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J. theor. Biol. (1999) 200, 19}37 Article No. jtbi.1999.0974, available online at http://www.idealibrary.com on Epigenetic Inheritance, Genetic Assimilation and Speciation CSABA PAD L* AND ISTVAD N MIKLOD S Department of Plant ¹axonomy and Ecology, ¸oraH nd EoK tvoK s ;niversity, Budapest, ¸udovika 2, H-1083, Hungary (Received on 1 December 1998, Accepted in revised form on 20 May 1999) Epigenetic inheritance systems enable the environmentally induced phenotypes to be transmitted between generations. Jablonka and Lamb (1991, 1995) proposed that these systems have a substantial role during speciation. They argued that divergence of isolated populations may be "rst triggered by the accumulation of (heritable) phenotypic di!erences that are later followed and strengthened by genetic changes. The plausibility of this idea is examined in this paper. At "rst, we discuss the &&exploratory'' behaviour of an epigenetic inheritance system on a one peak adaptive landscape. If a quantitative trait is far from the optimum, then it is advantageous to induce heritable phenotypic variation. Conversely, if the genotypes get closer to the peak, it is more favorable to canalize the phenotypic expression of the character. This process would lead to genetic assimilation. Next we show that the divergence of heritable epigenetic marks acts to reduce or to eliminate the genetic barrier between two adaptive peaks. Therefore, an epigenetic inheritance system can increase the probability of transition from one adaptive state to another. Peak shift might be initiated by (i) slight changes in the inducing environment or by (ii) genetic drift of the genes controlling epigenetic variability. Remarkably, drift-induced transition is facilitated even if phenotypic variation is not heritable. A corollary of our thesis is that evolution can proceed through suboptimal phenotypic states, without passing through a deep adaptive valley of the genotype. We also consider the consequences of this "nding on the dynamics and mode of reproductive isolation. 1999 Academic Press 1. Introduction Mostly we "nd that variation is not continuous in a single geographical region. There are separated species with genetical, morphological, behavioural di!erences. Why should this be so? One possible answer is that species are adapted to discontinuous ecological niches, and the less "t intermediate forms are selectively eliminated. Imagine that there is a bimodal distribution of some resource combined with di!erences in resource use between the genotypes. The resource * Author to whom correspondence should be addressed. E-mail: [email protected] 0022}5193/99/017019#19 $30.00/0 is assumed to be unlimited, so that individual "tnesses do not depend on the current distribution of the genotypes (no frequency dependent selection). These assumptions generate a bimodal relation between individual "tness and the value of a quantitative trait (Barton, 1989b). How can traits evolve from one adaptive state to another, if intermediates are at a selective disadvantage? This problem can be visualized by the adaptive landscape, originally addressed by Sewall Wright (1932, 1980). Adaptive landscape is a surface in a multidimensional space, showing the mean "tness of the population as a function of the possible genetical states (e.g. gene frequency). Adaptation is a &&hill-climbing'' process of the 1999 Academic Press 20 C. PAD L AND I. MIKLOD S population reaching a local maximum. The peaks of the adaptive landscape separated by valleys de"ne the possible stable states of the populations. Peak shift is the transition between these states. The possible mechanisms driving the population to a new peak are of interest from two points of view. First, the rate of evolution in a given lineage may crucially depend on, how it can proceed through suboptimal stages. Second, speciation can also be envisaged as a movement on a "tness surface. In the classic view, speciation occurs by accident. If populations are isolated from each other, they diverge genetically from each other because of the appearance and spread of di!erent mutations. Reproductive isolation evolves gradually in these populations as a side e!ect of this process. The evolutionary dynamics of divergence depends on the strength of the barrier to gene #ow and the rate of transitions between the peaks (Barton & Charlesworth, 1984). Remarkably, both depend on the depth of the valley separating two peaks: weak selection makes the transition more probable, but the degree of isolation is also lowered. An obvious limitation of this approach is that it does not include any explicit ecological factors, so it cannot say anything about the coexistence of diverged populations. Nevertheless, the population genetic approach may highlight the genetic and environmental factors that can potentially facilitate speciation. What drives divergence if selection opposes any shift from the status quo? Genetic drift has attracted substantial attention as a possible candidate. Genetic drift can presumably knock out a population from a local optimum, suggesting that it can cause reproductive isolation even under unchanging ecological circumstances. Nevertheless, the probability of stochastic transitions quickly diminishes with increasing population size (Barton & Charlesworth, 1984; Charlesworth & Rouhani, 1987; Lande, 1985). How can we remove this problem? Wright argued for the importance of population structure, which enables the species to continually "nd its way from lower to higher peaks by the combination of stochastic and selective forces (Wright, 1932). His analysis included population subdivision, "nite population size and inbreeding. In spite of the intellec- tual merit of his work, recent theoretical issues and empirical facts do not support the shiftingbalance theory (Coyne et al., 1997). Because of the ambiguities of drift-induced transition, it is necessary to look for other possibilities that could initiate peak-shift alone or with the combination of stochastic forces. Changing environmental conditions can be one candidate. Environment has two di!erent e!ects on phenotypic pattern: it generates variation and selects the variants. As "rst emphasized by Fisher (1930), under a new selective environment, isolated populations can evolve to di!erent adaptive peaks. The environment can also induce di!erent phenotypic variants. According to Kirkpatrick (1982), the mean of a quantitative trait may be shifted to the neighboring optimal state deterministically, provided that the phenotypic variance exceeds a critical value. Is this change in phenotypic variance solely due to changes in extrinsic conditions? We know that there is a genetic variation for the degree to which phenotypes are a!ected by environmental di!erences (Waddington, 1957; Scheiner, 1993; Via et al., 1995). Therefore, the genes specifying the genetic}environmental interaction can also evolve, and as a by-product they can in#uence the probability of peak shift. It is particularly interesting when the induced phenotypes are transmitted between generations. Because dual inheritance of a trait separates the e!ects of selection on phenotypes and genotypes (PaH l, 1998), divergence might be a hierarchical process (see below). In this paper, we examine the consequences of inherited phenotypic variation on the probability of peak shift. 2. Epigenetic Inheritance Systems and Speciation We have got a growing number of experimental facts showing the inheritance of non-DNA variation (for details and references see Jablonka & Lamb 1995; Russo et al., 1996). The systems enabling the transmission of phenotypic information have been recently called epigenetic inheritance systems (EIS), (Maynard Smith, 1990). The epigenetic variants are mostly not reversed in a predictable manner, and are not as stable as genetic mutations. Environment can induce DIVERGENCE OF DUAL INHERITANCE SYSTEMS di!erent phenotypes, and the changed phenotypic pattern can be inherited even if the original stimuli are absent (for references see Jablonka & Lamb 1995, Table 6.1). Several types of cellular epigenetic inheritance systems are known (for di!erent classi"cations see Jablonka & Lamb, 1995; SzathmaH ry, 1999). The most prominent of them is the chromatin marking system which is carried and transmitted to the chromosomes during DNA replication. Chromatin marks*including DNA methylation pattern, or proteins attached to the DNA*do not change the coding properties of the genes, they rather a!ect the nature and long-term stability of gene expression (e.g. Holliday, 1987). Recent studies convincingly demonstrated that epigenetic states are sometimes meiotically heritable (e.g. Grewal & Klar, 1996; Hollick & Chandler, 1998; Cavalli & Paro, 1998; Kakutani et al., 1999). According to Jablonka & Lamb (1991, 1995), both genetic and epigenetic variation can contribute to speciation and reproductive isolation. They stated that while two parts of the population are geographically or ecologically separated, they accumulate di!erent epigenetic marks that are followed by the "xation of DNA mutations. They proposed that epigenetic changes alone can lead to some reduction of hybrid viability or fertility. This idea seems to be very plausible, because (i) the rate of new epigenetic variation produced can be much higher, than the rate of genetic changes. Therefore, an EIS can initiate evolutionary divergence. (ii) Selection can force the stabilization of (favourable) epigenetic variants by genetic changes, ensuring more reliable transmission of information. In the following, we examine the theory in the light of recent experiments: (1) Changes in epigenetic marks of related species can cause inviability or sterility of hybrids. Jablonka & Lamb (1995, 1998) claimed that divergence of epigenetic marks could alone lead to some "tness reduction in hybrids. They considered the nuclear transplantation experiments in mouse (Mcgrath & Solter 1984, Surani et al., 1984) as a support of their ideas. These studies demonstrated that zygotes with two maternal or parental set of genomes of the same species show abnormal development. Although these results are very impressive, we cannot conclude that 21 divergence of epigenetic marks would solely cause "tness reduction in hybrids (Johnson, 1998). Nevertheless, they do show that epigenetic compatibility is necessary for normal development (Jablonka & Lamb, 1998). Sometimes, alleles or characters of interspecies hybrids show species-speci"c expression (e.g. Reeder, 1985; Zakian et al., 1991). In other cases, interspecies crosses display parent-of-origin defects (Forejt & Gregova, 1992; Signoret & David, 1986; Vrana et al., 1998). It is clear from these examples that species sometimes carry di!erent epigenetic marks that a!ect gene expression in hybrids. Nevertheless, these "ndings do not demonstrate directly that the epigenetic divergence anticipated genetic changes. It is possible that the epigenetic states are strictly determined by the genetic system, without any epigenetic variability. Accordingly, epigenetic changes might be only a manifestation of the accumulation of di!erent mutations in di!erent species. (2) Epigenetic marks change under a new selection pressure. We know that highly inbred populations sometimes exhibit high epigenetic variation (Ruvinsky et al., 1983). More remarkably, selection experiments showed the capacity of inbred strains to adapt to new environmental conditions (Brun, 1965; Fitch & Atchley, 1985; Shaposhnikov, 1966). Although other explanations can be given to these "ndings, it is conceivable that the environment induced heritable phenotypic variation, and the bene"cial variants spread in the population. We do not know, whether these results are the tip of an iceberg, or simple contamination errors, but the paradox can be resolved by a series of selection experiments using inbred strains. (3) Epigenetic divergence can occur prior to genetic changes. West-Eberhard (1986, 1989) argued that new ecological innovations begin with the bifurcation of developmental or behavioural programs giving rise to intraspeci"c alternative phenotypes. This enables the genotype to occupy a new niche without abandoning the old one. Some conditions may favour the "xation or exclusive expression of one of the alternatives that are followed by rapid genetic changes. WestEberhard documented rather good examples that make her idea highly plausible. Nevertheless, she 22 C. PAD L AND I. MIKLOD S did not consider the case when the di!erent phenotypes have heritable capacities. We know a few cases when we might suspect that the divergence of heritable epigenetic marks*e.g. methylation*has been responsible for the isolation. Shaposhnikov (1966) showed that a clone of aphids allowed to reproduce parthenogenetically on a novel host diverged rapidly and acquired a new host preference. If sexual reproduction was permitted, the derived clone failed to produce a viable o!spring with the original population. Even more remarkably, the new strain displayed close morphological resemblance to a conspeci"c species living on the host to which it had become adapted. Because of the genetical similarity of the ancient and the derived clones, it is probable that epigenetic marks diverged during the experiment. This proposition is strengthened by the "nding that heritable phenotypic variation in aphids correlates with changes in methylation (Field et al., 1989). (4) Epigenetic changes can be strengthened by genetic mutations. We know some cases when environmentally induced phenotypic variants resemble the e!ects of known genetic mutations (phenocopies; Scharloo, 1991). Series of selection experiments demonstrated that the phenotypic response became expressed even in the absence of the stimuli that were initially necessary to induce it (Scharloo, 1991). Waddington (1953) introduced the concept of genetic assimilation to highlight this phenomenon. According to his interpretation, there is a variation in the dependence of gene expression on environmental conditions. Selection can act on this variation, producing genetic variants with constitutive expression of the required phenotype. Unfortunately, we still do not know exactly whether EIS plays an important role in the assimilation of an environmentally induced trait. Nevertheless, there is at least one remarkable example. Ho et al. (1983) found that the assimilation of the Bithorax phenocopy can also occur in inbred lines of Drosophila. Maybe epigenetic variants were induced during the experiment, and they were transmitted for the subsequent generations (Jablonka & Lamb, 1995, p. 167). It has also been proposed that these epigenetic variants can be later "xed by genetic mutations. That epigenetic changes of a given gene might help the population to move across an adaptive valley is obvious from the following elementary example. Consider a diploid sexual organism with two di!erent possible alleles at a locus. We de"ne the "tness of genotypes (AA,Aa,aa) as =(AA)"1, =(Aa)"1!s and =(aa)"1, respectively. This "tness scheme can arise due to epistatic interaction between the alleles. Consider now a large panmictic population "xed for one of the alleles (A). Because of the selective disadvantage of heterozygotes, a rare a allele has negligible chance to spread. Nevertheless, gene expression might be suppressed on an epigenetic variant of the common allele (denoted by A). Now the rare a can be coupled either with A or A. It has a clear disadvantage as Aa. In contrast, the Aa type mimics a hemyzygous * (a0) stage. If di!erences in the amount of gene products do not cause serious phenotypic di!erences, then Aa may be indistinguishable from aa. Therefore, the "tness of the suboptimal (heterozygote) genotype is enhanced. Consequently, the probability of peak shift is also enlarged. In this paper, we improve the theory "rst suggested by Jablonka & Lamb (1991, 1995). Using the adaptive landscape metaphor, we consider the plausibility of epigenetically triggered peakshift of a quantitative trait. 3. Preliminary Assumptions Historically, the notions of multiple inheritance systems and Lamarckian evolution have been coupled. The mechanism of Lamarckian evolution can be summarized in three statements: (i) the change in a character can be induced by the environment, (ii) the induced change can be transmitted to the following generations, even if the original inducing stimuli are no longer present, and (iii) the induced phenotypes are adaptive in the given environment. The &&soft'' version of Lamarckism includes only the "rst two points, while the &&hard'' version covers all three points. If we agree with the former, we do not think that the environment induces the required phenotype. Rather good and bad phenotypes are induced alike. In this paper, we deal only with the latter possibility: we do not assume directed epigenetic DIVERGENCE OF DUAL INHERITANCE SYSTEMS variation. However, we heavily rely on the following assumptions: (1) The genetic system speci"es only the limiting conditions: the possible epigenetic states of the genotype. However, the genotype does not specify which transmissible state is actually present in a given individual. Therefore, we do not address directly the adaptive value of genomic imprinting (Hurst, 1997). Imprints are abolished and reset each generation in a parental-speci"c manner, and the expression states seem to be strictly determined by the genetic system. (2) The genetic factors controlling epigenetic variation do not act in a genome wise manner, but rather localize their action to particular regions in the genome. This enables certain traits to evolve rapidly, while also ensuring the conservation of other genes (similar to the e!ects of certain hypermutable loci, Moxon et al., 1994). In this way, the ratio of bene"cial to harmful variants is enhanced. We think that it is reasonable to assume locus-speci"c epigenetic state modi"er genes. In certain cases, multiprotein complexes assemble at speci"c sites in the genome to mediate the propagation of discrete states of gene expression by organizing heritable chromatin structures (Klar, 1998). Changes in the regulatory proteins or cis-acting control regions seem to have only local e!ects on the establishment or propagation of these states (Grewal and Klar, 1996; Cavalli and Paro, 1998). Nevertheless, we admit that there are known cases, when mutation of certain genes caused meiotically stable reduction in the methylation level of the whole genome (e.g. Kakutani et al., 1999). (3) The establishments of di!erent expression states are mechanistically (or genetically) separable from the heritable maintenance of expression states. In the light of recent experimental "ndings (e.g. Pirrotta, 1997; Cavalli and Paro, 1998) this assumption seems to be plausible. 4. The Basic Model In this section, we examine the e!ect of the environment on a quantitative, polygenic trait. 23 The environment modi"es the phenotypic pattern in two di!erent ways: it generates phenotypic variation and selects the variants. Throughout the paper, we restrict our attention to a constant environment. This means the following: only microenvironmental #uctuation is present, inducing di!erent phenotypes, but the individual "tness function remains unchanged. It is assumed that the reaction of phenotypes on environmental changes is determined genetically. According to Gavrilets and Scheiner (1993), the value of an arbitrary individual trait ( y) can be given by a polynomial of the form y"x#P m#P m#2#P mL, L (1) where the quantities x, and the P 's depend on the G genotype and m is a single environmental parameter. The P 's specify the reaction norm of the G genotype to microenvironmental changes. The distribution of m is assumed to be Gaussian with mean m and variance d. As a "rst approximation, K we omit the quadratic and higher-order term, so eqn (1) reduces to y"x#P m. (2) This linear relationship between phenotype and environment can be a good approximation if the e!ect of the microenvironments on the trait is small (Gavrilets & Hastings, 1994). The mean phenotype of a genotype across environments is yN "x#P m, and the phenotypic variance of the genotype is < "P d. Without the loss of genW K erality*possibly by rede"ning the scale*we set m"0, so yN "x. Therefore, the phenotypic distribution of a given genotype is also Gaussian 1 !(x!y) u (y)" exp , 2< (2n< W W (3) with mean x and variance < . For a simpler W denotation, we change P to P, and call it epi genetic variability. Epigenetic variability determines the capability of the genotype to generate phenotypic variants di!erent from the genetically hard-wired one (x), regardless of their adaptive states. It has nothing to do with adaptive 24 C. PAD L AND I. MIKLOD S phenotypic plasticity, because we do not assume speci"c responses to changing selective conditions. Presumably, epigenetic variability crucially depends on the possible number of epigenetic states of the trait. If an EIS is present, then phenotypes are not only generated, but are also inherited. The mean number of generations through which the phenotypes are transmitted is measured by the memory (m). In the model, epigenetic variability and memory characterize the EIS from a functional point of view. The mean phenotype (x), the epigenetic variability (P) and the memory (m) of a given trait are assumed (i) to be determined by the genetic system; (ii) to be genetically independent of each other and any other traits; and (iii) to evolve, like any other character. 5. The Adaptive Landscape of the Genotype The adaptive landscape metaphor has di!erent interpretations. In the "rst version, which is quite common although sometimes misleading, the adaptive landscape shows the relationship between the possible genetic state and the mean "tness of the population. This interpretation is based on a more fundamental version: the adaptive landscape of the genotype. This shows the "tness of the genotype as plotted against the possible genetic states of the characters (Gavrilets, 1997). A discrete point on the landscape does not represent a possible state of a population, but rather a single genotype. Population is given here as a cloud of points. We also have to make a clear distinction between genotypic and individual "tnesses. The latter shows the "tness of a given phenotype as a function of the possible phenotypic states. Due to phenotypic variation, the genotypic "tness is a weighted average over the "tnesses of all possible phenotypes. The "tness of the genotype with an epigenetic inheritance system is given as follows (see also PaH l, 1998). The "tness of an individual with phenotype y is =(y). This phenotype is induced with frequency u(y) in the "rst generation. The phenotypes are transmitted to subsequent generations speci"ed by the memory. After m generation selection the frequency of the y phenotype is proportional to u(y)=(y)K. Therefore, the frequency of the (x, P, m) genotype after m generation is proportional to u(y)=(y)K dy. The expected "tness of the (x, P, m) genotype is the geometric mean over the memory span. We obtain that =(x, P, m)" u(y)=(y)K dy K . (4) In the case of P'0 and m"1 the phenotypes are only induced but not inherited. The "tness of a genotype with epigenetic inheritance is reasonable only if memory is much shorter than the expected time it takes the trait to mutate. Because the transmission e$ciency of epigenetic information is much lower than that of genetic information, this assumption seems to be ful"lled. Accordingly, the time-scales of phenotypic variation and selection is di!erent from the time-scale of genetic evolution. The central question of our work is the following: when is it favourable for the genotype to enlarge or to suppress phenotypic variability? How can evolution of an EIS modify the probability of peak shift? Before exploring these questions on a multi-peak "tness surface, we have to examine the behaviour of these systems if the landscape has only one global optimum. 6. The One-peak Adaptive Landscape We assume Gaussian stabilizing selection on the character. Accordingly, the individual "tness function is =(y)"exp !(y !y) MNR , 2< Q (5) where y gives the optimal phenotype and MNR 1/< measures the intensity of selection. If we Q ignore the e!ect of linkage disequlibrium, or frequency}dependent selection, then the mean phenotype will tend to the maximum. How does selection change epigenetic variability during the evolution of the mean? Applying eqns (3) and (5), the evaluation of eqn (4) gives =(x, P, m)" < K !(x!y )2 Q MNR . exp 2(<#Pmd) < #P md Q K Q K (6) DIVERGENCE OF DUAL INHERITANCE SYSTEMS If phenotypic variation is suppressed (P"0), eqn (6) reduces to =(y), with y"x. We assume that the mean and the epigenetic variability evolve synchronously on the adaptive landscape, leaving the memory unaltered. Now we can examine the optimal epigenetic variability for a given point of the landscape (x) and for a given memory (m). Solving * ln =(x,P,m)/*P"0, we obtain < 1 . P " (x!y )! Q MNR MNR m d K (7) Increasing epigenetic variability is advantageous in the following cases [see eqn (7)]: (1) The character is far from the peak of the landscape. On the peak of the landscape (x"y ) selection forces to suppress epigenetic MNR variability. This is due to the fact that in this case phenotypic variance produces only suboptimal phenotypes. (2) The intensity of selection is high. It ensures that the possibility of "nding a much better epigenetic variant is high. (3) The microenvironmental variance (d) is K low. If d decreases, epigenetic variability has K to increase to ensure unchanging phenotypic variation. (4) The memory is long. Memory increases the strength of phenotypic selection. It allows the rarely induced, but advantageous phenotypes to spread. On the other hand, if the phenotypes are not heritable (m"1), the genotype &®enerates'' its variants in every generation, so that they are always present with the same frequency. Epigenetic inheritance systems enable phenotypic variability to be high, even if the intensity of selection is low, and the mean is close to its optimum. 7. Genetic Assimilation: Theory and Consequences By de"nition, we "nd the critical point of the landscape, where the optimal epigenetic variability is zero: x "y $ APGR MNR < Q. m (8) 25 If the phenotypes are generated but not inherited (m"1), the critical point coincides with the in#ection point of the landscape (x "y $(< ). This relationship can also be APGR MNR Q realized by elementary algebra using the Jensen inequality (see appendix A). As a result of a very di!erent theoretical framework, Rice (1998) arrived at the same conclusion. Increasing phenotypic variance is advantageous if the adaptive landscape is convex, and disadvantageous if it is concave. It has been widely demonstrated that environmental sensitivity (plasticity) is a trait that can respond to selection, and this response is partly independent of a change in the mean of that trait (Scheiner, 1993). Our model predicts that (i) directional selection pressure on the mean could lead to changes in plasticity. (ii) In contrast, selection on plasticity is expected to leave the mean of the trait unchanged. (iii) Increasing intensity of selection on the mean would lead to higher plasticity [eqn (7)]. The "rst two, (i) and (ii), predictions have been already con"rmed by the experiment of Scheiner & Lyman (1991). Nevertheless, the third point suggests some new empirical tests. In a truncation selection experiment similar to that mentioned above, it would be quite easy to manipulate the intensity of selection by resetting the threshold (see also Rice, 1998). It has also been shown that as long as the trait is far from the adaptive peak, heritable epigenetic variation is especially favourable. Near the optimum, the stabilizing e!ect of selection dominates, so it forces to suppress epigenetic variation. The capacity of the organisms to ensure the production of a standard phenotype in spite of environmental disturbances is called canalization (Waddington, 1942). Under stabilizing selection, canalization can bu!er against phenotypic variation, without changing the average expression of the character. If the trait is under directional selection, the optimal phenotype appears with a very low frequency at the beginning, and it is probably induced by the environment. The frequency of the optimal phenotype increases, as epigenetic variability increases, and the mean evolves closer to the optimal phenotype. Reaching the critical point of the landscape, the suppression of phenotypic variation is selectively advantageous. Genetic assimilation occurs much 26 C. PAD L AND I. MIKLOD S closer to the peak, if an epigenetic inheritance system is present [m'1, eqn (8)]. Waddington (1953) showed how phenotypic variation can be converted to genetic changes without hurting the central dogma. However, Williams (1966) claimed that genetic assimilation cannot contribute signi"cantly to adaptive evolution, because the e!ects of environmental challenges are mostly harmful for the organisms. Under a new selection pressure never experienced before, we cannot expect the organisms to give immediate speci"c responses. Adaptive phenotypic plasticity can only be achieved under regularly changing selective conditions, through the slow process of natural selection. It is obviously true, but irrelevant for us. We considered the evolution of phenotypic variability under directional selection forces, and not the evolution of speci,c responses under changing (selective) environments. We do not assume any correlation between selective demands and inducing factors (for a similar, detailed argument see Eshel & Matessi, 1998). Thus, changing inducing conditions do not necessarily lead to "tter phenotypes. It is enough, if only a fraction of the induced phenotypes is adaptive: directed epigenetic variation is not necessary for the process to go on. Genetic assimilation is an inevitable consequence of the selection acting on the mean and epigenetic variability. The inheritance of phenotypic variation seems to be especially favourable if new favourable genetic variants turn up very rarely. This can occur under a wide range of circumstances: if the size of the population is low, recombination is limited or absent, etc. 8. Multiple Adaptive Peaks Using the results of the previous sections, we address the question, how can epigenetic inheritance modify the evolution of a trait if there are multiple "t states for the individuals. We will see that even if there is a bimodal relation between individual "tness and the value of a quantitative trait, the "tness surface of the genotype can be both uni- or bimodal, depending on the phenotypic variance and the memory. We also examine the selection pressure acting on epigenetic variability. Kirkpatrick (1982) designed a bimodal "tness function of the individuals. According to his formula, the "tness of an individual with phenotype y is !(y!y ) MNR 2< Q !(y!y ) MNR , #H exp 2< Q =(y)"exp (9) where the two phenotypic maxima are located at y and y , 1/< measures the intensity of MNR MNR Q selection, and H gives the relative heights of the peaks. Using eqns (3) and (9) the integration in eqn (4) can be evaluated to give =(x, P, m)" < K K m K Q , HI exp(b/4a!c) <#m< k Q W I (10a) where a"m/2< #1/2< , Q W b"x/< #(y k#y (m!k))/< , W MNR MNR Q (10b) (10c) c"x/2< #(y k#y (m!k))/2< . (10d) W MNR MNR Q [For the derivation of eqn (10) see Appendix B.] One of the disadvantages of this formula is that the derivatives with respective to x and P are transcendental functions, hence the minima and maxima cannot be solved analytically. Therefore, we used numerical simulations to see the behaviour of the model. 8.1. THE TRANSITION BETWEEN PEAKS Generally, peak shift is described as a movement along the population "tness surface. However, because population clusters around one genotype, it is enough to envisage the e!ects of an EIS on the adaptive landscape of the genotype. We considered a bimodal, asymmetric individual "tness function; therefore two isolated optimal states are present in the absence of phenotypic variation (Fig. 1). With epigenetic variability, the DIVERGENCE OF DUAL INHERITANCE SYSTEMS 27 "tnesses of the intermediate genotypes are enhanced. This is a result of the environmental deviations producing individuals with phenotypes of the other peak (Kirkpatrick 1982). This e!ect is especially pronounced with large memory [Fig. 1(a) and (b)]. Because the transition rate primarily depends on the valley depth, an EIS is expected to facilitate the shift between adaptive states. Why is it so? Phenotypes between the optimal states still su!er selective elimination. Nevertheless, a genotype with intermediate mean (x) need not reside in a "tness valley, because it may contain a mixture of more or less "t phenotypes (Coyne et al., 1997). The transmission of phenotypic information ensures that the rarely induced favorable variants are not lost in the next generation, and they can increase in frequency. The e$ciency of phenotypic selection depends on the epigenetic variability (P) and the memory (m). This process leads to the epigenetic divergence of a given genotype: the intermediates are selectively eliminated, and the more "t phenotypes accumulate in the population. We can conclude that an EIS enables the genotypes to "nd a new adaptive zone, without abandoning the old one. As phenotypic variation increases, the adaptive valley shallows, until the lower peak vanishes (Fig. 2). This would lead to the evolution of the FIG. 1. The e!ects of an epigenetic inheritance system on the adaptive landscape of the genotype. The dashed line shows the individual "tness function. Increasing memory reduces the depth of the adaptive valley (a). Nevertheless, memory must be quite high, to be favourable in the vicinity of the lower adaptive peak (a) and (b). The genotypes with epigenetic inheritance (P'0, m'1) su!er selective disadvantages on the higher adaptive peak (c). Therefore, near the global optimum, canalization of phenotypic expression is favourable. Data: m"10, 100, < "0.1, < "1, y "0, W Q MNR y "2.4, H"1.2. MNR genotypic optima do not coincide with the peaks of the individual "tness function, but are displaced towards the other peak. The attraction of the higher peak is extended. In addition, the FIG. 2. Changing positions of the minima and maxima on the adaptive landscape as phenotypic variance of the genotype increases. Epigenetic variability diminishes the distance between the peak and the bottom of the valley. Reaching the critical phenotypic variance, the lower peak vanishes, and only the upper maximum remains. Data: m"10, < "1, y "0, y "2.4, H"1.2. Q MNR MNR 28 C. PAD L AND I. MIKLOD S mean by mass selection alone. As the mean gets to the vicinity of the neighboring peak, it is disadvantageous to generate epigenetic variation. Once the optimal genotype is found, exploration away from this optimum only reduces "tness [Fig. 1(c), Sections 6 and 7]. 8.2. THE EFFECT OF SELECTION ON EPIGENETIC VARIABILITY Previously, we argued that a given epigenetic inheritance system can in#uence the evolutionary dynamics of a trait. However, another problem arises. Generally, evolution is regarded to be myopic, an event happening now would not be selected for just because it will turn out to be advantageous in the distant future. Therefore, it is not enough to show that dual inheritance of a trait enhances the transition towards a higher peak. We also have to know how the genes controlling non-zero epigenetic variability are maintained in the population. The possible answer is either (a) the presence of an EIS is a side e!ect of other evolutionary forces or structural constraints, or (b) the genetic control of phenotypic variation has been directly shaped by natural selection. Although, we may never know whether (a) holds in general, we put forth an argument in favour of (b). Consider now the genotypic "tness as a function of epigenetic variability and the mean. If the mean (x) coincides with a maximum, the mutants with phenotypic variance slightly higher than zero necessarily su!er "tness reduction (Fig. 3.) In this case, the phenotypes of the new adaptive zone do not appear frequently enough to compensate the disadvantage of generating suboptimal variants. Thus, although the population may evolve from y to y along various paths on MNR MNR the x}P plane, it still has to move across an adaptive valley. We propose two di!erent ways to solve the problem: (1) Changes in the inducing e!ect of the environment, leaving the selective conditions unaltered can initiate peak shift (Fig. 4). If microenvironmental variance (d) temporarily inK creases, the population may cross the adaptive valley of epigenetic variability without genetic changes. Consequently, the mean (x) evolves to the neighbouring peak purely by selection. This FIG. 3. Fitness as a function of phenotypic variance on the lower "tness peak. The adaptive valley shallows, as memory increases. x:0, < "1, y -y "2.5, H"1.2. Q MNR MNR FIG. 4. A possible way on the x}P plane in which two isolated peaks may be connected. The contours of the adaptive landscape are also indicated. The most likely path for transition passes the saddle point. Changes in the inducing e!ects of the environment (represented by mM and d) may K help the population overcome suboptimal phenotypic states. Note that the zone of attraction extends to the valley as the epigenetic variability increases. mechanism has been originally suggested by Kirkpatrick (1982). Nevertheless, he did not consider the possibility of heritable phenotypic variation. =ith growing memory both the depth of the valley and the distance of the neighbouring adaptive zone are lowered (Figs 3 and 4, Table 1). This e!ect is especially pronounced, if there is a large discrepancy in peak height in favour of the new peak [Table 1(b)]. This is due to the increased strength of phenotypic selection. It is also possible that the mean (m) of the environmental distribution changes. The epigenetic e!ect would di!er from genetic mutations in an important aspect: it 29 DIVERGENCE OF DUAL INHERITANCE SYSTEMS TABLE 1 ¹he position of the saddle point and its ,tness value. P"0 corresponds to the situation, when epigenetic variability is not allowed to evolve. Data: (a) H"1.2, y "0, y "3, <"1, MNR MNR Q d"1. (b) H"1.4, y "0, y "3, < "1, K MNR MNR Q d"1 K (a) Position of the saddle point P"0 m"1 m"10 m"50 x"1.390 x"0.799, P"0.871 x"0.413, P"0.840 x"0.166, P"0.549 Fitness in the saddle point ="0.709 ="0.857 ="0.927 ="0.988 (b) P"0 m"1 m"10 m"50 Position of the saddle point Fitness in the saddle point x"0.760 x"0.677, P"0.757 x"0.33, P"0.690 x"0.131, P"0.422 ="0.759 ="0.890 ="0.945 ="0.996 yields novel phenotypes in many individuals of the population at the same time (Jablonka & Lamb, 1995, pp. 222}224). If the suboptimal phenotypic state is passed through, then the transformed population "nds itself under a new selective regime. After phenotypic inducement, the population can evolve by genetic changes to the other adaptive state. Because increasing memory enlarges the attraction domain of the higher peak, the initial increase of phenotypic variation can be driven even by slight changes in the environment. Summarizing, the rate of evolution may depend on the speci"c distribution of the (inducing) environment. (2) In Section 3, we assumed that the epigenetic variability and the mean are genetically independent, evolving traits. Therefore, the rate of stochastic shift may be augmented by the simultaneous changes of x and P. Each path on the x}P plane has certain probabilities. In a multidimensional adaptive surface, the most probable #ux between alternative states passes a single saddle point (Barton & Rouhani, 1987). The frequency of stochastic shifts is proportional to the ratio = M /= where = and = are the popuQ N N Q lation "tnesses on the peak and saddle, respec- tively. With an EIS, genotypic "tness value of the saddle point is highly enhanced (Table 1). Thus, the rate of drift-induced transition is also expected to increase. We in turn examine the details of this possibility. 9. The Expected Time for Peak Shift Under Finite Population Size and Evolving Genetic}environmental Interaction In the preceding section, we stated that an EIS substantially reduces the depth of the valley separating peaks (Fig. 3, Table 1). Because the probability of drift-induced transition primarily depends on the valley depth (Barton & Charlesworth 1984, Lande 1985), the evolutionary dynamics of divergence is expected to increase. This statement is in turn investigated explicitly. We restrict our attention to the case when phenotypes are induced but not inherited (P'0, m"1). Intuitively, this seems to be the least hopeful situation, because any initial increase of epigenetic variability only reduces "tness (Fig. 3). In spite of this, the simulation experiments demonstrated that the evolution of epigenetic variability substantially enhances the rate of stochastic shifts (see below). Because valley depth diminishes with increasing memory (Fig. 3, Table 1), we may extrapolate our "nding to those cases, when epigenetic inheritance is present (m'1). We plan to study the e!ect of phenotypic transmission on the probability of drift-induced peak shift in a future work. 9.1. MATHEMATICAL TREATMENT If phenotypic variation is induced, but not inherited (m"1), then eqn (10) reduces to =(x, P, 1)" u (y) =(y) dy " < !(x!y ) Q MNR exp < #Pd 2(< #Pd) Q K Q K #H exp !(x!y ) MNR . 2(< #Pd) Q K (11) Presumably, the mean (x) and the epigenetic variability (P) are genetically independent of each 30 C. PAD L AND I. MIKLOD S other. Accordingly, we neglect the possibility of pleiotropic mutations and the e!ects of linkage disequilibrium. The latter might be generated directly under strong selection or genetic drift (Barton, 1989a). Nevertheless, this problem does not seem to be too severe. There is evidence from similar simulation experiments (Phillips, 1996) that the assumption of linkage equilibrium does not jeopardize the qualitative picture. Thus, x and P in the population in the t-th generation may be described by two independent distributions, denoted by p(x, t) and p(P, t), respectively. Accordingly, the mean "tness of the population becomes = (t)" p(x, t)p(P, t) =(x, P, 1) dP dx. (12) The strength of stabilizing selection on the mean (x) is generally higher than that on canalization (P) (for details see Rice, 1998; Wagner et al., 1997). We also know that the genetic variance of phenotypic variation is generally lower than the genetic variance of the mean (Scheiner, 1993). Therefore, we have a good reason to think that the distribution of epigenetic variability does not have a substantial e!ect on the mean "tness. Therefore, we approximate eqn (12) by (t): p(x, t) =(x, P , 1) dx, = (13) where P is the population mean of epigenetic variability. Generally, it is assumed that continuously varying traits have normal distributions. Therefore, the frequency of x in generation t may then be written as 1 !(x!xN (t)) p(x, t)" exp , 2< (2n< EV E V (14) where x (t) is the population mean and < is the EV genetic variance of x. Using (14), eqn (13) can be evaluated to give < !(xN (t)!y ) Q MNR = (t): ; exp < #< #< 2(< #< #< ) Q W EV Q W EV #H exp !(xN (t)!y ) MNR , 2(< #< #< ) Q W EV (15) where < "( PM (t)d ). We used Monte Carlo W K simulations to examine the evolutionary dynamics of x and PM under a given selection pressure and random genetic drift. The simulations were semi deterministic (Kimura, 1980), and expressed the changes of x and PM as the sum of deterministic and random variable terms: DxN (t)"< EV * ln = M (t) #d , V *x (t) (16a) DxN (t)"< E. * ln = M (t) #d , . *PM (t) (16b) where < is the genetic variance of P in the E. population. The deterministic part expresses that selection causes the population to climb the surface of the mean "tness along the greatest slope (Lande, 1976). The stochastic terms (d and d ) V . represent the e!ects of genetic drift. They are generated as pseudo-random Gaussian deviates with mean 0, variances < /2N and < /2N , EV C E. C respectively. (N is the e!ective population size). C This method provides an excellent approximation of the Fisher}Wright individual sampling process, and it also takes several magnitudes shorter to complete. The genetic variances were taken constant throughout a simulation run. The validity of this presumption is not straightforward, because the genetic variances are likely to change in the long term (Barton, 1989a). Nevertheless, increasing genetic}environmental interaction diminishes the strength of selection on the mean (x), so the genetic variance (< ) under EV mutation-selection equilibrium is expected to increase. Because growing genetic variation usually facilitates the transition from one adaptive state to another (Charlesworth & Rouhani, 1988, Kirkpatrick, 1982; Whitlock, 1995), our treatment overestimates the expected time for peak shift. 9.2. SIMULATION METHODS All simulations were conducted by starting the population close to the smaller peak (xN (0): y ), showing well canalized phenotypic expresMNR sion (P(0)"0). Each run was continued until the population mean (xN ) reached the vicinity of the higher peak and epigenetic variability was DIVERGENCE OF DUAL INHERITANCE SYSTEMS approximately zero (x (¹):y , P (¹):0). In MNR all cases, 200 replicates were run for each set of parameters. The expected time before shift (¹M ) was the mean of the individual replicate times. We used this procedure with preset parameter values (Fig. 5). If < "0, epigenetic variability E. remained zero throughout the simulation. This special case has been analysed previously by Lande (1985). He gave an approximate formula [Lande, 1985, eqn (7)] for the expected time until shift of the mean phenotype. To shorten computer time, we used this approach, when < "0 E. and N 53000. C 9.3. RESULTS The results of the simulations can be summarized in three statements (Fig. 5): (i) The expected time (¹ ) is de"nitely lower when epigenetic variability is allowed to evolve (< '0), compared with the case, when phenoE. typic variation is suppressed (< "0). (ii) The E. di!erences increase exponentially with the e!ective size of the population. (iii) Moderate changes of < do not cause substantial di!erences in the E. expected time. Clearly, this is what we expected. According to Barton & Charlesworth (1984) and Lande (1985), ¹ increases exponentially with the population size and the valley depth. The latter is signi"cantly lowered if epigenetic variability is an evolving trait. 31 FIG. 5. The expected time for peak shift as a function of the e!ective population size at di!erent genetic variances of phenotypic variability (< ). See text for details. Note that E. the expected time is on a logarithmic scale. Data: < "0.13, Q < "0.7, < "0, 0.3 and 0.7, y !y "2, H"1.1, EV E. MNR MNR m"1, d"1. Key: < "0 (simulation results) (䊏); < "0 K E. E . (analytic approach) (;); < "0.7 (䉱); < "0.3 (䊏 ). E . E . FIG. 6. The self-reinforcing e!ects of selection on phenotypic variance. Increasing epigenetic variability (P) has dual e!ects on the mean. At "rst, it reduces the strength of selection on the trait, therefore the mean (x) is allowed to evolve, leading to further increase of P. This direct e+ect has been analysed explicitly in Section 9. However, epigenetic variability might also have a variance directed e!ect. Increasing P enhances the genetic variance of the trait (< ), allowEV ing further displacement of the mean. 9.4. THE SELF-REINFORCING EFFECT OF SELECTION ON EPIGENETIC VARIABILITY Why should evolving genetic}environmental variance cause such a huge di!erence? One possible answer is that it is an extra dimensional bypass, where the population can evolve. However, we think that this is only a partial answer. The strength of stabilizing selection on the mean (x) is generally higher than that on canalization (P). Therefore, the genes controlling some slight increase in epigenetic variability can be "xed in the population without signi"cant "tness reduction. Epigenetic variability produces some individuals whose phenotypes place them in the domain of the other peak. This results in the displacement of the population optimum towards the other peak. Thus, selection acts to change the mean, so that further increase in epi- genetic variability might be selectively favourable (or only slightly detrimental). Accordingly, selection is expected to push the mean further. The initial slight increase of phenotypic variation could lead to subsequent genetic changes of the mean (Fig. 6). We can conclude that evolution of the genes specifying the reaction norm can help the population to pass through suboptimal phenotypic states. In the vicinity of the neighbouring peak, phenotypic variation is suppressed, leading to relatively high ,tness reduction of the suboptimal (hybrid) genotypes. 10. The Evolution of Reproductive Isolation Mostly, phenotypic variability is limited by some structural reasons. Therefore, we propose 32 C. PAD L AND I. MIKLOD S that reproductive isolation evolves gradually in a series of successive shallow peak shifts rather than a large single one. As long as new adaptations are available, selection forces the epigenetic variability to enhance, triggering new peak shifts. This chain reaction is expected to stop, if the population reaches a global optimum*or a highly isolated peak. Once the optimal genotype is found, exploration away from this optimum only reduces "tness. This raises an intriguing possibility*the reversible evolution of the genes controlling epigenetic variability (Fig. 7). The presence of an EIS is favourable during the transition from one adaptive state to another, but is expected to be selectively eliminated near the global optimum. This is in agreement with the results of sections 5 and 6. Inducing epigenetic variation is favorable under directional selection, but it is supposed to remain suppressed as the stabilizing e!ect of selection dominates. Reproductive isolation arises as a side-e!ect of genetic and epigenetic divergence, and it is completed by the canalization of phenotypic expression. After the suppression of phenotypic variation, "t phenotypic states are not available for the hybrids. The expected properties of hybrids with an EIS di!er from those formed when such a system is not present and when the adaptive peaks FIG. 7. The adaptive landscape of the genotype as a function of the phenotypic variance and the mean. Two separated peaks are connected by the ridge of epigenetic variability. Consequently, no deep valley should be crossed if P and x evolve synchronously. Data: m"30, < "1, Q y "0, y "2.4, H"1. MNR MNR are isolated. In the latter case, any deviation from a coadapted combination of genes leads to signi"cant reduction in "tness (Barton, 1989b). In contrast, a hybrid genotype with epigenetic inheritance is expected to have viable phenotypic variants resembling one of the parent species. Perhaps the most widely cited example of one factor peak shift is the evolution of new karyotypes. There are many cases of related species "xed for di!erent chromosome rearrangements. The interpopulation hybrids often show improper segregation, or abnormal chromosome pairing leading to reduced fertility. Therefore, the "xation of a new karyotype has been advocated as an evidence that evolution sometimes proceeds through an adaptive valley. Nevertheless, these examples do not show directly that the new rearrangements are "xed when drift overcomes the strong underdominance of heterozygotes. Sometimes, heterokaryotypes vary widely in "tness (Coyne et al., 1991, 1993, 1997), and there are clear-cut examples of reduction in hybrid viability or fertility being non-detectable (Coyne et al., 1991). Is this only due to the genetic heterogeneity of the hybrids? It has been previously proposed that equivalent phenotypic e!ects can be obtained either by genetic or epigenetic changes (Zuckerlandl & Villet, 1988). Maybe, the "tness reduction of hybrids depends on the ability of an EIS to compensate for genetic changes and mask the incompatibilities of the parental genomes. Of course, we do not argue that this mechanism would be a special adaptation for the hybrids, rather it seems to be an accidental by-product of the EIS initiating divergence. For example, pericentric inversions could theoretically lead to aneuploid gametes through crossing over. However, recombination is often nearly absent in the inverted region for some unknown reason. There are experimental evidences that the epigenetic state of chromatin affects the frequency of recombination (Ashley, 1988). In general, chiasma cannot form in highly condensed regions (John, 1988; Wu & Lichten, 1994). Therefore, the fertility of heterozygotes might be due to the accumulation of chromatin marks in the inverted regions, leading to heterochromatinization and inhibition of crossing overs. Finally, we must emphasize that we do not want to give an alternative explanation for the DIVERGENCE OF DUAL INHERITANCE SYSTEMS Haldane's rule (the preferential sterility or inviability of hybrids of the heterogametic sex), which is usually explained by speci"c properties of the sex chromosomes or sexual selection (Orr, 1997). Our speciation theory is not limited to the action of the sex chromosomes. 11. Di4erences Between Epigenetic and Behavioural Inheritance Throughout the paper, we restricted our attention to cellular epigenetic inheritance systems, and we did not mention behavioural inheritance of animals and man. The analogy seems to be straightforward: new patterns of behaviour, "rst acquired by chance or through individual learning in a new condition, may be transmitted to the following generations through social learning (Heyes & Galef, 1996; Jablonka et al., 1998). Surely, there is an increasing acceptance that learning precedes mutations, and the acquired behaviour may be augmented by genetic assimilation (Hinton & Nowlan, 1987). It is also known that behavioural #exibility can play a crucial role in evolution and speciation (Wyles et al., 1983; Baker and Jenkins, 1987; Laland, 1994; Whitehead, 1998). However, we do not think that our results would be directly applicable in the context of behavioural inheritance. First, it is often argued that animals acquire bene"cial traits by learning. In this paper, we avoided discussing the possibility of directed phenotypic variation*the possibility of &&hard'' Lamarckism. Second, we only investigated the vertical*parent}o!spring* transmission of phenotypic information, which is naturally true in the case of cellular inheritance. On the contrary, in the case of behavioural inheritance animals may acquire information from (i) their parents (strict vertical transmission), (ii) relatives or (iii) anyone else in the population with given probabilities (strict horizontal transmission). In the third case, naive animals copy the behaviour of individuals with di!erent genetic backgrounds, so the linked transmission of genetic and phenotypic information is not ful"lled. We suspect that the horizontal transmission of phenotypic information greatly alters the dynamics of evolution and the possibility of genetic assimilation (PaH l, unpublished data). 33 12. Discussion The dual inheritance of a character separates the e!ects of selection on genotypes and phenotypes. Therefore, adaptation can be viewed as a hierarchical process. At "rst, new epigenetic variants of a given genotype appear in the population, that are later followed by the "xation of DNA mutations. The bene"t of phenotypic selection depends on the possibility of "nding much better phenotypes than the genetically hard-wired ones. Accordingly, genetic adaptation limits the e$ciency of the &&exploratory behaviour'' of epigenetic inheritance systems. This paper showed that an EIS can have a temporary e!ect on adaptation and divergence, but it is selectively eliminated at the global optimum. Thus, the question remains: How can an EIS be permanently maintained? Are these side e!ects of structural constraints, other selective forces, or do they have an important role in evolution? Previously, it has been argued that random (Jablonka et al., 1995), or periodically #uctuating (Lachmann & Jablonka, 1996) environment favours the spread of an EIS. Our framework can also consider this e!ect. We have to incorporate that both the inducing and the selective conditions may change in time and space. In this paper, we investigated the behaviour of an epigenetic inheritance system, if there are multiple "t states for the individuals. We arrived at the conclusion that evolution can proceed through suboptimal phenotypic states, without passing through a deep adaptive valley of the genotype. We suggest that epigenetic changes might contribute to the evolutionary divergence of populations. This "nding may alter our view on the dynamics and mode of reproductive isolation. In the following, we summarize our speciation scenario: (1) The preliminary condition to epigenetically induced divergence is the presence of the genes controlling non-zero epigenetic variability. The initial increased frequency of an epigenetic modi"er may be driven by slight changes in the inducing environment or by genetic drift. (2) New epigenetic variants appear in the population. Some of them can be selectively favourable and can accumulate in the population. This enables the genotypes to occupy a new 34 C. PAD L AND I. MIKLOD S adaptive zone without abandoning the old one. The extent of epigenetic divergence crucially depends on the transmission e$ciency of phenotypic information. Low e$ciency prevents the spread of these variants. (3) Selection forces the favourable epigenetic variants to be conserved by genetic changes. As a result, the favourable variants are transmitted to the o!spring in a more reliable way. Genetic assimilation of phenotypic variants leads to the genetic divergence of the populations. It is possible that this process*epigenetic changes followed by genetic "xation*is reiterated as long as new adaptive peaks are available. (4) As the derived population is genetically close to an optimal state, and no higher adaptive peaks are available, selection acts to suppress epigenetic variability. Canalization of phenotypic expression strengthens the reproductive barrier, because no "t epigenetic states are available for the hybrids. In one of his papers, Wright (1932) said that there must be some trial and error mechanism by which the species may explore a small portion of the "eld (genotypic space) which it occupies. He thought that this mechanism works at the level of subpopulation (demes). We rather think that epigenetic inheritance systems enable this localized search at the phenotypic level, guiding the genetical system to new optima. Epigenetic variability opens new dimensions on the adaptive landscape, where the population can evolve. Therefore, our work is in a good accordance with Fisher's view. Fisher (1930) rejected the existence of isolated peaks, arguing that as the number of traits considered increases, the probability that a given equilibrium would be stable to perturbations decreases. He thought that our imagination is limited by viewing the population at equilibrium, and by neglecting the complicated "tness surface that channels evolution. Dobzhansky's (1937) model also laid special emphasis on the &&ridges'' of quasi well-"t genotypes, connecting reproductively isolated genotypes. The existence of an adaptive valley separating current genotypes is not an evidence that this valley has been crossed during divergence. This idea has become a point of renewed interest (for a review see Gavrilets, 1997). Based on a theoretical framework, Gavrilets & Gravner (1997) stated that ridges must be a general property of multidimensional adaptive landscapes. According to this view, populations become reproductively isolated when they are on the opposite sides of a &&hole'' on the landscape (Gavrilets, 1997). This strict geometrical view of evolution gives ample scope to any special genetic trait (dimensions) that could routinely facilitate divergence. In this work, we argued that heritable non-DNA changes open new routes for evolution. 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A function = de"ned on some interval I is said to be strictly convex if = p y ( p =(y ), G G G G (A.1) where y , y , 2 , y 3l, the p 's are positive num L G bers and p "1. We apply eqn (A.1) to show G that increasing phenotypic variation is advantageous if the adaptive landscape is convex, and disadvantageous if it is concave. The p 's may G de"ne the frequency by which an y phenotype is G induced, and = may give the "tness function of the phenotypes. In the main text we approximated the distribution of phenotypes by Gaussian. Because of the symmetry of this distribution around the mean (x), any x!e , x#e G G phenotype is generated with the same p freG quency. Therefore, in this special case, p y "x. G G Accordingly, inequality (A.1) can be rearranged to give =(x)( p =( y ) G G (A.2) We can conclude that it is bene"cial to generate y phenotypes, instead of the genetically deterG mined x, if the arithmetic mean "tness of the induced phenotypes is higher than the "tness of x. Inequality (A.2) is satis"ed if and only if the "tness function is convex on the given interval. APPENDIX B The "tness of the genotype with an epigenetic inheritance system is =(x, P, m)" u (y) =(y)K , (B.1) 37 DIVERGENCE OF DUAL INHERITANCE SYSTEMS where u ( y) gives the initial phenotypic distribution of the genotypes, and =(y)"exp !(y!y ) MNR 2< Q #H exp !(y!y ) MNR 2< Q ;exp (B.2) gives the "tness of individuals with y phenotype. If m is an integer, then we can use the binomial expansion K m AK\I BI (A#B)K" k I in eqn (B.1) to give u =(x, P, m)" K m (y) k I ;HI exp !(m!k) (y!y ) MNR 2< Q K !k(y!y ) MNR dy . 2< Q (B.3) Changing the order of summation and integration in eqn (B.3) leads to K m HI u (y) =(x, P, m)" k I ;exp !(m!k)(y!y ) MNR 2< Q ;exp K !k(y!y ) MNR dy . (B.4) 2< Q Evaluating the integration in eqn. (B.4) and after some rearrangements we get eqn (10).