In November 2007, the Society for Acupuncture Research (SAR) held an international symposium to mark the 10th anniversary of the 1997 NIH Consensus Development Conference on Acupuncture. The symposium presentations revealed the considerable maturation of the field of acupuncture research, yet two provocative paradoxes emerged. First, a number of well-designed clinical trials have reported that true acupuncture is superior to usual care, but does not significantly outperform sham acupuncture, findings apparently at odds with traditional theories regarding acupuncture point specificity. Second, although many studies using animal and human experimental models have reported physiological effects that vary as a function of needling parameters  the extent to which these parameters influence therapeutic outcomes in clinical trials is unclear. This White Paper, collaboratively written by the SAR Board of Directors, identifies gaps in knowledge underlying the paradoxes and proposes strategies for their resolution through translational research. We recommend that acupuncture treatments should be studied (1) “top down” as multi-component “whole-system” interventions and (2) “bottom up” as mechanistic studies that focus on understanding how individual treatment components interact and translate into clinical and physiological outcomes. Such a strategy, incorporating considerations of efficacy, effectiveness and qualitative measures, will strengthen the evidence base for such complex interventions as acupuncture. The 1997 National Institutes of Health (NIH) Consensus Development Conference on Acupuncture was a landmark event in the history of acupuncture research [In November 2007, the Society for Acupuncture Research (SAR) held an international symposium aimed at presenting and discussing the progress made in acupuncture research during the decade following the NIH Consensus Development Conference. The symposium presentations, as well as their published summaries , unequivThe emergent potential paradoxes that frame this paper are theA large number of well-designed clinical trials have reported that true acupuncture is superior to usual care, but does not significantly outperform sham acupuncture, findings apparently at odds with traditional theories regarding acupuncture point specificity and needling technique.While many studies in animal and human experimental models have reported physiological effects that vary as a function of needling parameters , the extent to which these parameters influence therapeutic outcomes in clinical trials is unclear.The goal of this White Paper is not to present a detailed review of the literature. Rather, this paper aims to identify gaps in current knowledge that underlie these paradoxes and to recommend research strategies that can help deconstruct and resolve them. We propose that resolution of the paradoxes lies in applying a bidirectional translational research approach, in which clinical trial design is informed by knowledge of mechanisms underlying acupuncture and by results from pragmatic trials of “real world” clinical practice. With this approach, findings from clinical trials will better inform the role of acupuncture in our evolving health care system.see ).The field of acupuncture research includes a number of terms that have created confusion and impeded progress, the word “acupuncture” notwithstanding. The term “acupuncture” has been used in the literature to convey not only the simple process of inserting a needle but also to describe a more complex intervention that includes a traditional exam, diagnosis, and the incorporation of additional modalities such as massage and moxibustion. “Sham acupuncture” and “acupuncture points” are other examples of terms fraught with confusion in the literature  rather than needling or other components specific to acupuncture. If this is the case, the implication would be that the mechanisms underlying the therapeutic effects of acupuncture are essentially placebo-related. A central thrust of this White Paper is that there are other explanations that are just as plausible since a number of factors that could lead to Type II errors  remain poorly understood: (1) both verum and sham needling may have therapeutic effects, (2) acupuncture-specific nonneedle and/or needle components may be retained in sham treatments, and/or (3) the therapeutic effects of the provision of verum acupuncture treatments in the context of clinical trials may be suboptimal.A major factor potentially underlying Paradox 1 is the lack of consensus on which component(s) of acupuncture needling are therapeutically important. There has been little systematic investigation regarding the importance of needle placement and depth, type and intensity of stimulation, needle number, and so forth , as wellSham acupuncture treatments have consistently been shown to have greater therapeutic effects than conventional placebos –10. Placintention for a therapeutic outcome, traditionally described as yi. Some research has begun to experimentally investigate the therapeutic effects of healing intention as well as possible mechanisms associated with these effects [An additional factor that may contribute to the therapeutic effect of sham acupuncture treatments is the strong practitioner-patient interactions that commonly take place during interventions . One spe effects , 15. As In contrast to drug trials that evaluate well-defined and consistent pharmaceutical agents, acupuncture treatment protocols in clinical trials are rarely validated using objective, standardized approaches. Moreover, studies attempting to develop valid protocols based on consensus of multiple acupuncturists have revealed significant pluralism regarding what constitutes appropriate treatment . Unlike In summary, possible explanations for the generally small effect sizes observed in many sham-controlled trials are the following Verum and sham treatments are equivalent, because the therapeutic effects of both are due to nonspecific effects .Specific therapeutic effects of acupuncture treatments do exist but are not detected in sham-controlled trials becausesham acupuncture treatments are not inert , and/or verum acupuncture treatments are less than optimal.As with Paradox 1, a possible explanation for Paradox 2 is that the needle-related physiological effects observed in basic science experiments are unrelated to therapeutic effects. Another possible explanation is that at least some of the physiological effects of needling are therapeutic, but these effects are difficult to demonstrate in the context of a clinical trial.acute pain models  may have limited value for informing clinical trial design and practice-based treatment of chronic pain conditions given the emerging view of chronic pain as a qualitatively distinct neurological dysfunction from acute pain [As in most medical research, experiments in healthy humans and animal models have limitations in their applicability to clinical research. To date, the large majority of basic science research on acupuncture needling has been performed in healthy humans and animals; it is likely that physiological responses to acupuncture needling are different in pathological states . In partute pain . Furtherute pain , it is lor humans for functional concepts that are key to the traditional rationale underlying acupuncture treatments, such as acupuncture points, meridians, and “qi”.Mechanistic studies of acupuncture treatments and needling are also seriously hampered by three additional factors. First, similar to biomedical research, a significant proportion of acupuncture research evaluates conditions for which the pathophysiology is largely unknown, therapeutic outcomes are mostly subjective, and no good animal models or biomarkers exist . Second, there are no current animal models that investigate the mechanisms of acupuncture treatments within the explanatory framework of traditional East Asian medicine. Third, we do not understand what, if any, physiological equivalents exist in animals As discussed above, clinically relevant physiological effects  demonstrated in experimental models may be masked in clinical trials by the magnitude and variability of the multiple components of verum acupuncture treatment. In a sham-controlled clinical trial for chronic pain, an anti-inflammatory effect of verum needling may indeed be present, but the magnitude of this effect may be proportionally small compared with large and variable therapeutic effects produced by both verum and sham nonneedling components . A potentially important difference between basic science studies and clinical trials is that the former tend to use small numbers of needles, inserted at the same locations in every animal or human subject, while the latter, for the sake of reflecting clinical practice, often use 10 to 20 needles per session with at least some degree of practitioner-permitted choice of needling location. Thus, a small but specific therapeutic “signal” due to needling may be detectable in a physiological experiment but not readily detectable in clinical trials due to the additional “noise” in the system. In summary, possible explanations for why specific effects of needling are clearly demonstrable in basic science studies, but not as evident in sham-controlled clinical trials are as follows: Some or all of the needle-related physiological effects observed in basic science experiments are unrelated to therapeutic effects.  Acupuncture needling does have therapeutic effects, but these effects are proportionately small compared with nonneedle and nonspecific effects in clinical trials.  Sham-controlled clinical trials are compromised methodologically because the influence of needling parameters for verum and sham needling are poorly understood, andobjective biomarkers relevant to the commonly treated clinical conditions are lacking.needling” and “acupuncture treatment”  style of acupuncture, “pattern differentiation” is purported to be paramount for effective treatments and influences choice of needle location, needle stimulation techniques, and treatment frequency and duration. Consensus among acupuncturists regarding TCM diagnosis has ranged from good to poor in various interrater reliability studies –33 but wA particularly controversial question is the issue of “point specificity” with respect to clinical and/or physiological effects of acupuncture needling. It remains unclear whether effects observed at certain acupuncture points  are due to proximity of the needle to a major nerve, rather than its location at a specific acupuncture points. Few studies, for example, have included nonacupuncture point controls located along the same nerve as the tested acupuncture point. More studies including such controls therefore are urgently needed to advance our understanding of this important issue. A better understanding of potential differences in physiological responses elicited at acupuncture points and nonacupuncture points also is important for the future design of sham controls in clinical trials. Development of a modern scientific rationale for the traditional theories related to insertion of needles at specific locations is particularly challenging. Although it is possible that acupuncture points are truly unique anatomical structures, so far this has not been sufficiently and rigorously corroborated by research. In particular, a recent systematic review found little evidence in support of using skin conductance or impedance to identify acupuncture points . HoweverStudies are needed to identify the physiological effects associated with individual components of acupuncture, both needling and nonneedling. These include more in-depth and systematic investigations regarding the importance of such parameters as needle placement and depth , stimulaBiomarkers that relate to a given pathological condition and improve with treatment are needed to objectively evaluate treatment efficacy both in humans with these conditions and in animal models. If an understanding of disease pathophysiology is lacking, patients may be studied who have similar symptoms but who may have quite different disease mechanisms. Thus, the question “is acupuncture effective for chronic low back pain?” is likely not precise enough since low back pain is heterogeneous and may include subgroups with pathology that responds to treatment and subgroups that do not , 40. Morimmediate and delayed (hours to years) physiological responses to needling in humans be developed to facilitate assessments of effective needling parameters. This can be designed as a two-phase program in which the outcome measures are developed and correlated first in healthy volunteers, then repeated for assessment of clinical validity in patients. For example, quantification of changes in pulses (informed by Chinese and/or Japanese acupuncture theory) could not only qualify as an immediate response measure but would have an added advantage of beginning to develop biomarkers from within the acupuncture explanatory model. Further, suggestions as to why some parameters of needle stimulation appear to have physiological effects in basic science studies but are not associated with specific therapeutic effects in clinical trials (Paradox 2) may emerge from assessing immediate and/or delayed responses to needling  of healthy volunteers in comparison with medically diagnosed patients using the same specific needling parameters, for example, needle insertion depths, mode and amount of stimulation.An additional recommendation is that reliable biomarkers and clinical outcome measures of both and “what to avoid”. Most sham designs  are based mainly on the former consideration (aspects of treatment the patient sees) with insufficient attention to the latter. Thus, development of sham controls should be based on a systematic basic science evaluation of the components of acupuncture treatments  that the sham procedure should avoid. In order to gain clarity on the interpretation of these studies, we urge the scientific community to clearly state the hypothesis that is being tested and select a control procedure(s) that will most specifically test this hypothesis. For example, if a given physiological response in a basic science experiment is hypothesized to be due to electrical stimulation of acupuncture needles, an appropriate control would be needle insertion at the same location and depth but without electrical stimulation. If, on the other hand, the hypothesis is that there is an interaction between electrical stimulation, and needle location , then four experimental groups are needed . Thus, the frequent practice of comparing needle stimulation at acupuncture points to no stimulation at nonacupuncture points should be interpreted with caution as it risks confounding the two variables . Similarly, the common practice of comparing acupuncture points to nonacupuncture points in a different body region  risks confounding point type  and body region unless additional controls are included. Rational design of a sham needling procedure requires knowing “what to mimic” We thus encourage future clinical trials including both usual care (to test the effectiveness of acupuncture) as well as appropriate control group(s) designed to test the efficacy of specific treatment components . Such trials would be further strengthened by the inclusion of objective measurements of pathology relevant to the condition being treated. A summary of combined recommendations based on the paradoxes identified in this paper is shown in The experience gained by acupuncture researchers over the last ten years is invaluable both for the field of acupuncture itself and as an example for the study of other complex treatments that may wrestle (now or later) with similar issues. It is therefore vital to recognize the lack of clear difference between verum and sham acupuncture treatments in clinical trials, as well as the documented physiological effects of acupuncture needling in basic science studies. It is now the important responsibility of acupuncture researchers to face these results squarely and move the field forward. It is conceivable, as stated above, that the therapeutic effects of acupuncture treatments are all or largely based on nonspecific, placebo-related responses. If this is true, the implications for the investigation and practice of acupuncture are profound as this would certainly raise interesting questions as to the value of specific acupuncture point location and needling techniques. Before we consider such a perspective, however, it is important that the issues outlined in this White Paper are addressed.An opinion commonly voiced in the acupuncture community is that “acupuncture cannot be studied using randomized controlled trials”. We strongly disagree. Rather, we encourage clinicians and scientists to recognize that much patience and hard work will be needed before we fully understand how best to use the tools at our disposal, as well as develop new ones, to study this complex form of treatment. As has been shown in the field of ecology, complex phenomena are best tackled using a pluralistic strategy including both “top-down” approaches that evaluate the behavior of a whole “intact” system, as well as “bottom-up” approaches that evaluate smaller-scale cause-and-effect links. Acupuncture treatments thus should be studied both from the “top down” as multicomponent “whole-system” interventions employing a pragmatic systems perspective, as well from the “bottom up” as mechanistic studies that focus on understanding individual treatment components and how the effects of these components interact and translate into clinical outcomes. Viewed through the lens of translational research , the schCreation of an acupuncture research agenda involves more than formulating lists of individualized goals for basic science and clinic research. In addition to providing a framework for addressing the paradoxes, the translational research approach will deepen our understanding of complex systems of care and refine our perspectives of human health and disease. Further, in calling for bidirectional translational research, SAR emphasizes the importance and value of building clinical relevance into the design of clinical as well as basic science research. The idea that research design should be framed in the context of ecological validity is an imperative, especially in light of the limited funds available for acupuncture research and its primary raison d'être, which is improved patient care. A bidirectional translational research strategy, incorporating considerations of efficacy, effectiveness, and qualitative measures, will result in a broader and stronger basis of support for the study of complex treatments in general, including acupuncture and other CAM interventions.


Lethal mutagenesis is a promising new antiviral therapy that kills a virus by raising its mutation rate. One potential shortcoming of lethal mutagenesis is that viruses may resist the treatment by evolving genomes with increased robustness to mutations. Here, we investigate to what extent mutational robustness can inhibit extinction by lethal mutagenesis in viruses, using both simple toy models and more biophysically realistic models based on RNA secondary-structure folding. We show that although the evolution of greater robustness may be promoted by increasing the mutation rate of a viral population, such evolution is unlikely to greatly increase the mutation rate required for certain extinction. Using an analytic multi-type branching process model, we investigate whether the evolution of robustness can be relevant on the time scales on which extinction takes place. We find that the evolution of robustness matters only when initial viral population sizes are small and deleterious mutation rates are only slightly above the level at which extinction can occur. The stochastic calculations are in good agreement with simulations of self-replicating RNA sequences that have to fold into a specific secondary structure to reproduce. We conclude that the evolution of mutational robustness is in most cases unlikely to prevent the extinction of viruses by lethal mutagenesis. The high mutation rate of RNA viruses, such as HIV, allows them to rapidly evolve resistance to host defenses and antiviral drugs. A new approach to treating these viruses—lethal mutagenesis—turns the mutation rate of these viruses against them. It uses mutagens to increase the viruses' mutation rates so much that the accumulation of harmful mutations drives viral populations to extinction. Is there any way that a virus could adapt to a drug that increases its mutation rate? One way is that the virus could evolve so that mutations tend to be less harmful. In previous experimental work, there have been reports that virus populations can differ in robustness. Yet, the evolution of mutational robustness did not seem to inhibit extinction by lethal mutagenesis. In this work, we model viral populations under lethal mutagenesis in order to see when viruses might escape extinction by evolving robustness to mutations. We find that viruses can benefit from robustness only at relatively low mutation rates because the extent to which robustness increases fitness is rapidly drowned out by the extent to which higher mutation rates decrease fitness. The implication is that the evolution of mutational robustness is not a fundamental impediment to lethal mutagenesis therapy. Lethal mutagenesis is a proposed therapy for patients with viral infections. The general approach is to increase the deleterious viral mutation rate enough so that the viral population will go extinct Research on lethal mutagenesis and the question of how much mutational robustness can affect mutagenesis are of practical importance. In support of the promise of lethal mutagenesis as a treatment for many human and agricultural viruses, there are reports of the addition of a mutagen severely reducing or extinguishing populations of coxsackievirus B3 An important limitation to any pathogen treatment is the ability of the pathogen to develop resistance. Since lethal mutagenesis introduces deleterious mutations throughout the genome of viruses, it seems that there are only two types of effective resistance mechanisms. First, the virus could evolve a mechanism to reduce the number of mutations that the therapeutic mutagen introduces. Ref. Empirical studies of lethal mutagenesis appear to yield conflicting results. While Ref. The organization of this paper parallels our line of inquiry. First we ask, when will a population at equilibrium go extinct? We find with a deterministic model that an approximation for the critical mutation rate, i.e. the mutation rate beyond which the population goes extinct, is the log of reproductive capacity divided by the non-neutrality of the population at equilibrium. The implication is that small increase in the mutation rate can compensate for relatively large increases in neutrality. Next, we ask, how will elevating the mutation rate increase the rate at which populations move to areas of a neutral network with higher equilibrium neutrality? We find with a semi-deterministic model that the time it takes for a population undergoing mutagenesis to find the optimal area of the network grows exponentially with the size of the barrier to it. The implication is that we can usually disregard these shifts of the virus population, since the population will quickly shift to the optimal area if the barrier is small and the population will stay where it begins if the barrier is large. Finally, we ask, when will a population that is not at equilibrium go extinct? We show with a stochastic analytical model and simulations based on RNA-secondary structure networks both the critical mutation rate in these more complex models and the probability of stochastic extinction at mutation rates below the critical mutation rate. The implication is that the initial robustness of the population can be important in some cases, but not when the mutation rate exceeds the critical mutation rate.First, we consider the effects of mutational robustness in a deterministic model of lethal mutagenesis. In general, virus extinction is guaranteed if We can write In general, we can write the deleterious mutation rate as Throughout the remainder of this paper, we consider populations evolving on neutral networks. All sequences on the neutral network have the same reproductive capacity In the case where neutral sequences are distributed at equal density throughout the mutational network, Under the assumption that Of course, the critical mutation rate may be far above unity and the assumption that entropic barrier means that the probability to jump from one network to another with one mutational event is low.) In other words, there may be few possible paths in the network from one area to another. In this case, there is the risk that increasing mutation rates will increase the rate at which virus populations find rare paths to other areas of the neutral network in which it is possible to evolve greater neutrality. This process is comparable to that of demes drifting between equilibria (adaptive peaks) in the context of shifting-balance theory In general, a neutral network may be broken into separate areas of differing neutrality and separated by entropic barriers. . The bits are organized into The neutral-staircase landscape can be solved analytically, and the full derivation can be found in Ref. If we sum is large . This isThe prospect of the equilibrium neutrality increasing raises the question of how much increases in equilibrium neutrality may increase on rate  .When barriers are small, we can expect that the area of the neutral network with the greatest connection density can be found in a reasonable number of generations. In this case, the main question is whether the population can find areas with high connection density before it goes extinct under mutagenesis. In the following subsections, we will address this question using fully stochastic models.According to Equation (1), extinction is guaranteed if the mutation rate is so high that the equilibrium mean fitness of the population is less than 1. But lethal mutagenesis is not an equilibrium process. Therefore, we next explore how extinction occurs in a population out of equilibrium, using the mathematical framework of multi-type branching processes. Because this approach is a stochastic one, we calculate not only the mutation rate at which extinction is guaranteed but more generally the probability that extinction happens at any given mutation rate. Our main question here is how the extinction probability changes if the population resides initially in regions of the neutral network with particularly low or high connection density.The mathematical framework we use to calculate the extinction probability under lethal mutagenesis is that of multi-type branching processes. This framework has been used previously to calculate the fixation probability of a rapidly mutating virus on a neutral network Consider a population where all offspring are identical to their parents. A sequence produces a random number of offspring in the next generation. All these offspring sequences produce their own random number of offspring according to the same probability distribution. The number of progeny that a sequence has in two generations, then, is the sum of these random variables. The use of a probability generating function (p.g.f.) allows for convenient expression of these sums. We useWhen there is a finite number Extinction probabilities can easily be found numerically from Equation (12), but we next present two approximations to illuminate how extinction probabilities follow from offspring distributions.First, we need an explicit expression for the multivariate p.g.f.s in the fixed-point equation. If the number of offspring of type When extinction probabilities When extinction probabilities The previous subsection developed the general theory of stochastic extinction under lethal mutagenesis. We will now apply this theory to the special case of a neutral network of RNA sequences. To this end, we will first describe a model that links a sequence's location in a neutral network with the sequence's neutrality. This model yields the rates Consider how the probability-density function of the offspring distribution smaller the probability that offspring are neutral (see next paragraph). Instead of We define Our approach is inspired by Ref. Equations (19) and (20) say that, since there are only so many neutral sequences, if a sequence is in an area of the neutral network with a high connection density, then the connection density of neutral sequences must generally decline as we move away from it, and vice versa. This reasoning implies that re that  .We can use this framework to determine the Putting everything together, the probability that any one offspring of a parent of type The matrix tion of  . Since  by  and  . The relFirst, we present results based on the assumption that populations initially consist of a single sequence. This case is relevant to a scenario in which a patient is inoculated with a small dose of virus while on lethal mutagenesis therapy or a virus is establishing itself in a new tissue of a patient's body. With this assumption, we found that the probability of extinction declined with the initial sequence's neutrality, but also that the gradient in extinction probabilities rapidly leveled as the mutation rate increased . In agreNext we used the analytic calculations to study the effect of the size of the neutral network. When going from a smaller neutral network to a larger neutral network, the extinction threshold r values . Extinctpulation .Since lethal mutagenesis is intended to eliminate virus populations that have grown to high levels, we also considered the effect of the initial population size. We considered an initial population that was uniformly composed of sequences with a given initial robustness  changed . The gra was one .We verified our branching-process model by carrying out simulations with individual RNA sequences  prevailed in competition against a strain that was more fit in the absence of a mutagen when 5-FU doses were 20, 40, 60, and 80 While our models do show that the initial neutrality of a population can affect its probability of extinction, this relationship may be overshadowed in practice. For example, the models neglect the effect of defective interfering particles, which may contribute to extinction by lethal mutagenesis This work has provided quantitative support for the statement that the evolution of mutational robustness will have only a minor effect on lethal mutagenesis. In an extreme case, half of all non-beneficial mutations could evolve to become neutral. In this case, doubling the mutation rate will be sufficient to cause extinction . For lesWe evaluated the convergence times given by Equation (8), numerically derived We obtained the fixed point Sequences that folded into a target shape were considered neutral, and all others were considered inviable. The neutrality of a sequence was the fraction of neighbors at a Hamming distance of one that also had the target phenotype. The RNAfold function in the Vienna package ((((....)))).............................(((..........(((((.....))))))))........Here, positions that form base pairs are indicated with matching parentheses, and unpaired positions are indicated with dots. For the first target, which was used to generate the results in The extinction probability of a sequence was determined by simulation of a branching process on the RNA secondary-structure neutral network. Simulations began with a single neutral sequence. These sequences were selected from the sample of sequences used to estimate the size of the neutral network so as to get the full range of initial neutralities. At each iteration, each sequence in the population had a Poisson distributed number of offspring. Each letter of the sequence changed to any of the other three possible letters with a probability equal to the genomic mutation rate divided by the sequence length. Mutation rates ranged from zero to fifteen. Each sequence was tested to see if it folded into the target, and sequences that did not were removed. Simulation was continued until the population size reached zero or 10,000. Simulations were replicated 100 times for each of 500 initial sequences and the extinction probability was the number of simulations in which extinction occurred divided by the total number of simulations. A local polynomial fitting function  and (20) predict that as the parameter α of a sequence increases, the epistasis parameter β decreases. The panels are labeled with the sequence lengths of 40, 5000, and 10000. For each sequence length, the lines, from highest to lowest, are numerical solutions where log(0.02 MB EPS)Click here for additional data file.Figure S2p. Parameters: sequence length l = 40, neutral-network size 29, reproductive capacity R = 50, initial population size = 1 sequence.Extinction probability as a function of initial neutrality and deleterious genomic mutation rate. Panel A displays results from simulations where sequence neutrality was determined by RNA folding. Panel B displays results from a branching process model derived from the correlation between sequence neutrality and epistasis. Only in a band of intermediate mutation rates does the extinction probability depend on initial neutrality 1-(0.26 MB EPS)Click here for additional data file.Dataset S1Raw data and computer code necessary to reproduce all results reported in this paper.(0.46 MB ZIP)Click here for additional data file.Text S1Supplementary text.(0.06 MB PDF)Click here for additional data file.


Despite the emerging experimental techniques for perturbing multiple genes and measuring their quantitative phenotypic effects, genetic interactions have remained extremely difficult to predict on a large scale. Using a recent high-resolution screen of genetic interactions in yeast as a case study, we investigated whether the extraction of pertinent information encoded in the quantitative phenotypic measurements could be improved by computational means. By taking advantage of the observation that most gene pairs in the genetic interaction screens have no significant interactions with each other, we developed a sequential approximation procedure which ranks the mutation pairs in order of evidence for a genetic interaction. The sequential approximations can efficiently remove background variation in the double-mutation screens and give increasingly accurate estimates of the single-mutant fitness measurements. Interestingly, these estimates not only provide predictions for genetic interactions which are consistent with those obtained using the measured fitness, but they can even significantly improve the accuracy with which one can distinguish functionally-related gene pairs from the non-interacting pairs. The computational approach, in general, enables an efficient exploration and classification of genetic interactions in other studies and systems as well. The systematic mapping of genetic interactions in biological systems has the potential to provide a better understanding of how genes function as networks to carry out and regulate cellular processes. In particular, recent advances in the experimental technologies which allow for the large-scale screening of the effects of combinatorial gene deletions are providing an exciting glimpse into the organization of complex genetic networks in terms of revealing novel interacting cellular components and compensatory pathways involved in many cell functions. Comprehensive maps of genetic interactions in model organisms, such as yeast, may also provide a valuable template for understanding the basic principles underlying the relationships between genotype and phenotype in other populations Saccharomyces cerevisiae, have already identified a number of synthetic lethal interactions, in which a combination of two individually non-lethal mutations results in lethality vs. non-interacting pairs), quantitative phenotypes, such as the relative growth rate of yeast colonies, have recently been explored systematically using high-throughput screening approaches like epistatic miniarray profiling (E-MAP) and genetic interaction mapping (GIM) Several large-scale studies, especially in yeast i and j can be defined by the deviation (ijε) of an observed double-mutant phenotype (ijP) from the expected neutral phenotype of an organism's fitness (ijE) under the hypothesis that it carries two non-interacting mutations (the null hypothesis). If the fitness ijP is evaluated in terms of the growth rate of double-mutant ijw, relative to the wild-type growth rate, and ijE is a function g of the relative single-mutant fitness values iw and jw, this definition can be formulated as:ijε| provides evidence for genetic interaction, while deviations close to zero indicate non-interacting gene pairs. Significant genetic interactions can further be classified into so-called synergistic interactions (ijε<0) and alleviating interactions (ijε>0). Synergistic interactions occur when a double-mutant has a more extreme effect on the fitness than would be expected from independent single mutants alone, and can therefore identify e.g. complementary pathways, with synthetic lethality being the extreme case (ijw = 0). Alleviating interactions, in which the double-mutant phenotype is less severe than expected, can occur, for example, when the first mutation already impairs the function of a whole pathway and thereby masks the effect of the second mutation in the same pathway.Regardless of the experimental technology used, the screening strategies aim to quantify the extent to which a mutation in one gene modulates the phenotype (or fitness) associated with altering the second gene, either by explicitly measuring or analytically comparing the observed fitness of double-mutants to those of single-mutants. More formally, a genetic interaction between mutants g = iwjw provides a convenient null model in the sense that it yields a distribution with location close to zero and low dispersion over all of the measured deviations. The comparison was based on the principle that, as the vast majority of gene pairs should be non-interacting, the rare gene pairs sharing a specific function should be distinguishable from this background distribution as outlying cases with extreme absolute deviations. Accordingly, it was shown that the observed deviations based on the multiplicative null model were indeed most accurate at identifying functional relationships between the genes Recently, Mani et al. w is a 26-dimensional column vector and W is a 26×26-dimensional symmetric matrix. Under the multiplicative null model, Eq. 1 takes the form:w with itself, and E is the n×n-matrix comprising the ijε values as its elements for each gene pair i, j = 1,2,…,n. In the ideal case, when there are no measurement inaccuracies, the approximation problem of Eq. 2 could easily be solved using the well-established machinery of linear algebra. More precisely, using the spectral decomposition theorem, one can represent any symmetric real matrix as λ is the largest eigenvalue of W and e is the corresponding eigenvector E equals zero. However, as the genetic interaction screens are bound to present with experimental variation, missing data, and hopefully also with significant genetic interactions, the estimation problem must in practice be solved by numerical means.As an initial study objective, we sought to assess the accuracy to which the single-mutant fitness vector W to provide a series of estimates for w as solutions of the weighted least squares problem In the present work, we developed a sequential matrix approximation procedure, which uses increasingly larger subsets of mutation pairs in ijε and the ranked deviations p<10−11), whereas the functional enrichment was exceptionally low among the 50 pairs with the lowest p = 0.998). These results show that the sequential matrix approximation procedure gives as its by-product a ranking of the mutation pairs that is in good agreement with their likelihood of sharing a specific function.Beyond the dynamic variability in the estimates of the single-mutant fitness values, the behavior of the approximation procedure with different subsets of mutation pairs revealed another interesting observation: the order in which the mutation pairs were added into the approximation process reflects on average the relative order of their actual measured deviations, even if the measurements of single-mutant fitness were not employed . In factp = 0.003). The prediction capability was improved systematically at each specificity-level, demonstrating that the procedure could distinguish with high accuracy the functionally related gene pairs over the whole spectrum of the exceptional deviations (p = 0.002). The prediction accuracy of the double-mutant fitness values alone was similar to that of a random classifier (AUC = 0.506), demonstrating that the normalization by the measured or estimated single-mutant fitness values can, in any case, improve the prediction of the functional links. The estimates achieved using different subsets of mutation pairs can also lead to different degrees of prediction accuracy . Although the different measures of location and dispersion gave relatively similar results, the trimmed mean and interquartile range identified the distribution with least background bias and variability at k = 136  with lth step of the procedure. Although the general formulation in Eq. 3 allows for an element of Cl) is used at step l. To select increasingly large subsets of non-missing and non-diagonal mutation pairs from the double-mutant fitness matrix W, we adapted the floating search method of Pudil et al. The subset of mutation pairs used at a particular step of the approximation process was encoded in the binary weight matrix l←0 and initialize the weight matrix C(0) (see the next subsection)Set ŵl)Estimate While there are non-diagonal and non-missing entries with i, j) with select pair (ŵl) with if there exists a pair (l←l+1 and repeat step 3set tl) = 2×10−5 in the present work) or from a poor initialization of the weight matrix (see the next subsection for details), and it also made it possible to increase the size k of the subsets up to the maximal size K. K = 323 mutation pairs in the dataset of St Onge et al. We modified the general subset search method by making the deletion of pairs an adaptive process over the evolution of the subset search. More specifically, the threshold Cl) = 0 for the dataset under analysis. After calculating the initial residuals i) for each i, we set i, j) with the smallest residual C(0) non-zero; (ii) we made an additional setting of C(0); and finally (iii) we set pairs  with C(0) became non-zero. Every time a pair  was added in the initialization steps (i)–(iii), we also set its transpose entry to one, that is, C(0) with 28 entries of ones.The approximation algorithm requires the weight matrix ijmeasured deviations ε = ijw−iwjw were estimated in two different ways. The sequential approximation procedure gives as its by-product a surrogate for the deviations in the form of the residual errors of Eq. 4. More specifically, we defined the ranked deviationi, j) as its residual error l during which the pair was included into the approximation subset. In this way, even when there are multiple inclusions and deletions of a particular pair during the procedure, we obtained an unambiguous ranking of the pairs according to their The k, and by using the estimate of Eq. 3 in place of the measured single-mutant fitness vector w in the definition of the deviation in Eq. 1. For a mutation pair , this resulted in a sequence of estimated deviationsl at which the size of the approximation subset equals k. Note that since the residual errors in Eq. 4 are updated each time that a new pair is added, the estimated deviations can vary considerably as a function of k. These estimates are not generally congruent with the ranked deviations. The distributions of the estimated deviations with different subset sizes k are illustrated in The second set of estimates was obtained by stopping the sequential approximation procedure at a given subset size, S. cerevisiae. The detailed time course fitness measurements were performed in the presence and absence of the DNA-damaging agent methyl methaneusulfonate (MMS). The results of the present study were based on the growth measurements in the absence of MMS. The prediction of functionally related gene pairs was in fact more challenging in this case than in the data measured in the presence of MMS. The measured and estimated single-mutant fitness values and the double-deletion deviations in the dataset are shown in To evaluate the performance of the sequential matrix approximation procedure in practice, we applied it to a recent high-resolution genetic interaction study of St Onge et al. Functional links among the 26 genes were defined using the same approach as in many previous genetic interaction studies t is the number of top mutation pairs selected according to their residual errors, and M is the total number of the functionally-related links, then the probability of obtaining at least m functionally-related pairs when selecting pairs at random from the set of K mutation pairs can be calculated using the cumulative distribution function:Statistical enrichment of the specific functional links among a set of mutation pairs selected by the sequential approximation procedure was assessed using the standard hypergeometric test. Briefly, if M = 35 functionally-linked pairs among the t = 50 mutation pairs selected on the basis of either small (m = 1) or large (m = 22) residual errors is shown in M/K = 35/323 = 0.108.The enrichment for the U-statistics to calculate an estimated covariance matrix and hence it can also take into account the correlated nature of the data. The ROC curves and the corresponding AUC values for the prediction of the 35 functional links in the dataset of St Onge et al. The predictive power of the measured and estimated deviations was assessed using the receiver operating characteristic (ROC) curves that characterize the relative trade-off between the true positive rate (sensitivity) and the false positive rate (1 - specificity). The overall predictive performance was summarized using the area under the ROC curve (AUC). The statistical significance of the difference in the AUC values between two genetic interaction measures was assessed using a custom written algorithm based on the method of DeLong et al. Figure S1Estimated vs. measured single-mutant fitness values. The comparison is shown both as histogram and scatter-plot. The two fitness values were highly correlated . The estimated values were calculated at the cut-off point k = 317, in which the approximation procedure used all but the diagonal and missing entries of the double-mutant fitness matrix and also omitted those six pairs with the most extreme residual errors . This point can approximately be identified from the sharp increase in the trace of approximation error ((0.01 MB PDF)Click here for additional data file.Figure S2Scatter-plot between the measured and ranked deviations. The ranked deviations were highly correlated with the true measured deviations over all of the mutation pairs . The inset shows the five synthetic lethal mutation pairs. The dotted diagonal line corresponds to the one-to-one correspondence between the two deviations.(0.01 MB PDF)Click here for additional data file.Figure S3Distributions of the measured and estimated deviations. The non-scaled version of the (0.02 MB PDF)Click here for additional data file.


Poecilia formosa) is a small unisexual fish that has been suspected of being threatened by extinction from the stochastic accumulation of slightly deleterious mutations that is caused by Muller's ratchet in non-recombining populations. However, no detailed quantification of the extent of this threat is available.The Amazon molly  asexual species. Poecilia formosa, the Amazon molly.It is a general observation that asexual lineages do not last over very long periods of time, but the precise reasons for this are less clear . AncientQuantifications of the threat of extinction from Muller's ratchet are often not trivial theoretical work that requires either challenging mathematics  or complPoecilia formosa was the first unisexual vertebrate that was discovered . Howd models ,19. To sPoecilia mexicana or Poecilia latipinna as a mechanical trigger to start development . Wh. Whs )dominant mutations as follows. If and only if the ratchet can fix the first mutation at a site  then it will eventually also fix the second mutation . If a heterozygote cannot be fixed, there is no way the homozygote state can be fixed and the effective mutation rate used to compute the speed of the ratchet should be reduced accordingly to exclude such sites.We can treat recessive deleterious mutations, there are the two following extremes. If s is small (most mutations), then click times are hardly affected and almost no correction is necessary. If s is large enough to prevent fixation of heterozygotes, then again the effective mutation rate may be reduced, as fixation of the homozygous state is impossible. Only few intermediate s can become heterozygous but not homozygous, as they have to be close to the switch-like transition between 'can accumulate' and 'cannot accumulate' . This equation assumes that (i) recombination is free between sites, so that the ratchet only operates at homozygous sites, since selfing can restore the best class on a heterozygous site, (ii) Usdm, d is constant and independent from the number of mutations per individual, (iii) mutations at heterozygous sites are negligible, since selfing produces many more homozygotes, (iv) the standard model of Muller's ratchet without back mutations is valid [N0, can be computed by adding up all heterozygotes that are free from homozygous deleterious mutations and then scaling by the effective population sizewhere is valid ,104. TheUsing the well known result thatequations (4) – (6) yieldh = 0) this reduces to the haploid asexual case  that the degree of dominance of mutations with small selection coefficients does not significantly affect numerical values and hence the rate of the ratchet. Since the ratchet does not click for large s, one might as well neglect dominance altogether.For complete recessivity . To arrive at such an estimate we obtain corresponding diversity data from the control region of mtDNA, since it is linked to the rest of the genome by unisexual inheritance (see the Models section above for a justification of this approach). We use the DNA sequence diversity data from Möller , where Ne is the effective population size of mitochondrial DNA in the presence of background selection, μbp is the mutation rate per site/generation and a = 1 + 1/2 + 1/3 + ... + 1/(n-1) for n sequences in the sample. Here we assume an mtDNA divergence rate of 3.6% ± 0.46%/site/Myr as estimated in the species groups of the fish Centropomus [Ne between 153,000 and 198,000 that will have to be scaled downward, if actual mutation rates are higher, like in human pedigrees [For our simulations we need an estimate of the effective population size in the presence of background selection (nd 55 in ), who obterson's  equationtropomus  from divtropomus . This suedigrees . The nucedigrees ,116.Nt may be used to place upper bounds on Ne in most situations. Some mark and recapture experiments could only estimate that Nt > 10,000 in that local study area . One may speculate that about 10 million individuals of P. formosa may exist in total. If this is combined with the typical finding that vertebrates frequently have Ne-values that are about 10% of their census population sizes [Ne would be about a million. Given that a species as abundant as the fruitfly Drosophila melanogaster has effective population sizes of little more than a million [Ne < 106. On the other extreme it is highly unlikely that the Amazon molly would have an effective population size smaller than that of primates, suggesting Ne > 104.Comparative analyses suggest that the resulting values are reasonable. The total population size on sizes , then Ne million , suggestNe = 104 to 107 to demonstrate that population size has a rather small influence on the operation of Muller's ratchet in the presence of a distribution of mutational effects.We use deliberately wide upper and lower bounds of Tgen is between 4 month (min) and 2.5 years (max), so we assume 1 year as our middle value. The maximal reproductive capacity in the wild with a non-degraded genome Rmax is assumed to be 50 (min) 500 (mid) and 2000 (max) offspring per lifetime (see below). While Tgen scales extinction times linearly, Rmax has a much smaller effect. It is possible that our values for Rmax are too large since they are more based on the highest possible values conceivable under laboratory conditions than on any real life survival situation. Once such data are available, results from Figure Based on observations we estimate that generation time P. formosa comes from extensive aquarium cultures. In captivity the Amazon molly reaches maturity between 4 and 6 months, produces more offspring at larger body sizes and may live for up to 3 years under optimal laboratory conditions. Here we report that females caught in the wild contain a mean of 28 embryos  with the maximal number of embryos per period observed in the wild .Unfortunately there is very little information about life history from natural habitats. Most of the knowledge about  the lab ). Femalehis size ). We didPoecilia mexicana female and a Poecilia latipinna male [Tage for this event may be derived from comparing alleles in the Amazon molly with alleles in their corresponding parent species using a molecular clock. At the moment there are two datasets that are large enough to infer an evolutionary age that is different from zero. Comparisons of 1,377 bp in various nuclear genes (mostly introns) lead to a point estimate of Tage = 100,000 years assuming a divergence rate of 2%/Myr from standard mitochondrial clocks [Tage = 8,930 to Tage = 209,000 years due to the small number of substitutions observed [Ne above [P. formosa and P. mexicana and that this haplotype is at the center of P. formosa diversity ; lower and upper limits range from 63,000 to 104,000 years). This time will have to be scaled downward, if actual mutation rates are higher, as has been found in human pedigrees [Tage = 40 Kyr, 70 Kyr and 100 Kyr as estimates of the age of the Amazon molly in Figure The ancestor of the Amazon molly clone known to us was formed by a hybridisation event between a nna male ,12 and il clocks . This agobserved ,70. Thus. Since many of these mutations will be synonymous or affect only non-functional DNA, we have to scale this rate by the effective genome size [Drosophila melanogaster as a lower limit .-10/site/cell replication, respectively [P. formosa experience about 25 germ line cell divisions per adult generation (as female mice [Utot ≈ 0.4 potentially deleterious mutations/haploid genome/generation. Obviously, P. formosa is not a mouse and there is much room for improving the precision of such mutation rate estimates, even in mice. Recent high-precision measurements of the deleterious mutation rate in mutation accumulation experiments in Drosophila found Utot ≈ 0.6 deleterious mutations/haploid genome/generation [Utot ≈ 0.7.These estimates do not change much, even if we use a completely different way of obtaining the mutation rate. Observations in mice have lead to estimates of 1.8 × 10ectively . If an eale mice  that havneration . ScalingIt is encouraging to find such general agreement between these estimates that are based on very different approaches. This suggests that we can have some confidence in our estimates, even in the absence of direct observations in the Amazon molly. Since mutations are the inevitable consequence of DNA replication errors and the number of mitotic cell divisions per generation is a key determinant of mutation rates per generation ,155, we KA/KS, to infer that the fraction 1-KA/KS of all mutations must be more deleterious than about s = 1/Ne, since mutations with effects below that threshold accumulate as if they were neutral [Ne ≈ 10-6 [fsdm, could be perhaps around 30%, probably larger than 10%, but probably not much larger than 50%. We use these values as a point estimate with lower and upper bound. Thus Usdm probably shares the same order of magnitude with Utot. As this calculation has large errors, an array of other mutation rates is included as well when quantifying the ratchet. We assume that selection coefficients did not change much for most genes in the time since asexuality arose in the Amazon molly. This is compatible with the assumption that most genes in the Amazon molly are generic to similar fish species and thus well adapted, while only a small fraction is actually responsible for the specific adaptations of the Amazon molly.Recent progress has shown that the distribution of non-synonymous mutational effects is very leptokurtic and spans many orders of magnitude in many different species ,110-112. neutral ,156. If e ≈ 10-6 . Robust e ≈ 10-6 ,47. FromUtot ≈ 0.4 and Utot ≈ 0.9, while our best estimate of the fraction of mutations with critical slightly deleterious effects is expected to be between fsdm ≈ 10% and fsdm ≈ 50%. Combining these values makes us expect Usdm ≈ 0.2 slightly deleterious mutations with critical effects/haploid genome/generation with a lower limit of Usdm ≈ 0.04 and an upper limit of Usdm ≈ 0.45. These values may have to be doubled, depending on the genome model that is used for accounting for the effects of diploidy. To err on the side of caution and provide a better feeling for the effects of mutation rates we plot values from Usdm = 0.01 to Usdm = 1 in Figure As discussed above, our best estimate for the total mutation rate at potentially deleterious sites per haploid genome per generation is expected to be between LL conceived the study, wrote the software, managed the computation of simulation results, developed the theoretical aspects of the paper, estimated many parameters, analyzed the results and wrote the manuscript. DKL provided most details on the Amazon molly biology, contributed to parameter estimation and helped writing the manuscript. Both authors read and approved the final manuscript.


A novel method to infer the number and fitness effect of beneficial mutations reveals that the bulk of adaptive evolution is attributable to a few mutations with variable effects on fitness. Aspergillus nidulans evolving for approximately 800 generations at two population sizes using a novel maximum likelihood framework, the results of which were confirmed experimentally using sexual crosses. We find that adaptive walks do indeed tend to be short, and fitness increases become smaller as successive mutations fix. Moreover, we show that these patterns are associated with a decreasing supply of beneficial mutations as the population adapts. We also provide empirical distributions of fitness effects among mutations fixed at each step. Our results provide a first glimpse into the properties of multiple steps in an adaptive walk in asexual populations and lend empirical support to models of adaptation involving selection towards a single optimum phenotype. In practical terms, our results suggest that the bulk of adaptation is likely to be accomplished within the first few steps.The rarity of beneficial mutations has frustrated efforts to develop a quantitative theory of adaptation. Recent models of adaptive walks, the sequential substitution of beneficial mutations by selection, make two compelling predictions: adaptive walks should be short, and fitness increases should become exponentially smaller as successive mutations fix. We estimated the number and fitness effects of beneficial mutations in each of 118 replicate lineages of  adaptive walk, and used this to test predictions about the properties of adaptive walks in experimental populations of fungus. Our work shows that, in contrast to the gradualist view of adaptation dominant since the 1930s, adaptive walks tend to be fast and short, with beneficial mutations of large effect substituted first, followed by those of smaller effect.Adaptation is one of the least understood processes in biology because it relies on beneficial mutations, which are often too rare to study. We developed a method to infer the number and size of beneficial mutations substituted during adaptation, a process called an  Although recent work with microbial populations has shed light on the properties of a single bout of adaptation The rate and extent of adaptive evolution over long time periods depends ultimately on the sequential substitution of beneficial mutations by natural selection, a process termed an N) and mutation rate (μ), is high, because clonal interference becomes increasingly important The dominant view championed since the 1930s by Fisher through his geometric model of adaptation has been that adaptive walks are driven by the substitution of many mutations with exceedingly small fitness effects With these caveats in mind, models of adaptation make three compelling predictions about the nature of the distributions of mutational effects over the course of an adaptive walk. First, the combined effects of genetic drift, which tends to eliminate small-effect mutations, and clonal interference transforms the typically L-shaped distribution of fitness effects available to selection into a distribution of fitness effects among fixed mutations that is unimodal with a substantially higher mean Aspergillus nidulans. This species forms spatially structured colonies on solid surfaces, making it relatively easy to detect beneficial mutations  of replicate lineages utations . We founAssumptions of the modelWe assume a strictly clonal population growing exponentially at rates that do not vary in time or depend on the frequencies of other clones. We assume that during propagation, clones are serially transferred to fresh medium so there is no density dependence. We assume that the minimum population size during each bottleneck is large enough to not affect the trajectories of allele frequencies between bottlenecks see .Derivation of the likelihood functionr =  and times of origin t =  of up to n clones  from data consisting of fitness estimates (observations) collected at k spaced time points during the adaptation of a single population. The likelihood of the data D =  is thenE[ . ] is the expectation over all possible clones present at each time point in the population assayed and prob denotes the probability of observing a fitness iw at time point iT given a collection of n clones with associated fitness r1, r2,…, nr. The model is very general and allows for clonal competition and could readily be extended to include a stochastic component for extreme bottlenecking or various forms of density dependence.In general, we wish to estimate a vector of Malthusian parameters Parameter estimation and hypothesis testingr and t  is used to decide which model (nM), and thus how many beneficial mutations, best fit the data in each population. Note that although the AIC criterion does not ensure that beneficial mutations in the final model have reached strict fixation (frequency = 1), simulations evaluating the time to reach quasifixation (defined as 1−1/(e sN); see We maximize numerically this likelihood with respect to nd t see  and thert110 = 2.10, p = 0.038). The variance in fitness among lineages follows a similar diminishing returns pattern but accumulates faster in the small bottleneck treatment , reaching comparable levels in both treatments by the end of the experiment . These results lend support to the idea that natural selection in populations of different size tends to lead to different effective fitness plateaus The fitness trajectories of all 118 lineages are shown in t-tests for all lineages between times t and t+1 using a false discovery rate criterion of α = 0.3 to determine significance), as would be expected if mildly deleterious mutations had a higher probability of increasing in frequency following each transfer.Occasional fitness decreases occurred in both treatments although these were modest in size relative to the initial fitness increase at the first transfer and not sustained over multiple transfers. This variation is likely due to uncontrollable microenvironmental variation in environmental conditions arising during the selection experiment. Alternative explanations include frequency-dependent selection or drift causing a transient increase in the frequency of mildly deleterious mutations. Frequency-dependent selection is a priori unlikely because spatial structure, which facilitates frequency dependent selection by allowing persistent interactions among genotypes In previous work, the fungicide-sensitive strain from which the founder of our experiments was derived showed little evidence of adaptation to the same growth conditions used here, suggesting that the sensitive strain resides close to a fitness optimum n clones with fitness values r1, r2,…, rn substituting in the population. Comparing the fit of the models to the data allowed us to estimate how many beneficial mutations arose in each lineage and the fitness effects associated with each new mutation. This procedure assumes an exponential model of population growth, and we provide experimental evidence to support this assumption in i+1 does not necessarily arise in the genetic background of clone i.Instead of relying on the fitting of fitness trajectories by step functions n = 100,000 permutations comparing the absolute value of the difference in coefficients of variation, d, between two distributions: d = 0.387, p<0.0001), the small bottleneck treatment appearing more L-shaped than in the large bottleneck treatment. This result suggests that weaker clonal competition arising from smaller population sizes leads to the fixation of more small-effect mutations and the fixation of mutations with a wider range of effects. Interestingly, the differences between treatments at steps 2 and 3 are not significant , implying that clonal competition becomes less important in shaping the distributions of fixed effects as the population adapts. Further support for this interpretation comes from the fact that significant differences were observed between the first and second steps in the large bottleneck treatment , but not between the second and third steps , nor between any of the steps in the small bottleneck treatment . Inspection of Z = 3.15, p = 0.0016, nlarge = 58, nsmall = 60), consistent with the higher probability of losing beneficial mutations by drift in smaller populations. Mutations restoring sensitivity to fludioxonil were rare in our experiment, being observed in just five of the 118 lineages. In two lineages from the small bottleneck treatment, sensitivity was restored by a single mutation. The remaining three fludioxonil-sensitive lineages  substituted multiple mutations that resulted in restored sensitivity. Removing these lineages from the analysis does not change our results. Thus, the short walks we observed are not merely due to the predominance of back mutations of the resistance mutation (fldA1). Moreover, we were able to detect significant genetic (GV) and genotype-by-environment interaction (GEV) variation among evolved lineages in both bottleneck treatments  when grown across a concentration gradient of fludioxonil  and a long right tail representing putatively accessible beneficial mutations. More striking is the relationship between mean fitness of the evolving population and the fraction of beneficial mutations available to selection, which is negative . Importantly, even in our small bottleneck treatment, N is unlikely to be so low as to prevent sampling of a large number of single-step mutations, as required by theory Interestingly, our conclusions are not much altered by the bottleneck size used between serial transfers. This result is attributable to the opposing effects of drift and clonal competition in small and large populations. In small populations, only mutations of moderate to large effect are likely to escape stochastic loss when rare, and those that do are nearly guaranteed to fix due to the absence of clonal competition. In large populations, more mutations of small effect will reach appreciable frequencies but are eventually lost due to clonal competition. The result is that the mean fitness effect of mutations that fix under different population sizes remains approximately the same at each step throughout the course of an adaptive walk. By contrast, population size does affect the mean fitness among lineages achieved by the end of the experiment, with small populations having lower fitness than large populations by the end of the experiment. This observation is consistent with our finding of fewer fixed mutations in small compared to large populations, which likely stems from changes in the mutation supply rate, As with most selection experiments, we have little direct information about the topography of the adaptive landscape; however, two results suggest that it may be fairly rugged. First, fitness tends to plateau at different levels in most lineages by about generation 500, and these fitness differences are maintained throughout the experiment The shortness of the adaptive walks is a striking feature of our results that stands in contrast to modestly longer walks of three to six steps documented in bacteria Second, it may be argued that short adaptive walks are expected a priori because we are studying compensatory evolution, that is, the recovery of fitness due to the presence of a costly resistance mutation Why did we observe shorter adaptive walks than previous experiments? One possibility is that our founding strain was already well adapted to the conditions of growth and so had only a few beneficial mutations available to it. This explanation is difficult to reconcile, however, with the observation of substantial genotype-by-environment interaction mentioned above, the apparently large fraction of beneficial mutations available to the ancestor  and a gaThat the relationship between the mean fitness increases with each successive step in an adaptive walk is negative suggests two important insights into the mechanics of adaptive evolution. First, the distribution of fitness effects among beneficial mutations prior to selection is expected to lie in the Weibull domain of attraction Our results have two important practical implications. First, if the environment changes regularly, populations may be expected to adapt perpetually, and it will be difficult to make strong predictions as to the outcome, for example, of long-term changes to the environment such as those expected to occur under many climate change scenarios. Second, compensatory evolution in response to losses in fitness due to mutations conferring resistance to fungicides, antibiotics, or other chemotherapies, is likely to be fast, requiring only a few mutational steps. More fundamentally, the picture emerging from these experiments is that adaptive evolution in mutation-limited populations can be readily understood from rather simple models that, because they are largely based on the statistical properties of extreme events, are likely to be robust to biological details.A. nidulans strain WG615  for the selection experiment, kindly provided by Fons Debets and Marijke Slakhorst at Wageningen University, the Netherlands. This strain is resistant to the fungicide fludioxonil (fldA1). Resistance confers a cost of around 46% relative to the sensitive strain from which it was derived when growing in the absence of fungicide yA1, veA1), which has the same genetic background as WG615. In the selection experiment, spores and mycelium were washed from the plate using 5 ml of saline-Tween (water containing NaCl 0.8% and Tween-80 0.05%), reserving 5 µl for transfer to fresh medium and storing 0.8 ml of the mixture mixed with 0.3 ml 80% glycerol at −80°C. Bottleneck sizes were adjusted by dilution. For all experiments, we used solid Complete Medium (CM) set at pH 5.8, consisting of NaNO3 6.0 g/l; KH2PO4 1.5 g/l; MgSO4.7H2O 0.5 g/l; NaCl 0.5 g/l; 0.1 ml of a saturated trace element solution containing FeSO4, ZnSO4, MnCl2, and CuSO4; tryptone 10 g/l; and yeast extract 5 g/l and (added after autoclaving) glucose 4.0 g/l. Cultures were incubated at 37°C.We used A. nidulans strain WG615 by placing 5 µl of a dense spore suspension  in the center of a Petri dish containing solid CM medium. We propagated these replicate lineages by serial transfer for a total of 800 generations. Every 5 d,  as the colony diameter  after 5 d of incubation, which is widely used as a measure of fitness and is highly correlated with other fitness measures such as total spore production and biomass r = 0.81, t5 = 3.09, p = 0.027). Selection coefficients were calculated as s = (iw−wancestor)/wancestor. Each lineage was assayed in triplicate at multiple time points  in a single assay. Two lineages from the large bottleneck treatment were eliminated from the maximum likelihood analysis due to missing data. For 26 evolved strains that had a high relative fitness, we reassayed the fitness at 800 generations using at least seven replicates together with the fungicide-sensitive strain WG638. We estimated fitness of all lineages at the end of the selection experiment in CM supplemented with 0, 0.05, 0.2, or 0.4 ppm of fludioxonil in duplicate and used an analysis of covariance to test the main effect of lineage (genetic variance) and the interaction between lineage and fludioxonil concentration (genotype-by-environment interaction variance).We measured fitness of the fldA1, lysB5, veA1). We estimated the fitness of at least 50 progeny for each of 15 crosses and used the Castle-Wright estimator We performed sexual crosses 2 of mycelium including spores at three preassigned locations on the edge of the colony. Fitness was estimated for each spore sample, and the fraction of samples with fitness greater than that of the parent genotype was calculated for each time point. We assayed fitness in blocks and used strains WG615 and WG638 as reference strains to account for variation between blocks. Logistic regressions were performed in R version 2.6.1 using generalized linear models with binomial errors. We controlled for false positives, which would inflate our estimates of the fraction of beneficial mutations, using a false discovery rate analysis We initiated replicate populations of genotypes of interest by placing 5 µl of a dense spore suspension in the center of a Petri dish. After 5 d of colony expansion, we sampled approximately 3 mmFigure S1Fitness of 28 evolved lineages after 800 generations (grey bars), the ancestor that founded the selection experiment, and a strain that is fungicide sensitive but has the same genetic background. Error bars show 95% confidence intervals. The founding strain of the evolved lineages carries a fungicide resistance mutation that is costly under the growth condition used here  revealed that at least three strains have significantly higher fitness than the sensitive strain. (0.56 MB TIF)Click here for additional data file.Figure S2Relationship between the number of nuclei present in a mycelium (ln transformed number of colony forming units [CFU]) as a function of the size of the fungal colony (colony diameter in millimeters).(0.00 MB EPS)Click here for additional data file.Figure S3Mutations available to selection. Fraction of mutations sampled available to selection, scaled to the fraction available to the ancestor. Click here for additional data file.Figure S4Average time to quasifixation of beneficial mutations as a function of their selection coefficient and the size of the population see . Dots denote the average of simulations tracking the time to fixation or loss of 10,000 new mutations with selection coefficient s. Continuous lines are a diffusion-based approximation for the average time to quasifixation, assuming the strong selection limit .(0.02 MB EPS)Click here for additional data file.Table S1Comparison of number of loci contributing to adaptation estimated with Caste-Wright estimator and maximum likelihood. Number of segregating loci (en) in a cross between evolved genotypes and the non-evolved ancestor, using the Castle-Wright estimator (0.04 MB DOC)Click here for additional data file.Text S1Likelihood framework for estimating selection coefficients of beneficial mutations spreading in each population from fitness trajectory data.(0.06 MB DOC)Click here for additional data file.Text S2Experimental evidence for exponential population growth.(0.03 MB DOC)Click here for additional data file.Text S3Estimation of the number of segregating loci using the Castle-Wright estimator.(0.03 MB DOC)Click here for additional data file.Text S4Comparison between mycelial growth rate and competitive fitness.(0.03 MB DOC)Click here for additional data file.