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W3035883615 abstract "Article Figures and data Abstract eLife digest Introduction Results Discussion Materials and methods Data availability References Decision letter Author response Article and author information Metrics Abstract Fitness effects of mutations depend on environmental parameters. For example, mutations that increase fitness of bacteria at high antibiotic concentration often decrease fitness in the absence of antibiotic, exemplifying a tradeoff between adaptation to environmental extremes. We develop a mathematical model for fitness landscapes generated by such tradeoffs, based on experiments that determine the antibiotic dose-response curves of Escherichia coli strains, and previous observations on antibiotic resistance mutations. Our model generates a succession of landscapes with predictable properties as antibiotic concentration is varied. The landscape is nearly smooth at low and high concentrations, but the tradeoff induces a high ruggedness at intermediate antibiotic concentrations. Despite this high ruggedness, however, all the fitness maxima in the landscapes are evolutionarily accessible from the wild type. This implies that selection for antibiotic resistance in multiple mutational steps is relatively facile despite the complexity of the underlying landscape. eLife digest Drug resistant bacteria pose a major threat to public health systems all over the world. Darwinian evolution is at the heart of this drug resistance: a mutation that allows bacteria to divide in the presence of a drug appears initially in a single cell. This mutation makes this cell and its descendants more likely to survive, so they can end up taking over the population. The evolution of resistance can be thought of in terms of ‘bacterial fitness landscapes’. These landscapes visualise the relationship between the mutations present in a population of bacteria and how quickly the bacteria divide or reproduce. They are called landscapes because they can be represented as a series of mountains and valleys. The peaks of this landscape represent combinations of mutations that give bacteria the greatest chance of dividing (the greatest fitness). In a landscape with multiple peaks, some peaks will be higher than others. If the landscape is smooth, bacteria can easily acquire mutations for drug resistance. However, in a rugged landscape, bacteria may get stuck at sub-optimal peaks, because the mutations that would enable them to reach a higher peak would first lead them to losing fitness. Several studies on the evolution of antibiotic resistance exist for specific bacteria and specific drugs, but relatively little is known about the general properties of the underlying fitness landscapes. Do these landscapes have features that can help explain the rapid evolution of high levels of resistance? Antibiotic resistance often comes at a cost – more resistant strains of bacteria tend to grow more slowly when the drug is absent. To build a model of antibiotic resistance landscapes, Das et al. performed growth experiments on several strains of Escherichia coli exposed to a drug called ciprofloxacin. They measured how the rate at which the bacteria divided changed at different antibiotic concentrations, and combined this with the observation about resistant strains growing slower to formulate a mathematical model of antibiotic resistance landscapes. The landscapes that resulted were found to be very rugged, but unexpectedly, the bacteria could still evolve to access all fitness peaks. This means that landscape ruggedness does not constrain the evolution of resistance. Understanding how and when resistance evolves is important both for the design of new drugs and the development of treatment protocols. A specific prediction of the model is that resistance evolution in fitness landscapes where resistant strains divide more slowly is reversible. This implies that the bacteria could regain their susceptibility to treatment when the drug concentration decreases, but this would depend on the specific bacteria and drug in question. More broadly, the model provides a framework for addressing the evolution of resistance in clinical and environmental settings, where drug concentrations vary widely in time and space. Introduction Sewall Wright introduced the concept of fitness landscapes in 1932 (Wright, 1932), and for decades afterwards it persisted chiefly as a metaphor, due to lack of sufficient data. This has changed considerably in recent decades (de Visser and Krug, 2014; Hartl, 2014; Kondrashov and Kondrashov, 2015; Fragata et al., 2019). There are now a large number of experimental studies that have constructed fitness landscapes for combinatorial sets of mutations relevant to particular phenotypes, such as the resistance of microbial pathogens to antibiotics (Weinreich et al., 2006; DePristo et al., 2007; Marcusson et al., 2009; Lozovsky et al., 2009; Brown et al., 2010; Schenk et al., 2013; Goulart et al., 2013; Mira et al., 2015; Palmer et al., 2015; Knopp and Andersson, 2018), and the genomic scale of these investigations is rapidly growing (Wu et al., 2016; Bank et al., 2016; Domingo et al., 2018; Pokusaeva et al., 2019). Mathematical modeling of fitness landscapes has also seen a revival, motivated partly by the need to quantify and interpret the ruggedness of empirical fitness landscapes (Szendro et al., 2013; Weinreich et al., 2013; Neidhart et al., 2014; Ferretti et al., 2016; Blanquart and Bataillon, 2016; Crona et al., 2017; Hwang et al., 2018; Kaznatcheev, 2019; Crona, 2020). Conceptual breakthroughs, such as the notion of sign epistasis (where a mutation is beneficial in some genetic backgrounds but deleterious in others), have shed light on how ruggedness can constrain evolutionary trajectories (Weinreich et al., 2005; Poelwijk et al., 2007; Franke et al., 2011; Lobkovsky and Koonin, 2012; Zagorski et al., 2016). Despite this progress, a limitation of current studies of fitness landscapes is that they focus mostly on G×G (gene-gene) interactions, and little on G×G×E (where E stands for environment) interactions, that is on how changes in environment modify gene-gene interactions. A few recent studies have begun to address this question (Flynn et al., 2013; Taute et al., 2014; Gorter et al., 2018; de Vos et al., 2018). In the context of antibiotic resistance, it has been realized that the fitness landscape of resistance genes depends quite strongly on antibiotic concentration (Mira et al., 2015; Stiffler et al., 2015; Ogbunugafor et al., 2016). This is highly relevant to the clinical problem of resistance evolution, since concentration of antibiotics can vary widely in a patient’s body as well as in various non-clinical settings (Kolpin et al., 2004; Andersson and Hughes, 2014). Controlling the evolution of resistance mutants thus requires an understanding of fitness landscapes as a function of antibiotic concentration. Empirical investigations of such scenarios are still limited, and systematic theoretical work on this question is also lacking. In the present work, we aim to develop a theory of G×G×E interactions for a specific class of landscapes, with particular focus on applications to antibiotic resistance. The key feature of the landscapes we study is that every mutation comes with a tradeoff between adaptation to the two extremes of an environmental parameter. For example, it has been known for some time that antibiotic resistance often comes with a fitness cost, such that a bacterium that can tolerate high drug concentrations grows slowly in drug-free conditions (Andersson and Hughes, 2010; Melnyk et al., 2015). While such tradeoffs are not universal (Hughes and Andersson, 2017; Durão et al., 2018), they certainly occur for a large number of mutations and a variety of drugs. Tradeoffs can also arise in complex scenarios involving multiple drugs. It has been reported in Stiffler et al., 2015 that certain mutations in TEM-1 β-lactamase are neutral at low ampicillin concentration but deleterious at high concentration, and that a number of the latter mutations also confer resistance to cefotaxime. Therefore in a medium with cefotaxime and a moderately high concentration of ampicillin, it is possible that these mutations will be deleterious at low cefotaxime concentrations but beneficial at high cefotaxime concentration. Fitness landscapes with adaptational tradeoffs are therefore also of potential relevance to evolution in response to multi-drug combinations. Our starting point for investigating fitness landscapes induced by tradeoffs is the knowledge of two phenotypes that are well studied – the drug-free growth rate (which we call the null-fitness) and the IC50 (the drug concentration that reduces growth rate by half), which is a measure of antibiotic resistance. These two phenotypes correspond to the two extreme regimes of an environmental parameter, that is zero and highly inhibitory antibiotic concentrations. The function that describes the growth rate of a bacterium for antibiotic concentrations between these two extremes is called the dose-response curve or the inhibition curve (Regoes et al., 2004). When tradeoffs are present, the dose-response curves of different mutants must intersect as the concentration is varied (Gullberg et al., 2011). This is schematically shown in Figure 1. The intersection of dose-response curves of the wild type and the mutant happens at point A, swapping the rank order between the two fitness values. The intersection point is known as the minimum selective concentration (MSC), and it defines the lower boundary of the mutant selection window (MSW) within which the resistance mutant has a selective advantage relative to the wild type (Khan et al., 2017; Alexander and MacLean, 2018). Figure 1 Download asset Open asset Schematic showing dose response curves of a wild type and a mutant. To the left of the intersection point A the wild type is selected over the mutant, whereas to the right of A the mutant is selected. When there are several possible mutations and multiple combinatorial mutants, a large number of such intersections occur as the concentration of the antibiotic increases. This leads to a succession of different fitness landscapes defined over the space of genotype sequences (Maynard Smith, 1970; Kauffman and Levin, 1987). Whenever the curves of two mutational neighbors (genotypes that differ by one mutation) intersect, there can be an alteration in the evolutionary trajectory towards resistance, whereby a forward (reverse) mutation now becomes more likely to fix in the population than the corresponding reverse (forward) mutation. These intersections change the ruggedness of landscapes and the accessibility of fitness maxima. In this way a rich and complex structure of selective constraints emerges in the MSW. To explore the evolutionary consequences of these constraints, here we construct a theoretical model based on existing empirical studies as well as our own work on ciprofloxacin resistance in E. coli. Specifically, we address two fundamental questions: (i) How does the ruggedness of the fitness landscape vary as a function of antibiotic concentration? (ii) How accessible are the fitness optima as a function of antibiotic concentration? We find that even when the null-fitness and resistance values of the mutations combine in a simple, multiplicative manner, the intersections of the curves produce a highly epistatic landscape at intermediate concentrations of the antibiotic. This is an example of a strong G×G×E interaction, where changes in the environmental variable drastically alter the interactions between genes. Despite the high ruggedness at intermediate concentrations, however, the topology of the landscapes is systematically different from existing oft-studied random landscape models, such as the House-of-Cards model (Kauffman and Levin, 1987; Kingman, 1978), the Kauffman NK model (Kauffman and Weinberger, 1989; Hwang et al., 2018) or the Rough Mt. Fuji model (Neidhart et al., 2014). For example, most fitness maxima have similar numbers of mutations that depend logarithmically on the antibiotic concentration. Importantly, all the fitness maxima remain highly accessible through adaptive paths with sequentially fixing mutations. In particular, any fitness maximum (including the global maximum) is accessible from the wild type as long as the wild type is viable. As a consequence, the evolution of high levels of antibiotic resistance by multiple mutations (Hughes and Andersson, 2017; Wistrand-Yuen et al., 2018; Rehman et al., 2019) is much less constrained by the tradeoff-induced epistatic interactions than might have been expected on the basis of existing models. Results Mathematical model of tradeoff-induced fitness landscapes The chief goal of this paper is to develop and explore a mathematical framework to study tradeoff-induced fitness landscapes. We consider a total of L mutations, each of which increases antibiotic resistance. A fitness landscape is a real-valued function defined on the set of 2L genotypes made up of all combinations of these mutations. A genotype can be represented by a binary string of length L, where a 1 (0) at each position represents the presence (absence) of a specific mutation. Alternatively, any genotype is uniquely identified as a subset of the L mutations (the wild type is the null subset, that is the subset with no mutations). In this paper, unless mentioned otherwise, we define the fitness f as the exponential growth rate of a microbial population. The fitness is a function of antibiotic concentration. This function has two parameters of particular interest to us – the growth rate at zero concentration, which we refer to as the null-fitness and denote by r, and a measure of resistance such as IC50 which we denote by m. Each single mutation is described by the pair (ri,mi), where ri and mi are the null-fitness and resistance values respectively of the ith single mutant. We further rescale our units such that for the wild type, r=1 and m=1. We consider mutations that come with a fitness-resistance tradeoff, that is a single mutant has an increased resistance (mi>1) and a reduced null-fitness (ri<1) compared to the wild type. To proceed we need to specify two things: (i) how the fitness of the wild type and the mutants depend on antibiotic concentration, and in particular if this dependence exhibits a pattern common to various mutant strains; (ii) how the r and m values of the combinatorial mutants depend on those of the individual mutations. To address these issues we take guidance from two empirical observations. Scaling of dose-response curves Marcusson et al., 2009 have constructed a series of E. coli strains with single, double and triple mutations conferring resistance to the fluoroquinolone antibiotic ciprofloxacin (CIP), which inhibits DNA replication (Drlica et al., 2009). In their study they measured MIC (minimum inhibitory concentration) values and null-fitness but did not report dose-response curves. Some of the present authors have recently shown that the dose-response curve of the wild-type E. coli (strain K-12 MG1655) in the presence of ciprofloxacin can be fitted reasonably well by a Hill function (Ojkic et al., 2019). Here we expand on this work and determine dose-response curves for a range of single- and double-mutants with mutations restricted to five specific loci known to confer resistance to CIP (Marcusson et al., 2009) (see Materials and methods). Figure 2A shows the measured curves for the wild type, the five single mutants, and eight double-mutant combinations. The genotypes are represented as binary strings, where a 1 or 0 at each position denotes respectively the presence or absence of a particular mutation. If we rescale the concentration c of CIP by IC50 of the corresponding strain, x=c/I⁢C50, and the growth rate by the null-fitness f⁢(0), the curves collapse to a single curve w⁢(x) that can be approximated by the Hill function (1+x4)-1 (Figure 2B). The precise shape of the curve is not important for further analysis in this paper. However, the data collapse suggests that we can assume that the dose-response curve of a mutant with (relative) null-fitness r and (relative) resistance m is (1) f⁢(c)=r⁢w⁢(c/m), that is, it has the same shape as the wild-type curve w except for a rescaling of the fitness and concentration axes. Similar scaling relations have been reported previously by Wood et al., 2014 and Chevereau et al., 2015. A good biological understanding of the conditions underlying this feature is presently lacking, but it seems intuitively plausible that the shape w⁢(x) would be robust to changes that do not qualitatively alter the basic physiology of growth and resistance. Figure 2 Download asset Open asset Dose-response curves for E. coli in the presence of ciprofloxacin. Each binary string corresponds to a strain, where the presence (absence) of a specific mutation in the strain is indicated by a 1 (0). The five mutations in order from left to right are S83L (gyrA), D87N (gyrA), S80I (parC), ΔmarR, and ΔacrR. The names of the strains are given in Table 1. (A) Dose-response curves of the wild type, the five single mutants and eight double mutants. Unlike the experiments reported in Marcusson et al., 2009, the mutants were grown in isolation rather than in competition with the wild type. (B) The same curves, but scaled with the null-fitness and IC50 of each individual genotype. The dashed black line is the Hill function (1+x4)-1. Table 1 Epistasis in null-fitness and MIC for E. coli in the presence of ciprofloxacin The table contains a combinatorially complete subset of the data reported by Marcusson et al., 2009, composed of the four mutations S83L (gyrA), D87N (gyrA), marR, and acrR. The names of the strains and values of null-fitness (in competition assays with the wild type) in the third column and MIC (of ciprofloxacin) in the fifth column are obtained from Marcusson et al., 2009. The binary representations follow the same convention as given in the caption of Figure 2. The fourth and sixth columns are respectively the null-fitness and MIC values expected in the absence of epistasis. NA denotes the cases where this is not applicable. The values in parentheses are error estimates. In the third and fifth columns, the errors in the log⁡(x) are calculated as |Δ⁢x|x, where |Δ⁢x| are the standard error as calculated from the standard deviations reported in the paper. The errors in columns four and six were estimated as ∑i|Δ⁢xi|xi where the sum is over the mutations present in the combinatorial mutants. The detectable cases of epistasis are marked in blue. Negative epistasis is found in all these cases. Also, all the cases with epistasis correspond to two or more mutations that affect the same chemical pathways. StrainStringLog null-fitnessNon-epistaticLog MICNon-epistaticMG1655000000.00 (±0.004)NA0.00 (±0.35)NALM378100000.01 (±0.016)NA3.17 (±0.70)NALM53401000−0.01 (±0.018)NA2.75 (±0.70)NALM20200010−0.19 (±0.020)NA0.69 (±0.70)NALM35100001−0.094 (±0.014)NA1.08 (±0.70)NALM62511000−0.030 (±0.011)0.0 (±0.038)3.17 (±0.70)5.92 (±1.1)LM42110010−0.15 (±0.019)−0.18 (±0.040)4.13 (±0.70)3.56 (±1.1)LM64710001−0.051 (±0.013)−0.084 (±0.034)3.44 (±0.70)4.65 (±1.1)LM53801010−0.19 (±0.020)−0.20 (±0.042)4.13 (±0.70)3.46 (±1.1)LM59201001−0.083 (±0.015)−0.10 (±0.036)3.16 (±0.70)3.83 (±1.1)LM36700011−0.20 (±0.026)−0.28 (±0.038)2.06 (±0.70)1.77 (±1.1)LM69511010−0.24 (±0.017)−0.19 (±0.058)3.85 (±. 70)6.61 (±1.1)LM69111001−0.073 (±0.013)−0.094 (±0.052)3.85 (±. 70)7.00 (±1.4)LM70910011−0.24 (±0.027)−0.274 (±0.054)4.54 (±. 70)4.94 (±1.4)LM59501011−0.51 (±0.051)−0.294 (±0.056)4.54 (±. 70)4.52 (±1.4)LM70111011−0.42 (±0.037)−0.284 (±0.072)4.83 (±. 70)7.69 (±1.8) Limited epistasis in r and m An interesting recent finding reported by Knopp and Andersson, 2018 is that chromosomal resistance mutations in Salmonella typhimurium mostly alter the null-fitness as well as the MIC of various antibiotics in a non-epistatic, multiplicative manner, that is if a particular mutation increases (decreases) the resistance (null-fitness) by a factor k1, and another mutation does the same with a factor k2, then the mutations jointly alter these phenotypes roughly by a factor of k1⁢k2 (with a few exceptions). We have done a similar comparison for the data on the null-fitness and MIC for E. coli strains in Marcusson et al., 2009. We have analyzed a subset of 4 mutations for which the complete data set for all combinatorial mutants is available from Marcusson et al., 2009. The data are shown in Table 1. Out of 11 multiple-mutants, only 2 show epistasis in r and 4 show epistasis in m. Moreover, in all cases where significant epistasis occurs it is negative, that is the effect of the multiple mutants is weaker than expected from the single mutation effects. Formulation of the model The above observations suggest a model where one assumes, as an approximation, that all the r and m values of individual mutations combine multiplicatively. A genotype with n mutations (r1,m1),(r2,m2),…,(rn,mn) has a null-fitness r and a resistance value m given by (2) r=∏i=1nri⁢and⁢m=∏i=1nmi. Moreover, the dose-response curves of the genotypes are taken to be of the scaling form (Equation 1), where the function w⁢(x) does not depend on the genotype. As indicated before, and without any loss of generality, we choose units such that, for the wild type, r=1 and m=1. Therefore the dose-response curve of the wild type is w⁢(x) with w⁢(0)=1, and choosing IC50 as a measure of resistance, we have w⁢(1)=12. Henceforth, we refer to x simply as the concentration. We also recall that the condition of adaptational tradeoff means that ri<1 and mi>1 for all mutations. If the ri and mi values combine non-epistatically, and if the shape of the dose-response curve is known, it is thus possible to construct the entire concentration-dependent landscape of size 2L from just 2⁢L measurements (of the ri and mi values of the single mutants) instead of the measurement of 2L fitness values at every concentration. In practice we do not expect a complete lack of epistasis among all mutations of interest, and the dose-response curve is also an approximation obtained by fitting a curve through a finite set of fitness values known only with limited accuracy. However, the fitness rank order of genotypes, and related topographic features such as fitness peaks, are robust to a certain amount of error in fitness values (Crona et al., 2017), and our model may be used to construct these to a good approximation. Lastly, we require that the dose-response curves of the wild type and a mutant intersect at most once, which implies that the equation w⁢(x)=r⁢w⁢(xm) with r>1 and m<1 has at most one solution. This then also implies that the curves of any genotype σ and a proper superset of it (i.e. a genotype which contains all the mutations in σ and some more) intersect at most once. This property holds for all functions that have been used to represent dose-response curves in the literature, such as the Hill function, the half-Gaussian or the exponential function, as well as for all concave function with negative second derivate (see Materials and methods for details). Properties of tradeoff-induced fitness landscapes To understand the evolutionary implications of our model, we first describe how the fitness landscape topography changes with the environmental parameter represented by the antibiotic concentration. Next we analyze the properties of mutational pathways leading to highly fit genotypes. Intersection of curves and changing landscapes We start with a simple example of L=2 mutations and a Hill-shaped dose-response curve w⁢(x)=11+x2 (Figure 3). At x=0, the rank ordering is determined by the null-fitness. The wild type has maximal fitness, and the double mutant is less fit than the single mutants. As x increases, the fitness curves start to intersect, and each intersection switches the rank of two genotypes. In the present example we find a total of six intersections and therefore seven different rank orders across the full range of x. This is actually the maximum number of rank orders that can be found by scanning through x for L=2, see Materials and methods. The final fitness rank order (in the region G in Figure 3A) is the reverse of the original rank order at x=0. Figure 3B depicts the concentration-dependent fitness landscape of the 2-locus system in the form of fitness graphs. A fitness graph represents a fitness landscape as a directed graph, where neighboring nodes are genotypes that differ by one mutation, and arrows point toward the genotypes with higher fitness (de Visser et al., 2009; Crona et al., 2013). Figure 3 Download asset Open asset Crossing dose-response curves and fitness graphs. (A) An example of dose-response curves of four genotypes – the wild type (00), two single mutants (10 and 01), and the double mutant (11). The parameters of the two single mutants are r1=0.8, m1=1.3, r2=0.5, m2=2.5. Null-fitness and resistance combine multiplicatively, which implies that the parameters of the double mutant are r12=r1⁢r2=0.4 and m12=m1⁢m2=3.25. (B) Fitness graphs corresponding to antibiotic concentration ranges from panel (A). The genotypes in red are the local fitness peaks. The purple arrows are the ones that have changed direction at the beginning of each segment. All arrows eventually switch from the downward to the upward direction. A fitness graph does not uniquely specify the rank order in the landscape (Crona et al., 2017). For example, the three regions C, D and E have different rank orders but the same fitness graph. Because selection drives an evolving population towards higher fitness, a fitness graph can be viewed as a roadmap of possible evolutionary trajectories. In particular, a fitness peak (marked in red in Figure 3B) is identified from the fitness graph as a node with only incoming arrows. Fitness graphs also contain the complete information about the occurrences of sign epistasis. Sign epistasis with respect to a certain mutation occurs when the mutation is beneficial in some backgrounds but deleterious in others (Weinreich et al., 2005; Poelwijk et al., 2007). It is easy to read off sign epistasis for a mutation from the fact that parallel arrows (i.e. arrows corresponding to the gain or loss of the same mutation) in a fitness graph point in opposite directions. For example, in the graph for the region B there is sign epistasis in the first position, since the parallel arrows 00 → 10 and 01 ← 11 point in opposite directions. Notice that in the current example, we start with a smooth landscape at x=0 (as seen in the fitness graph for region A), and the number of peaks and the degree of sign epistasis both reach a maximum in the intermediate region C+D+E. This fitness graph displays reciprocal sign epistasis, which is a necessary condition for the existence of multiple fitness peaks (Poelwijk et al., 2011). Beyond the region E, the landscape starts to become smooth again, with only one fitness maximum and a lower degree of sign epistasis. In the last region G, the landscape is smooth with only one peak (the double mutant 11) and no sign epistasis. These qualitative properties generalize to larger landscapes. To show this, we consider a statistical ensemble of landscapes with L mutations, where the parameters ri, mi of single mutations are independently and identically distributed according to a joint probability density P⁢(r,m). Figure 4 shows the result of numerical simulations of these landscapes for L=16. The mean number of fitness peaks with n mutations reaches a maximum at xmax⁢(n) where to leading order log⁡xmax⁢(n)∼n⁢⟨log⁡m⟩, independent of any further details of the system, as argued in Materials and methods. The asymptotic expression works well already for L=16 (see inset of Figure 4A). Figure 4B shows the mean number of mutations in a fitness peak. This is well approximated by the curve n=log⁡x⟨log⁡m⟩, showing that the mean number of mutations in a fitness peak grows logarithmically with the concentration. This is consistent with what we would expect from the variation in the number of peaks with n mutations as shown in Figure 4A. The existence of a typical number of mutations in a fitness peak is one of the distinctive features of our landscape, a feature typically lacking in other well-studied random landscape models. This property arises from the existence of adaptational tradeoffs. Since a high number of mutations is beneficial at higher concentrations but deleterious at lower concentrations, it is clear that there must be an optimal number of mutations at some intermediate concentration. Figure 4 Download asset Open asset Fitness landscape ruggedness changes with drug concentration. (A) Number of fitness peaks as a function of concentration for different numbers of mutations in the peak, n, and L=16. The dashed green curve is the total number of fitness peaks, summed over n. The peaks were found by numerically generating an ensemble of landscapes with individual effects distributed according to the joint distribution (8). For this distribution, ⟨log⁡m⟩=1.19645. Inset: The maximal number of peaks for a given value of n occurs at log⁡xmax⁢(n)=n⁢⟨log⁡m⟩, and grows exponentially with L. (B) Mean number of mutations in a fitness peak as a function of concentration x for the same model. The black circles are the mean number of mutations in the fittest genotype. The green dashed line is log⁡(x)⟨log⁡m⟩, where ⟨log⁡m⟩=1.19645 as before. As another indicator of ruggedness, we consider the number of backgrounds in which a mutation is beneficial as a function of x. At x=0, any mutation is deleterious in all backgrounds, whereas" @default.
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W3035883615 title "Author response: Predictable properties of fitness landscapes induced by adaptational tradeoffs" @default.
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