Various definitions of biological complexity have been proposed: the number of genes, cell types, or metabolic processes within an organism. As knowledge of biological systems has increased, it has become apparent that these metrics are often incongruent.Here we propose an alternative complexity metric based on the number of genetically uncorrelated phenotypic traits contributing to an organism's fitness. This metric, phenotypic complexity, is more objective than previous suggestions, as complexity is measured from a fundamental biological perspective, that of natural selection. We utilize a model linking the equilibrium fitness (drift load) of a population to phenotypic complexity. We then use results from viral evolution experiments to compare the phenotypic complexities of two viruses, the bacteriophage X174 and vesicular stomatitis virus, and to illustrate the consistency of our approach and its applicability.Because Darwinian evolution through natural selection is the fundamental element unifying all biological organisms, we propose that our metric of complexity is potentially a more relevant metric than others, based on the count of artificially defined set of objects. A persistent question in biology is how organismal complexity changes through the course of evolution Phenotypic complexity quantifies the number of genetically uncorrelated phenotypic traits contributing to an organism's fitness. A phenotypic trait contributes to an organism's fitness only to the extent that natural selection acts upon that trait. Thus an organismal phenotype that is no longer under selection , although expressed by the organism, contributes nothing to organismal complexity. Secondly, if two phenotypes contribute to complexity, they must be genetically separable: some mutations must exist that affect one phenotype but not the other. If no such mutations exist, then although we may perceive two phenotypes under selection, these phenotypes contribute only a single trait toward determining phenotypic complexity. As an example consider the affinity of an enzyme for a substrate, and the rate at which that substrate is converted to product. If there are no mutations that affect one of these traits but not the other, then these two phenotypes are considered one, until the organism gains the genetic complexity to generate variation in one phenotype without affecting the second, for example by evolving functionally separate domains in the enzyme. Phenotypic complexity is thus a combined description of how natural selection perceives organismal phenotypes and how phenotypic variation is generated by the organism. This concept was first articulated by Orr, and followed later by others An important aspect of measuring complexity in this manner is that both the organism and the environment affect the metric. An organism with many phenotypes, but living in simple environment could thus be just as complex as a simpler organism in the same environment. For example, if one organism is capable of metabolizing both lactose and glucose, while second can metabolize only glucose, the first organism will only be designated as more complex when there is a possibility that lactose will be present in the environment.Recent population genetic theory The phenotypic model used to link drift load to phenotypic complexity was first formalized by R. A. Fisher A population of individuals can be represented as a collection of points in FGM and the phenotypic values of each point allow ascertainment of the fitness of each individual. Individual fitness then determines the probability of each individual surviving and reproducing the next generation. Evolution is thus described in FGM by following the collection of points over many generations. To generate novel genetic variation, mutations are drawn from an assumed distribution that is centered on the phenotypic position of each individual . By neveFGM makes a set of theoretical predictions about how adaptation tends to occur, and many of these have been corroborated by experimental results. The greater frequency of small-sized beneficial mutations The utility of FGM lies in the fact that it does not require any particular assumptions about the map between phenotype and genotype, and that the specific predictions about how fitness changes during evolution appear to be robust. In the present paper, we further refine previous predictions derived from FGM The link between drift load and phenotypic complexity under FGM was first investigated by Hartl and Taubes e−1 in the diploid case and 2⋅Ne−2 in the haploid, and ρ(f), the density function, i.e. the size of fitness f isocline, is dependent on the number of dimensions and of the fitness function used.In which ν = 2⋅Neq is the equilibrium fitness (drift load) expressed as a fraction of the maximum attainable fitness of the organism, ne is the effective number of dimensions of the phenotypic space , and Ne is the effective population size. This confirms the results obtained by Poon and Otto who approximated Feq as 2Ne/(2Ne+ne) If fitness is assumed to be a linearly decreasing function of the phenotypic distance to the optimum, then we find that the average fitness is given by :in the dAlthough earlier studies on FGM model have used such linear fitness functions , recent experimental studies do not seem to support the use of such a function Q)), in which fitness is an exponentially decaying function of the distance to the optimum to the power of Q. Q is a parameter that modifies the concavity of the fitness decline. As organisms move away from the optimum the effect of the mutation tend to have bigger effect if Q>1 and smaller effect if Q<1. In such a case the fitness equilibriums are  = exp (ixiQ), where X =  is the coordinate of an individual in FGM, and αi are positive parameters  suggests, the fitness equilibrium is independent of the mutational properties. As long as mutation is assumed to be isotropic, only the convergence time to equilibrium, and not the equilibrium fitness value, is affected by the distribution of mutational effects (data not shown). Second, although the results depend on the shape of the fitness function , they are independent of the slope: equilibrium values will be the same if f(d) = exp(−αdQ) . Third, rameters . FinallyThe equilibrium drift load seems to be a robust property of FGM that is determined by the number of dimensions of phenotypic space, the population size and the fitness function . An accurate estimate of phenotypic complexity can thus be obtained if it is possible to estimate equilibrium fitness values (drift load) for several population sizes, as well as the amount of curvature in the fitness function.Escherichia coli C. We increased the mutation rate of the phage to hasten convergence to fitness equilibrium with the use of hydroxylamine; this resulted in a mean mutation rate of 0.1 per genome per generation through the 450 generations of evolution. Population size-dependent fitness equilibriums were observed, confirming that evolution occurred in a manner compatible with the assumptions of FGM. From these populations we obtained 21 measures of equilibrium fitness at five different population sizes. Every transfer corresponded roughly to five phage generations and effective population size was then approximated to five times the number of plaques transferred (the harmonic mean).We used two sets of evolution experiments in which both fitness equilibrium values and fitness curvature have been investigated . In the To estimate the curvature of the fitness function we performed a mutation accumulation analysis for high and low fitness clones and showed that the distribution of deleterious mutations was similar at both ends of the fitness range spanning a 300-fold difference. This suggests that there is very little curvature of the fitness function.We used a second set of data from the literature, in which populations were evolved for 20 transfers at different effective sizes. Novella et al. Additionally, an impressive set of data using site directed mutagenesis in VSV suggests that the concavity of fitness function is slightly upward e/Q, fref), in which ne is the phenotypic complexity, Q a parameter of the curvature of the fitness surface, and fref the maximum attainable fitness. We also take into account the noise in our experimental assessment of fitness values; especially for high fitness populations, noise in the estimates of fitness can alter the estimation of fref, as this parameter is by definition higher than all fitness measures. Thus rather than using the probability of the point estimate of fitness, we integrated the probability between plus (f+) and minus (f−) one standard deviation of the point estimate. In Appendix C we show that We wish to use the previous mathematical results to estimate phenotypic complexity from experimental data. However, there are two unknown parameters in the experimental system that affect the equilibrium drift load in a population: phenotypic complexity and the maximum attainable fitness that can be reached by the viruses in the laboratory environment . Using methods from statistical physics, we can find the distribution of population fitness at equilibrium (shown above), and thus derive a likelihood model that gives the probability of the observed data for each couplet (ne/Q = 45 (42−49), and fref = 1.245 (1.23−1.26), whereas for VSV we found ne/Q = 10 (8−12) and fref = 1.98 (1.94−2.05) , which we list here in parentheses. For ΦX174 we found n94−2.05) Fig. 3. e). We have quantified this metric in the viruses ΦX174 and VSV by utilizing a population genetic model that describes how phenotypic complexity affects the drift load that a population experiences.To understand how biological complexity changes during the course of evolution, a metric is needed. Previously, measures such as the number of genes, cell types or metabolic processes have been proposed, but they often lead to incongruent results. Organisms with more cell types do not necessarily have more genes. Here we suggest that a metric unifying biological systems has not been appropriately identified. To circumvent this problem, we have developed a metric of biological complexity termed phenotypic complexity (ne) is a measure of the number of genetically uncorrelated phenotypes that are acted upon by natural selection. Because Darwinian evolution through natural selection is the fundamental element unifying all biological organisms, we propose that ne is potentially a more relevant metric than those previously suggested.Phenotypic complexity  on the effective population size and phenotypic complexity. The linear fitness function that has been employed previously to simplify the mathematical analyses is no longer necessary. Such a function makes strong assumptions about the form of the fitness landscape; specifically, mutational effects become very large as fitness is reduced. Thus at low fitness most mutations are either lethal or of very large effect, a scenario which is incompatible with what we have previously observed Q)), and found that Feq = (1−(2⋅Ne−1)−1)(ne/Q). It appears that this equation remains valid over a much wider range of conditions than those used in the canonical FGM, in which mutations are required to be isotropic and fitness isoclines are symmetric about the origin. An interesting feature of this formula is that it does not require a model in which mutations can affect all phenotypic traits simultaneously. In the initial formulation of FGM, all phenotypic axes intersect at the origin of each axis. This original FGM can be modified slightly such that some phenotypes are grouped into separate phenotypic modules, and within a module, all phenotypes again intersect at each other's origin. Any mutation that occurs within a module can affect only other phenotypes within that module, and none that lie outside of it  = exp predicted a 0.002 mean fitness effect per mutation, while more recent and accurate estimates  brought the estimate it to 14% (a 70-fold difference). Secondly, neutral mutations are not considered in the model developed by Martin and Lenormand. However, part of the mutation produced by transposable elements might affect genes useless in laboratory conditions and will therefore be taken into account in the calculation of mean fitness effects of mutations even if they do not affect any phenotype in the laboratory environment. Third, contrary to our model, if phenotypes are organized in different modules (as many data suggest), their model will provide different estimates; in other words, their model requires that all traits can be simultaneously affected by a single mutations. All these considerations suggest that a second independent model should be used to estimate phenotypic complexity.Recently, another theoretical study developed a framework to estimate phenotypic complexity One of the central FGM hypotheses that we have so far not addressed is the single-peaked nature of the landscape. Although FGM contains few assumptions about the nature of the genotypic landscape, the model explicitly requires a phenotypic landscape containing a single peak; without this, then the fitness function, f(d), cannot be described by a decreasing function. However, recent experimental evidence over large evolutionary time scales strongly suggests that while the genotypic landscape may contain multiple peaks, the phenotypic landscape is generally much less complex. Several experimental studies using microbes have shown that a considerable amount of phenotypic convergence occurs during evolution e is the number of genetically uncorrelated phenotypes that are influenced by the action of natural selection. The dimensions enumerated by ne are thus genetically orthogonal to each other, and analogous to the axes needed to describe the variation among multiple phenotypes measured on a collection of individuals and mutants in a principal component analysis. However, the number of axes enumerated by ne is filtered by natural selection, while in a PCA analysis the number of axes is limited only by the number of independent phenotypes that are measured. Because each phenotype is optimized at a value determined by each organism's ecological environment, there is a dependence of phenotypic complexity on the complexity of the ecological niche experienced by each organism; if natural selection does not act on a phenotype, then that phenotype does not contribute to the complexity metric. Finally, although the estimates of ne arise from an idealized model of phenotypic evolution; as Orr suggested previously, estimates of phenotypic complexity using FGM can be viewed as “effective” estimates of phenotypic complexity e, in which two populations with different numbers of individuals and different sex ratios might have the same effective population size and therefore respond similarly to the different population genetic forces. Thus two organisms, although they may differ in both the underlying genetic mechanisms and in the complexity of the environment in which they live, may have similar phenotypic complexities. The utility of the concept lies not in the implications it makes about specific phenotypes or genetic details, but in that it enables a general quantification of how an organism is affected by natural selection , and furthermore, how this action affects the evolutionary dynamics of that organism As discussed previously, the quantity denoted by ni.e. they are genetically separable phenotypes), it is clearly unlikely that each nucleotide or amino acid is independently acted up by natural selection. Secondly, each estimate of complexity is greater than the number of genes encoded by each virus . The presence of multiple functional domains within a single protein is consistent with an estimate of complexity that is greater than the number of encoded proteins.Unsurprisingly, our estimates of phenotypic complexity are orders of magnitude smaller than either the number of nucleotides or even the number of amino acids encoded by the genomes of these organisms . This agrees with the concept of phenotypic complexity that we have defined. Although mutations that occur at one nucleotide or amino acid do not affect those at another , and a quantitative level (that the number of phenotypic dimensions are reasonable) that the predictions from FGM theory are borne out. An assessment of drift load and phenotypic complexity in a greater number of organisms is needed before further conclusions can be drawn.Although the theory presented here appears to be quite robust, it is too early to conclude that it is an accurate reflection of the underlying biology. To be studied in an FGM framework, organisms need to present at least one phenotypic property to selection. Additionally, populations, even those of very small size, should evolve towards a fitness equilibrium that is explicitly dependent on population size. We found data in the literature consistent with this expectation for one organism, VSV. We now provide further support for population size-dependent fitness equilibria by evolving populations of the bacteriophage ΦX174. Together, these two data sets suggest that evolutionary analyses using an FGM framework are a valid approach. Moreover, the use of very simple organisms such as viruses is useful for gaining insight into metrics of complexity, as for such simple organisms, gene number is likely to be a very good correlate of organismal complexity, and this should be reflected by the metric. Although our observations are currently limited to two viral species, it is clear that from both a qualitative level . Two important conceptual differences separate these two approaches. Most importantly, phenotypic complexity is dependent on both the organism and the environmental context. An organism is not complex because it has many measurable phenotypes; it is complex because it has many phenotypes on which natural selection acts. Secondly, phenotypic complexity does not rely on artificially constructed concepts such as genes However, phenotypic complexity remains an inherently abstract metric. It cannot aid in identifying the specific characteristics contributing to the complexity of an organism. Instead, it addresses the complexity with which natural selection views an organism, and the complexity with which an organism is capable of generating novel phenotypic variation. For this reason, testing how phenotypic complexity compares to more traditional metrics of complexity  may provide significant insight into biological systems. Finally, phenotypic complexity (and the resulting equilibrium drift load) affords a unique opportunity to contrast the action of natural selection between different organisms or different environments in a very general and unconstrained manner.The details of experimental evolution of ΦX174 have been described previously e−1 in the diploid case and 2Ne−2 in the haploid, and ρ(f) is the density function of fitness value f.Sella and Hirsh f) df is derived from the hyper-sphere surface of radius x, with f = f(x), f(x) being the fitness function describing the dependency of fitness on the distance to the optimum, x. The surface of the hyper-sphere of radius x is Ω(n)xn−1) is the unit radius hyper-sphere surface, Ω(n) = 2⋅πn/2)(/Γ(n/2). We have therefore ρ(f)df = Ω(n) xn−1) = 1−x, then Ω(n) xn−1)( dx = −Ω(n) (1−f)n−1 df andIf fitness is defined as f(x) = exp(−xQ) we have If fitness is defined as: f = exp(−α xQ)Note that if f(RQ) where R is defined by x =  is the position in the n-dimensional space and αi are positive numbers. We then have ellipsoidal fitness isoclines of semi-axes R/αi. As the volume of such an ellipsoid is f) df = dV(f) df is similar to the one found in the previous case Let us assume that f = exp = exp(−xQ), we find the probability that f lies between a and b isUsing the previous derivations with f(ref, we must estimate it and therefore fitness b and a be used relative to fref.Because we do not know maximum fitness f


Mutational (genetic) robustness is phenotypic constancy in the face of mutational changes to the genome. Robustness is critical to the understanding of evolution because phenotypically expressed genetic variation is the fuel of natural selection. Nonetheless, the evidence for adaptive evolution of mutational robustness in biological populations is controversial. Robustness should be selectively favored when mutation rates are high, a common feature of RNA viruses. However, selection for robustness may be relaxed under virus co-infection because complementation between virus genotypes can buffer mutational effects. We therefore hypothesized that selection for genetic robustness in viruses will be weakened with increasing frequency of co-infection. To test this idea, we used populations of RNA phage φ6 that were experimentally evolved at low and high levels of co-infection and subjected lineages of these viruses to mutation accumulation through population bottlenecking. The data demonstrate that viruses evolved under high co-infection show relatively greater mean magnitude and variance in the fitness changes generated by addition of random mutations, confirming our hypothesis that they experience weakened selection for robustness. Our study further suggests that co-infection of host cells may be advantageous to RNA viruses only in the short term. In addition, we observed higher mutation frequencies in the more robust viruses, indicating that evolution of robustness might foster less-accurate genome replication in RNA viruses. RNA phage viruses evolved under high co-infection of host cells are less robust to mutations than those propagated under low co-infection, suggesting that co-infection may be advantageous to RNA viruses only in the short term. Mutational (genetic) robustness can be defined as constancy of phenotype in the face of mutational perturbation . GeneticHigh mutation rate is perhaps the most important prerequisite for adaptive genetic robustness , so mutaPseudomonas syringae pathovar (pv) phaseolicola bacteria for hundreds of virus generations. Three of the populations were evolved at low multiplicity of infection , and three at high MOI. Co-infection level was controlled by mixing viruses and bacteria in liquid medium at a given MOI, allowing sufficient time for virus attachment to cells, and then plating a dilution of the mixture onto agar with a superabundance of cells. During overnight incubation, viruses formed distinct plaques in the lawn, which resulted from lysis of infected cells and the release of viral progeny that infected neighboring cells. The passage cycle was repeated by harvesting plaques, removing the bacteria by filtration, and mixing viruses and naive bacteria at the controlled MOI. A total of 60 passage cycles were conducted, which equaled roughly 300 viral generations  = e−MOI × MOI. Therefore, at MOI = 0.002, only approximately 0.1% of all infected (n > 0) cells are co-infected and clonal infections predominate. In contrast, approximately 97% of infected cells should be multiply infected at MOI = 5, and co-infection predominates, generally with two to three viruses . Every ormation , the proon in φ6 ) infection in φ6 .) Thus, on in φ6 .Here we examined whether evolution of mutational robustness occurs differently for viruses evolved at low and high levels of co-infection. We tested the hypothesis by randomly isolating clones from each of the six previously evolved populations, and using these clones to found lineages that were subjected to a mutation accumulation experiment –19. MutaP. phaseolicola for 20 d (see n = 3) fitness assays against a common virus competitor for the focal genotypes  of each lineage. In this way, we were able to measure the mean change in log10 fitness (Δlog10W) for each lineage as a result of mutation accumulation; Δlog10W = log10Wpost-bottleneck − log10Wpre-bottleneck. This design resulted in 360 total fitness estimates , in which focal genotypes of a lineage were always assayed within the same temporal block.We isolated ten clones at random from each replicate population in the low co-infection and high co-infection level treatments at generation 300, and used these to found 60 independent virus lineages (ten clones × three populations × two co-infection treatments). We then conducted a mutation accumulation experiment ,19, in w20 d see . Plaque 10W values estimate the fitness effect of adding roughly one random non-lethal mutation to the founding genotype of a lineage. Mutational robustness is defined as decreased phenotypic variability in the face of mutational change. Thus, one set of genotypes can be considered more robust than another collection of genotypes if the first group has a significantly lower variance in the fitness change brought on by addition of mutation(s) to the genome  = 0.00, p = 0.9940; P. tomato: F = 0.17, p = 0.7175). This result is intriguing because the robust viruses seem to feature higher mutation frequencies on average. In turn, the data suggest that evolution of mutational robustness (whatever the underlying molecular mechanism) allows RNA viruses to tolerate less-accurate genome replication, perhaps explaining why these viruses remain highly mutable. Unraveling the exact molecular mechanism(s) for these results would be of great interest. Furthermore, our study implies a need for population genetic models that consider the impact of adaptive robustness on the evolution of replication fidelity, theory that has not been previously explored.One possibility is that the rare occurrence of complementation at low co-infection caused these viruses to adapt by evolving greater robustness than the wild-type φ6 ancestor; but this is unknown. If so, our data suggest it is unlikely that this greater robustness was accompanied by more accurate RNA replication; ANOVA shows mutation frequencies of the low co-infection genotypes do not differ from the ancestor on either host . Most important, we found that mutation frequency in these assays was significantly lower in the high co-infection viruses , which again argues that our robustness conclusions are not confounded by elevated mutation rates in these viruses. Overall, our data suggest that mutation rate governed by the accuracy of the viral replicase is a trait under selection, especially in the viruses evolved at high co-infection. It is unclear why viruses that frequently experience complementation would evolve higher fidelity of replication, and this may be a pleiotropic effect of selection occurring elsewhere in the genome. The possible adaptive significance of this result is unclear and merits future exploration.For the wild-type φ6 ancestor and evolved genotypes in our study, mutation frequencies were lower than expected given the ease with which host-range–marked mutants of φ6 were obtained in previous studies. That is, host-range mutants occurred at frequencies ranging between 10aligenes . To explosts see . The datosts see  supporteF = 0.05, p = 0.84). Thus, we rejected the possibility that diminishing-returns epistasis, or any argument that hinged on differing fitness among the starting clones, provided an alternative explanation for our results.We considered the potential relevance of two other possible confounding factors in our study. First, we examined whether the two groups of viruses differed in fitness prior to mutation accumulation; the alternative explanation is that the low co-infection viruses were already of low fitness (in comparison to the high co-infection strains), and that fixation of one or more additional mutations via bottlenecking did not lead to a further reduction in their fitness. That is, lower fitness genotypes of φ6 are shown to be less affected by addition of further deleterious mutations, a result demonstrating that diminishing-returns epistasis can operate in phage φ6 . This phWe also considered that erroneous conclusions may be drawn from mutation accumulation experiments if conditions impose a change in selective environment for only a subset of test lineages. A similarity between the mutation accumulation experiment and the low MOI treatment is that viruses must infect cells alone, whereas in every fifth generation of the high MOI treatment viruses are forced to undergo co-infection. Thus, it might be argued that the high co-infection lineages (but not low co-infection lines) experienced a change in growth conditions, and that the magnitude and variability in their performance in the bottlenecking experiment results from adaptation to the new conditions. However, we believe that this could not have been a confounding factor in our study for several reasons.First, the pre-bottleneck clones drawn from both MOI treatments performed equally well prior to mutation accumulation (see above), as measured using fitness assays conducted at low MOI. These data may seem surprising, given that at generation 200 the treatment populations showed significantly different fitness under low-MOI conditions . HoweverSecond, the mutation accumulation experimental conditions can be considered novel for all test lineages. In particular, the experimental evolution necessitated that viruses attached to host cells in liquid medium prior to overnight growth on agar plates, whereas the mutation accumulation habitat did not. Even if phage φ6 attachment to cells in liquid versus on agar surfaces is fundamentally the same (this is not well described), it is plausible that these two habitats differ substantially in the dispersal of phages between infected cells .8 virus particles within a plaque. However, the mutation accumulation design prevents virus adaptation because the action of genetic drift overwhelms that of natural selection; i.e., lineages evolve at an effective population size of N ≍ 2, the harmonic mean of the serial passage fluctuating between 1 and 108 viruses.Third, it is highly unlikely that the test lineages could have adapted to the mutation accumulation conditions at all. Plaque formation on agar necessarily involves expansion of the bottlenecked population to large size, which allows for some positive selection. That is, a plaque forms from a single virus experiencing five generations of growth on the plate, where the average number of progeny made by an infecting virus is 100 particles, yielding ~10Our study suggests several intriguing possibilities for future research in φ6 and other RNA viruses. Identifying whether the low co-infection populations, the high co-infection populations, or both have changed in robustness relative to their common ancestor could help shed light on the genetic mechanism(s) underlying this difference in the φ6-derived viruses.  The 300-generation experiment  that preP. phaseolicola was purchased from American Type Culture Collection . L. Mindich  kindly provided host strain P. pseudoalcaligenes ERA (East River isolate A); G. Martin  kindly providedP. syringae pv.tomato and P. syringae pv.atrofaciens. Bacterial stocks were stored in 4:6 glycerol/LC (v/v) at −80 °C. Viruses were grown on lawns made from overnight bacterial cultures. Agar concentrations in plates were 1.5% and 0.7% for bottom and top LC agar, respectively. Plates contained 3 ml of top agar and a 200 μl bacterial lawn. Phage lysates were prepared by growing viruses on a P. phaseolicola lawn for 24 h; plaques were then collected and filtered  to remove bacteria. Phage lysates were stored at −20 °C in 4:6 glycerol/LC (v/v).Phages and bacteria were cultured at 25 °C in LC medium, Luria-Bertani broth (pH 7.5) . OvernigP. phaseolicola cells in a test tube containing liquid LC medium, incubated for 40 min at 25 °C. These infected cells were then diluted and plated on a P. phaseolicola lawn on which viruses form distinct plaques. Population size of N ≍ 500 was held constant across treatment populations by controlling the number of harvested plaques [N ≍ 500 was controlled by harvesting 500 plaques from a low MOI population and 100 plaques from a high MOI population. After the experimental evolution, ten clones were chosen at random from each population. Each of the 60 clones was then used to found a single lineage subjected to 20 d of mutation accumulation via bottlenecking [P. phaseolicola lawn, placing it in sterile liquid medium, and vortexing gently to disperse the progeny viruses contained within the plaque. A sterile platinum loop was then used to streak the viruses onto agar containing a naive host lawn. After 24-h incubation, the viruses formed distinct plaques, and the process was repeated for 20 consecutive days. Because each plaque grows from a single virus, daily passage caused a test lineage to be forced through an extreme bottleneck , in which the intense drift allows fixation of non-lethal mutations, the majority of which are presumed to be deleterious. Because the phage expand to approximately 108 phage in five generations within a plaque, this creates the opportunity for selection to operate within a population propagated by such one-plaque transfers. Thus, positive selection on the plate should bias against fixation of highly deleterious mutations, and it is possible to observe the fixation of a rare beneficial mutation in a bottlenecked lineage [P. phaseolicola lawn containing 8 × 108 cells; MOI on the plate of 5 × 10−7).Virus populations were experimentally evolved at MOI of 0.002 or 5 for 250 generations , followe plaques . On averenecking ,19. Bott lineage . However lineage –19. AlsoP. phaseolicola of a focal genotype relative to a common competitor of φ6 bearing a genetic marker  and in the harvested lysate (R1) were obtained by plating on mixed lawns of P. phaseolicola and P. pseudoalcaligenes (200:1 mixture) on which ordinary and host-range genotypes form turbid and clear plaques, respectively. Fitness (W) was defined as the relative change in ratio of ordinary to host-range virus, or W =R1/R0.Fitness assays consisted of paired-growth experiments , which cbacteria ). The coP. tomato or P. atrofaciens. A high-titer lysate  of a virus genotype was grown and titered on P. phaseolicola, and a sample of the lysate was then screened on a selective plate. Mutant frequency was calculated as the number of plaque-forming mutants per viruses in the inoculum; when no mutants were found, we used the limit of detection as a conservative estimate. Acquisition of a second host-range mutation was examined by rearing phage mutants on P. atrofaciens, followed by selective plating on P. tomato.We measured the appearance of host-range mutants formed on selective plates: bacterial lawns of A mixed general linear model controlling for experimental assay day was run , testingTable S1(91 KB DOC).Click here for additional data file.Table S2(26 KB DOC).Click here for additional data file.


Can a single unifying mathematical framework help to explain robustness - the ability of organisms to persist in the face of changing conditions - at all biological scales, from biochemical to ecological? One of the most important features of biology is the ability of organisms to persist in the face of changing conditions. Consider the remarkable fact that every organism alive today is the product of billions of generations in which its progenitors, without fail, managed to produce progeny that survived to reproduce. To achieve this consistency, organisms must have a balance between robustness and evolvability, that is, between resisting and allowing change in their own internal states . MoreoveExamples of robust biological systems are found at many scales, from biochemical to ecological. At each scale, robustness may reflect the properties of individual elements or, alternatively, the dynamic feedbacks between interacting elements. The expression of some metabolic function, for example, may be robust in the face of temperature change, because an enzyme maintains its shape and specificity across a range of temperatures or because an interconnected network of reactions sustains the supply of product, even when some enzyme fails. A genome may be robust because it encodes proofreading and repair systems that reduce replication errors or because it is organized such that many mutations have little effect on its phenotype. An ecosystem might be robust if it resists the extinction of some keystone species or, if extinction does occur, because surviving species can compensate over physiological, demographic, or evolutionary time scales.One important question is whether there exists a single unifying mathematical framework that can encompass such diverse examples of biological robustness. Might new insights come from such a conceptual unification, or will future understanding require detailed analyses of specific cases? Across the different scales, recurring mechanisms for achieving robustness—including redundancy of component parts and negative feedbacks—might serve as organizing principles. Yet, similarities in mechanism could mask important differences in the evolutionary origins of those mechanisms. At the level of genes in genomes or of cells in multicellular organisms, it is reasonable to suggest that redundancy evolved by natural selection to maintain some functional capacity in the face of perturbation . But wheAnd if robustness has evolved to maintain performance, what prevents systems from becoming ever more robust? We will focus on genomic robustness to mutations, because it provides a concrete example, although many ideas are speculative and much work is needed to formalize and test them. Two mechanisms that could make a genome more robust are genetic redundancy, so that many otherwise deleterious mutations are masked, and proofreading during replication, so that fewer mutations occur. Redundancy imposes a cost of replicating the additional gene copies , whereasMutational robustness can also arise in more subtle ways. Populations evolving at high mutation rates may settle in regions of genotypic space where mutations are less deleterious, on average, than those regions that attract populations that experience low mutation rates. The idea is that evolution at low mutation rates favors populations that achieve high fitness peaks, even if they are surrounded by steep cliffs, because mutations that push progeny off those cliffs are rare. By contrast, at high mutation rates, most offspring carry mutations, and selection favors populations that find lower fitness peaks surrounded by less precipitous mutational chasms. Experiments with digital organisms (self-replicating computer programs) provide direct support for “survival of the flattest” at high mutation rates . RNA virBut generalizing to other organisms presents some difficulties. The strength of selection for robustness should be weaker in larger genomes if the advantage to a mutation that increases robustness locally is correspondingly smaller. According to one alternate hypothesis, mutational robustness is not so much a directly evolved property as it is a correlated benefit of selection for robustness in the face of variable environments . The essTwo recent studies with evolving computational systems have shown, unexpectedly, that sexual reproduction promotes the evolution of mutational robustness ,14. The Another important issue revolves around the tension between robustness and evolvability. Are genomes that are more robust to mutations less evolvable in the face of changing environments? In other words, does canalizing the phenotype to minimize perturbations—including biochemical and environmental as well as mutational—lead to an evolutionary conservatism that inhibits the discovery of new adaptive solutions? Some mechanisms of robustness, such as proofreading and repair, must inhibit evolvability because they reduce the production of new beneficial mutations. But are robustness and evolvability inversely correlated more generally? In the case of redundancy, the presence of multiple gene copies might mask the beneficial effects of some new mutations, thus suppressing evolvability. But redundancy can also promote adaptation by allowing duplicated genes to evolve distinct functions ,17.Evolving populations can also become robust by finding regions of genotypic space that are flat because they contain a high proportion of neutral mutations . As showTheoretical population genetics has historically emphasized models with one or two loci, whereas quantitative genetics has relied on a sort of statistical mechanics that ignores underlying detail. Richer mathematical representations of genotypic spaces and fitness landscapes may be required to understand the balance between robustness and evolvability. Meanwhile, empiricists must push ahead to obtain data about evolvability and robustness. Experimental evolution, in which populations are monitored while they evolve under defined conditions, offers the potential to observe changes in these properties as a function of environmental and genetic manipulations . For exa


After 20 cycles of random substitution at sites 12–130 of the initial random polypeptide and selection for infectivity, the selected phage showed a 1.7×104-fold increase in infectivity, defined as the number of infected cells per ml of phage suspension. Fitness was defined as the logarithm of infectivity, and we analyzed (1) the dependence of stationary fitness on library size, which increased gradually, and (2) the time course of changes in fitness in transitional phases, based on an original theory regarding the evolutionary dynamics in Kauffman's n-k fitness landscape model. In the landscape model, single mutations at single sites among n sites affect the contribution of k other sites to fitness. Based on the results of these analyses, k was estimated to be 18–24. According to the estimated parameters, the landscape was plotted as a smooth surface up to a relative fitness of 0.4 of the global peak, whereas the landscape had a highly rugged surface with many local peaks above this relative fitness value. Based on the landscapes of these two different surfaces, it appears possible for adaptive walks with only random substitutions to climb with relative ease up to the middle region of the fitness landscape from any primordial or random sequence, whereas an enormous range of sequence diversity is required to climb further up the rugged surface above the middle region.The fitness landscape in sequence space determines the process of biomolecular evolution. To plot the fitness landscape of protein function, we carried out  In vitro molecular evolution can be considered an adaptive walk on a fitness landscape in sequence space, where “fitness” is a quantitative measure of a certain physicochemical property of a biopolymer, such as thermostability or enzymatic activity n-k landscape model, in which substitutions occurring on one of n sites affect the contribution of residues at k other sites to fitness, was proposed as a model of the fitness landscape k and the difference in altitude between the global peak and the foot, defined as the region in the sequence space where random sequences are located. If k = 0, all amino acid sites are independent, and so the effects of substitutions on fitness are additive. As fitness changes gradually with substitutions, the landscape is smooth with a single global peak, which is referred to as a “Mt. Fuji-type” profile. In this case, adaptive walks of the search with single substitutions gradually reach the global peak. On the other hand, larger values of k are associated with more rugged landscapes. If substitutions at a single amino acid site affect residues on k other sites, the effects of double substitutions on two different sites may not be equal to the sum of the effects of the two independent single substitutions k is an essential determinant for the fitness landscape structure. There have been a number of theoretical studies of evolutionary dynamics on both smooth and rugged landscapes n-k model to affinity maturation of the immunoglobulin V region; based on the number of steps in the adaptive walk up to the local optima, the value of k was estimated to be about 40 in this case The e.g., 10 different random sequence for esterase activity, are sufficient to allow Darwinian selection of random polypeptides composed of about 140 amino acid residues Escherichia coli by the coliphage fd is mediated by the minor coat protein g3p via flexible glycine-rich linker sequences Although a great deal is known about the landscape structure near the fitness peaks of native proteins in vitro molecular evolution by increasing the library size gradually from 102 to 106. By applying the experimental data to an original theory of the adaptive walk on the n-k fitness landscape model k value and other parameters to plot a protein fitness landscape and discussed its implications regarding the primordial stages of protein evolution and in vitro evolutionary molecular engineering.Although it was shown to be possible for a single arbitrarily chosen polypeptide to evolve infectivity, the evolution stagnated after the 7th generation, which was probably due to the small mutant library size at each generation. Therefore, we have extended in vitro molecular evolution, which we previously carried out up to the 7th generation, with the addition of an enrichment process by which the fittest phage clone(s) becomes dominant through several cycles of infection and growth in E. coli. As stagnation occurred due to small library size in our previous experiments, we gradually increased the library size N if the time course of infectivity reached a plateau  fitness in the stationary phase and the library size, and (2) the time course of changes in fitness in the transitional phase toward the stationary phase. The method applied here was based on the findings of theoretical studies and its applicability to the experimental data and more detailed analyses are described elsewhere .The parameters determining the structure of the N = 10, 103, 105, and 106, respectively. These stagnations can be explained by the balance between mutation, selection, and random drift due to the limited library size. If a mutant clone with greater fitness than the parent clone appears with a limited library size N, it has a high chance of increasing the fitness of the parent clone for the next generation. On the other hand, if most but not all of N mutants have lower fitness than the parent clone, one of these mutants can be selected by chance as only a limited number of cells are chosen at random after fitness-dependent growth. Thus, the fitness of the parent clone for the next generation can decrease. Therefore, with a limited library size, N, there will be a certain fitness value of the parent clone at stationary phase, given by Eqn. (11) (d is the mutation rate (2.4 per generation) and n is the length of the amino acid sequence subjected to mutation (119), which corresponds to amino acid residues 12–130 of RP3-42. Note that n = 119 implicitly assumes that the region contributes to phage infectivity with little epistatic effect with other regions of the phage genome. Using the mean fitness in the stationary phase for each N value in the evolutionary dynamics shown in W* at each stationary phase followed Eqn. (11) (N = 1 as the random sequence RP3-42 in fd-RP was assumed to be “selected” among N = 1 arbitrarily chosen sequences. This assumption was confirmed by the observation that the infectivity of fd-RP was comparable to that of the deletion mutant phage lacking the D2 domain. Fitting Eqn. (11) to the plot of the W* values against the respective values of N (O+ε as 6.0 and 4νn/d(1+k) as 5.8. We adopted the estimated values of W* given above, and Eqn. (13) was fitted to the plot of the time course of changes in fitness for the 1st–7th and 8′th generations and for 8th–13th generations, as shown in d(1+k)/n = 0.5 for N = 10 and 0.39 for N = 103, and with n = 119 and d = 2.4, we estimated the values of k as 24 and 18, respectively. Thus, we adopted an average of k = 21 for the fitness landscape of phage infectivity. By combining the values determined above and Eqns. (4) and (5) with k = 21, we estimated the following parameters: O = 21.5, ε = −15.5, ν = 0.64. It should be noted that the fitness of the wild-type fd-tet phage, which was not used for determination of the parameters, was as high as the global peak.We assumed that the adaptive walk shown in qn. (11)  where d qn. (11) . Note thues of N , we obtan-k model with the parameters estimated as described above, we depicted the structure of the fitness landscape of the g3p minor coat protein D2 domain for phage infectivity as follows. First, the frequency of sequences that take the fitness values of W obeys the Gaussian distribution given in Eqn. (3) with average O+ε = 6.0 and variance ν = 0.64 2/2ν).Based on the ν = 0.64 . The regr-th order as a function of fitness r-th order that takes r-fold point mutants around it have fitness values lower than r+1-fold point mutants has a value greater than r = 1, 2, 3, 4, 5, 6). The vertical broken lines represent the fitness values, the relative fitness of W* (N with N(d)all in Eqn. (11), where the mutation-selection-random drift balance sets when all conceivable d-fold point mutants of N(d)all , given by Eqn. (14), are explored in each generation. In the range from i.e., they have at least one ascending path. The fraction of the local optima in the first order (r = 1) rises above (1)all of all single point mutants will find only lower fitness, whereas around the latter sequences, the search will find at least one with higher fitness. Thus, the adaptive walk with N(1)all of all single point mutants will lead to small fluctuations around r = 1) reach their maximal frequency of 60% at (2)all of all double point mutants ride and fluctuate due to the mutation-selection-random drift balance. With further increases in fitness, the local optima tend to have a larger basin size and the frequency of sequences located on the slopes decreases.We found that there are no local optima between the foot and the middle altitude on the landscape, whereas above the middle region the number of local optima is very large. Hereafter, we indicate the altitude on the landscape by the relative fitness, .40 A in . At  = 0.47 B in , about 4.47 B in . That is.49 C in . At  = 00.5 D in , the loc4-fold increase in infectivity as compared to the original phage carrying a random polypeptide of 139 amino acids in place of the D2 domain of the g3p minor coat protein. We applied the fitness data of selected clones at each generation to our theory regarding adaptive walking on the n-k landscape model to estimate several parameters of the model and calculate the frequencies of local optima depending on their fitness values.We have extended our previous experimental evolution based on phage infectivity, which showed stagnation after the 7th generation. Increasing the library size overcame the stagnations to some extent and resulted in a 1.7×10k, representing the number of amino acid sites the contributions of which to fitness are affected by a single substitution on a given site, was 21 indicating that an arbitrary residue interacts with about 21 amino acid residues through its mutational effects. The k value around the level of fitness for native proteins was estimated to be 40 for immunoglobulin k of 40 is around the native protein on the fitness landscape and is not necessarily close to our value of 21, but may be an overestimate because two assumptions were made in computer simulations to estimate the k value: a relatively small population was used, particularly at the beginning, and the cloned immunoglobulins were assumed to be at local peaks. If these assumptions are not true, the estimate of k for the native protein will be smaller and may be closer to approximately 20. The interaction of amino acid residues with approximately 20 other residues through mutational effects suggests that a single amino acid residue may interact with 20 other residues through direct or indirect contacts. Clustering the interactions by a size of 20 may allow the clusters to evolve as “modules” The estimated value of the epistasis parameter (1)all For example, a local peak at relative fitness r, the ridge declines down to the fitness given by Eqns. (9) and (10) with N(1)all recursively r times before meeting ridges of other peaks. All ridges meet with others above (1)all.The fitness landscape was depicted schematically but semi-quantitatively based on the frequency of local optima calculated from the experimental data . The lanMore than one such mountain exists in the fitness landscape of the function for the D2 domain in phage infectivity. The sequence selected finally at the 20th generation has The landscape structure has a number of implications for initial functional evolution of proteins and for molecular evolutionary engineering. First, the smooth surface of the mountainous structure from the foot to at least a relative fitness of 0.4 means that it is possible for most random or primordial sequences to evolve with relative ease up to the middle region of the fitness landscape by adaptive walking with only single substitutions. In fact, in addition to infectivity, we have succeeded in evolving esterase activity from ten arbitrarily chosen initial random sequences d)/F′ traD36 + lacIproABqlacZΔM15], JM109 [recA1 supE44 endA1 hsdR17 gyrA96 relA1 thiΔ(lac-proAB)/F′ traD36 proAB+lacIqlacZM15], and HB2151 [Δ(lac-proAB) aranalrthi/F′ traD36 proAB+lacIqlacZΔM15], . The fd-tet The SfiI fragment of the fd-7 genome encoding the target sequence corresponding to the D2 domain of the g3p minor coat protein as follows. The SfiI fragments of phage genomes in the library obtained after the enrichment process of the previous generation were amplified under error-prone PCR conditions SfiI, and the SfiI fragments were cloned into the corresponding region of the fresh fd-RP vector digested with SfiI. Fresh fd-RP vector is the shorter vector prepared by first digesting the fd-RP DNA with BamHI and ligating the resultant vector fragment, yielding a construct with a truncated SfiI fragment E. coli JM109 cells by electroporation E. coli HB2151 was used instead of E. coli JM109.One generation of our evolutionary study consisted of one cycle of mutation and enrichment processes. Random mutations were introduced in the region of the g for 10 minutes to remove the bacterial cells, and the supernatants containing the phage particles were filtered through Dismic 0.45 µm membranes  to ensure the elimination of any remaining bacterial cells. The filtrate containing the phage particles was stored at 4°C as the phage library of the generation for the enrichment process and the phage infectivity assay described below.Aliquots of 10 µl of the mutant library cell suspension described above were dispensed into 10 ml of 2×YT medium containing 20 µg/ml tetracycline and grown at 37°C overnight. The cultures were centrifuged at 6000×E. coli JM109 cells at OD600 0.8–0.9 (900 µl) for 40 minutes at 37°C. The bacteria-phage mixtures were spread onto 2×YT agar medium containing 40 µg/ml tetracycline by gently swirling the plates. Cells in the colonies grown on the plates after incubation overnight at 37°C were collected as described above and stored at −80°C as the 1st-round enriched library of this generation. The enrichment was repeated until the infectivity of the phage library stopped increasing. Aliquots of 10 µl of the last-round enriched library were dispensed into 10 ml of 2×YT medium containing 20 µg/ml tetracycline and grown at 37°C overnight. The phage genomes in the cells collected by centrifugation of the cultures were purified as the replication form for generating the mutant library in the next generation.The enrichment process consisted of several rounds of phage preparation and infection. The phage particles contained in aliquots of 100 µl of the phage library suspension prepared as described above were allowed to infect freshly grown E. coli with phage particles contained in the library was used as a proxy measure of infectivity, as described previously E. coli JM109 cells as described previously E. coli cells was about threefold greater than that of fd-RP phage, although the major change in CFU was attributed to the change in infectivity The number of tetracycline-resistant colonies that grew after infection of W, was defined here as ln(CFU) under the assumption that CFU is approximately proportional to exp(−ΔG/kT), where ΔG is the free energy change of phage infection. Thus, W can be handled as an apparent energy as described below in the n-k fitness landscape. Actually, this definition of W satisfied some theoretical predictions on the n-k landscape . Experimentally, it has been shown that the mutational effect on protein properties can be described well by taking the logarithmic scale of the equilibrium constant or rate constant Fitness was originally defined as the relative growth rate for predicting population dynamics E. coli for evaluation of infectivity in the selection process: TG1 for generations 0–7, JM109 for generations 8–15, and HB2151 for generations 16–20. For systematic analysis of the lineage through generations 0–20, we used the CFU values for the JM109 strain as the standard measure of infectivity. Then, the infectivity of the best phage clone or clones selected in each of the 0th–20th generations was evaluated simultaneously with the CFU values for JM109. A strong correlation was found between the CFU values for JM109 and those for the other strains, with differences of less than one order of magnitude. The landscape climbed is the infectivity landscape for TG1 strain through generations 0–7 or HB2151 strain through generations 16–20. We converted the time series of the CFU values for TG1 through generations 0–7 or for HB2151 through generations 16–20 to the time series of the CFU values for JM109 as follows:XXX denotes the CFU value for E. coli strain “XXX.” These converted values were then used in the analysis. The fitness, W, was defined as ln(CFUJM109) for all phages.We used three different strains of n be the number of amino acid residues in a variable region subjected to random mutagenesis through error-prone PCR. The fitness, W, for a given amino acid sequence, “α1α2…αn,” is defined by:i.e., the fitness contribution from a particular amino acid residue, jα, at the j-th site when the k sites {j1, j2,…,kj} are occupied by particular residues {jα1, jα2,…,jkα}. The k sites {j1, j2,…,kj} are chosen randomly from all n-1 sites except the j-th site. The site-fitness of an arbitrary amino acid residue, α , at each site with a given set {jα1, jα2,…,jkα} is assigned randomly from the following set of 20 values, but degeneracy of assignment is not allowed: Let σ2 is approximately ε2/3. As the first term on the right-hand side of Eqn. (1) is ∼0 for the globally optimal sequence , the second term O is determined as the fitness for the global peak. The fitness landscape resulting from this model is called the “n-k landscape.” Note that the original n-k landscape proposed by Kauffman et al. is slightly different from the model defined above ε(≤0) is a negative constant equivalent to the mean of the site-fitness over all available amino acid residues. Note that there is no significant effect on the theoretical conclusion when the ε value is different by sites n-k landscape. ε(≤0) is defined as the expectation of the first term in Eqn. (1) and represents the difference in fitness from the peak to the foot of the landscape, where the “foot” is the region in which random sequences are located in the sequence space. Then, O+ε is the fitness at the foot of the landscape, and corresponds to the expected fitness of an arbitrarily generated random sequence. ν is the variance of fitness over all possible sequences in the whole sequence space. The probability density of the fitness over all possible sequences in the whole sequence space approximately follows the Gaussian distribution:n-k fitness landscape, with its precise derivation and justification reported elsewhere . We consider the following rule of the adaptive walk: the clone(s) with the highest fitness in the previous generation generate N mutants in the t-th generation, and subsequently the fittest clone(s) among the N clones will become new parental clone(s) in the t+1-th generation. N is the “library size” of mutants to be screened for the next generation. The Hamming distance between a parent and each of its children or mutation rate is d. In the n-k model described above, d-fold point substitutions cause changes in site-fitness at about d(k+1) sites, because a single mutation causes site-fitness changes on the mutated site and k other sites. Let tW be the fitness of the parent in the t-th generation. In addition, let ΔW be the fitness change from the parent to an arbitrary mutant in the mutant population. The probability density of ΔW with tW fixed is described by: Here, we present the essence of the theory regarding evolutionary dynamics on the et al., in preparation). Using extremal statistics of normal distribution, the expectation of the change in fitness from the parent to a new parent after a single generation is given as follows:ζ is defined as the expectation of the greatest value among the N random numbers from the standard Gaussian probability density, ζ is approximately given by transforming N via:The average and variance of the fitness over the generated mutants were roughly consistent with Eqn. (7) and Eqn. (8), respectively. In addition, the validity of Eqn. (9) was confirmed experimentally. The details are reported elsewhere  indicates that the evolution rate increases with increases in N. Substituting Eqns. (7) and (8) into Eqn. (9), we find that, as the adaptive walker climbs the fitness landscape, the evolution rate decreases gradually and finally becomes zero. By solving Eqn. (9) under E[tW+1]−tW = 0, we obtained the fitness value in the stationary phase, W*, as follows:The change in fitness, WN, was confirmed experimentally, as described elsewhere .This stationary phase is caused by the mutation-selection-random drift balance W*, Eqn. (9) can be rewritten as follows:Using the stationary fitness value, t can be obtained approximately by the following function of t:W0 is the fitness of the initial sequence.Then, the expectation of the fitness in generation Nd), the probability that all conceivable d -fold point mutants take fitness values less than W is given by:Next, we refer to the existence of local optima. Let r-fold point mutants take fitness values less than W but at least one of the r+1-fold point mutants takes a fitness value greater than W, then the parent sequence is designated the “local optimum in the r-th order.” That is, r represents the basin size for the local optimum. The probability that a parent with fitness W is the local optimum in the r-th order is given by:Here, we define the local optima as follows. If all conceivable single point mutants, double point mutants,…,


As the mutation rate was increased within a moderate range, deleterious mutation accumulation and mean fitness improvement both increased. The fixation rates were higher than predicted by many population-genetic models. This seemingly paradoxical result was resolved in part by the observation that, during the time to fixation, the selection coefficient (s) of initially deleterious mutations reversed to confer a selective advantage. Significantly, more than half of the fixations of initially deleterious mutations involved fitness reversals. These fitness reversals had a substantial effect on the total fitness of the genome and thus contributed to its success in the population. Despite the relative importance of fitness reversals, however, the probabilities of fixation for both initially beneficial and initially deleterious mutations were exceedingly small .Deleterious mutations are considered a major impediment to adaptation, and there are straightforward expectations for the rate at which they accumulate as a function of population size and mutation rate. In a simulation model of an evolving population of asexually replicating RNA molecules, initially deleterious mutations accumulated at rates nearly equal to that of initially beneficial mutations,  Mutations are the fuel of natural selection. It is widely believed that most mutations are deleterious, that is, they harm the organisms in which they occur. Thus, biologists would like to understand how deleterious mutations impact evolution. Most of the theoretical work on this problem makes an important assumption: mutations that start bad stay bad. It may be possible, however, for an initially bad mutation to become good  by interacting with subsequent mutations. In this study, Cowperthwaite, Bull, and Meyers show that such “fitness reversals” are surprisingly common and can lead to the fixation of initially deleterious mutations. Perhaps mutations that undergo such changes serve as stepping stones for greater evolutionary progress. Modern evolutionary theory recognizes that deleterious mutations may reduce fitness and retard adaptation –5. AccumIn the standard infinite population experiencing a combination of natural selection and random mutation, deleterious mutations should not fix, but accumulate to a level perfectly balanced by mutation and selection. Some processes can lead to deleterious mutations fixing in infinite populations, however. For example, in Eigen's quasispecies model, high rates of mutation can overwhelm selection and shift the mutation–selection balance such that deleterious mutations accumulate to exceedingly high levels ,8. In fiThe fixation of deleterious mutations certainly reduces the fitness of populations. It may be possible, however, for the fitness effect of an initially deleterious mutation to change over time. In particular, compensatory mutations may evolve that reduce the negative impact of deleterious mutations or, in extreme cases, the resulting fitness may be even higher than the fitness of the ancestor in which the deleterious mutation arose . Such coEvolutionary geneticists have long considered mutations that ameliorate or compensate for the deleterious effect of a prior mutation. The literature on this subject, however, focuses almost exclusively on compensatory mutations occurring after the fixation of the initial deleterious mutation, and therefore does not address the likelihood that the initial mutation will fix in the first place –15. One If the mutation rate is relatively large, however, additional mutations may arise in the genome carrying the initially deleterious mutation before it fixes or is lost. Such secondary mutations change the genetic background and thus potentially change the fitness effect of the initial deleterious mutation. The background selection literature has frequently considered the scenario in which a good mutation is driven to extinction by bad mutations  mutations.We used a computational simulation of a population of replicating and evolving RNA molecules. Similar simulation models have been extensively used in previous studies of evolutionary dynamics –24. The L = 76 nucleotides, which is similar in size to a typical tRNA molecule. The focal phenotype is RNA secondary structure , which provides the scaffold for functional tertiary structure and has been highly conserved during evolution . A comparison of the observed and expected rates of fixation suggests that, under U = 0.32, fitness reversals may lead to rates of deleterious mutation fixation that are higher than expected by drift alone, while under U = 0.08 the rates of deleterious fixation do not exceed the expected rates from Kimura's model. We stress, however, that our populations are significantly different from the idealized ones Kimura envisioned and thus there may be multiple reasons for the observed discrepancies.We finally ask whether the number of fixation events that we attribute to fitness-effect reversals and hitchhiking might be within the range predicted to occur by drift alone. Populations experiencing genomic mutation rates of ectively . Kimura'U = 0.01 to U = 0.08 to U = 0.32), perhaps contributing to the evolution of higher mean fitnesses across these mutation rates. Fitness-effect reversals are only part of the story, however, as the initial deleterious effects of fixed deleterious mutations were much larger in the U = 0.32 populations than in the U = 0.08 populations.In summary, a complicated mix of forces allowed initially deleterious mutations to occasionally rise to fixation. Hitchhiking may work in concert with fitness-effect reversals, and therefore our estimates of the contributions of these two processes may be low . FurtherIn this study, we offer a new perspective on the effect and role of deleterious mutations in adaptation. We simulated the adaptation of asexual populations of 1,000 individual RNA genomes that each coded for a phenotype, which consisted of a set of thermodynamically probable secondary structures. In turn, fitness depended on the overall similarity of a molecule's phenotype to a target shape. The effect of a mutation was determined by measuring its impact on the shape of the molecule (its phenotype), and thus the distribution of fitness effects behaved as might be expected of a biological system.The novel result is that nearly one-third of the mutations that evolve in the MRCA lineage  arise with deleterious effects, yet this apparent load of deleterious mutations does not impede adaptation. This can be explained by the frequent occurrence of fitness reversals, that is, more than half of these deleterious mutations do not stay deleterious, but become neutral or beneficial through interactions with compensatory mutations. Importantly, the compensatory mutation(s) arise and reverse the deleterious effect well before the deleterious mutation fixes, and the beneficial combination of mutations then ascends together to fixation.s. He showed that under continuous mutation pressure, the double mutant can fix relatively rapidly, even in large populations. The fixation time for the double mutant was not unreasonably long, being slightly longer than the fixation time for a pair of neutral mutations and much shorter than the fixation time for a pair of unconditionally deleterious mutations.Kimura described a special case of our process in a model of neutral compensatory mutations . He deriA major question is whether this process occurs in nature. There is abundant experimental evidence that the fitness effect of a mutation can depend on genetic background –39. TherTwo factors may be necessary for this process to occur: a high mutation rate and epistasis. The mutation rate must be high enough that a second, interacting mutation arises in the genome before the first mutation is lost or fixed. While background selection typically refers to pairs of mutations  that have net negative fitness effects and no epistatic interactions, here we focus on pairs of mutations (with at least one deleterious) that epistatically interact to yield net positive fitness effects. We conjecture that, to the extent background selection is occurring, fitness reversals may likewise be important to the evolutionary dynamics. Furthermore, the mutation rate of interacting sites must be high enough to have a reasonable probability of creating the right combinations. Some natural systems are characterized by high mutation rates, including RNA viruses. Additionally, there is a sense that the self-replicating molecules present at the origin of life may have had high error rates, and so may fit this model. In the early stages of the process, a small population size may be important to the extent that it affects the rate of drift.Ascent via fitness reversals also requires a rugged (epistatic) fitness landscape. Although epistasis is widely recognized in genetics and evolution, the process described here requires an extreme form of it: the fitness effect of a mutation actually reverses (from bad to good) in the presence of a second mutation. Most studies of epistasis focus on the weaker form in which the fitness effect of a first mutation undergoes small changes in response to a second interacting mutation.Recent theoretical and experimental efforts, however, are beginning to elucidate additional details of these stronger epistatic interactions  ,39. For Other studies suggest that compensatory mutations occur at relatively high frequencies ,43. For prior to fixation (or loss) of the deleterious mutation, and thus have a fundamentally different evolutionary implication: they alter the fitness effect of a deleterious mutation sufficiently early to sway its ultimate evolutionary fate.Our results are a natural extension of previous work examining compensatory evolution in viruses and bacteria. As noted above, those studies almost exclusively considered compensatory beneficial mutations appearing after the fixation of a deleterious mutation and demonstrated that the compensatory effect depends on the presence of the initial deleterious mutation –15,44. TWhile it is widely recognized that asexuality poses several problems to adaptation through processes such as clonal interference, background selection, and Muller's ratchet , the relIn our study, deleterious mutations accumulated rapidly without impeding adaptation—a result counter to most theoretical predictions. We attribute our results, at least in part, to the fact that the fitness effect of a mutation can change dramatically and rapidly upon additional mutations. It remains unclear whether these reversions are sufficient not only to ensure fixation of the original mutation, but also to constitute major adaptive steps.


A number of recent papers have cast doubt on the applicability of the quasispecies concept to virus evolution, and have argued that population genetics is a more appropriate framework to describe virus evolution than quasispecies theory.I review the pertinent literature, and demonstrate for a number of cases that the quasispecies concept is equivalent to the concept of mutation-selection balance developed in population genetics, and that there is no disagreement between the population genetics of haploid, asexually-replicating organisms and quasispecies theory.Since quasispecies theory and mutation-selection balance are two sides of the same medal, the discussion about which is more appropriate to describe virus evolution is moot. In future work on virus evolution, we would do good to focus on the important questions, such as whether we can develop accurate, quantitative models of virus evolution, and to leave aside discussions about the relative merits of perfectly equivalent concepts. I findquasispecies in this 1971 paper; he coined this term in a later paper coauthored with Peter Schuster [Quasispecies theory has its origin in a seminal paper written by Eigen in 1971 , in whicSchuster . These eSchuster ) were soSchuster ,12-14.s, and all other sequences have inferior fitness 1. As a result, much of the generality of Eigen's original work, as well as its connection to population genetics, have been obscured, and the conclusions of these special-case studies are frequently taken to be general predictions of quasispecies theory.Eigen's papers also generated substantial interest among theoreticians (mostly physicists), who found the description of highly error-prone replication an interesting theoretical challenge. Unfortunately, much of the theoretical follow-up work -21 has fBecause of the development of quasispecies theory independently from population genetics, and because of the widespread emphasis on a single fitness landscape in quasispecies theory, many authors now hold a set of beliefs about quasispecies theory that do not correspond to the actual predictions of the theory. These beliefs are:1. Quasispecies theory is at odds with population genetics.2. Quasispecies theory is inapplicable if populations are finite and there is neutral drift.3. Quasispecies theory predicts an error threshold.In the next three sections, I will address each of these points in detail. However, first I have to define what exactly I mean by quasispecies theory.Throughout this paper, by quasispecies theory I mean specifically Eq. (6) in Ref. ,xi(t) is the concentration of sequence i, Wij = AjQij is the product of the replication rate (fitness) Aj of sequence j and the mutation probability Qij from sequence j to i, and E(t) is the total production of new sequences,where In my definition of quasispecies theory, I also include straightforward generalizations of the above equation that have been used in the quasispecies literature, such as the discrete-time quasispecies equation, which can be written as ,23Wij. In both cases, the steady-state solution is given by the dominant eigenvector of Wij.and leads to the same steady-state solution as Eq. (1). Both Eqs. (1) and (3) can be mapped onto linear equations, and then solved by diagonalizing the matrix Wij, which consist of the fitness landscape (as given by the Aj) and the mutation landscape (as given by the Qij), are constants. In the most general case, fitness will depend on the mutant frequencies xi, as different mutants may make use of different resources, and the relative resource concentrations change as the mutant frequencies change. It turns out that the mapping onto a linear system is still valid if resource abundances change due to external factors [xi. In this latter case, which corresponds to frequency-dependent selection, the conclusions drawn from quasispecies theory do not apply.The mapping onto a linear system assumes that the  factors , but notSeveral recent papers present quasispecies theory as a theory that is alternative to (and maybe even contradictory to) standard population genetics ,7. Is ths over the a allele. Further, assume that allele a mutates into allele A, and likewise allele A into allele a, with probability μ. Then, in Eq. (1), we have WAA = (1 + s)(1 - μ), WaA = (1 + s)μ, WAa = μ, Waa = 1 - μ, and hence (note that xa(t) = 1 - xA(t))Let us investigate what form the quasispecies equations take in a simple example. Consider a single locus with two alleles a and A, and assume that the A allele has a selective advantage μ to zero, then this equation turns intoIf we set the mutation rate , and findthat is, into the standard logistic equation that describes the rise of a beneficial allele in an otherwise homogeneous population. Thus, we can recover standard population dynamics from the quasispecies equations. Now, let us calculate the steady state solution of Eq. (4) for an arbitrary mutation rate. We set xa = 1 - xA. For μ = 0, this expression becomes xA = 1, which simply means that the A allele will reach fixation in the absence of any mutation pressure. As μ increases, xA decreases, and xa increases. For a positive μ, even though the a allele is removed from the population by selection, it is constantly regenerated from the A allele by mutation pressure, and thus reaches a positive equilibrium frequency. If the mutation rate is sufficiently high, then the equilibrium frequency of the a allele, maintained by the balance of selection and mutation pressure, can be substantial. In summary, we find that for the case of a single locus with two alleles, the quasispecies model predicts logistic growth of the beneficial allele in the absence of mutations, and mutation-selection balance in the presence of mutations.and of course xi of a sequence with i mutations changes from one generation to the next according to (Eq. (3.1) in Ref. [Now consider the multi-locus case. A classic paper on mutation-selection balance is the one by Kimura and Maruyama, written in 1966 . In this in Ref. ):wi is the fitness of a sequence with i mutations, , and μ is the mutation rate (note that Kimura and Maruyama use fi instead of xi and 2M instead of μ). Now, define the mutation matrix Qij aswhere Wij in Eq. (3) as Wij = wjQij. Then, we see that E(t) as defined in Eq. (2) becomes . Furthermore, the sum ∑jWijxj in Eq. (3) runs from j = 0 to j = i, since Qij = 0 for i <j. After introducing a new index k = i - j, we can rewrite the sum asand write the matrix L of a population described by the quasispecies model is in many  cases approximately given by L= 1 - eμ-.  follows directly from the quasispecies equation Eq. (3). As a consequence, the quasispecies model is in agreement with the Haldane-Muller principle , which mutations ,28.)Now that we have seen that quasispecies theory and mutation-selection balance are equivalent, the question remains whether Eigen just reinvented parts of population genetics, or actually contributed to the development of the field. While Eigen was not the first to consider mutation-selection balance (this concept goes back to Wright and Fisher in the early 20th century), by studying multi-locus mutation-selection equations at arbitrary mutation rate he was certainly at the forefront of theoretical population genetics in the late 1970s and early 1980s. The first analytic solutions to equations of the form Eq. (1) were found by Thompson and McBride in 1974  and indeWij, is a stable mixture of closely related mutants. This mixture of mutants, also called a mutant cloud or quasispecies, does not necessarily have to contain the fastest-replicating individual sequence that exists in the fitness landscape. In other words, sequences with high fitness can lose out against sequences with lower fitness that have better support from their mutational neighbors [survival of the flattest  that the equilibrium state of the population, which is given by the dominant eigenvector of the matrix eighbors ,33, an et Figure .It is important to understand that the emergence of a quasispecies is not something that has been put into the model ad hoc, but is a necessary consequence of the mutation-selection equations. We see in Eqs. (1) and (3) that the model is built on reproduction of individual sequences, but that mutations provide coupling between the different sequence types. When the mutation rate is low, then the quasispecies model predicts that the fastest-replicating sequence takes over the population, as we witness from the emergence of the logistic equation Eq. (5) at zero mutation rate. However, when the mutation rate is high, then the coupling between sequences caused by mutations can become stronger than the individual selection coefficients, and a quasispecies forms. Note that this effect will arise in any model of mutation-selection balance that correctly takes into account the coupling of different mutants at high mutation rate.quasispecies effects, I mean that the population behaves in a way that can only be explained through strong mutational coupling between genotypes. An example would be the observation of the survival of the flattest effect.)In the previous section, I have established that the quasispecies model is equivalent to the theory of mutation-selection balance in an infinite, haploid, asexual population. However, this equivalence does not necessarily imply that the quasispecies model applies to populations of RNA viruses, because these populations are finite. Jenkins et al.  argue thFirst, let us have a look at some theoretical studies of finite populations that have been carried out within the context of the quasispecies model ,34-38. IHowever, one could still object that these models may be describing idealized and simplified situations that differ too much from the reality of RNA viruses to be of any relevance. To counter this argument, we have to ask whether there is a more general way to determine the relevance of quasispecies theory to finite populations of RNA viruses. The hallmark of quasispecies theory  is that for a sufficiently high mutation rate, we must take into account the formation of a quasispecies to obtain a faithful description of the population dynamics. Therefore, the question is under what conditions does mutational coupling become so strong that we can observe quasispecies effects in a finite population. Can theory help us to address this question?evolution of mutational robustness [μ and effective population size Ne was significantly larger than one, μNe ≫ 1 [In the formation of a quasispecies, the population minimizes the mutational load by accumulating sequences that have a reduced probability to suffer from deleterious mutations ,28,39,40bustness . Van Nim μNe ≫ 1 . I have  μNe ≫ 1 . What is μNe ≫ 1 . These r μNe ≫ 1 ,43. Note μNe ≫ 1 ,45.47 and 1068 sequences large. Clearly, a population of size 250  cannot even come close to complete coverage of such a huge sequence space.Finally, the recent paper by Comas et al.  also proThe previous paragraphs show that on purely theoretical grounds, there is no reason to assume that quasispecies effects cannot play a role in finite populations of RNA viruses. Nevertheless, to date we have no experimental evidence that unequivocally demonstrates such effects in a specific experimental system. On the other hand, selection for specific, individual mutants is common (see e.g. Ref. ). What dIn summary, we currently have no evidence  that contradicts the existence of quasispecies effects in finite populations of RNA viruses, but we also have no experimental evidence in favor of it. Theoretical studies and computer simulations indicate that quasispecies effects should become important when the product of effective population size and genomic mutation rate is significantly larger than one. Since for many RNA viruses the genomic mutation rate is already on the order of one ,49, evenThe error threshold is probably one of the most misunderstood concepts of quasispecies theory. Eigen described the error threshold in his 1971 paper as a limit to the amount of information a genome can store at a given mutation rate . If the The first important point to understand about the error threshold is that it is a deterministic effect. This means that the position of the error threshold depends only weakly on the population size, and that even in an infinite population the error threshold occurs at a finite mutation rate ,35. In tper se makes no statements about population extinction. The alternative model is hard selection, where the population size will decline if the mean fitness of the population is too low. Extinction due to mutation pressure can occur under hard selection, and is usually called mutational meltdown [There is certainly no lack of lethal mutants in viruses ,58, and meltdown -65. Mutaerror cascade [Finally, it is interesting to note that in certain fitness landscapes, several error-threshold-like transitions can occur one after the other as the mutation rate increases . At each cascade , and in  cascade ,33,46 caTo summarize, I have provided arguments for the following conclusions: Quasispecies theory is in perfect agreement with population genetics, it can make usefull predictions for finite populations if the product of population size and mutation rate is large, and it predicts an error threshold only for fitness landscapes that lack lethal mutants, and which therefore have little relevance for virus evolution.However, these arguments do not imply that quasispecies theory is the final answer to all questions of virus evolution. Quasispecies theory has its short-comings that need to be addressed in future modeling work. Ironically, the biggest shortcoming of quasispecies theory, as far as I can see, does not have its origin in quasispecies theory being at odds with population genetics, but rather in quasispecies theory being too similar to the population genetics theory of asexual, haploid organisms. Viruses differ from other forms of life in that they don't have a well-defined ploidy. When a single virus particle infects a cell, the virus can be considered a haploid organism, and indeed the quasispecies model makes this assumption. However, frequently several virus particles coinfect the same cell, in which case the ploidy is given by the number of coinfecting particles. Standard population genetics has no model for such a variable-ploidy organism, and only a handful of authors have considered the theoretical implications of viral coinfection in detail -81.COW carried out all aspects of this work.