evolutionary Process, Oxford University Press 4 Gibson, G. and Wagner, G.P. (2000) Canalization in evolutionary genetics: a stabilizing theory? Bioessays 22, 372–380 5 Lynch, M. et al. (1993) The mutational meltdown in asexual populations. Heredity 84, 339–344 6 Kimura, M. and Maruyama, T. 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