Fitness landscapes emerging from pharmacodynamic functions in the evolution of multidrug resistance

Adaptive evolution often involves beneficial mutations at more than one locus. In this case, the trajectory and rate of adaptation is determined by the underlying fitness landscape, that is, the fitness values and mutational connectivity of all genotypes under consideration. Drug resistance, especially resistance to multiple drugs simultaneously, is also often conferred by mutations at several loci so that the concept of fitness landscapes becomes important. However, fitness landscapes underlying drug resistance are not static but dependent on drug concentrations, which means they are influenced by the pharmacodynamics of the drugs administered. Here, I present a mathematical framework for fitness landscapes of multidrug resistance based on Hill functions describing how drug concentrations affect fitness. I demonstrate that these ‘pharmacodynamic fitness landscapes’ are characterized by pervasive epistasis that arises through (i) fitness costs of resistance (even when these costs are additive), (ii) nonspecificity of resistance mutations to drugs, in particular cross‐resistance, and (iii) drug interactions (both synergistic and antagonistic). In the latter case, reciprocal drug suppression may even lead to reciprocal sign epistasis, so that the doubly resistant genotype occupies a local fitness peak that may be difficult to access by evolution. Simulations exploring the evolutionary dynamics on some pharmacodynamic fitness landscapes with both constant and changing drug concentrations confirm the crucial role of epistasis in determining the rate of multidrug resistance evolution.     

Properties of selected mutations and genotypic landscapes under Fisher's geometric model

The fitness landscape—the mapping between genotypes and fitness—determines properties of the process of adaptation. Several small genotypic fitness landscapes have recently been built by selecting a handful of beneficial mutations and measuring fitness of all combinations of these mutations. Here, we generate several testable predictions for the properties of these small genotypic landscapes under Fisher's geometric model of adaptation. When the ancestral strain is far from the fitness optimum, we analytically compute the fitness effect of selected mutations and their epistatic interactions. Epistasis may be negative or positive on average depending on the distance of the ancestral genotype to the optimum and whether mutations were independently selected, or coselected in an adaptive walk. Simulations show that genotypic landscapes built from Fisher's model are very close to an additive landscape when the ancestral strain is far from the optimum. However, when it is close to the optimum, a large diversity of landscape with substantial roughness and sign epistasis emerged. Strikingly, small genotypic landscapes built from several replicate adaptive walks on the same underlying landscape were highly variable, suggesting that several realizations of small genotypic landscapes are needed to gain information about the underlying architecture of the fitness landscape.     

The Effect of Bacterial Recombination on Adaptation on Fitness Landscapes with Limited Peak Accessibility

There is ample empirical evidence revealing that fitness landscapes are often complex: the fitness effect of a newly arisen mutation can depend strongly on the allelic state at other loci. However, little is known about the effects of recombination on adaptation on such fitness landscapes. Here, we investigate how recombination influences the rate of adaptation on a special type of complex fitness landscapes. On these landscapes, the mutational trajectories from the least to the most fit genotype are interrupted by genotypes with low relative fitness. We study the dynamics of adapting populations on landscapes with different compositions and numbers of low fitness genotypes, with and without recombination. Our results of the deterministic model (assuming an infinite population size) show that recombination generally decelerates adaptation on these landscapes. However, in finite populations, this deceleration is outweighed by the accelerating Fisher-Muller effect under certain conditions. We conclude that recombination has complex effects on adaptation that are highly dependent on the particular fitness landscape, population size and recombination rate.     

Evolution in alternating environments with tunable interlandscape correlations

Natural populations are often exposed to temporally varying environments. Evolutionary dynamics in varying environments have been extensively studied, although understanding the effects of varying selection pressures remains challenging. Here, we investigate how cycling between a pair of statistically related fitness landscapes affects the evolved fitness of an asexually reproducing population. We construct pairs of fitness landscapes that share global fitness features but are correlated with one another in a tunable way, resulting in landscape pairs with specific correlations. We find that switching between these landscape pairs, depending on the ruggedness of the landscape and the interlandscape correlation, can either increase or decrease steady‐state fitness relative to evolution in single environments. In addition, we show that switching between rugged landscapes often selects for increased fitness in both landscapes, even in situations where the landscapes themselves are anticorrelated. We demonstrate that positively correlated landscapes often possess a shared maximum in both landscapes that allows the population to step through sub‐optimal local fitness maxima that often trap single landscape evolution trajectories. Finally, we demonstrate that switching between anticorrelated paired landscapes leads to ergodic‐like dynamics where each genotype is populated with nonzero probability, dramatically lowering the steady‐state fitness in comparison to single landscape evolution.     

The rank ordering of genotypic fitness values predicts genetic constraint on natural selection on landscapes lacking sign epistasis

Sewall Wright's genotypic fitness landscape makes explicit one mechanism by which epistasis for fitness can constrain evolution by natural selection. Wright distinguished between landscapes possessing multiple fitness peaks and those with only a single peak and emphasized that the former class imposes substantially greater constraint on natural selection. Here I present novel formalism that more finely partitions the universe of possible fitness landscapes on the basis of the rank ordering of their genotypic fitness values. In this report I focus on fitness landscapes lacking sign epistasis (i.e., landscapes that lack mutations the sign of whose fitness effect varies epistatically), which constitute a subset of Wright's single peaked landscapes. More than one fitness rank ordering lacking sign epistasis exists for L > 2 (where L is the number of interacting loci), and I find that a highly statistically significant effect exists between landscape membership in fitness rank-ordering partition and two different proxies for genetic constraint, even within this subset of landscapes. This statistical association is robust to population size, permitting general inferences about some of the characteristics of fitness rank orderings responsible for genetic constraint on natural selection.     

Population dependent Fourier decomposition of fitness landscapes over recombination spaces: Evolvability of complex characters

The effect of recombination on genotypes can be represented in the form of P-structures, i.e., a map from the set of pairs of genotypes to the power set of genotypes. The interpretation is that the P-structure maps the pair of parental genotypes to the set of recombinant genotypes which result from the recombination of the parental genotypes. A recombination fitness landscape is then a function from the genotypes in a P-structure to the real numbers. In previous papers we have shown that the eigenfunctions of (a matrix associated with) the P-structure provide a basis for the Fourier decomposition of arbitrary recombination landscapes. Here we generalize this framework to include the effect of genotype frequencies, assuming linkage equilibrium. We find that the autocorrelation of the eigenfunctions of the population-weighted P-structure is independent of the population composition. As a consequence we can directly compare the performance of mutation and recombination operators by comparing the autocorrelations on the finite set of elementary landscapes. This comparison suggests that point mutation is a superior search strategy on landscapes with a low order and a moderate order of interaction p < n/3 (n is the number of loci). For more complex landscapes 1-point recombination is superior to both mutation and uniform recombination, but only if the distance among the interacting loci (defining length) is minimal. Furthermore we find that the autocorrelation on any landscape is increasing as the distribution of genotypes becomes more extreme, i.e., if the population occupies a location close to the boundary of the frequency simplex. Landscapes are smoother the more biased the distribution of genotype frequencies is. We suggest that this result explains the paradox that there is little epistatic interaction for quantitative traits detected in natural populations if one uses variance decomposition methods while there is evidence for strong interactions in molecular mapping studies for quantitative trait loci.     

Evolutionary accessibility of mutational pathways

Functional effects of different mutations are known to combine to the total effect in highly nontrivial ways. For the trait under evolutionary selection ('fitness'), measured values over all possible combinations of a set of mutations yield a fitness landscape that determines which mutational states can be reached from a given initial genotype. Understanding the accessibility properties of fitness landscapes is conceptually important in answering questions about the predictability and repeatability of evolutionary adaptation. Here we theoretically investigate accessibility of the globally optimal state on a wide variety of model landscapes, including landscapes with tunable ruggedness as well as neutral 'holey' landscapes. We define a mutational pathway to be accessible if it contains the minimal number of mutations required to reach the target genotype, and if fitness increases in each mutational step. Under this definition accessibility is high, in the sense that at least one accessible pathway exists with a substantial probability that approaches unity as the dimensionality of the fitness landscape (set by the number of mutational loci) becomes large. At the same time the number of alternative accessible pathways grows without bounds. We test the model predictions against an empirical 8-locus fitness landscape obtained for the filamentous fungus Aspergillus niger. By analyzing subgraphs of the full landscape containing different subsets of mutations, we are able to probe the mutational distance scale in the empirical data. The predicted effect of high accessibility is supported by the empirical data and is very robust, which we argue reflects the generic topology of sequence spaces. Together with the restrictive assumptions that lie in our definition of accessibility, this implies that the globally optimal configuration should be accessible to genome wide evolution, but the repeatability of evolutionary trajectories is limited owing to the presence of a large number of alternative mutational pathways.     

Detecting epistasis from an ensemble of adapting populations

The role that epistasis plays during adaptation remains an outstanding problem, which has received considerable attention in recent years. Most of the recent empirical studies are based on ensembles of replicate populations that adapt in a fixed, laboratory controlled condition. Researchers often seek to infer the presence and form of epistasis in the fitness landscape from the time evolution of various statistics averaged across the ensemble of populations. Here, we provide a rigorous analysis of what quantities, drawn from time series of such ensembles, can be used to infer epistasis for populations evolving under weak mutation on finite‐site fitness landscapes. First, we analyze the mean fitness trajectory—that is, the time course of the ensemble average fitness. We show that for any epistatic fitness landscape and starting genotype, there always exists a non‐epistatic fitness landscape that produces the exact same mean fitness trajectory. Thus, the presence of epistasis is not identifiable from the mean fitness trajectory. By contrast, we show that two other ensemble statistics—the time evolution of the fitness variance across populations, and the time evolution of the mean number of substitutions—can detect certain forms of epistasis in the underlying fitness landscape.     

Dynamic RNA Fitness Landscapes of a Group I Ribozyme during Changes to the Experimental Environment

Fitness landscapes of protein and RNA molecules can be studied experimentally using high-throughput techniques to measure the functional effects of numerous combinations of mutations. The rugged topography of these molecular fitness landscapes is important for understanding and predicting natural and experimental evolution. Mutational effects are also dependent upon environmental conditions, but the effects of environmental changes on fitness landscapes remains poorly understood. Here, we investigate the changes to the fitness landscape of a catalytic RNA molecule while changing a single environmental variable that is critical for RNA structure and function. Using high-throughput sequencing of in vitro selections, we mapped a fitness landscape of the Azoarcus group I ribozyme under eight different concentrations of magnesium ions (1–48 mM MgCl₂). The data revealed the magnesium dependence of 16,384 mutational neighbors, and from this, we investigated the magnesium induced changes to the topography of the fitness landscape. The results showed that increasing magnesium concentration improved the relative fitness of sequences at higher mutational distances while also reducing the ruggedness of the mutational trajectories on the landscape. As a result, as magnesium concentration was increased, simulated populations evolved toward higher fitness faster. Curve-fitting of the magnesium dependence of individual ribozymes demonstrated that deep sequencing of in vitro reactions can be used to evaluate the structural stability of thousands of sequences in parallel. Overall, the results highlight how environmental changes that stabilize structures can also alter the ruggedness of fitness landscapes and alter evolutionary processes.     

Steering Evolution with Sequential Therapy to Prevent the Emergence of Bacterial Antibiotic Resistance

The increasing rate of antibiotic resistance and slowing discovery of novel antibiotic treatments presents a growing threat to public health. Here, we consider a simple model of evolution in asexually reproducing populations which considers adaptation as a biased random walk on a fitness landscape. This model associates the global properties of the fitness landscape with the algebraic properties of a Markov chain transition matrix and allows us to derive general results on the non-commutativity and irreversibility of natural selection as well as antibiotic cycling strategies. Using this formalism, we analyze 15 empirical fitness landscapes of E. coli under selection by different β-lactam antibiotics and demonstrate that the emergence of resistance to a given antibiotic can be either hindered or promoted by different sequences of drug application. Specifically, we demonstrate that the majority, approximately 70%, of sequential drug treatments with 2–4 drugs promote resistance to the final antibiotic. Further, we derive optimal drug application sequences with which we can probabilistically ‘steer’ the population through genotype space to avoid the emergence of resistance. This suggests a new strategy in the war against antibiotic–resistant organisms: drug sequencing to shepherd evolution through genotype space to states from which resistance cannot emerge and by which to maximize the chance of successful therapy.     

Adaptation in Tunably Rugged Fitness Landscapes: The Rough Mount Fuji Model

Much of the current theory of adaptation is based on Gillespie’s mutational landscape model (MLM), which assumes that the fitness values of genotypes linked by single mutational steps are independent random variables. On the other hand, a growing body of empirical evidence shows that real fitness landscapes, while possessing a considerable amount of ruggedness, are smoother than predicted by the MLM. In the present article we propose and analyze a simple fitness landscape model with tunable ruggedness based on the rough Mount Fuji (RMF) model originally introduced by Aita et al. in the context of protein evolution. We provide a comprehensive collection of results pertaining to the topographical structure of RMF landscapes, including explicit formulas for the expected number of local fitness maxima, the location of the global peak, and the fitness correlation function. The statistics of single and multiple adaptive steps on the RMF landscape are explored mainly through simulations, and the results are compared to the known behavior in the MLM model. Finally, we show that the RMF model can explain the large number of second-step mutations observed on a highly fit first-step background in a recent evolution experiment with a microvirid bacteriophage.     

The population genetics of adaptation on correlated fitness landscapes: the block model

Several recent theoretical studies of the genetics of adaptation have focused on the mutational landscape model, which considers evolution on rugged fitness landscapes (i.e., ones having many local optima). Adaptation in this model is characterized by several simple results. Here I ask whether these results also hold on correlated fitness landscapes, which are smoother than those considered in the mutational landscape model. In particular, I study the genetics of adaptation in the block model, a tunably rugged model of fitness landscapes. Considering the scenario in which adaptation begins from a high fitness wild-type DNA sequence, I use extreme value theory and computer simulations to study both single adaptive steps and entire adaptive walks. I show that all previous results characterizing single steps in adaptation in the mutational landscape model hold at least approximately on correlated landscapes in the block model; many entire-walk results, however, do not.     

Environmental change exposes beneficial epistatic interactions in a catalytic RNA

Natural selection drives populations of individuals towards local peaks in a fitness landscape. These peaks are created by the interactions between individual mutations. Fitness landscapes may change as an environment changes. In a previous contribution, we discovered a variant of the Azoarcus group I ribozyme that represents a local peak in the RNA fitness landscape. The genotype at this peak is distinguished from the wild-type by four point mutations. We here report ribozyme fitness data derived from constructing all possible combinations of these point mutations. We find that these mutations interact epistatically. Importantly, we show that these epistatic interactions change qualitatively in the three different environments that we studied. We find examples where the relative fitness of a ribozyme can change from neutral or negative in one environment, to positive in another. We also show that the fitness effect of a specific GC-AU base pair switch is dependent on both the environment and the genetic context. Moreover, the mutations that we study improve activity at the cost of decreased structural stability. Environmental change is ubiquitous in nature. Our results suggest that such change can facilitate adaptive evolution by exposing new peaks of a fitness landscape. They highlight a prominent role for genotype-environment interactions in doing so.     

ON THE FINDABILITY OF GENOTYPES

Can we define a measure that describes how easy or difficult it is for a population to evolve to a specific genotype? For populations evolving under weak mutation on a time‐invariant fitness landscape, I argue that one appropriate measure is the expected waiting time, starting from equilibrium, for a population to become fixed for a given genotype. Under this definition for the “findability” of genotypes, I show that for any pair of genotypes (1) a population at equilibrium is always more likely to fix at the more findable before the less findable genotype and (2) the expected time to evolve from the more findable to the less findable genotype is always greater that the expected time to evolve in the opposite direction. Although increasing the fitness of a genotype always increases its findability, in general there is no simple relationship between the rank ordering of genotypes by fitness and the rank ordering of genotypes by findability. I also present a method for quantifying the relative contributions of mutation, selection, substitution rate, and probability of reversion to a genotype's findability.     

Directionality of evolution at molecular and organismic levels.

The molecular evolution theories of Eigen and Kimura are compared and their difference is explained. In terms of Eigen's theory for the evolution of macromolecules, the selection of genotypes occurs directly. The physical meaning of the neutral theory is the degeneracy of the correlation between a phenotype and a genotype at the molecular level. A model theory of evolution on a fitness landscape is proposed. The theory shows that the constraints of selection determined by the structure and dynamics of previous evolution stages increases its rate strongly.     

Reciprocal sign epistasis between frequently experimentally evolved adaptive mutations causes a rugged fitness landscape

The fitness landscape captures the relationship between genotype and evolutionary fitness and is a pervasive metaphor used to describe the possible evolutionary trajectories of adaptation. However, little is known about the actual shape of fitness landscapes, including whether valleys of low fitness create local fitness optima, acting as barriers to adaptive change. Here we provide evidence of a rugged molecular fitness landscape arising during an evolution experiment in an asexual population of Saccharomyces cerevisiae. We identify the mutations that arose during the evolution using whole-genome sequencing and use competitive fitness assays to describe the mutations individually responsible for adaptation. In addition, we find that a fitness valley between two adaptive mutations in the genes MTH1 and HXT6/HXT7 is caused by reciprocal sign epistasis, where the fitness cost of the double mutant prohibits the two mutations from being selected in the same genetic background. The constraint enforced by reciprocal sign epistasis causes the mutations to remain mutually exclusive during the experiment, even though adaptive mutations in these two genes occur several times in independent lineages during the experiment. Our results show that epistasis plays a key role during adaptation and that inter-genic interactions can act as barriers between adaptive solutions. These results also provide a new interpretation on the classic Dobzhansky-Muller model of reproductive isolation and display some surprising parallels with mutations in genes often associated with tumors.     

Microbial experiments on adaptive landscapes

The adaptive landscape is one of the most widely used metaphors in evolutionary biology. It is created by plotting fitness against phenotypes or genotypes in a given environment. The shape of the landscape is crucial in predicting the outcome of evolution: whether evolution will result in populations reaching predictable end points, or whether multiple evolutionary outcomes are more likely. In a more applied sense, the landscape will determine whether organisms will evolve to lose 'costly' resistance to antibiotics, herbicides or pesticides when the use of the control agent is stopped. Laboratory populations of microbes allow evolution to be observed in real time and, as such, provide key insights into the topology of adaptive landscapes.     

Percolation on fitness landscapes: effects of correlation, phenotype, and incompatibilities

We study how correlations in the random fitness assignment may affect the structure of fitness landscapes, in three classes of fitness models. The first is a phenotype space in which individuals are characterized by a large number n of continuously varying traits. In a simple model of random fitness assignment, viable phenotypes are likely to form a giant connected cluster percolating throughout the phenotype space provided the viability probability is larger than 1/2(n). The second model explicitly describes genotype-to-phenotype and phenotype-to-fitness maps, allows for neutrality at both phenotype and fitness levels, and results in a fitness landscape with tunable correlation length. Here, phenotypic neutrality and correlation between fitnesses can reduce the percolation threshold, and correlations at the point of phase transition between local and global are most conducive to the formation of the giant cluster. In the third class of models, particular combinations of alleles or values of phenotypic characters are "incompatible" in the sense that the resulting genotypes or phenotypes have zero fitness. This setting can be viewed as a generalization of the canonical Bateson-Dobzhansky-Muller model of speciation and is related to K-SAT problems, prominent in computer science. We analyze the conditions for the existence of viable genotypes, their number, as well as the structure and the number of connected clusters of viable genotypes. We show that analysis based on expected values can easily lead to wrong conclusions, especially when fitness correlations are strong. We focus on pairwise incompatibilities between diallelic loci, but we also address multiple alleles, complex incompatibilities, and continuous phenotype spaces. In the case of diallelic loci, the number of clusters is stochastically bounded and each cluster contains a very large sub-cube. Finally, we demonstrate that the discrete NK model shares some signature properties of models with high correlations.     

A system for studying evolution of life-like virtual organisms

Background  

Fitness landscapes, the dependences of fitness on the genotype, are of critical importance for the evolution of living beings. Unfortunately, fitness landscapes that are relevant to the evolution of complex biological functions are very poorly known. As a result, the existing theory of evolution is mostly based on postulated fitness landscapes, which diminishes its usefulness. Attempts to deduce fitness landscapes from models of actual biological processes led, so far, to only limited success.