Phenotype of biological systems needs to be robust against mutation in order to sustain themselves between generations. On the other hand, phenotype of an individual also needs to be robust against fluctuations of both internal and external origins that are encountered during growth and development. Is there a relationship between these two types of robustness, one during a single generation and the other during evolution? Could stochasticity in gene expression have any relevance to the evolution of these types of robustness? Robustness can be defined by the sharpness of the distribution of phenotype; the variance of phenotype distribution due to genetic variation gives a measure of ‘genetic robustness’, while that of isogenic individuals gives a measure of ‘developmental robustness’. Through simulations of a simple stochastic gene expression network that undergoes mutation and selection, we show that in order for the network to acquire both types of robustness, the phenotypic variance induced by mutations must be smaller than that observed in an isogenic population. As the latter originates from noise in gene expression, this signifies that the genetic robustness evolves only when the noise strength in gene expression is larger than some threshold. In such a case, the two variances decrease throughout the evolutionary time course, indicating increase in robustness. The results reveal how noise that cells encounter during growth and development shapes networks' robustness to stochasticity in gene expression, which in turn shapes networks' robustness to mutation. The necessary condition for evolution of robustness, as well as the relationship between genetic and developmental robustness, is derived quantitatively through the variance of phenotypic fluctuations, which are directly measurable experimentally. Robustness is ability to function against changes in the parameter of a systemIndeed, phenotype of isogenic individual organisms is not necessarily identical. ChemotaxisNoise, in general, can be an obstacle in tuning a system to the fittest state and maintaining it there. Phenotype of an organism is often reproducible even under fluctuating environment or under molecular fluctuationsWhether these two types of robustness emerge under natural selection have long been debated in the context of developmental dynamics and evolution theoryipV, gives an index for robustness to noise in developmental dynamics. On the other hand, robustness to mutation is measured from the fitness distribution over individuals with different genotypes. An index for it is given by variance of phenotypic fluctuation arising from diversity of genotypes in a populationgV. This variance gV increases as the fraction of low-fitness mutants increases.Under the presence of noise in gene expression, phenotype as well as fitness, of isogenic organisms is distributed, usually following a bell-shaped probability function. When the phenotype is less robust to noise, this distribution is broader. Hence, the variance of this distribution, i.e., variance of isogenic phenotypic fluctuation denoted as ipV≥gV is satisfied. Since the isogenic phenotypic fluctuation ipV increases with noise, this means that evolution of robustness is possible only when the amplitude of phenotypic noise is larger than some critical value as derived by ipV≥gV, implying a positive role of noise to evolution. We demonstrate that both the two variances ipV and gV decrease in the course of evolution, while keeping the proportionality between the two. This proportionality is consistent with an observation in a bacterial evolution experiment Here we show that evolution to increase both types of robustness is possible only when the inequality We explain the origin of the critical noise strength, by noting that smooth dynamical behavior free from a rugged potential landscape evolves as a result of phenotypic noise. When the noise amplitude is smaller than the threshold, we observe that low-fitness mutants are accumulated, so that robustness to mutation is not achieved. Generality and relevance of our results to biological evolution are briefly discussed.P where x is a variable characterizing a phenotype and a is that for the corresponding genotypex is responsible for the fitness of an individual, and the selection depending on x is considered as an evolutionary process. Since the phenotype differs even among isogenic individuals, the distribution P for a fixed genotype a0 has some variance. This isogenic phenotypic variance ipV, defined as the variance over clones, is written as a, namely ipV works as a measure of robustness of the phenotype against noise.In natural population, both the phenotype and genotype differ among individuals. Let us consider population distribution gV here, is then defined as the variance of the average P(a) is the distribution of the genotype a and <x̅ > is the average of a. While ipV is defined as variance over clones, i.e., individuals with the same genotype, gV comes from those with different genotypes. As gV is smaller, the phenotypic change by genetic variation is smaller. Hence gV gives a measure of robustness of the phenotype against mutation.On the other hand, the standard evolutionary genetics ip>VgV between the two variances, by assuming evolutionary stability of the population distribution P, that is preservation of single-peakedness through the course of evolution ipV approaches gV, where the distribution is extended to very low values of x (fitness). In other words, error catastrophe occurs at g≈VipV;  is based on the existence of two-variable distribution function P, and the postulate that single-peaked distribution is maintained throughout evolution, which is not trivial. Hence the above relationships need to be examined by some models for evolution. In addition, ip is smaller than Vgwhy does the population distribution extend to low-fitness values when the phenotypic fluctuation V? Or, put it another way, why do systems with small phenotypic noise run into “error catastrophe”? In fact, the emergence of error catastrophe as a result of decreasing isogenic phenotypic fluctuation below gV may look rather counter-intuitive, since in general one expects fluctuation to perturb a system from the fittest state. The necessity of fluctuation for evolution to increase robustness to noise and to mutation needs theoretical examination.Note, however, that the derivation of these relationships  from a given initial state. In general, this dynamic process is complex so that the process may not reach the identical phenotype due to the noise through this developmental process. This leads to the isogenic variance of the phenotype ipV. On the other hand, the expression pattern varies by mutation in the network, and gives rise to variation in the average fitness, resulting in gV.We consider a simple model to satisfy the requirement on ‘development’ above. It consists of a complex dynamic process to reach a target phenotype under a noise which may alter the final phenotypic state. We do not choose a biologically realistic model that describes a specific developmental process, but instead take a model as simple as possible, to satisfy a minimal requirement for our study. Here we take a simplified model, borrowed from a gene regulatory network, where expression of a gene activates or inhibits expression of other genes under noise. These interactions between genes are determined by the network. The expression profile changes in time, and eventually reaches a stationary pattern. This gene expression pattern determines fitness. Selection occurs after introduction of mutation at each generation in the gene network. Among the mutated networks, we select a network with a higher fitness value. Since there is a noise term in the gene expression dynamics, fitness fluctuates even among the individuals with an identical gene network, which leads to the isogenic fluctuation ix is described byijJ = −1,1,0, and η(t) is Gaussian white noise given by <η(t)η(t′)> = δ(t−t′). M is the total number of genes, and k is the number of output genes that are responsible for fitness to be determined. The value of σ represents noise strength that determines stochasticity in gene expression . By following a sigmoid function tanh, ix has a tendency to approach either 1 or −1, which is regarded as “on” or “off” of gene expression. Even though x is defined over , it is attracted to the range  (or slightly above or below the range due to the noise term). We consider a developmental process leading to a matured phenotype from a fixed initial state, which is given by ; i.e., all genes are off, unless noted otherwise. .This simplified gene expression follows a typical switch-like dynamics with a sigmoid input-output behavior ix) for genes i = 1,2,…,k(<M) would reach an “on” state, i.e., ix>0. The fitness is maximum if all k genes are on after a transient time span iniT, and minimum if all are off. To be specific, we define the fitness function byS(x) = 1 for x>0, and 0 otherwise, […]temp is time average between init = T and ft = T . Adoption of the value (S(jx)−1) after initial time iniT leads to the same result  is always “on” and takes the minimum (F = −k) when all k genes are always off. Note that fitness is calculated only after time iniT, which is chosen sufficiently large so that the temporal average can be computed after the gene expression dynamics has fallen on an attractor. This initial time can be considered as the time required for developmental dynamics.Let us define a fitness function so that gene expression levels , for a given network “g”, whose variance gives isogenic phenotypic fluctuation. At each generation, we compute the fitness F over L runs, to obtain the average fitness value F̅ of a given network.As the model contains a noise term, fitness fluctuates at each run, which leads to the distribution in ijJ which determines the ‘rule’ of the dynamics, it is natural to treat ijJ as a measure of genotype. Individuals with different genotype have a different set of ijJ At each generation there are N individuals with different sets of ijJ For each individual network, we compute the average fitness F̅. Then we select the top sN(<N) networks that have higher fitness values. .Now we consider the evolutionary process of the network. Since the network is governed by ijJ at a given mutation rate µ. We rewire the network at a given rate so that changes in ijJ produce N new networks. At each generation, mutation changes the network, i.e., changes sN/N mutants from each of the top sN networks, so that there will be N networks again for the next generation. From this population of networks we repeat the process of the developmental dynamics, fitness calculation, selection, and mutation .Here we make ijJ are nonzero). At each generation, the N individuals have slightly different networks ijJ, so that the values of F̅ are different. We denote the fitness distribution over individuals with different genotype as P(F̅ ). On the other hand, the fitness distribution for an identical network (“g”) is computed to obtain p.Simulations start from a population of random networks with a given fraction of paths  are invariant.Developmental dynamics and selection process in our model are too simple. Still, this model is relevant to examine general statement on phenotypic fluctuations, as the model at least captures complex dynamics giving rise to a phenotype, stochasticity in dynamics, mutation, and selection according to a given phenotype. Indeed, real gene expression dynamics depend on environmental conditions, and the fitness is defined as expression patterns to adapt each environmental condition. We have also carried out some simulations by imposing such fitness but the results to be discussed  increases, so that the top of F̅ in the population approaches the highest value 0. Indeed, in all cases, the top group quickly evolves to the highest fitness state F̅ = 0 . The time necessary for the system to reach this state becomes shorter as the phenotypic noise decreases (see P(F̅) depends drastically on the noise strength σ. When σ is small, the distribution is broad and the existing individual with the lowest F̅ remains at the low fitness state, while for large σ, even the individuals with the lowest fitness approach F̅ = 0 (see cσ(≈0.02), below which the distribution P(F̅) is broadened, as is discernible in the data of the variance of the distribution, gV in F̅> and the lowest fitness over individuals F̅min, after a sufficiently large number of generations, are plotted against σ in cσ, below which low-fitness mutants always remain in the distribution. For cσ>σ, the distribution P(F̅) takes a sharp peak at F̅ ∼0, where the variance is rather small. Distribution below and above cσ are displayed in cσ>σ).Let us first see how the evolutionary process changes as a function of the noise strength ases see . On the  = 0 see . There igV and ipV Here ipV is defined as variance from the distribution p, i.e., over individuals with the same genotype. As the distribution p depends on each individual with different genotype, the variance changes accordingly. Naturally, the top individual has a smaller variance, and the individual with lower fitness has a larger variance. As a measure of ipV, we used either the average of the variance over all individuals or the variance of phenotype from a gene network that is located closest to the peak in the distribution P(F̅). Both estimates of ipV do not differ much, and the following conclusion is drawn in both cases. gV, on the other hand, is simply the variance of the distribution P(F̅), i.e., over individuals having different genotypes present.Let us study the relationship between gV and ipV thus evaluated is plotted in cσ>σ, where we note:The relationship between ipV>gV for cσ>σ.(i) gV∝ipV during the evolutionary time course under a fixed noise strength σ(>cσ) and a fixed mutation rate. As σ is lowered toward cσ, gV increases so that it approaches ipV.(ii) gV≈ipV at cσ≈σ, where error catastrophe occurs.(iii) ipV>gV is satisfied. As σ is decreased to cσ, ipV approaches gV, error catastrophe occurs and a considerable fraction of low-fitness mutants accumulates. Hence, the relationships proposed theoretically assuming evolutionary stability of a two-variable distribution function P is confirmed. Here, without introducing phenomenological assumptions, the three relationships are observed in a general stochastic gene-network model.In other words, the fittest networks maintaining a sharp distribution around the peak dominate only when cσ<σ? To study the difference in dynamics evolved for different values of σ, we choose the top individual (network) that has evolved at σ = σ0, and place it under a different noise strength σ = σ′. In F = 0 under such circumstance. As shown, the successful fraction decreases when σ′ goes beyond σ0. In other words, the network evolved under a given noise strength successfully reaches the target gene expression up to that critical noise level, while it begins to fail doing so at a higher noise strength. Accordingly, the distribution p extends to lower values in fitness, when a network evolved under small phenotypic noise is developed under a higher noise level. On the other hand, the network evolved under high level noise maintains a high fitness value, even when the noise is lowered.Why didn't the system maintain the highest fitness state under small phenotypic noise ix>0 for i = 1,…,k). Hence the basin of attraction for this target attractor is expected to be larger for the dynamics evolved under higher level noise. We have simulated the dynamics (1) for the evolved fittest network under zero noise, starting from a variety of initial conditions over the entire phase space, and measured the distribution of F at each attractor. The distribution is shown in jx = 1↔jx = −1 in the model, the distribution is symmetric around F = −k/2 when all initial conditions are taken. In fact, by starting from ix = 1 for all i, the orbit reaches an attractor jx<0 for j = 1,…,k, resulting in F = −k). For the network evolved under cσ>σ, the distribution has a sharp peak at F = 0 (and F = −k due to the symmetry), with more than 40% attraction to each. On the other hand, for the networks evolved under cσ<σ, the peak height at F∼0 is very small, i.e., the basin for the attractor with F = 0 is tiny. There exist many small peaks corresponding to attractors with −k<F<0, having similar sizes of basin volumes. In fact, the basin volumes for attractors with −k<F<0 grow as σ is decreased, and are dominant for c.σ<σNext we study the basin structure of attractors in the present system. Note that an orbit of the network with the highest fitness, starting from the prescribed initial condition, is within the basin of attraction for an attractor corresponding to the target state  which is defined by Δ here. The distance Δ remains small for the dynamics evolved under a low noise strength cσ<σ, and increases for those evolved under a higher noise. It is interesting to note that evolution influences the basin structure globally over the phase space, although the fitness condition is applied locally to an orbit starting from a specific initial condition.The difference in the basin structure suggested by jx for j = 1,…,k changes its sign. Recall that the fixed point solution ix = tanh(ΣjJijxj) changes its sign when ΣjJijxj in the sigmoid function changes its sign. This change may occur during the developmental dynamics by noise, and we call such points in the phase space ‘turning points’. When an orbit of eq.(1) passes over turning points, jx takes a negative value for some j (for 1≤i≤k) at the attractor  changes its sign at several points in the phase space {jx} generally. Hence there can be many turning points in the phase space. The fittest network with F̅≈0 chooses such orbits having no turning points within the noise range from the original orbit. An orbit from the fittest individual evolved under low-level noise encounters many turning points when subjected to noisy environment.The results in The average distance between the turning points and an orbit that has reached the target gene expression pattern is estimated by the distance Δ defined above. Recall that the distance Δ is small for the dynamics evolved under a low noise strength. Such dynamics, if perturbed by a higher level of noise, are easily caught in the turning points, which explains the behavior shown in gV and ipV. Noise disturbs an orbit so that it may go across the basin boundary (turning points) with some probability. We denote the standard deviation of the location of the orbit due to noise as pδ, which is proportional to the noise strength σ. Since the distance between the orbit and the basin boundary is deviated by pδ, and the fitness value drops when the orbit crosses the basin boundary, the variance ipV is estimated to be proportional to (pδ/Δ)2.Let us now discuss the relationship between ijJ, there exists a variation of the order of gδ, which increases with the mutation rate. The variance gV is estimated to be proportional to (gδ/Δ)2, with the same proportion coefficient as that between ipV and (gδ/Δ)2.Next, we discuss how the mutation in the network influences the dynamics. When the network is altered, i.e., a path is added or removed as a result of mutation in pδ from the “target” orbit. This leads to an increase in Δ. However, if pδ is larger than pδ, the memory of this distance between the target and the boundaries will not be propagated to the next generation, due to large perturbation to the original network by the mutation. Hence increase in Δ  is expected only if p>δgδ. Since pδ and gδ increase with the noise strength σ and the mutation rate µ respectively, there exists a critical noise strength cσ beyond which this inequality is satisfied. From the relationship between p,gδ and ip,gV, the condition for robust evolution is rewritten as ip>VgV.Under the presence of noise, there is a selective pressure to avoid the turning points (basin boundaries) that exist within the distance ip>VgV  is satisfied, the system increases Δ during evolution. We have computed the temporal evolution of basin distribution. With generations, the distribution evolves from the pattern at a low level noise in σ characterized by enhanced peak at F = 0. Accordingly Δ increases with generations. Recall that ipV∼(pδ/Δ)2, and gV∼(gδ/Δ)2, both variances decrease with generations, while ip/VgV is kept constant.When the condition gV and ipV, through numerical evolution experiment of a gene network. As phenotypic noise is decreased and the inequality ip>VgV is broken, low-fitness mutants are no longer eliminated. This is because the mutants fail to reach the target gene expression pattern, by crossing the boundary of the basin of attraction to the target. When the amplitude of the noise is larger, on the other hand, the networks of the dynamics with a large basin volume for the target attractor are selected and thus mutants with lower fitness are removed successively through the selection process. Hence noise increases developmental robustness through evolution, together with genetic robustness.We have demonstrated the inequality and proportionality between ipV and gV and the error catastrophe, we expect this relationship to be generally applicable to systems satisfying the following conditions:Although we used a specific example to demonstrate the relationship between (i) Fitness is determined through developmental dynamics.(ii) Developmental dynamics is sufficiently complex so that its orbit, when deviated by noise, may fail to reach the state with the highest fitness.(iii) There is effective equivalence between mutation and noise in the developmental dynamics with regards to phenotype change.Note that the present system as well as the previous cell modelg<VipV must be satisfied in order for the low-fitness mutants to be eliminated. Violation of the inequality leads to accumulation of low-fitness (or deleterious) mutants, a phenomenon known as error catastrophegV of given phenotype in mutants, and comparing it with that of isogenic bacteria, ipV, one can examine the validity of our conclusion between gV and ipV.When developmental process fails due to phenotypic noise, the fitness function takes a low value. Evolution under noise acts to prevent such failure within the range of noise. On the other hand, due to the condition (iii), mutation may also lead to such lethality. When the effect of mutation goes beyond the range given by the phenotypic noise, mutants with very low fitness values begin to accumulate. Hence there appears a threshold level of phenotypic noise below which low-fitness (or deleterious) mutants accumulate (or threshold mutation rate beyond which such mutants accumulate). In this sense, we expect that for robust evolution, the inequality It is worthwhile to mention that in a class of theoretical models, fitness landscape is given as an explicit continuous function of a gene sequence  is formed through a mutation and selection process. As a result, dynamics over the entire phase space are simplified to those having only a few attractors, even though the fitness function is given locally without scanning over the entire phase space. When the time-course is represented as a motion along a potential surface, our results suggest that the potential landscape becomes smoother and simpler through evolution, and loses ruggedness after generations. Indeed, existence of such global attraction in an actual gene network has recently been reported in yeast cell-cycleijJ between spins (corresponding to gV) and the temperature  is formulated. Such relationship may be relevant to understand the relationships between gV and ipV in our study.Such smooth landscape was also studied in protein foldinggVipV and gV then suggests that the evolution speed is proportional to the isogenic phenotypic fluctuation, as is also supported by an experiment on bacterial evolution in a laboratoryAccording to established Fisher's theorem on natural selection, evolution speed of phenotype is proportional to the phenotypic variance by genetic variation, Isogenic phenotypic fluctuation is related to phenotypic plasticity, which is a degree of phenotype change in a different environment. Positive roles of phenotypic plasticity in evolution have been discussedText S1Derivation of General Relationship on Fluctuations. Mathematical derivation on general relationships on phenotypic variances is presented.(0.05 MB PDF)Click here for additional data file.


Using a transparent mathematical model, we illustrate the concept of genetic potential and show that as environmental variability decreases, the evolving population reaches three distinct steady state conditions: (1) organismal flexibility, (2) genetic potential, and (3) genetic robustness. As a specific example of this concept we examine fluctuating selection for hydrophobicity in a single amino acid. We see the same three stages, suggesting that environmental fluctuations can produce allele distributions that are distinct not only from those found under constant conditions, but also from the transient allele distributions that arise under isolated selective sweeps.Organisms employ a multitude of strategies to cope with the dynamical environments in which they live. Homeostasis and physiological plasticity buffer changes within the lifetime of an organism, while stochastic developmental programs and hypermutability track changes on longer timescales. An alternative long-term mechanism is “genetic potential” Variation is the fuel of natural selection. Understanding the mutational processes that underlie evolution has long been a central objective of population genetics. Today, amidst a computational revolution in biology, such understanding is pivotal to progress in many biological disciplines. For example, neutral mutations make the molecular clock tick, and this clock is fundamental to reconstructing phylogenies, measuring recombination rates, and detecting genetic functionality. In this manuscript, the researchers provide an original perspective on a long-standing question in evolutionary biology: to what extent do mutation rates evolve? They argue that to cope with environmental fluctuation, populations can evolve their phenotypic mutation rate without changing their genetic mutation rate. That is, populations can evolve “genetic potential”—a heightened sensitivity to the effects of mutation. The researchers use a simple mathematical model of amino acid evolution to illustrate the evolution of genetic potential, and show that as environmental variability decreases, evolving populations reach three distinct states. In a rapidly fluctuating environment, organisms evolve the flexibility to cope with variation within an individual lifetime; in moderately variable environments, populations evolve the ability to evolve rapidly; and in fairly constant environments, populations evolve robustness against the adverse effects of mutation. Recent work in evolutionary biology has highlighted the degeneracy of the relationship between genes and traits . For anyA separate body of evolutionary theory addresses adaptation under fluctuating conditions ,8. The rIn this paper we show that genetic degeneracy may give rise to an alternative outcome under fluctuating conditions: the evolution of genotypes with heightened sensitivity to mutation. We introduce the term “genetic potential” to describe this state. Metaphorically, populations with genetic potential lie near the edge of neutral plateaus. Although the rate of mutation is unchanged, the likelihood that mutations produce beneficial variation increases. Heightened sensitivity to mutations has been recognized as a critical and transient phase of adaptive evolution ,15,16. HWe then present a more biological example of this phenomenon using a model of amino acid evolution. There is evidence that, within viral pathogens, the physiochemical properties of amino acids found within epitopes—regions of proteins that directly interact with the host immune system—can rapidly evolve ,18. LikeUnder infrequent environmental change, populations evolve from one mutationally robust phenotype to another, briefly passing through genotypes that can easily mutate to either state. One might therefore be tempted to equate genetic potential with confinement to the intermediate steps on a path from robustness for one phenotype to robustness for another . While tA and EB), spending exactly λ generations per state between shifts. The simple model includes three phenotypes—one optimal phenotype for each of the two environments (A and B) and a third that has intermediate quality in both environments (V)—and a minimal amount of degeneracy in the relationship between the genotype and the phenotype. In particular, there is a single genetic locus, and five allelic possibilities at that locus  mod 5( or gi + 1) mod 5 in its preferred environment, and 0 ≤ k ≤ 1 determines the intermediacy of the V phenotype.where We can write the full model as a set of difference equationsi ∈ {0,1,2,3,4}, where μ is the mutation rate and tw denotes the fitness in the current environment , Φi is the set of three transition point mutations of codon i, and Γi is the set of six transversion point mutations of codon i.for 1 ≤ A, the equilibrium relative frequencies of g0, g1, g2, g3, and g4 are 0.291, 0.412, 0.292, 0.003, and 0.002, respectively, and in environment EB, they are 0.005, 0.000, 0.000, 0.010, and 0.985, respectively. When there is degeneracy, as there is for phenotype A, the populations evolve genetic robustness, that is, more mutationally protected genotypes appear in higher frequency. In particular, g1, which lies in the center of the three genotypes that code for A, appears in higher frequency than either genotype on the edge of the neutral network for A (g0 and g2) at equilibrium in EA. In the absence of degeneracy (phenotype B), we observe a mutation–selection balance around the single optimal genotype. These findings are consistent with and provide a transparent example of the extensive theory on mutation–selection balance, quasi-species, and the evolution of genetic robustness in neutral networks [We provide an intuitive perspective on evolution in fluctuating environments using the simple model and then demonstrate the generality of the results in the codon model. The first results assume a mutation rate μ = 0.01, and fitnesses 1, 1.5, and 2 for the unfavored, intermediate, and favored phenotypes, respectively. In a constant environment, a population will equilibrate on genotypes that encode the optimal phenotype. In environment Enetworks –24.A and EB, thereby alternating between genetic robustness for A and a mutation–selection balance around the single allele for B. At intermediate rates of fluctuation, populations hover near g4 and g0, where the genotypes for A abut the genotype for B. Thus, mutation between the two phenotypes occurs frequently. We call this outcome genetic potential because of the enhanced potential for mutations to give rise to novel  phenotypes. Finally, for highly variable environments, the populations converge on the phenotype V, which has unchanging, intermediate fitness in both environments. Phenotype V corresponds to organismal flexibility—individual organisms tolerate both conditions, but neither one exceptionally well. There are a variety of mechanisms that can give rise to an intermediate phenotype including homeostasis, somatic evolution, physiological plasticity, and behavioral plasticity [Under rapid environmental fluctuations, populations do not have time to reach a stable allele distribution. As the environment becomes more variable, the distributions of alleles go through three distinct phases. asticity ,8. As orasticity , the asc−5, and fitnesses 1, 1.5, and 2 for the unfavored, intermediate, and favored phenotypes, respectively. (We address the impact of mutation rate in the Discussion.) Whereas in the simple model only one of the three phenotypes had multiple genotypes, in this model all three phenotypic classes have genetic degeneracy, and thus can evolve genetic robustness  if and when the environment reverts. For populations that have equilibrated in a constant environment and have recently experienced an environmental shift, genetic potential will decrease as the population becomes increasingly robust to the effects of mutation . For pop2 generations. A and EB. We then compared these two distributions to the evolving amino acid distribution in a population that equilibrates in one of the two environments and then faces an environmental shift. This difference also appears in the distributions of amino acids. We calculated the genetic potential in each generation of a population experiencing fluctuations every λ = 10−5 in the codon model, populations evolve genetic potential when environment varies at rates of 101 < λ < 106 generations, approximately K [aK/sK, such genes should employ a distinct set of codons with high genetic potential. Note that this type of genetic potential is not equivalent to codon bias, but rather implies changes in the actual distribution of amino acids.A long-standing technique for identifying selected genes is to compare the frequencies of nonsynonymous and synonymous substitutions (Ka/Ks) . Genes e(Ka/Ks) ,41 or ge(Ka/Ks) . In the A similar argument also underlies the recent use of codon distributions for detecting genetic loci under directional selection . Codon vaK/sK ratios. Such sites have been identified, for example, in human class I MHC genes, the HIV envelop gene, and a gene from a human T cell lymphotropic virus (HTLV-1) [We would like to emphasize that the goal of this study was not to develop a new method for detecting positive (or fluctuating) selection, but rather to develop a theoretical framework for considering the multiple outcomes of evolution under fluctuating conditions. We conclude by suggesting an empirical method to identify loci that have evolved genetic potential under such conditions as distinct from those that have experienced a recent selective sweep. Suppose that a gene experiences fluctuations at a characteristic rate across many species. Furthermore, suppose that multiple sites within the gene are influenced by such fluctuations. For example, there may be fluctuating selection for molecular hydropathy, charge, size, or polarity, and several sites within the gene may contribute to these properties. Such sites should evolve in tandem and equilibrate on similar levels of genetic potential, and thus exhibit similar codon (and amino acid) distributions across species. In contrast, if a gene experiences isolated selective sweeps, then the variation at all sites should correspond to both the history of selective events and the species phylogeny, and the amino acid distributions at sites should correlate only when sites functionally mirror each other. Thus, one can seek evidence for the evolution of genetic potential as follows. First, identify genes that are rapidly evolving, perhaps by calculating (HTLV-1) ,32. WithMA and MB be the normalized transition matrices that govern changes in the allele frequencies in EA and EB epochs, respectively. The entries in these matrices are defined by equations 2 and 4. The left leading eigenvectors for MA and MB give the equilibrium frequency distributions of alleles in each of the two constant environments, respectively. Under fluctuating conditions with epoch duration of λ generations, we iteratively apply the matrices, and then compute the left leading eigenvector of vB, gives the allele frequency distribution at the end of an EA epoch followed by an EB epoch.For the two models, we calculate the deterministic, infinite population allele frequency distributions in constant and fluctuating environments. Let A epoch byWe are interested not only in the final allele distributions, but also in the dynamics throughout each epoch. Thus, we calculate the average frequency of each allele across a single EG is the total number of alleles in the model (G = 5 for the simple model and G = 64 for the codon model) and the subscript k indicates the kth entry in the vector. Similarly, the average distribution across an EB epoch is given bywhere vA is the allele frequency distribution at the end of an EB epoch followed by an EA epoch and is equal to the left leading eigenvalue of where For the codon model, we compare these calculations that assume a regularly fluctuating environment to numerical simulations that assume a Poisson distribution of epoch lengths. In each generation of the simulations, the environmental state switches with probability 1/λ and the codon frequencies are then multiplied by the appropriate transition matrix.We use an elementary group theoretic result known as Burnside's Lemma  to proveX be the set of all pentagons with vertices labeled {A, B, V} having at least one A vertex and at least one B vertex. The size of X is the number of all pentagons with labels {A, B, V} minus the number of pentagons with labels {A, V} or {B, V}, that is, |X| = 35 − (2 · 25 − 1) = 180.Let G of all actions on X that produce equivalent pentagons (as specified above). G is made up of (1) the identity, (2) the four rotations and five reflections of the pentagon, (3) interchanging all As and Bs, and (4) all the combinations of the above actions. Thus G is equal to the 20-member group {ι, ρ, ρ2, ρ3, ρ4, σ0, σ1, σ2, σ3, σ4, α, αρ, αρ2, αρ3, αρ4, ασ0, ασ1, ασ2, ασ3, ασ4} where ι is the identity, ρ is a single (72°) rotation, σi is a reflection through vertex i, and α is replacement of all As with Bs and all Bs with As. We define the group G on X. Burnside's Lemma tells us that this number isThe number of distinct mutational networks is equal to the number of orbits of F(g) = {x ∈ X | gx = x} is the set of fixed points of g. For each of the twenty elements of G, we exhaustively count F(g).where X, that is, F(ι) = X. Each of the various rotations of a pentagon  has the property that its iterations move a given vertex to every other vertex of the pentagon without changing the letter assigned to that vertex. The same is true of the square of the product of any rotation and an A–B flip. Hence, any fixed point of one of these elements of the group G would necessarily have the same label at each vertex of the pentagon. Since every labeled pentagon in X has at least one A label and at least one B label, then no element of X has the same label at each vertex. Thus, the fixed point set of every rotation and of every product of a rotation and an A–B flip must be empty, that is, F(ρn) = F(αρn) = ∅ for all n. By a similar argument, the simple A–B flip also has no fixed points. Every reflection fixes 12 elements of X, for example,The identity fixes all elements of X, for example,and every product of a reflection and an A–B flip fixes eight elements of X, all five reflections fix 12 elements of X, and all five combinations of a reflection and an A–B exchange fix eight elements of X. Thus,In sum, all eight group elements that involve rotations fix no elements of


Quasispecies are clouds of genotypes that appear in a population at mutation–selection balance. This concept has recently attracted the attention of virologists, because many RNA viruses appear to generate high levels of genetic variation that may enhance the evolution of drug resistance and immune escape. The literature on these important evolutionary processes is, however, quite challenging. Here we use simple models to link mutation–selection balance theory to the most novel property of quasispecies: the error threshold—a mutation rate below which populations equilibrate in a traditional mutation–selection balance and above which the population experiences an error catastrophe, that is, the loss of the favored genotype through frequent deleterious mutations. These models show that a single fitness landscape may contain multiple, hierarchically organized error thresholds and that an error threshold is affected by the extent of back mutation and redundancy in the genotype-to-phenotype map. Importantly, an error threshold is distinct from an extinction threshold, which is the complete loss of the population through lethal mutations. Based on this framework, we argue that the lethal mutagenesis of a viral infection by mutation-inducing drugs is not a true error catastophe, but is an extinction catastrophe. The concept of a mutation–selection balance is one of the oldest and most fundamental pillars of population genetics: natural selection increases the frequency of fit variants while mutations introduce unfit variants, giving rise to an equilibrium distribution balanced between these two effects. Mutation–selection balance has been invoked to explain the persistence of undesirable genes, for example, those underlying inbreeding depression, genetic diseases, and even senescence. Despite the long history of the concept, some of its consequences were only realized in 1971, when Manfred Eigen studied mutation–selection dynamics in long genomes . This coThe error catastrophe has been applied liberally as a metaphor for complications of high mutation rates, as likely plagued primordial life  and currEigen's insights were developed in the context of genomes with many loci, each of which suffered mutation. Appropriately, the quasispecies has since been considered in this large-genome context. Yet many of its concepts are easily illustrated in the much simpler case of few genotypes, which is our approach here. Our results are not new, per se, but our models should convey quasispecies and error-catastrophe concepts to a broad audience and correct some common misunderstandings.A1 has fitness w1, and of those w1 offspring a fraction 1 − μ1 retain the A1 genotype  or the mutation-free fraction of offspring (1 − μ1) of A1 relative to that of A2, then the equilibrium distribution will move to progressively lower values on the vertical axis, until it lacks A1 entirely. This brings us to the error catastrophe.If a mutation rate or fitness is changed, the equilibrium distribution changes. Thus, the analogous figure for a different equilibrium would have the curves converging to a different value on the vertical axis. In particular, if we continually lower the fitness (w1(1 − μ1) is the number of A1 offspring of an A1 parent; we will refer to this product as the replacement rate of the genotype—the product of fitness times the mutation-free fraction of offspring. For mathematical simplicity, we assume that this number is an absolute quantity greater than one and that it does not change with population density. Hence, the population is forever expanding in unlimited space (this assumption is convenient but not necessary). For convenience, we will continue to treat A1 and A2 as single genotypes, but later in the paper, we will explain how they can be treated as sets of genotypes and how additional types can be included .Error thresholds and error catastrophes are properties of mutation–selection equilibria, and an error catastrophe is a specific type of change in the equilibria. Consider again the genotypes and fitnesses illustrated in A1 has a higher fitness than A2, it follows that, for small mutation rates μ1, the replacement rate of A1 is higher than the replacement rate of A2, or w1 (1 − μ1) > w2 (1 − μ2). As long as the replacement rate of A1 exceeds that of A2, both genotypes will be maintained at equilibrium: A1 because of its higher replacement rate, and A2 because of mutation from A1. If the mutation rate of A1 increases such that the mutation-free fraction of A1 drops relative to that of A2, however, the replacement rate inequality will eventually reverse, yielding a simple error catastrophe: A1 is no longer maintained. The error threshold is the point at which both genotypes have identical replacement rates. Beyond this point, A2 has the higher replacement rate, and A1 is absent at all equilibria—the lack of back mutations ensures that it is not recreated from A2.Since A1 and A2 replacement rate functions approaching each other as the mutation rate increases toward the error threshold. The functions cross at the error threshold, giving rise to an error catastrophe (complete loss of A1) for higher mutation rates. The right part of the figure illustrates that, despite the suggestive terminology (threshold and catastrophe), an error catastrophe is the culmination of a gradual decrease in the equilibrium frequency of A1 rather than a dramatic one. An apt analogy is the geological transition from mountains to plains: starting from the mountains, the plains are approached gradually until they are reached, at which point the plains continue in the absence of any trace of mountains.  TheA1 experiences a potentially much greater mutational loss than A2. Beyond the error threshold this deleterious effect of mutation is retarded: by replacing a mutation-sensitive genotype with a mutation-robust genotype, the error catastrophe reduces the speed at which the replacement rate drops in response to increases in the mutation rate  < 1 for all genotypes). The most trivial example of the distinction between an error threshold and an extinction threshold is the case in which only one genotype exists, and all mutations are lethal. No error catastrophe is possible because no other genotypes exist, but extinction is obviously possible. Our model is one genotype more complicated: mutant offspring of A1 survive as A2, but mutants of A2 are inviable. Thus, there is not an additional error threshold beyond A2, because the model does not allow a third inferior genotype. An A2 population, however, will go extinct if its net growth rate drops below one  can disappear through either an error catastrophe or a population extinction, but A2 can disappear only through extinction. Depending on the values of w1, w2, and μ2, the rate of mutation μ1 at which A1 would be lost to A2 in an error catastrophe may be less than or greater than the value of μ1 at which the population would go extinct. If the error threshold lies beyond the extinction value, then as μ1 increases, the A1 + A2 population will go extinct of its own accord before a true error catastrophe can happen, because the population has disappeared before the mutation rate rises high enough for the error catastrophe . The interplay between extinction catastrophes and error catastrophes can likewise add interesting complexity to the model.The model implies that multiple error thresholds may exist in a fitness landscape. For example, a fitness landscape may include multiple i and μi . With annd so on . It is and so on . For exaA1 and A2 or to A2 exclusively. This equilibration strictly requires an infinite amount of time in an infinite population. In a finite population, as long as the effective population size times the mutation rate is significantly bigger than one (Nμ ≫ 1), the finite population behaves similarly to the infinite population, except that the error threshold is not as sharp. In particular, the lack of back mutations ensures that random events will eventually cause the loss of genotype A1 from the population, so that only genotype A2 remains. If A2 represents multiple types of mutants, then this random loss of A1 is Muller's ratchet [A2 to A1, as described in Our model assumes deterministic dynamics and ignores the stochastic effects of finite population sizes. We have argued that populations will equilibrate either to a combination of  ratchet ,15. BeyoA1 could not be regenerated once lost from the population . Two equations describe the system:At this point we use simple matrix algebra to explore the problem. Assuming discrete time steps, let 3, which allows for back mutation and cannot exceed μ2. If we set μ3 = 0, then these equations describe our earlier model exactly (in n1(t) and n2(t) as t becomes arbitrarily large; that is, we calculate  tM for large t. The standard method for evaluating long-term behavior in such a linear system is to consider the eigenvalues of the system, found as the values of λ satisfyingso that actly in . To eval3 = 0). Equation 4 then yields the solutionsTo establish a baseline, we first assume no back mutation  and y = w2(1 − μ2).where 1 to λ2 as the dominant eigenvalue occurred at x = y in the absence of back mutation, now λ1 remains superior at all values of x and y. As μ3 approaches arbitrarily close to zero, the eigenvalues approach each other but do not cross. Thus, there is no strict error threshold unless μ3 equals zero exactly. Yet as μ3 approaches zero, the system appears more and more like that with no back mutation  leads to a higher net replacement rate of the phenotype.Error thresholds require that some genotypes or phenotypes are more sensitive to mutation than others. Biologically, many phenotypes have the property that they can be produced by multiple, mutationally related genotypes, and mutational robustness is known to vary among protein and nucleic acid sequences ,18–22. THere we illustrate how the breadth of a neutral network—the number of and mutational connectivity among genotypes contained within it—affects the competitiveness of the phenotype. In particular, “survival of the flattest” refers to the ability of inferior phenotypes with large neutral networks to displace superior phenotypes with smaller neutral networks. This phenomenon was recently demonstrated in digital life simulations and simulated RNA molecules ,24.A2 (3 = 0), the error threshold is unaffected by the “network” and is the same as with a single A2 genotype: A2 here has exactly the same properties as the A2 in A1/A2 error threshold is the same as in A2 genotype (A2). Back mutation provides a mutational “connectivity” among both genotypes in the neutral network and renders the network more competitive against A1. This connectivity serves to bolster the frequency of the left A2 genotype at the expense of the right A2. Since only the right peak experiences mutations off the neutral network (at rate μ2), this shift increases the overall replacement rate of A2, allowing A2 to more easily supplant A1. Higher rates of back mutation increase this effect We explore the impact of neutrality on the error threshold with a simple extension of our previous model, this time allowing two genotypes to have the phenotype A2 . This crs effect , top andA2 genotypes into the middle of the neutral network, each with forward mutation rate μ2 and back mutation rate μ3. A2 network also increases, thus shifting the error threshold in favor of A2 and buffering A2 against extinction.This model can also be extended to demonstrate the impact of neutral network size on the error threshold. In this case, we add In summary, the error threshold decreases as the fitness landscape around the inferior phenotype broadens, either via the extension of the neutral network to include additional genotypes or via a net decrease in mutation off the network. The error threshold above which a particular wild-type is lost will depend on its fitness relative to the fitness and breadth of the inferior network. The broader and more mutationally connected the inferior neutral network, the greater its tolerance for mutation and the more likely an error catastrophe that tips the population in its favor. Nonetheless, a sufficiently high fitness can compensate for a high rate of mutational loss.These trade-offs also apply to more complex fitness landscapes consisting of multiple neutral networks with differing fitnesses. In this case, the sequence of error thresholds becomes difficult to intuit and has not previously been addressed quantitatively. L loci with a per locus mutation rate of ν. A single mutant-free, wild-type genotype A1 has highest fitness w1, and all other genotypes with one or more mutations fall into the same neutral network in which each genotype has the same fitness w2. Eigen's classic error threshold result is that, depending on L and w1/w2, there is a specific per base mutation rate ν beyond which the wild-type is lost or maintained only through back mutation and the w2 neutral network prevails, as in our For those familiar with Eigen's original quasispecies model . Base analogs are incorporated into the viral genome as it is being copied from the parent strand. In general, incorporation of a base analog can have two types of effects: chain termination, resulting in immediate death of the progeny strand, or mutation, whereby the base analog is later miscopied when the genome gets replicated. The latter mechanism underlies the antiviral activity of the drug ribavirin for some RNA viruses [Quasispecies error thresholds have been invoked in an important medical application, the extinction of RNA virus populations by lethal mutagenesis—elevating mutation rates with base analog drugs to the point that the virus population disappears. In 2005, an entire issue of  viruses ,29,30. I viruses ,5,30–32)volume 10, issue 2From a practical perspective, it does not matter whether lethal mutagenesis is referred to as an error catastrophe or extinction catastrophe. There may be a benefit, however, to distinguishing between these two concepts and, more generally, to clarifying the concept of a quasispecies. For example, consider a protocol of sublethal mutagenesis, whereby a virus mutation rate is artificially elevated close to an extinction threshold but the virus population persists. Conventional wisdom suggests that such a strategy should be disastrous from a host health perspective, because the increased mutation rate may accelerate viral adaptation. Yet two components of the quasispecies framework suggest that a permanently elevated mutation rate may be deleterious to the virus even when extinction is not achieved. First, a higher mutation rate reduces mean fitness around any local fitness peak. Second, a high mutation rate “flattens” the fitness landscape, so that some narrow peaks cannot be attained. Thus, a sustained high mutation rate that does not drive the virus to extinction may enforce a low mean fitness. Holmes  has likeIn light of this potential impact of high mutation rates, it seems paradoxical that a new class of drugs is being touted for its ability to reduce mutation rates, and thereby block the evolution of resistance to other drugs . Yet mutWe have used a variety of simplifications to explain and depict error catastrophes from a mathematical perspective. Yet it may not be obvious how an error catastrophe would manifest itself in a population, especially how it would appear in a sample of genome sequences taken before and after crossing the error threshold. One possible scenario is similar to that of A second scenario is that of a population in which each individual's genome belongs to either of two “disjoint” neutral networks from very different parts of the fitness landscape. This type of polymorphism could arise if one type was present in the starting population and the other was introduced from a different population through migration. For this scenario, we assume that the population size is sufficiently small relative to the mutational distance between the two neutral networks, so that it is essentially impossible for the population to reach one network from the other via mutation . If one of the networks is more mutationally robust but of lower fitness than the other, an error catastrophe can cause the loss of the higher-fitness network. This type of error catastrophe might reveal itself in genomic data as the loss of polymorphism through the independent mutational meltdown and subsequent disappearance of one of the original types. Back mutation complicates the previous scenario by continually re-introducing the fitter nucleotides, which are then maintained at only a slightly higher level than neutral nucleotides. Back mutation is not a factor in the disjoint network model because the first network to disappear is exceedingly unlikely to be re-introduced via point mutations from the remaining population.Given the difficulty of demonstrating the relatively simple properties of quasispecies dynamics and equilibrium properties, it is not surprising that the empirical demonstration of an error threshold/catastrophe has not been achieved. An interesting step in this direction was provided recently in a study that estimated the error threshold mutation rate that might apply to a primitive organism consisting of RNA molecules—a riboorganism . The stuThe systems offering the best hope for empirically demonstrating an error threshold are in fact naked RNA molecules evolved by directed evolution: ribozymes  and aptamers (binding molecules) evolved in vitro from initial pools of random-sequence molecules. In this work, mutational robustness is recognized as a key property of molecules affecting evolution , but thiw) than a short motif. By virtue of differences in mutational robustness among the different motifs, there should be an error threshold mutation rate, above which the long motifs will be displaced by the short motifs. If one knows the fitnesses i), than the class II and class III ligases . KinetiThe ligase ribozymes will permit an easier demonstration of an error threshold than most other empirical systems, because the alternative functional networks have been identified and can be seeded into the experiment. We thus know in advance what to compare. If, instead, we have information about only the wild-type but not the suboptimal type that dominates beyond the error threshold (as would be the case when working with viruses or bacteria), then it becomes far more complicated to identify an error threshold empirically. Even with the ligase ribozymes, if a population was initiated with only the class I molecule, the class II or III molecules might not evolve at high mutation rates because their sequences might be too distant in sequence space to arise in a quasispecies of class I molecules. All three classes arose in the initial experiment because they were selected from a random sequence pool rather than evolved from a single type.Identification of an error threshold is certainly facilitated by use of a previously characterized genotype–fitness map, but that kind of information is not usually available. In ignorance of the alternative neutral networks for a phenotype, the obvious protocol to use in blindly testing for an error catastrophe is to impose stabilizing selection for the wild-type trait while progressively increasing the mutation rate. One would then test for characteristics of an error catastrophe, such as a population shift in genotype space or increased mutational robustness of the genotypes evolved at high mutation rate. The key difficulty in demonstrating an error catastrophe without a priori knowledge of alternative networks is that there are population genetic processes not considered in the simple quasispecies models that give rise to superficially similar outcomes. Thus, under our suggested experimental protocol, shifts in genotype space may stem from (i) an error catastrophe—the deterministic fixation of deleterious mutations, (ii) stochastic fixation of deleterious and neutral mutations, (iii) the higher mutation rate allowing the population to find higher adaptive peaks, or (iv) adaptive evolution because mutagenesis has altered the selective environment (other than through higher mutation rate). Likewise, increases in mutational robustness may, in fact, suggest an error catastrophe, or may simply be a by-product of a general decline in fitness under high mutation rates. Recent work suggests that genotypes of reduced fitness generally experience a higher fraction of mutations with beneficial effect than genotypes of high fitness ,41. ThisThe theory of quasispecies error thresholds appears sound. The major avenues for new developments are twofold: (i) obtaining empirical support and (ii) incorporating additional realism into the basic models. There are several papers that make progress on this second point, by including recombination , stochas


E. coli may have the key.Why have sex? It seems that genetic studies in the normally non-sexual The evolution of organisms that mix their genes during reproduction is considered one of the major transitions in evolution , becauseNineteenth-century German biologist August Weismann  was the Drosophila melanogaster that could not recombine and hence evolved asexually. This trick allowed the authors to study the effect of recombination on the fate of an introduced beneficial mutation without the obscuring effects of differences in initial fitness or environmental conditions used to induce sex in some populations. In another recent attempt, Goddard et al. [Several recent laboratory evolution experiments have found support for Weismann's proposal that sex facilitates the response to selection . Howeverd et al.  construcd et al.  to compaPLoS Biology, Tim Cooper [Escherichia coli—which, like all bacteria, reproduce asexually—to test the relative contribution of the models mentioned above. The choice of E. coli may seem odd but is an excellent one for several reasons. First, by introducing the F plasmid carrying the genes for conjugation , Cooper could study the consequence of sex—i.e., genetic recombination—while excluding obscuring effects from changed conditions or fitness. Second, by manipulating the mutation rate, he could create conditions where competition among different beneficial mutations is known to limit the rate of adaptation [In this issue of m Cooper  takes thaptation , a prereaptation , Cooper spoT), which was found in rec+ and rec− lines alike. This allowed him to test two specific predictions of the FM model. First, competition with other clones carrying beneficial mutations should slow down the fixation of the focal mutation in rec− lines, but not in rec+ lines. Second, this same interference should lower the fitness of clones carrying the focal mutation relative to contemporary clones in rec− lines, but not in rec+ lines, once competing clones reach appreciable frequencies  as primary explanation, and recombining (rec+) lines  had improved more than nonrecombining (rec−) lines. To distinguish between the explanations offered by the FM model  and the ML model , Cooper looked at the competitive dynamics of a focal beneficial mutation  and from the more efficient removal of multiple deleterious mutations (MD model) may have contributed to the higher fitness of the rec+ populations. Another complicating factor is that the support for the FM model based on the dynamics of the focal mutation depends on the assumption that beneficial mutations go to fixation one-by-one in the rec− populations. A recent study of asexual adaptation in yeast [spoT mutations from rec− and rec+ populations relative to the ancestor and to contemporary clones (along the lines suggested in How strongly do these data reject alternate explanations for the evolutionary maintenance of sex? Parasites or otherwise-fluctuating environmental conditions can be ruled out from the experimental conditions set during evolution. Similarly, the faster removal of deleterious mutations, proposed by the MD model, is unlikely to have been very important, because both of its prerequisites—a rate of deleterious mutations of at least one per genome per generation and prevailing synergistic interactions among mutations—are not met by in yeast  indicatein yeast . A systeHow relevant are these bacterial results for understanding the evolutionary maintenance of sex in higher organisms? In other words, does the bacterial support for the FM model mean that recombination between beneficial mutations is the universally best explanation for the existence of sex and recombination? Not necessarily. The benefits of sex depend on factors that are likely to differ among species, including the variability of environmental conditions, the population size, and the rate, fitness effects, and interactions of mutations. We are gathering information on some of these factors for micro-organisms, but they are hardly known for higher organisms. Only the patient collection of such data from a variety of different organisms may ultimately reveal whether organisms share general unifying principles that allow a universal explanation for sex, or alternatively whether different explanations are required for different species . PossiblUndoubtedly, the study by Cooper  will sti


Understanding interactions between mutations and how they affect fitness is a central problem in evolutionary biology that bears on such fundamental issues as the structure of fitness landscapes and the evolution of sex. To date, analyses of fitness landscapes have focused either on the overall directional curvature of the fitness landscape or on the distribution of pairwise interactions. In this paper, we propose and employ a new mathematical approach that allows a more complete description of multi-way interactions and provides new insights into the structure of fitness landscapes.Escherichia coli obtained by Elena and Lenski. The genotypes were constructed by introducing nine mutations into a wild-type strain and constructing a restricted set of 27 double mutants. Despite the absence of mutants higher than second order, our analysis of this genotypic space points to previously unappreciated gene interactions, in addition to the standard pairwise epistasis. Our analysis confirms Elena and Lenski's inference that the fitness landscape is complex, so that an overall measure of curvature obscures a diversity of interaction types. We also demonstrate that some mutations contribute disproportionately to this complexity. In particular, some mutations are systematically better than others at mixing with other mutations. We also find a strong correlation between epistasis and the average fitness loss caused by deleterious mutations. In particular, the epistatic deviations from multiplicative expectations tend toward more positive values in the context of more deleterious mutations, emphasizing that pairwise epistasis is a local property of the fitness landscape. Finally, we determine the geometry of the fitness landscape, which reflects many of these biologically interesting features.We apply the mathematical theory of gene interactions developed by Beerenwinkel et al. to a fitness landscape for A full description of complex fitness landscapes requires more information than the average curvature or the distribution of independent pairwise interactions. We have proposed a mathematical approach that, in principle, allows a complete description and, in practice, can suggest new insights into the structure of real fitness landscapes. Our analysis emphasizes the value of non-independent genotypes for these inferences. The image of populations evolving on fitness landscapes traces to Sewall Wright's seminal work in the thirties . Since tIf all mutations were strictly additive or multiplicative in their effects on fitness, then it would be rather easy to describe the structure of fitness landscapes and understand the resulting dynamics of adaptation by natural selection. However, many mutations interact with one another in complex ways. For example, two or more mutations may interact such that their combined effect on fitness is much greater or much less than predicted from their individual effects; their combined effect may even be opposite in sign to the expectation based on their individual effects. These deviations from simple expectations are called epistasis ,9, and tE. coli that was obtained and first analyzed by Elena and Lenski [Over the last decade, several studies have sought to examine the form and prevalence of epistatic interactions by measuring the fitness effects of numerous mutations alone and in combination in viruses, bacteria, fungi, and animals ,27-31. Ta, b, c), , and , where all genotypes in each of the three groups share the same antibiotic-resistance markers. Notice that double mutants, each denoted by a pair of letters, were produced except for those consisting of pairs from the same set of three adjacent letters. For example, a was paired with r, s, t, x, y, and z, but not with either b or c. Table We organized the 37 genotypes in the symmetric 10-by-10 matrix with missing values as shown in Table genotopes of all three-locus subsystems of the nine-locus system. A genotope is the set of all possible allele frequencies for a collection of genotypes. As a reference point, Figure a, r, and x, induces the genotope in Figure a, b, and r, further prunes the cube to a triangular prism as in Figure a, b, and c, results in a tetrahedron such as the one shown in Figure The experimental design is illustrated geometrically in Figure E. coli fitness landscape obtained by Elena and Lenski [w·ar - a·r, where w is the fitness of the wild-type, ar is the fitness of a double mutant, and a and r are fitness values for each associated single mutant. They found many significant deviations from zero, including several cases of both positive and negative epistasis. The key messages of our paper are that additional types of gene interactions exist even within this fitness landscape, and that geometric methods can be used to describe and analyze the system more exhaustively.The goal of our analysis is to describe the geometry of the d Lenski . The fitOur analysis of this landscape is based on the approach developed by Beerenwinkel et al. . We idenw·ar - a·r compares two genotypes having zero and two mutations with two others each having one mutation. The minimal Markov basis  of the space of tables of the type displayed in Table ar·bs - as·br. Second, there are another 108 orthogonal "single-double" tests that compare one single mutant and one double mutant with another single mutant and another double mutant, again holding constant the allele frequencies, for example, a·br - b·ar. For both types of non-standard tests, the two genotypes on the right-hand side can be regarded as the products of recombination between the two genotypes on the left-hand side. Notice that the same relationship also holds true for the standard tests. Our first point is that the genotype space in Table Experimental biologists (including two authors of this paper) are likely to raise three concerns about these non-standard tests. What biological insights can non-standard tests provide beyond those obtained using standard tests? Are these additional tests independent of the standard tests? What computational tools are available to perform such tests on other datasets?As we show in the sections that follow, the non-standard tests are potentially useful in at least three respects.  First, they allow one to focus attention on features of epistasis that are not quantifiable by the standard tests.  For example, we perform non-standard tests of the "single-double" type to explore whether some mutations are better overall mixers than others.  Second, non-standard tests span greater genetic distances than do pairwise tests, allowing more powerful analyses of the structure of fitness landscapes.  For example, we use the "double-double" tests to test curvature at genetic distances of four, whereas standard tests allow curvature to be examined only at distances of two.  Third, non-standard tests are an integral part of the complete geometric description of a fitness landscape.  While this high-dimensional geometry is abstract and even foreign, we describe how biological features of gene interactions, such as mixing ability, are embedded in the geometric shape of the landscape.    The non-standard tests are not independent, in a statistical sense, of the pairwise tests or of one another because all the tests are calculated from the same underlying data.  Nonetheless, this complication can be addressed by employing appropriate statistical methods  to ensure that significance levels reflect the data structure.  Regarding the availability of computational tools, we provide references to programs that automate the calculations of the Markov basis and perform the triangulations necessary to describe the geometry of landscapes, and these tools can be applied to other datasets.  In supplementary material, we illustrate the use of these computational tools and show their output .Figure w·ar - a·r) the average fitness decrement Δ = 1 - (a + r)/2 of the two single mutants relative to the wild-type. We then plot the epistasis values as a function of Δ for all 27 equations. In this way, we can test the hypothesis that epistatic deviations tend more toward positive values when mutations in the wild-type background are more deleterious (large Δ) than when mutations are less deleterious . The scope of this analysis can be extended substantially by also including the non-standard "double-double" epistatic equations (such as ar·bs - as·br). We anchored each double-double test with the fittest genotype (say ar) on the left-hand side, and calculated the average fitness decrement, Δ = ar - (as + br)/2, of the two genotypes that appear on the right-hand side of the equation. These tests ask more generally for the relationship between epistasis and the average fitness loss relative to any local fitness peak rather than only the one centred on the wild-type. Moreover, these non-standard tests may have greater power because their "reach" extends over mutational distances of four, rather than the two allowed by standard tests. .To test this prediction for the fitness peak at the wild-type, we calculated for each standard test of epistasis , but highly significant for the larger set of 108 double-double tests , if each of the deviations is viewed as independent. However, these values all rest on 37 genotypes, whose fitness values were estimated with error , and hence the errors are not independent for those epistatic terms that share a genotype. To take this complication into account, we performed Tukey's jackknife test [r = 0.440, ts = 3.761, 26 d.f., one-tailed p = 0.0004) and non-standard tests . It is important to understand that this inference concerns the relationship between average fitness decrements and the form of epistasis around any local peak within the particular landscape represented by these 37 genotypes. In other words, it is an inference about a restricted region of the complete E. coli genotypic space.Figure E. coli genome. In that case, the unit of independent observation becomes the nine single mutations, and the resulting inference concerns the landscape of all possible double mutations derived from the same parental strain. We again applied Tukey's jackknife, this time eliminating in turn all tests containing a particular mutation in any of the four double mutants, rather than eliminating a particular genotype as before. In this case, the association is only marginally significant , but still supports the trend for increasingly positive epistasis with increasing average fitness loss. Given that these tests aim to infer a complex genome-wide relationship between average mutational effects and the form of epistasis from a small sample of mutations, it is encouraging to find such a strong trend.If we want to make the same type of inference about the complete genotype space, rather than the specific subset sampled by Elena and Lenski, we can apply a similar, but not identical, test. More precisely, we want to investigate the correlation between average fitness decrease and epistasis among any single and double mutants of the We present these alternative analyses to emphasize the subtly different hypotheses that can be addressed by using our mathematical approach. Comparing the last two analyses suggests that individual mutations might have pervasive effects on the shape of the local landscape. While pervasive effects of certain mutations can make it more difficult to test broader generalizations, the precise nature of these pervasive effects is of biological interest. In the next section, we follow the same general approach, but focusing on a different set of epistatic interactions, to examine differences between individual mutations in greater detail.a·br - b·ar. This test asks, in effect, whether mutation a or b mixes better with a third mutation r. An individual test of this form might be interesting when one has specific knowledge about the identity of the three mutated genes and the position of their products in a metabolic pathway, for example. By contrast, Elena and Lenski emphasized the statistical properties of epistatic interactions. In this context, we can examine related sets of these single-double equations to ask whether one mutation is a better mixer than another in the context of the sample of mutations with which they were each tested.In this section, we use non-standard tests of the "single-double" type to explore a particular aspect of the fitness landscape, specifically whether certain mutations are better mixers than others. The mixing ability of any particular mutation indicates whether its epistatic interactions with other mutations tend to be positive or negative. We can then measure the relative mixing ability of two mutations by holding constant the identity of other mutations with which the two of interest are mixed. Consider the polynomial a, b, c}, {r, s, t}, or {x, y, z}) was tested with the exact same set of six mutations belonging to the other two sets. For example, the following six equations examine the relative mixing ability of "focal mutations" a and b with respect to "tester mutations" r, s, t, x, y, and z: a·br - b·ar; a·bs - b·as; a·bt - b·at; a·bx - b·ax; a·by - b·ay; and a·bz - b·az. All in all, there are nine groups of six equations each that compare the general mixing abilities of two focal mutations from the same set: a versus b, a versus c, b versus c, r versus s, r versus t, s versus t, x versus y, x versus z, and y versus z. There are another 18 groups of three equations each that compare the mixing abilities of two focal mutations from different sets. Any pair of mutations belonging to the same set of three  of focal mutations, a was the better mixer with two tester mutations whereas b mixed better with four others. For each focal pair, we performed a t-test to ask whether the average epistatic deviation was significantly different from zero. The focal pairs  and  were both significant (p ≤ 0.027), with y being the better mixer in both cases. Two other focal pairs,  and , were marginally non-significant with p = 0.083 and p = 0.068, respectively, and b was the better mixer in both of these cases. The  comparison survives even a stringent Bonferroni correction that accounts for the multiplicity of related tests (p = 0.0054·9 < 0.05). Therefore, we can conclude that mutations are variable in their general mixing ability. The molecular identities of mutations were not identified by Elena and Lenski [In our analysis, we focus on the nine comparisons of mixing ability that each involves six tester mutations, because these provide more statistical power that might reveal differences between focal mutations. Figure t p ≤ 0.07, with yOur final set of results is concerned with the geometric shape of the fitness landscape. Since fitness landscapes are high-dimensional and complicated objects, it is desirable to classify them into a finite set of distinct shapes; with the general idea that fitness landscapes with the same shape are likely to share biological properties. This approach generalizes the classification of bi-allelic two-locus landscapes into those with positive epistasis versus those with negative epistasis. This appealing binary classification has been linked, for example, to the advantage of sex, but it does not extend to higher-dimensional genotype spaces. We present here a notion of the shape of a fitness landscapes for any genotypic space. This concept is intimately related to the interaction tests discussed so far, because the shape is determined by a certain subset of the gene interactions that includes the Markov basis. Thus, the proposed classification of landscapes into shapes can be regarded as a formal summary of all the various standard and non-standard tests.E. coli genotypes and their fitness values as shown in Table The fitness landscape studied in this paper consists of the 37 w, a, r, and ar.  Its genotope is the unit square in -space, and a generic fitness landscape has exactly one of two possible shapes corresponding to either negative or positive epistasis, i.e., to w  ar - a r being either negative or positive.  Negative epistasis induces the triangulation of the square consisting of the two triangles {w, a, r} and {a, r, ar}, whereas positive epistasis results in the other possible triangulation with {w, a, ar} and {w, r, ar}.Consider the bi-allelic two-locus system with genotypes E. coli fitness landscape investigated in this paper.  The general idea is to investigate and describe the interaction space.  This space has finite dimension, but infinitely many elements, and each element of this space represents one interaction.  There are many different ways of extracting potentially interesting subsets of the interaction space.  We suggest looking at the circuits, which generate the space but are not linearly independent.  The signs of the circuits determine the triangulation of the genotope and thus the shape of the fitness landscape.  However, the number of circuits grows fast with the size of the fitness landscape; the present E. coli landscape has 772,731 circuits.  Therefore, we restrict our attention to the much smaller subset provided by the Markov basis.  The circuits and the Markov basis provide a natural generalization of the concept of pairwise epistasis.For larger genetic systems, the role of the triangles is played by simplices.  The shape of the present fitness landscape on 37 genotypes is a triangulation of the genotope into 362 nine-dimensional simplices . This graph provides the basis for statistical inference about the three-way interactions in the given fitness landscape. We find the following shapes among all 27 of the three-dimensional genotopes of type (b) that appear as subsets of the complete dataset: shape 56 (frequency: 6), 7 (5), 25 (4), 8 (3), 52 (3), 21 (2), 23 (2), 2 (1), and 20 (1). A closer inspection of this shape distribution reveals many deviations from linearity in the fitness landscape, but no single dominating shape. This result therefore confirms and extends the earlier finding of Elena and Lenski [The three-locus subsystems that occur in this dataset are represented in Figure The analogous graph of possible shapes for the three-locus subsystem corresponding to the triangular prism Figure  is a hexw, a, b, r, ar, br}, is determined by the signs of the three tests w·ar - a·r, w·br - b·r, and b·ar - a·br. Therefore, the shape summarizes the information about the standard pairwise interactions between mutations a and r and between b and r, as well as the relative mixing ability of a and b with respect to r. For example, five of the six subsystems of type (c) involving mutations y and z have shape 21, while the subspace {w, y, z, b, by, bz} has shape 52. Shape 21 is defined by a negative sign for the second of the three tests above and a positive sign for the other two, whereas shape 52 is defined by positive signs for all three tests. Hence, these shapes reflect positive epistasis involving y (6/6 tests), positive epistasis involving z (5/6 tests), and superior mixing ability of y over z (6/6 tests). As such, the geometric shapes provide more details about the form of mutational interactions than the average values of the tests analyzed in the previous section. On the other hand, these shapes summarize the data by classifying continuous fitness values into discrete shape classes which reflect the sign pattern of the interaction tests.The shape of fitness landscapes on type (c) spaces, such as {Epistasis occurs whenever mutations interact non-linearly with one another, and it represents a major challenge in describing the mathematical structure of real fitness landscapes. With epistatic interactions, the combined effect of two or more mutations on fitness may be greater than, less than, or opposite in sign to expectations obtained by combining their separate effects. A growing body of empirical research indicates that epistasis is very common in nature ,23,26-34To date, two different aspects of epistasis have served as summary statistics of these interactions. First, studies have used the overall directional curvature of mean fitness as a function of the number of random mutations introduced into the genome of some wild-type organism ,27-31. AThe objective of this paper is to introduce biologists to a new mathematical framework for characterizing epistatic interactions between mutations, which goes beyond both overall directional curvature and pairwise interactions by providing a complete geometrical description of the epistatic interactions that define an empirically determined fitness landscape. To that end, we have re-analyzed the dataset from the pairwise experimental design performed by Elena and Lenski  using thE. coli that were constructed by introducing nine mutations into a wild-type strain and constructing a restricted set of 27 double mutants. Despite the absence of any triple or other higher-order mutants in the dataset, our analysis reveals complex new epistatic interactions, beyond the pairwise interactions reported previously. First, our analysis confirms and extends Elena and Lenski's inference that the fitness landscape is complex, such that an overall measure of curvature obscures a complex admixture of interaction types, some with positive and others with negative effects on fitness  on G. The genotope ΠG ⊂ R9 is the set of all allele frequencies that can be realized by populations on G. The genotope is a nine-dimensional polytope, defined as the convex hull of the 37 genotypes, i.e., as the set of all convex combinations λg ≥ 0 and G, all of which are contained in ΠG .Our analysis is based on the mathematical framework presented by Beerenwinkel et al. . The datG → ΠG that assigns to each population its allele frequency spectrum. The kernel of this map defines the interaction space . The elρ. The kernel of ρ is the space of integer tables of the same format with zero margins (row sums). A canonical set of generators for this kernel is the minimal Markov basis [ρ  affinely independent points in Rn [n + 1) points span an affine space of dimension n. For example, Figure w, a, b, and c. By a triangulation of a polytope we mean its decomposition into a set of simplices. A fitness landscape on G gives rise to a triangulation of the genotope ΠG  by algebraic computations.  However, there is no simple recipe for writing out the Markov basis.  We computed the Markov basis independently with the computer algebra systems Macaulay 2  and SingN.B., L.P., and B.S. are responsible for the development of the mathematical theory. S.F.E. and R.E.L. designed and performed the experiments with bacteria. N.B. and R.E.L. formulated the specific analyses reported here and wrote the paper. All the authors contributed to editing the paper and approve of its final form.Instructions for calculating the Markov basis and triangulation of a dataset with Macaulay 2.Click here for fileMacaulay 2 file for computing Markov bases of interaction spaces and triangulations of genotopes. Click here for fileExamples of fitness landscape files to be processed with the program fitness.m2 .Click here for fileMinimal Markov basis of the interaction space.Click here for fileGeometry of the fitness landscape.Click here for file