The evolution of epistasis and its links with genetic robustness, complexity and drift in a phenotypic model of adaptation

Learning from Protein Engineering by Deconvolution of Multi-Mutational Variants

This review analyzes a development in biochemistry, enzymology and biotechnology that originally came as a surprise. Following the establishment of directed evolution of stereoselective enzymes in organic chemistry, the concept of partial or complete deconvolution of selective multi-mutational variants was introduced. Early deconvolution experiments of stereoselective variants led to the finding that mutations can interact cooperatively or antagonistically with one another, not just additively. During the past decade, this phenomenon was shown to be general. In some studies, molecular dynamics (MD) and quantum mechanics/molecular mechanics (QM/MM) computations were performed in order to shed light on the origin of non-additivity at all stages of an evolutionary upward climb. Data of complete deconvolution can be used to construct unique multi-dimensional rugged fitness pathway landscapes, which provide mechanistic insights different from traditional fitness landscapes. Along a related line, biochemists have long tested the result of introducing two point mutations in an enzyme for mechanistic reasons, followed by a comparison of the respective double mutant in so-called double mutant cycles, which originally showed only additive effects, but more recently also uncovered cooperative and antagonistic non-additive effects. We conclude with suggestions for future work, and call for a unified overall picture of non-additivity and epistasis.

Allele Frequencies of Genetic Variants Associated with Varroa Drone Brood Resistance (DBR) in Apis mellifera Subspecies across the European Continent

Implementation of marker-assisted selection (MAS) in modern beekeeping would improve sustainability, especially in breeding programs aiming for resilience against the parasitic mite Varroa destructor. Selecting honey bee colonies for natural resistance traits, such as brood-intrinsic suppression of varroa mite reproduction, reduces the use of chemical acaricides while respecting local adaptation. In 2019, eight genomic variants associated with varroa non-reproduction in drone brood were discovered in a single colony from the Amsterdam Water Dune population in the Netherlands. Recently, a new study tested the applicability of these eight genetic variants for the same phenotype on a population-wide scale in Flanders, Belgium. As the properties of some variants varied between the two studies, one hypothesized that the difference in genetic ancestry of the sampled colonies may underly these contribution shifts. In order to frame this, we determined the allele frequencies of the eight genetic variants in more than 360 Apis mellifera colonies across the European continent and found that variant type allele frequencies of these variants are primarily related to the A. mellifera subspecies or phylogenetic honey bee lineage. Our results confirm that population-specific genetic markers should always be evaluated in a new population prior to using them in MAS programs.

Early detection of emerging viral variants through analysis of community structure of coordinated substitution networks

The emergence of viral variants with altered phenotypes is a public health challenge underscoring the need for advanced evolutionary forecasting methods. Given extensive epistatic interactions within viral genomes and known viral evolutionary history, efficient genomic surveillance necessitates early detection of emerging viral haplotypes rather than commonly targeted single mutations. Haplotype inference, however, is a significantly more challenging problem precluding the use of traditional approaches. Here, using SARS-CoV-2 evolutionary dynamics as a case study, we show that emerging haplotypes with altered transmissibility can be linked to dense communities in coordinated substitution networks, which become discernible significantly earlier than the haplotypes become prevalent. From these insights, we develop a computational framework for inference of viral variants and validate it by successful early detection of known SARS-CoV-2 strains. Our methodology offers greater scalability than phylogenetic lineage tracing and can be applied to any rapidly evolving pathogen with adequate genomic surveillance data.

An Amino Acid Substitution in Elongation Factor EF-G1A Alters the Antibiotic Susceptibility of Pseudomonas aeruginosa LasR-Null Mutants

The opportunistic bacterium Pseudomonas aeruginosa uses the LasR-I quorum-sensing system to increase resistance to the aminoglycoside antibiotic tobramycin. Paradoxically, lasR-null mutants are commonly isolated from chronic human infections treated with tobramycin, suggesting there may be a mechanism that permits the emergence of lasR-null mutants under tobramycin selection. We hypothesized that some other genetic mutations that emerge in these isolates might modulate the effects of lasR-null mutations on antibiotic resistance. To test this hypothesis, we inactivated lasR in several highly tobramycin-resistant isolates from long-term evolution experiments. In some of these isolates, inactivating lasR further increased resistance, compared with decreasing resistance of the wild-type ancestor. These strain-dependent effects were due to a G61A nucleotide polymorphism in the fusA1 gene encoding amino acid substitution A21T in the translation elongation factor EF-G1A. The EF-G1A mutational effects required the MexXY efflux pump and the MexXY regulator ArmZ. The fusA1 mutation also modulated ΔlasR mutant resistance to two other antibiotics, ciprofloxacin and ceftazidime. Our results identify a gene mutation that can reverse the direction of the antibiotic selection of lasR mutants, a phenomenon known as sign epistasis, and provide a possible explanation for the emergence of lasR-null mutants in clinical isolates. IMPORTANCE One of the most common mutations in Pseudomonas aeruginosa clinical isolates is in the quorum sensing lasR gene. In laboratory strains, lasR disruption decreases resistance to the clinical antibiotic tobramycin. To understand how lasR mutations emerge in tobramycin-treated patients, we mutated lasR in highly tobramycin-resistant laboratory strains and determined the effects on resistance. Disrupting lasR enhanced the resistance of some strains. These strains had a single amino acid substitution in the translation factor EF-G1A. The EF-G1A mutation reversed the selective effects of tobramycin on lasR mutants. These results illustrate how adaptive mutations can lead to the emergence of new traits in a population and are relevant to understanding how genetic diversity contributes to the progression of disease during chronic infections.

A null model for the distribution of fitness effects of mutations

The distribution of fitness effects (DFE) of new mutations is key to our understanding of many evolutionary processes. Theoreticians have developed several models to help understand the patterns seen in empirical DFEs. Many such models reproduce the broad patterns seen in empirical DFEs but these models often rely on structural assumptions that cannot be tested empirically. Here, we investigate how much of the underlying "microscopic" biological processes involved in the mapping of new mutations to fitness can be inferred from "macroscopic" observations of the DFE. We develop a null model by generating random genotype-to-fitness maps and show that the null DFE is that with the largest possible information entropy. We further show that, subject to one simple constraint, this null DFE is a Gompertz distribution. Finally, we illustrate how the predictions of this null DFE match empirically measured DFEs from several datasets, as well as DFEs simulated from Fisher's geometric model. This suggests that a match between models and empirical data is often not a very strong indication of the mechanisms underlying the mapping of mutation to fitness.

Tobramycin adaptation alters the antibiotic susceptibility of Pseudomonas aeruginosa quorum sensing-null mutants

The opportunistic bacterium Pseudomonas aeruginosa uses the LasR-I quorum sensing system to increase resistance to the aminioglycoside antibiotic tobramycin. Paradoxically, lasR-null mutants are commonly isolated from chronic human infections treated with tobramycin, suggesting there may be a mechanism allowing the lasR-null mutants to persist under tobramycin selection. We hypothesized that the effects of inactivating lasR on tobramycin resistance might be dependent on the presence or absence of other gene mutations in that strain, a phenomenon known as epistasis. To test this hypothesis, we inactivated lasR in several highly tobramycin-resistant isolates from long-term evolution experiments. We show that the effects of ΔlasR on tobramycin resistance are strain dependent. The effects can be attributed to a point mutation in the gene encoding the translation elongation factor fusA1 (G61A nucleotide substitution), which confers a strong selective advantage to lasR-null PA14 under tobramycin selection. This fusA1 G61A mutation results in increased activity of the MexXY efflux pump and expression of the mexXY regulator ArmZ. The fusA1 mutation can also modulate ΔlasR mutant resistance to two other antibiotics, ciprofloxacin and ceftazidime. Our results demonstrate the importance of epistatic gene interactions on antibiotic susceptibility of lasR-null mutants. These results support of the idea that gene interactions might play a significant role in the evolution of quorum sensing in P. aeruginosa.

Genetic complementation fosters evolvability in complex fitness landscapes

The ability of natural selection to optimize traits depends on the topology of the genotype-fitness map (fitness landscape). Epistatic interactions produce rugged fitness landscapes, where adaptation is constrained by the presence of low-fitness intermediates. Here, we used simulations to explore how evolvability in rugged fitness landscapes is influenced by genetic complementation, a process whereby different sequence variants mutually compensate for their deleterious mutations. We designed our model inspired by viral populations, in which genetic variants are known to interact frequently through coinfection. Our simulations indicate that genetic complementation enables a more efficient exploration of rugged fitness landscapes. Although this benefit may be undermined by genetic parasites, its overall effect on evolvability remains positive in populations that exhibit strong relatedness between interacting sequences. Similar processes could operate in contexts other than viral coinfection, such as in the evolution of ploidy.

Protein dynamics: The future is bright and complicated!

Biological life depends on motion, and this manifests itself in proteins that display motion over a formidable range of time scales spanning from femtoseconds vibrations of atoms at enzymatic transition states, all the way to slow domain motions occurring on micro to milliseconds. An outstanding challenge in contemporary biophysics and structural biology is a quantitative understanding of the linkages among protein structure, dynamics, and function. These linkages are becoming increasingly explorable due to conceptual and methodological advances. In this Perspective article, we will point toward future directions of the field of protein dynamics with an emphasis on enzymes. Research questions in the field are becoming increasingly complex such as the mechanistic understanding of high-order interaction networks in allosteric signal propagation through a protein matrix, or the connection between local and collective motions. In analogy to the solution to the "protein folding problem," we argue that the way forward to understanding these and other important questions lies in the successful integration of experiment and computation, while utilizing the present rapid expansion of sequence and structure space. Looking forward, the future is bright, and we are in a period where we are on the doorstep to, at least in part, comprehend the importance of dynamics for biological function.

MTHSA-DHEI: multitasking harmony search algorithm for detecting high-order SNP epistatic interactions

Genome-wide association studies have succeeded in identifying genetic variants associated with complex diseases, but the findings have not been well interpreted biologically. Although it is widely accepted that epistatic interactions of high-order single nucleotide polymorphisms (SNPs) [(1) Single nucleotide polymorphisms (SNP) are mainly deoxyribonucleic acid (DNA) sequence polymorphisms caused by variants at a single nucleotide at the genome level. They are the most common type of heritable variation in humans.] are important causes of complex diseases, the combinatorial explosion of millions of SNPs and multiple tests impose a large computational burden. Moreover, it is extremely challenging to correctly distinguish high-order SNP epistatic interactions from other high-order SNP combinations due to small sample sizes. In this study, a multitasking harmony search algorithm (MTHSA-DHEI) is proposed for detecting high-order epistatic interactions [(2) In classical genetics, if genes X1 and X2 are mutated and each mutation by itself produces a unique disease status (phenotype) but the mutations together cause the same disease status as the gene X1 mutation, gene X1 is epistatic and gene X2 is hypostatic, and gene X1 has an epistatic effect (main effect) on disease status. In this work, a high-order epistatic interaction occurs when two or more SNP loci have a joint influence on disease status.], with the goal of simultaneously detecting multiple types of high-order (k1-order, k2-order, …, kn-order) SNP epistatic interactions. Unified coding is adopted for multiple tasks, and four complementary association evaluation functions are employed to improve the capability of discriminating the high-order SNP epistatic interactions. We compare the proposed MTHSA-DHEI method with four excellent methods for detecting high-order SNP interactions for 8 high-orderepistatic interaction models with no marginal effect (EINMEs) and 12 epistatic interaction models with marginal effects (EIMEs) (*) and implement the MTHSA-DHEI algorithm with a real dataset: age-related macular degeneration (AMD). The experimental results indicate that MTHSA-DHEI has power and an F1-score exceeding 90% for all EIMEs and five EINMEs and reduces the computational time by more than 90%. It can efficiently perform multiple high-order detection tasks for high-order epistatic interactions and improve the discrimination ability for diverse epistasis models.

Hybridization alters the shape of the genotypic fitness landscape, increasing access to novel fitness peaks during adaptive radiation

Estimating the complex relationship between fitness and genotype or phenotype (i.e. the adaptive landscape) is one of the central goals of evolutionary biology. However, adaptive walks connecting genotypes to organismal fitness, speciation, and novel ecological niches are still poorly understood and processes for surmounting fitness valleys remain controversial. One outstanding system for addressing these connections is a recent adaptive radiation of ecologically and morphologically novel pupfishes (a generalist, molluscivore, and scale-eater) endemic to San Salvador Island, Bahamas. We leveraged whole-genome sequencing of 139 hybrids from two independent field fitness experiments to identify the genomic basis of fitness, estimate genotypic fitness networks, and measure the accessibility of adaptive walks on the fitness landscape. We identified 132 single nucleotide polymorphisms (SNPs) that were significantly associated with fitness in field enclosures. Six out of the 13 regions most strongly associated with fitness contained differentially expressed genes and fixed SNPs between trophic specialists; one gene (mettl21e) was also misexpressed in lab-reared hybrids, suggesting a potential intrinsic genetic incompatibility. We then constructed genotypic fitness networks from adaptive alleles and show that scale-eating specialists are the most isolated of the three species on these networks. Intriguingly, introgressed and de novo variants reduced fitness landscape ruggedness as compared to standing variation, increasing the accessibility of genotypic fitness paths from generalist to specialists. Our results suggest that adaptive introgression and de novo mutations alter the shape of the fitness landscape, providing key connections in adaptive walks circumventing fitness valleys and triggering the evolution of novelty during adaptive radiation.

What have we learned from genome-wide association studies (GWAS) in Parkinson's disease?

After fifteen years of genome-wide association studies (GWAS) in Parkinson's disease (PD), what have we learned? Addressing this question will help catalogue the progress made towards elucidating disease mechanisms, improving the clinical utility of the identified loci, and envisioning how we can harness the strides to develop translational GWAS strategies. Here we review the advances of PD GWAS made to date while critically addressing the challenges and opportunities for next-generation GWAS. Thus, deciphering the missing heritability in underrepresented populations is currently at the reach of hand for a truly comprehensive understanding of the genetics of PD across the different ethnicities. Moreover, state-of-the-art GWAS designs hold a true potential for enhancing the clinical applicability of genetic findings, for instance, by improving disease prediction (PD risk and progression). Lastly, advanced PD GWAS findings, alone or in combination with clinical and environmental parameters, are expected to have the capacity for defining patient enriched cohorts stratified by genetic risk profiles and readily available for neuroprotective clinical trials. Overall, envisioning future strategies for advanced GWAS is currently timely and can be instrumental in providing novel genetic readouts essential for a true clinical translatability of PD genetic findings.

On Deleterious Mutations in Perennials: Inbreeding Depression, Mutation Load, and Life-History Evolution

AbstractIn angiosperms, perennials typically present much higher levels of inbreeding depression than annuals. One hypothesis to explain this pattern stems from the observation that inbreeding depression is expressed across multiple life stages in angiosperms. It posits that increased inbreeding depression in more long-lived species could be explained by differences in the way mutations affect fitness, through the life stages at which they are expressed. In this study, we investigate this hypothesis. We combine a physiological growth model and multilocus population genetics approaches to describe a full genotype-to-phenotype-to-fitness map. We study the behavior of mutations affecting growth or survival and explore their consequences in terms of inbreeding depression and mutation load. Although our results agree with empirical data only within a narrow range of conditions, we argue that they may point us toward the type of traits capable of generating high inbreeding depression in long-lived species-that is, traits under sufficiently strong selection, on which selection decreases sharply as life expectancy increases. Then we study the role deleterious mutations maintained at mutation-selection balance may play in the joint evolution of growth and survival strategies.

SNPs in PRKCA-HIF1A-GLUT1 are associated with diabetic kidney disease in a Chinese Han population with type 2 diabetes

OBJECTIVE

To explore the relationship between SNPs in PRKCA-HIF1A-GLUT1 and diabetic kidney disease in Chinese Han people.

MATERIALS AND METHODS

A total of 2552 participants from Shanghai Diabetes Institute Inpatient Database of Shanghai Jiao Tong University Affiliated Sixth People's Hospital were involved in the stage 1 cross-sectional population. A total of 6015 subjects from the Hong Kong Diabetes Register were included for validation. Genotyping of participants was conducted by the MassARRAY Compact Analyzer (Agena Bioscience). The data were analysed by plink, SAS, Haploview.

RESULTS

We identified variants associated with diabetic kidney disease in stage 1. Rs1681851 (P = .0105, OR = 1.331) in GLUT1 as well as rs2301108 (P = .0085, OR = 1.289) and rs79865957 (P = .0204, OR = 1.263) in HIF1A were significantly associated with diabetic kidney disease. Regarding DKD-related traits, rs1681851 was associated with plasma creatinine levels (P = .0169, β = 4.822) and eGFR (P = .0457, β = -6.956). Moreover, the results showed the interactions between PRKCA-GLUT1 in the occurrence of DKD. We further sought validation of the 17 SNPs in a prospective cohort and found that rs900836 and rs844501 were associated with the percentage change in eGFR slope. We performed a meta-analysis of case-control analysis data from the Hong Kong samples together with the stage 1 data from Shanghai. Rs9894851 showed significant correlation with the serum creatinine level as well as eGFR and no SNP showed association with DKD after meta-analysis.

CONCLUSIONS

Our results suggest potential association between SNPs in PRKCA-HIF1A-GLUT1 and diabetic kidney disease in Chinese Han people.

Epistasis, inbreeding depression, and the evolution of self-fertilization

Inbreeding depression resulting from partially recessive deleterious alleles is thought to be the main genetic factor preventing self-fertilizing mutants from spreading in outcrossing hermaphroditic populations. However, deleterious alleles may also generate an advantage to selfers in terms of more efficient purging, while the effects of epistasis among those alleles on inbreeding depression and mating system evolution remain little explored. In this article, we use a general model of selection to disentangle the effects of different forms of epistasis (additive-by-additive, additive-by-dominance, and dominance-by-dominance) on inbreeding depression and on the strength of selection for selfing. Models with fixed epistasis across loci, and models of stabilizing selection acting on quantitative traits (generating distributions of epistasis) are considered as special cases. Besides its effects on inbreeding depression, epistasis may increase the purging advantage associated with selfing (when it is negative on average), while the variance in epistasis favors selfing through the generation of linkage disequilibria that increase mean fitness. Approximations for the strengths of these effects are derived, and compared with individual-based simulation results.

Predicting Evolution Using Regulatory Architecture

The limits of evolution have long fascinated biologists. However, the causes of evolutionary constraint have remained elusive due to a poor mechanistic understanding of studied phenotypes. Recently, a range of innovative approaches have leveraged mechanistic information on regulatory networks and cellular biology. These methods combine systems biology models with population and single-cell quantification and with new genetic tools, and they have been applied to a range of complex cellular functions and engineered networks. In this article, we review these developments, which are revealing the mechanistic causes of epistasis at different levels of biological organization-in molecular recognition, within a single regulatory network, and between different networks-providing first indications of predictable features of evolutionary constraint.

Moderate Amounts of Epistasis are Not Evolutionarily Stable in Small Populations

High mutation rates select for the evolution of mutational robustness where populations inhabit flat fitness peaks with little epistasis, protecting them from lethal mutagenesis. Recent evidence suggests that a different effect protects small populations from extinction via the accumulation of deleterious mutations. In drift robustness, populations tend to occupy peaks with steep flanks and positive epistasis between mutations. However, it is not known what happens when mutation rates are high and population sizes are small at the same time. Using a simple fitness model with variable epistasis, we show that the equilibrium fitness has a minimum as a function of the parameter that tunes epistasis, implying that this critical point is an unstable fixed point for evolutionary trajectories. In agent-based simulations of evolution at finite mutation rate, we demonstrate that when mutations can change epistasis, trajectories with a subcritical value of epistasis evolve to decrease epistasis, while those with supercritical initial points evolve towards higher epistasis. These two fixed points can be identified with mutational and drift robustness, respectively.

The distribution of epistasis on simple fitness landscapes

Fitness interactions between mutations can influence a population's evolution in many different ways. While epistatic effects are difficult to measure precisely, important information is captured by the mean and variance of log fitnesses for individuals carrying different numbers of mutations. We derive predictions for these quantities from a class of simple fitness landscapes, based on models of optimizing selection on quantitative traits. We also explore extensions to the models, including modular pleiotropy, variable effect sizes, mutational bias and maladaptation of the wild type. We illustrate our approach by reanalysing a large dataset of mutant effects in a yeast snoRNA (small nucleolar RNA). Though characterized by some large epistatic effects, these data give a good overall fit to the non-epistatic null model, suggesting that epistasis might have limited influence on the evolutionary dynamics in this system. We also show how the amount of epistasis depends on both the underlying fitness landscape and the distribution of mutations, and so is expected to vary in consistent ways between new mutations, standing variation and fixed mutations.

Match fitness landscapes for macromolecular interaction networks: Selection for translational accuracy and rate can displace tRNA-binding interfaces of non-cognate aminoacyl-tRNA synthetases

Advances in structural biology of aminoacyl-tRNA synthetases (aaRSs) have revealed incredible diversity in how aaRSs bind their tRNA substrates. The causes of this diversity remain mysterious. We developed a new class of highly rugged fitness landscape models called match landscapes, through which genes encode the assortative interactions of their gene products through the complementarity and identifiability of their structural features. We used results from coding theory to prove bounds and equalities on fitness in match landscapes assuming additive interaction energies, macroscopic aminoacylation kinetics including proofreading, site-specific modifiers of interaction, and selection for translational accuracy in multiple, perfectly encoded site-types. Using genotypes based on extended Hamming codes we show that over a wide array of interface sizes and numbers of encoded cognate pairs, selection for translational accuracy alone is insufficient to displace the tRNA-binding interfaces of aaRSs. Yet, under combined selection for translational accuracy and rate, site-specific modifiers are selected to adaptively displace the tRNA-binding interfaces of non-cognate aaRS-tRNA pairs. We describe a remarkable correspondence between the lengths of perfect RNA (quaternary) codes and the modal sizes of small non-coding RNA families.

BigPanDA: PanDA Workload Management System and its Applications beyond ATLAS

Modern experiments collect peta-scale volumes of data and utilize vast, geographically distributed computing infrastructure that serves thousands of scientists around the world. Requirements for rapid, near real-time data processing, fast analysis cycles and need to run massive detector simulations to support data analysis pose special premium on efficient use of available computational resources. A sophisticated Workload Management System (WMS) is needed to coordinate the distribution and processing of data and jobs in such environment. The ATLAS experiment at CERN uses PanDA (Production and Data Analysis) Workload Management System for managing the workflow for all data processing on over 150 data centers. While PanDAcurrently uses more than 250,000 cores with a peak performance of 0.3 petaFLOPS, it runs around 2 million jobs per day on hundreds of Grid sites and serving thousands of ATLAS users. In 2017 about 1.5 exabytes of data were processed with PanDA.In 2012 BigPanDA project project was started with aim to introduce new types of computing resources into ATLAS computing infrastructure, but also to offering PanDA features to different data-intensive applications for projects and experiments outside of ATLAS and High-Energy and Nuclear Physics. In this article we will present accomplishments and discuss possible directions for future work.

Power law fitness landscapes and their ability to predict fitness

Whether or not evolution by natural selection is predictable depends on the existence of general patterns shaping the way mutations interact with the genetic background. This interaction, also known as epistasis, has been observed during adaptation (macroscopic epistasis) and in individual mutations (microscopic epistasis). Interestingly, a consistent negative correlation between the fitness effect of beneficial mutations and background fitness (known as diminishing returns epistasis) has been observed across different species and conditions. We tested whether the adaptation pattern of an additional species, Schizosaccharomyces pombe, followed the same trend. We used strains that differed by the presence of large karyotype differences and observed the same pattern of fitness convergence. Using these data along with published datasets, we measured the ability of different models to describe adaptation rates. We found that a phenotype-fitness landscape shaped like a power law is able to correctly predict adaptation dynamics in a variety of species and conditions. Furthermore we show that this model can provide a link between the observed macroscopic and microscopic epistasis. It may be very useful in the development of algorithms able to predict the adaptation of microorganisms from measures of the current phenotypes. Overall, our results suggest that even though adaptation quickly slows down, populations adapting to lab conditions may be quite far from a fitness peak.

Mutation and Epistasis in Influenza Virus Evolution

Influenza remains a persistent public health challenge, because the rapid evolution of influenza viruses has led to marginal vaccine efficacy, antiviral resistance, and the annual emergence of novel strains. This evolvability is driven, in part, by the virus's capacity to generate diversity through mutation and reassortment. Because many new traits require multiple mutations and mutations are frequently combined by reassortment, epistatic interactions between mutations play an important role in influenza virus evolution. While mutation and epistasis are fundamental to the adaptability of influenza viruses, they also constrain the evolutionary process in important ways. Here, we review recent work on mutational effects and epistasis in influenza viruses.

Robustness and evolvability of heterogeneous cell populations

Biological systems are endowed with two fundamental but seemingly contradictory properties: robustness, the ability to withstand environmental fluctuations and genetic variability; and evolvability, the ability to acquire selectable and heritable phenotypic changes. Cell populations with heterogeneous genetic makeup, such as those of infectious microbial organisms or cancer, rely on their inherent robustness to maintain viability and fitness, but when encountering environmental insults, such as drug treatment, these populations are also poised for rapid adaptation through evolutionary selection. In this study, we develop a general mathematical model that allows us to explain and quantify this fundamental relationship between robustness and evolvability of heterogeneous cell populations. Our model predicts that robustness is, in fact, essential for evolvability, especially for more adverse environments, a trend we observe in aneuploid budding yeast and breast cancer cells. Robustness also compensates for the negative impact of the systems' complexity on their evolvability. Our model also provides a mathematical means to estimate the number of independent processes underlying a system's performance and identify the most generally adapted subpopulation, which may resemble the multi-drug-resistant "persister" cells observed in cancer.

Diminishing-returns epistasis among random beneficial mutations in a multicellular fungus

Adaptive evolution ultimately is fuelled by mutations generating novel genetic variation. Non-additivity of fitness effects of mutations (called epistasis) may affect the dynamics and repeatability of adaptation. However, understanding the importance and implications of epistasis is hampered by the observation of substantial variation in patterns of epistasis across empirical studies. Interestingly, some recent studies report increasingly smaller benefits of beneficial mutations once genotypes become better adapted (called diminishing-returns epistasis) in unicellular microbes and single genes. Here, we use Fisher's geometric model (FGM) to generate analytical predictions about the relationship between the effect size of mutations and the extent of epistasis. We then test these predictions using the multicellular fungus Aspergillus nidulans by generating a collection of 108 strains in either a poor or a rich nutrient environment that each carry a beneficial mutation and constructing pairwise combinations using sexual crosses. Our results support the predictions from FGM and indicate negative epistasis among beneficial mutations in both environments, which scale with mutational effect size. Hence, our findings show the importance of diminishing-returns epistasis among beneficial mutations also for a multicellular organism, and suggest that this pattern reflects a generic constraint operating at diverse levels of biological organization.

Evolution of drift robustness in small populations

Most mutations are deleterious and cause a reduction in population fitness known as the mutational load. In small populations, weakened selection against slightly-deleterious mutations results in an additional fitness reduction. Many studies have established that populations can evolve a reduced mutational load by evolving mutational robustness, but it is uncertain whether small populations can evolve a reduced susceptibility to drift-related fitness declines. Here, using mathematical modeling and digital experimental evolution, we show that small populations do evolve a reduced vulnerability to drift, or ‘drift robustness’. We find that, compared to genotypes from large populations, genotypes from small populations have a decreased likelihood of small-effect deleterious mutations, thus causing small-population genotypes to be drift-robust. We further show that drift robustness is not adaptive, but instead arises because small populations can only maintain fitness on drift-robust fitness peaks. These results have implications for genome evolution in organisms with small effective population sizes.

Genetic drift can reduce fitness in small populations by counteracting selection against deleterious mutations. Here, LaBar and Adami demonstrate through a mathematical model and simulations that small populations tend to evolve to drift-robust fitness peaks, which have a low likelihood of slightly-deleterious mutations.

Genotypic Complexity of Fisher's Geometric Model

Fisher's geometric model was originally introduced to argue that complex adaptations must occur in small steps because of pleiotropic constraints. When supplemented with the assumption of additivity of mutational effects on phenotypic traits, it provides a simple mechanism for the emergence of genotypic epistasis from the nonlinear mapping of phenotypes to fitness. Of particular interest is the occurrence of reciprocal sign epistasis, which is a necessary condition for multipeaked genotypic fitness landscapes. Here we compute the probability that a pair of randomly chosen mutations interacts sign epistatically, which is found to decrease with increasing phenotypic dimension n, and varies nonmonotonically with the distance from the phenotypic optimum. We then derive expressions for the mean number of fitness maxima in genotypic landscapes comprised of all combinations of L random mutations. This number increases exponentially with L, and the corresponding growth rate is used as a measure of the complexity of the landscape. The dependence of the complexity on the model parameters is found to be surprisingly rich, and three distinct phases characterized by different landscape structures are identified. Our analysis shows that the phenotypic dimension, which is often referred to as phenotypic complexity, does not generally correlate with the complexity of fitness landscapes and that even organisms with a single phenotypic trait can have complex landscapes. Our results further inform the interpretation of experiments where the parameters of Fisher's model have been inferred from data, and help to elucidate which features of empirical fitness landscapes can be described by this model.

Accelerated simulation of evolutionary trajectories in origin-fixation models

We present an accelerated algorithm to forward-simulate origin-fixation models. Our algorithm requires, on average, only about two fitness evaluations per fixed mutation, whereas traditional algorithms require, per one fixed mutation, a number of fitness evaluations of the order of the effective population size, Ne Our accelerated algorithm yields the exact same steady state as the original algorithm but produces a different order of fixed mutations. By comparing several relevant evolutionary metrics, such as the distribution of fixed selection coefficients and the probability of reversion, we find that the two algorithms behave equivalently in many respects. However, the accelerated algorithm yields less variance in fixed selection coefficients. Notably, we are able to recover the expected amount of variance by rescaling population size, and we find a linear relationship between the rescaled population size and the population size used by the original algorithm. Considering the widespread usage of origin-fixation simulations across many areas of evolutionary biology, we introduce our accelerated algorithm as a useful tool for increasing the computational complexity of fitness functions without sacrificing much in terms of accuracy of the evolutionary simulation.

Fisher's geometrical model and the mutational patterns of antibiotic resistance across dose gradients

Fisher's geometrical model (FGM) has been widely used to depict the fitness effects of mutations. It is a general model with few underlying assumptions that gives a large and comprehensive view of adaptive processes. It is thus attractive in several situations, for example adaptation to antibiotics, but comes with limitations, so that more mechanistic approaches are often preferred to interpret experimental data. It might be possible however to extend FGM assumptions to better account for mutational data. This is theoretically challenging in the context of antibiotic resistance because resistance mutations are assumed to be rare. In this article, we show with Escherichia coli how the fitness effects of resistance mutations screened at different doses of nalidixic acid vary across a dose-gradient. We found experimental patterns qualitatively consistent with the basic FGM (rate of resistance across doses, gamma distributed costs) but also unexpected patterns such as a decreasing mean cost of resistance with increasing screen dose. We show how different extensions involving mutational modules and variations in trait covariance across environments, can be discriminated based on these data. Overall, simple extensions of the FGM accounted well for complex mutational effects of resistance mutations across antibiotic doses.

The genetics of speciation: Insights from Fisher's geometric model

Research in speciation genetics has uncovered many robust patterns in intrinsic reproductive isolation, and fitness landscape models have been useful in interpreting these patterns. Here, we examine fitness landscapes based on Fisher's geometric model. Such landscapes are analogous to models of optimizing selection acting on quantitative traits, and have been widely used to study adaptation and the distribution of mutational effects. We show that, with a few modifications, Fisher's model can generate all of the major findings of introgression studies (including "speciation genes" with strong deleterious effects, complex epistasis and asymmetry), and the major patterns in overall hybrid fitnesses (including Haldane's Rule, the speciation clock, heterosis, hybrid breakdown, and male-female asymmetry in the F1). We compare our approach to alternative modeling frameworks that assign fitnesses to genotypes by identifying combinations of incompatible alleles. In some cases, the predictions are importantly different. For example, Fisher's model can explain conflicting empirical results about the rate at which incompatibilities accumulate with genetic divergence. In other cases, the predictions are identical. For example, the quality of reproductive isolation is little affected by the manner in which populations diverge.

Epistasis and the Structure of Fitness Landscapes: Are Experimental Fitness Landscapes Compatible with Fisher's Geometric Model?

The fitness landscape defines the relationship between genotypes and fitness in a given environment and underlies fundamental quantities such as the distribution of selection coefficient and the magnitude and type of epistasis. A better understanding of variation in landscape structure across species and environments is thus necessary to understand and predict how populations will adapt. An increasing number of experiments investigate the properties of fitness landscapes by identifying mutations, constructing genotypes with combinations of these mutations, and measuring the fitness of these genotypes. Yet these empirical landscapes represent a very small sample of the vast space of all possible genotypes, and this sample is often biased by the protocol used to identify mutations. Here we develop a rigorous statistical framework based on Approximate Bayesian Computation to address these concerns and use this flexible framework to fit a broad class of phenotypic fitness models (including Fisher's model) to 26 empirical landscapes representing nine diverse biological systems. Despite uncertainty owing to the small size of most published empirical landscapes, the inferred landscapes have similar structure in similar biological systems. Surprisingly, goodness-of-fit tests reveal that this class of phenotypic models, which has been successful so far in interpreting experimental data, is a plausible in only three of nine biological systems. More precisely, although Fisher's model was able to explain several statistical properties of the landscapes-including the mean and SD of selection and epistasis coefficients-it was often unable to explain the full structure of fitness landscapes.

Survival of the Curviest: Noise-Driven Selection for Synergistic Epistasis

A major goal of human genetics is to elucidate the genetic architecture of human disease, with the goal of fueling improvements in diagnosis and the understanding of disease pathogenesis. The degree to which epistasis, or non-additive effects of risk alleles at different loci, accounts for common disease traits is hotly debated, in part because the conditions under which epistasis evolves are not well understood. Using both theory and evolutionary simulation, we show that the occurrence of common diseases (i.e. unfit phenotypes with frequencies on the order of 1%) can, under the right circumstances, be expected to be driven primarily by synergistic epistatic interactions. Conditions that are necessary, collectively, for this outcome include a strongly non-linear phenotypic landscape, strong (but not too strong) selection against the disease phenotype, and "noise" in the genotype-phenotype map that is both environmental (extrinsic, time-correlated) and developmental (intrinsic, uncorrelated) and, in both cases, neither too little nor too great. These results suggest ways in which geneticists might identify, a priori, those disease traits for which an "epistatic explanation" should be sought, and in the process better focus ongoing searches for risk alleles.

Fisher's geometric model predicts the effects of random mutations when tested in the wild

Fisher's geometric model of adaptation (FGM) has been the conceptual foundation for studies investigating the genetic basis of adaptation since the onset of the neo Darwinian synthesis. FGM describes adaptation as the movement of a genotype toward a fitness optimum due to beneficial mutations. To date, one prediction of FGM, the probability of improvement is related to the distance from the optimum, has only been tested in microorganisms under laboratory conditions. There is reason to believe that results might differ under natural conditions where more mutations likely affect fitness, and where environmental variance may obscure the expected pattern. We chemically induced mutations into a set of 19 Arabidopsis thaliana accessions from across the native range of A. thaliana and planted them alongside the premutated founder lines in two habitats in the mid-Atlantic region of the United States under field conditions. We show that FGM is able to predict the outcome of a set of random induced mutations on fitness in a set of A. thaliana accessions grown in the wild: mutations are more likely to be beneficial in relatively less fit genotypes. This finding suggests that FGM is an accurate approximation of the process of adaptation under more realistic ecological conditions.

The rule of declining adaptability in microbial evolution experiments

One of the most recurrent observations after two decades of microbial evolution experiments regards the dynamics of fitness change. In a given environment, low-fitness genotypes are recurrently observed to adapt faster than their more fit counterparts. Since adaptation is the main macroscopic outcome of Darwinian evolution, studying its patterns of change could potentially provide insight into key issues of evolutionary theory, from fixation dynamics to the genetic architecture of organisms. Here, we re-analyze several published datasets from experimental evolution with microbes and show that, despite large differences in the origin of the data, a pattern of inverse dependence of adaptability with fitness clearly emerges. In quantitative terms, it is remarkable to observe little if any degree of idiosyncrasy across systems as diverse as virus, bacteria and yeast. The universality of this phenomenon suggests that its emergence might be understood from general principles, giving rise to the exciting prospect that evolution might be statistically predictable at the macroscopic level. We discuss these possibilities in the light of the various theories of adaptation that have been proposed and delineate future directions of research.

Genetic Interaction Scoring Procedure for Bacterial Species

A genetic interaction occurs when the phenotype of an organism carrying two mutant genes differs from what should have been observed given their independent influence. Such unexpected outcome indicates a mechanistic connection between the perturbed genes, providing a key source of functional information about the cell. Large-scale screening for genetic interactions involves measuring phenotypes of single and double mutants, which for microorganisms is usually done by automated analysis of images of ordered colonies. Obtaining accurate colony sizes, and using them to identify genetic interactions from such screens remains a challenging and time-consuming task. Here, we outline steps to compute genetic interaction scores in E. coli by measuring colony sizes from plate images, performing normalisation, and quantifying the strength of the effect.

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 Utility of Fisher's Geometric Model in Evolutionary Genetics

The accumulation of data on the genomic bases of adaptation has triggered renewed interest in theoretical models of adaptation. Among these models, Fisher Geometric Model (FGM) has received a lot of attention over the last two decades. FGM is based on a continuous multidimensional phenotypic landscape, but it is for the emerging properties of individual mutation effects that it is mostly used. Despite an apparent simplicity and a limited number of parameters, FGM integrates a full model of mutation and epistatic interactions that allows the study of both beneficial and deleterious mutations, and subsequently the fate of evolving populations. In this review, I present the different properties of FGM and the qualitative and quantitative support they have received from experimental evolution data. I later discuss how to estimate the different parameters of the model and outline some future directions to connect FGM and the molecular determinants of adaptation.

Modeling evolution using the probability of fixation: history and implications

Many models of evolution calculate the rate of evolution by multiplying the rate at which new mutations originate within a population by a probability of fixation. Here we review the historical origins, contemporary applications, and evolutionary implications of these "origin-fixation" models, which are widely used in evolutionary genetics, molecular evolution, and phylogenetics. Origin-fixation models were first introduced in 1969, in association with an emerging view of "molecular" evolution. Early origin-fixation models were used to calculate an instantaneous rate of evolution across a large number of independently evolving loci; in the 1980s and 1990s, a second wave of origin-fixation models emerged to address a sequence of fixation events at a single locus. Although origin fixation models have been applied to a broad array of problems in contemporary evolutionary research, their rise in popularity has not been accompanied by an increased appreciation of their restrictive assumptions or their distinctive implications. We argue that origin-fixation models constitute a coherent theory of mutation-limited evolution that contrasts sharply with theories of evolution that rely on the presence of standing genetic variation. A major unsolved question in evolutionary biology is the degree to which these models provide an accurate approximation of evolution in natural populations.

Effects of new mutations on fitness: insights from models and data

The rates and properties of new mutations affecting fitness have implications for a number of outstanding questions in evolutionary biology. Obtaining estimates of mutation rates and effects has historically been challenging, and little theory has been available for predicting the distribution of fitness effects (DFE); however, there have been recent advances on both fronts. Extreme-value theory predicts the DFE of beneficial mutations in well-adapted populations, while phenotypic fitness landscape models make predictions for the DFE of all mutations as a function of the initial level of adaptation and the strength of stabilizing selection on traits underlying fitness. Direct experimental evidence confirms predictions on the DFE of beneficial mutations and favors distributions that are roughly exponential but bounded on the right. A growing number of studies infer the DFE using genomic patterns of polymorphism and divergence, recovering a wide range of DFE. Future work should be aimed at identifying factors driving the observed variation in the DFE. We emphasize the need for further theory explicitly incorporating the effects of partial pleiotropy and heterogeneity in the environment on the expected DFE.

Mapping the fitness landscape of gene expression uncovers the cause of antagonism and sign epistasis between adaptive mutations

How do adapting populations navigate the tensions between the costs of gene expression and the benefits of gene products to optimize the levels of many genes at once? Here we combined independently-arising beneficial mutations that altered enzyme levels in the central metabolism of Methylobacterium extorquens to uncover the fitness landscape defined by gene expression levels. We found strong antagonism and sign epistasis between these beneficial mutations. Mutations with the largest individual benefit interacted the most antagonistically with other mutations, a trend we also uncovered through analyses of datasets from other model systems. However, these beneficial mutations interacted multiplicatively (i.e., no epistasis) at the level of enzyme expression. By generating a model that predicts fitness from enzyme levels we could explain the observed sign epistasis as a result of overshooting the optimum defined by a balance between enzyme catalysis benefits and fitness costs. Knowledge of the phenotypic landscape also illuminated that, although the fitness peak was phenotypically far from the ancestral state, it was not genetically distant. Single beneficial mutations jumped straight toward the global optimum rather than being constrained to change the expression phenotypes in the correlated fashion expected by the genetic architecture. Given that adaptation in nature often results from optimizing gene expression, these conclusions can be widely applicable to other organisms and selective conditions. Poor interactions between individually beneficial alleles affecting gene expression may thus compromise the benefit of sex during adaptation and promote genetic differentiation.

The pace and outcome of a series of adaptive steps in an evolving lineage depends upon how well different beneficial mutations stack on top of each other. We found that independent beneficial mutations that affected gene expression for a metabolic pathway did not work well together, and were often jointly deleterious. The most beneficial mutations interacted the most poorly with others, which was a trend we found common in other biological systems. Through generating a model that accounted for enzymatic benefits and expression costs, we uncovered that this antagonism was caused by a phenotype to fitness mapping that had an intermediate peak. This allowed us to predict the fitness effect of double mutants and to uncover that the single winning mutations tended to move straight to the peak in a single step. These findings demonstrate the importance of considering the phenotypic changes that cause nonlinear interactions between mutations upon fitness, and thus influence how populations evolve.

The reproducibility of adaptation in the light of experimental evolution with whole genome sequencing

A key question in evolutionary biology is the reproducibility of adaptation. This question can now be quantitatively analyzed using experimental evolution coupled to whole genome sequencing (WGS). With complete sequence data, one can assess convergence among replicate populations. In turn, convergence reflects the action of natural selection and also the breadth of the field of possible adaptive solutions. That is, it provides insight into how many genetic solutions or adaptive paths may lead to adaptation in a given environment. Convergence is both a property of an adaptive landscape and, reciprocally, a tool to study that landscape. In this chapter we present the links between convergence and the properties of adaptive landscapes with respect to two types of microbial experimental evolution. The first tries to reconstruct a full adaptive landscape using a handful of carefully identified mutations (the reductionist approach), while the second uses WGS of replicate experiments to infer properties of the adaptive landscape. Reductionist approaches have highlighted the importance of epistasis in shaping the adaptive landscape, but have also uncovered a wide diversity of landscape architectures. The WGS approach has uncovered a very high diversity of beneficial mutations that affect a limited set of genes or functions and also suggests some shortcomings of the reductionist approach. We conclude that convergence may be better defined at an integrated level, such as the genic level or even at a phenotypic level, and that integrated mechanistic models derived from systems biology may offer an interesting perspective for the analysis of convergence at all levels.

Epistasis, pleiotropy, and the mutation load in sexual and asexual populations

Mutation may impose a substantial load on populations, which varies according to the reproductive mode of organisms. Over the past years, various authors used adaptive landscape models to predict the long-term effect of mutation on mean fitness; however, many of these studies assumed very weak mutation rates, so that at most one mutation segregates in the population. In this article, we derive several simple approximations (confirmed by simulations) for the mutation load at high mutation rate (U), using a general model that allows us to play with the number of selected traits (n), the degree of pleiotropy of mutations, and the shape of the fitness function (which affects the average sign and magnitude of epistasis among mutations). When mutations have strong fitness effects, the equilibrium fitness W¯ of sexuals and asexuals is close to e(-U); under weaker mutational effects, sexuals reach a different regime where W¯ is a simple function of U and of a parameter describing the shape of the fitness function. Contrarily to weak mutation results showing that W¯ is an increasing function of population size and a decreasing function of n, these parameters may have opposite effects in sexual populations at high mutation rate.

Rates of fitness decline and rebound suggest pervasive epistasis

Unraveling the factors that determine the rate of adaptation is a major question in evolutionary biology. One key parameter is the effect of a new mutation on fitness, which invariably depends on the environment and genetic background. The fate of a mutation also depends on population size, which determines the amount of drift it will experience. Here, we manipulate both population size and genotype composition and follow adaptation of 23 distinct Escherichia coli genotypes. These have previously accumulated mutations under intense genetic drift and encompass a substantial fitness variation. A simple rule is uncovered: the net fitness change is negatively correlated with the fitness of the genotype in which new mutations appear—a signature of epistasis. We find that Fisher's geometrical model can account for the observed patterns of fitness change and infer the parameters of this model that best fit the data, using Approximate Bayesian Computation. We estimate a genomic mutation rate of 0.01 per generation for fitness altering mutations, albeit with a large confidence interval, a mean fitness effect of mutations of −0.01, and an effective number of traits nine in mutS⁻E. coli. This framework can be extended to confront a broader range of models with data and test different classes of fitness landscape models.

The strength of genetic interactions scales weakly with mutational effects

Background

Genetic interactions pervade every aspect of biology, from evolutionary theory, where they determine the accessibility of evolutionary paths, to medicine, where they can contribute to complex genetic diseases. Until very recently, studies on epistatic interactions have been based on a handful of mutations, providing at best anecdotal evidence about the frequency and the typical strength of genetic interactions. In this study, we analyze a publicly available dataset that contains the growth rates of over five million double knockout mutants of the yeast Saccharomyces cerevisiae.

Results

We discuss a geometric definition of epistasis that reveals a simple and surprisingly weak scaling law for the characteristic strength of genetic interactions as a function of the effects of the mutations being combined. We then utilized this scaling to quantify the roughness of naturally occurring fitness landscapes. Finally, we show how the observed roughness differs from what is predicted by Fisher's geometric model of epistasis, and discuss the consequences for evolutionary dynamics.

Conclusions

Although epistatic interactions between specific genes remain largely unpredictable, the statistical properties of an ensemble of interactions can display conspicuous regularities and be described by simple mathematical laws. By exploiting the amount of data produced by modern high-throughput techniques, it is now possible to thoroughly test the predictions of theoretical models of genetic interactions and to build informed computational models of evolution on realistic fitness landscapes.

Evolution of Escherichia coli rifampicin resistance in an antibiotic-free environment during thermal stress

BACKGROUND

Beneficial mutations play an essential role in bacterial adaptation, yet little is known about their fitness effects across genetic backgrounds and environments. One prominent example of bacterial adaptation is antibiotic resistance. Until recently, the paradigm has been that antibiotic resistance is selected by the presence of antibiotics because resistant mutations confer fitness costs in antibiotic free environments. In this study we show that it is not always the case, documenting the selection and fixation of resistant mutations in populations of Escherichia coli B that had never been exposed to antibiotics but instead evolved for 2000 generations at high temperature (42.2°C).

RESULTS

We found parallel mutations within the rpoB gene encoding the beta subunit of RNA polymerase. These amino acid substitutions conferred different levels of rifampicin resistance. The resistant mutations typically appeared, and were fixed, early in the evolution experiment. We confirmed the high advantage of these mutations at 42.2°C in glucose-limited medium. However, the rpoB mutations had different fitness effects across three genetic backgrounds and six environments.

CONCLUSIONS

We describe resistance mutations that are not necessarily costly in the absence of antibiotics or compensatory mutations but are highly beneficial at high temperature and low glucose. Their fitness effects depend on the environment and the genetic background, providing glimpses into the prevalence of epistasis and pleiotropy.

Gene functional trade-offs and the evolution of pleiotropy

Pleiotropy is the property of genes affecting multiple functions or characters of an organism. Genes vary widely in their degree of pleiotropy, but this variation is often considered a by-product of their evolutionary history. We present a functional theory of how pleiotropy may itself evolve. We consider genes that contribute to two functions, where contributing more to one function detracts from allocation to the second function. We show that whether genes become pleiotropic or specialize on a single function depends on the nature of trade-offs as gene activities contribute to different traits and on how the functionality of these traits affects fitness. In general, when a gene product can perform well at two functions, it evolves to do so, but not when pleiotropy would greatly disrupt each function. Consequently, reduced pleiotropy should often evolve, with genes specializing on the trait that is currently more important to fitness. Even when pleiotropy does evolve, not all genes are expected to become equally pleiotropic; genes with higher levels of expression are more likely to evolve greater pleiotropy. For the case of gene duplicates, we find that perfect subfunctionalization evolves only under stringent conditions. More often, duplicates are expected to maintain a certain degree of functional redundancy, with the gene contributing more to trait functionality evolving the highest degree of pleiotropy. Gene product interactions can facilitate subfunctionalization, but whether they do so depends on the curvature of the fitness surface. Finally, we find that stochastic gene expression favors pleiotropy by selecting for robustness in fitness components.

Genetic constraints on adaptation to a changing environment

Genetic correlations between traits can constrain responses to natural selection. To what extent such correlations limit adaptation depends on patterns of directional selection. I derive the expected rate of adaptation (or evolvability) under randomly changing selection gradients. When directional selection gradients have an arbitrary covariance matrix, the average rate of adaptation depends on genetic correlations between traits, contrary to the isotropic case investigated in previous studies. Adaptation may be faster on average with more genetic correlation between traits, if these traits are selected to change jointly more often than the average pair of traits. However, natural selection maximizes the long-term fitness of a population, not necessarily its rate of adaptation. I therefore derive the average lag load caused by deviations of the mean phenotype from an optimum, under several forms of environmental changes typically experienced by natural populations, both stochastic and deterministic. Simple formulas are produced for how the G matrix affects long-term fitness in these contexts, and I discuss how their parameters can be estimated empirically.