Fitness, Flux and Phantoms in Temporally Variable Environments

The evolutionary problem of selection in temporally variable environments is addressed by investigating a metabolic model describing the approach to steady state of a flux emanating from a simple linear pathway of unsaturated enzymes catalyzing reversible monomolecular reactions. Analysis confirms previous claims that steps having no influence on the steady state flux may influence transient behavior, and that enzymes immune to natural selection at steady state may become subject to selection when fitness is a function of individual transient metabolic events. Indeed, calculations show that the β-galactosidase of Escherichia coli, which exerts a negligible effect on the steady state lactose flux, controls the approach to steady state. However, after 6 sec the lactose flux is within 0.1% of steady state, and so an ever changing environment must be invoked to continually expose β-galactosidase to selection. Analysis of the metabolic model undergoing multiple transient events reveals that fitness differences remain unaffected if enzyme activities remain constant and become minimized if enzyme activities differ among environments. Until suitable data become available, claims that metabolic behavior away from steady state necessarily exposes a far greater proportion of allozymes to natural selection should be treated with great skepticism.     

Selection and Neutrality in Lactose Operons of Escherichia Coli

The kinetics of the permeases and beta-galactosidases of six lactose operons which had been transduced into a common genetic background from natural isolates of Escherichia coli were investigated. The fitnesses conferred by the operons were determined using chemostat competition experiments in which lactose was the sole growth-limiting factor. The cell wall is demonstrated to impose a resistance to the diffusion of galactosides at low substrate concentrations. A steady state model of the flux of lactose through the metabolic pathway (diffusion, uptake and hydrolysis) is shown to be proportional to fitness. This metabolic model is used to explain why an approximately twofold range in activity among the permease alleles confers a 13% range in fitness, whereas a similar range in activity among alleles of the beta-galactosidase confers a 0.5% range in fitness. This metabolic model implies that selection need not be maximized when a resource is scarce.     

Limits of Adaptation: The Evolution of Selective Neutrality

Many enzymes in intermediary metabolism manifest saturation kinetics in which flux is a concave function of enzyme activity and often of the Michaelis-Menten form. The result is that, when natural selection favors increased enzyme activity so as to maximize flux, a point of diminishing returns will be attained in which any increase in flux results in a disproportionately small increase in fitness. Enzyme activity ultimately will reach a level at which the favorable effect of an increase in activity is of the order 1/(4Ne) or smaller, where Ne is the effective population number. At this point, many mutations that result in small changes in activity will result in negligible changes in fitness and will be selectively nearly neutral. We propose that this process is a mechanism whereby conditions for the occurrence of nearly neutral mutations and gene substitutions can be brought about by the long-continued action of natural selection. Evidence for the hypothesis derives from metabolic theory, direct studies of flux, studies of null and other types of alleles in Drosophila melanogaster and chemostat studies in Escherichia coli. Limitations and complications of the theory include changes in environment or genetic background, enzymes with sharply defined optima of activity, overdominance, pleiotropy, multifunctional enzymes and branched metabolic pathways. We conclude that the theory is a useful synthesis that unites many seemingly unrelated observations. The principal theoretical conclusion is that the conditions for the occurrence of neutral evolution can be brought about as an indirect result of the action of natural selection.     

Do Deleterious Mutations Act Synergistically? Metabolic Control Theory Provides a Partial Answer

Metabolic control theory is used to derive conditions under which two deleterious mutations affecting the dynamics of a metabolic pathway act synergistically. It is found that two mutations tend to act mostly synergistically when they reduce the activity of the same enzyme. If the two mutations affect different enzymes, the conclusion depends on the way that fitness is determined by aspects of the pathway. The cases analyzed are: selection for (1) maximal flux, (2) maximal equilibrium concentration (pool size) of an intermediate, (3) optimal flux, (4) optimal pool size. The respective types of epistasis found are: (1) antagonistic, (2) partly synergistic, (3-4) synergism is likely to predominate over antagonism. This results in somewhat different predictions concerning the effect of metabolic mutations on fitness in prokaryotes and eukaryotes. The fact that bacteria are largely clonal but have often a mosaic gene structure is consistent with expectations from the model.     

The population genetics of alleles affecting enzyme activity.

It is possible to predict the population genetics of allozymes by assuming that fitness is proportional to flux through a biochemical pathway. The model presented here extends previous work by incorporating two additional features of biological realism. Firstly, that more than one biochemical route may exist between any two metabolites. The major routes have been identified as the classical biochemical pathways but in the event of a mutation blocking a major route, minor routes become significant. These minor routes are named "bypass fluxes" and have profound effects on the population genetics of allozymes. Secondly, recent work has suggested that a metabolic cost is associated with enzyme synthesis; this will constitute an additional selective pressure on alleles which affect the amount of enzyme synthesized. The model generates a fitness curve which predicts the fitness associated with any level of enzyme activity. It can utilize data on null or near-null, structural or regulatory, mutations in the presence or absence of bypass fluxes. When data from natural populations of Drosophila are investigated, it is concluded that selection pressures acting on enzyme variants may be much higher than previously thought.     

Validation-based model selection for 13C metabolic flux analysis with uncertain measurement errors

Accurate measurements of metabolic fluxes in living cells are central to metabolism research and metabolic engineering. The gold standard method is model-based metabolic flux analysis (MFA), where fluxes are estimated indirectly from mass isotopomer data with the use of a mathematical model of the metabolic network. A critical step in MFA is model selection: choosing what compartments, metabolites, and reactions to include in the metabolic network model. Model selection is often done informally during the modelling process, based on the same data that is used for model fitting (estimation data). This can lead to either overly complex models (overfitting) or too simple ones (underfitting), in both cases resulting in poor flux estimates. Here, we propose a method for model selection based on independent validation data. We demonstrate in simulation studies that this method consistently chooses the correct model in a way that is independent on errors in measurement uncertainty. This independence is beneficial, since estimating the true magnitude of these errors can be difficult. In contrast, commonly used model selection methods based on the χ²-test choose different model structures depending on the believed measurement uncertainty; this can lead to errors in flux estimates, especially when the magnitude of the error is substantially off. We present a new approach for quantification of prediction uncertainty of mass isotopomer distributions in other labelling experiments, to check for problems with too much or too little novelty in the validation data. Finally, in an isotope tracing study on human mammary epithelial cells, the validation-based model selection method identified pyruvate carboxylase as a key model component. Our results argue that validation-based model selection should be an integral part of MFA model development.     

Allozymes and fitness: Evolution of a problem

It was expected that studies of electrophoretic variability in natural populations would resolve longstanding controversies concerning the form of natural selection and its effect on genetic variance in fitness. Recent studies of fitness components for allozymes in E. coli and Drosophila, where genetic backgrounds have been rigidly controlled, and experiments designed to detect small selection coefficients, suggest that selection is much weaker than earlier investigations would indicate. However, perturbing the metabolic background associated with specific loci often allows functional differences to be amplified to an experimentally measurable level. Frequencies of null activity variants in natural populations indicate that the fitness consequences of reduced activity in heterozygoles are probably very small. These results are supported by recent theoretical considerations suggesting that the activity variation associated with electrophoretic variation will have little effect on overall flux in many pathways.     

Optimal selection of metabolic fluxes for in vivo measurement. I. Development of mathematical methods.

The measurement of uptake and secretion rates is often not sufficient to allow the calculation of all internal metabolic fluxes. Measurements of internal fluxes are needed and these additional measurements are used in conjunction with mass-balance equations to calculate the complete metabolic flux map. A method is presented that identifies the fluxes that should be selected for experimental measurement, and the fluxes that can be computed using the mass-balance equations. The criterion for selecting internal metabolic fluxes for measurement is that the values of the computed fluxes should have low sensitivity to experimental error in the measured fluxes. A condition number indicating the upper bound on this sensitivity, is calculated based on stoichiometry alone. The actual sensitivity is dependent on both the flux measurements and the error in flux measurements, as well as the stoichiometry. If approximate physiologic ranges of fluxes are known a realistic sensitivity can be computed. The exact sensitivity cannot be calculated since the experimental error is usually unknown. The most probable value of the actual sensitivity for a given selection of measured fluxes is estimated by selecting a large number of representative error vectors and calculating the actual sensitivity for each of these. A frequency distribution of actual sensitivities is thus obtained giving a representative range of actual sensitivities for a particular experimental situation.     

Dosage, deletions and dominance: simple models of the evolution of gene expression

Dominance of the wild-type allele over spontaneous null mutations, such as deletions, can be explained in terms of the effects of changes in enzyme dose on the flux of metabolic pathways. If ever increasing levels of enzyme activity have ever decreasing effects on the flux of the biochemical pathway, then halving of dosage will always have a lesser effect on flux than half the effect of complete removal of gene activity. Furthermore, if gene expression rates are high, then halving of dose can have a negligible effect on flux and dominance will be strong. Given that strong dominance appears to be common, this leaves open the issue of why enzyme activity levels are so high that a halving of expression rates is of minimal effect. Why produce so much surplus enzyme? One explanation, suggested by Haldane, is that selection favoured high expression levels as a defence against mutation. We model this scenario formally and show that protection from mutation is an extremely weak force determining expression levels. The selective coefficients are only of the order of the mutation rate. However, if we suppose a linear mapping of flux with fitness and a monotonic cost to increased gene expression, it follows simply that here exists an optimal level of gene expression. By contrast to the mutational model, doubling of gene expression rates when the system is distant from the optimum is associated with extremely high selective coefficients (orders of magnitude higher than the mutation rate). When the cost of gene expression is slight the optimal rate of expression is such that strong dominance will follow.     

Selection against toxic aggregation-prone protein sequences in bacteria

Despite genetic variation has the potential to arise new protein functions, spontaneous mutations usually destabilize the native fold. Misfolded proteins tend to form cytotoxic intracellular aggregates, decreasing cell fitness and leading to degenerative disorders in humans. Therefore, it is thought that selection against protein misfolding and aggregation constrains the evolution of protein sequences. However, obtaining experimental data to validate this hypothesis has been traditionally difficult. Here we exploit bacteria as a model organism to address this question. Using variants of the Alzheimer's related Aβ42 peptide designed to exhibit different in vivo aggregation propensities we show here that, in cell competition experiments, the most aggregation-prone variants are always purged out from the growing population. Flow cytometry analysis of cellular metabolism and viability demonstrates that this purifying effect responds to a clear correlation between physiological burden and intrinsic aggregation propensity. Interestingly, the fitness cost of aggregation appears to be associated with aggregation rates rather than with overall protein solubility. Accordingly, we show that, by reducing in vivo aggregation rates, the model osmolyte proline is able to buffer the metabolic impact of protein aggregation. Overall, our data provide experimental support for the role of toxic protein aggregation on the cell fitness landscape and the evolution of natural protein sequences.     

The shape of selection: using alternative fitness functions to test predictions for selection on flowering time

Selection gradient analysis examines the strength and direction of phenotypic selection as well as the curvature of fitness functions, allowing predictions on and insights into the process of evolution in natural populations. However, traditional linear and quadratic selection analyses are not capable of detecting other features of fitness functions, such as asymmetry or thresholds, which may be relevant for understanding key aspects of selection on many traits. In these cases, additional analyses are needed to test specific hypotheses about fitness functions. In this study we used several approaches to analyze selection on a major life-history trait—flowering time—in the annual plant Brassica rapa subjected to experimentally abbreviated and lengthened growing seasons. We used a model that incorporated a tradeoff between the time allocated to growth versus the time allocated to reproduction in order to predict fitness function shape. The model predicted that optimal flowering time shifts to earlier and later dates as the growing season contracts and expands. It also showed the flowering time fitness function to be asymmetrical: reproductive output increases modestly between the earliest and the optimal flowering date, but then falls sharply with later dates, truncating in a ‘tail of zeros’. Our experimental results strongly supported selection for early flowering in short season and selection for late flowering in long season conditions. We also found support for the predicted asymmetry of the flowering time fitness function, including a ‘tail of zeros’ at later flowering dates. The form of the fitness function revealed here has implications for interpreting estimates of selection on flowering time in natural populations and for refining predictions on evolutionary response to climate change. More generally, this study illustrates the value of diverse statistical approaches to understanding mechanisms of natural selection.     

Activity Variants of Acid Phosphatase-3 among Chromosome 3 Inversions of DROSOPHILA PSEUDOOBSCURA

Sexual selection is measured between two strains of Drosophila melanogaster: a wild strain and a strain mutant at the sepia locus. Frequency-dependent male mating was found to be successful, whereas the female genotype exerted no influence. The rarer the male genotype becomes, the greater is its mating success. A selection model is built for this behavior characteristic in which selection operates differently in the two sexes. The genetic consequencies of this model upon the maintenance of genetic polymorphism at the sepia locus are compared to experimental data from previous population cage studies. The fit obtained with this sexual selection model is compared to that of the larvel selection model previously investigated. A model composed of both sexual and larval components of fitness is presented. The role that each major selection component is expected to play in experimental populations as the gene frequency changes is discussed. Sexual selection leads to an equilibrium level higher than larval selection, and the combined model is very close to the experimental values.     

The Ability of Flux Balance Analysis to Predict Evolution of Central Metabolism Scales with the Initial Distance to the Optimum

The most powerful genome-scale framework to model metabolism, flux balance analysis (FBA), is an evolutionary optimality model. It hypothesizes selection upon a proposed optimality criterion in order to predict the set of internal fluxes that would maximize fitness. Here we present a direct test of the optimality assumption underlying FBA by comparing the central metabolic fluxes predicted by multiple criteria to changes measurable by a ¹³C-labeling method for experimentally-evolved strains. We considered datasets for three Escherichia coli evolution experiments that varied in their length, consistency of environment, and initial optimality. For ten populations that were evolved for 50,000 generations in glucose minimal medium, we observed modest changes in relative fluxes that led to small, but significant decreases in optimality and increased the distance to the predicted optimal flux distribution. In contrast, seven populations evolved on the poor substrate lactate for 900 generations collectively became more optimal and had flux distributions that moved toward predictions. For three pairs of central metabolic knockouts evolved on glucose for 600–800 generations, there was a balance between cases where optimality and flux patterns moved toward or away from FBA predictions. Despite this variation in predictability of changes in central metabolism, two generalities emerged. First, improved growth largely derived from evolved increases in the rate of substrate use. Second, FBA predictions bore out well for the two experiments initiated with ancestors with relatively sub-optimal yield, whereas those begun already quite optimal tended to move somewhat away from predictions. These findings suggest that the tradeoff between rate and yield is surprisingly modest. The observed positive correlation between rate and yield when adaptation initiated further from the optimum resulted in the ability of FBA to use stoichiometric constraints to predict the evolution of metabolism despite selection for rate.     

The principle of flux minimization and its application to estimate stationary fluxes in metabolic networks.

Cellular functions are ultimately linked to metabolic fluxes brought about by thousands of chemical reactions and transport processes. The synthesis of the underlying enzymes and membrane transporters causes the cell a certain 'effort' of energy and external resources. Considering that those cells should have had a selection advantage during natural evolution that enabled them to fulfil vital functions (such as growth, defence against toxic compounds, repair of DNA alterations, etc.) with minimal effort, one may postulate the principle of flux minimization, as follows: given the available external substrates and given a set of functionally important 'target' fluxes required to accomplish a specific pattern of cellular functions, the stationary metabolic fluxes have to become a minimum. To convert this principle into a mathematical method enabling the prediction of stationary metabolic fluxes, the total flux in the network is measured by a weighted linear combination of all individual fluxes whereby the thermodynamic equilibrium constants are used as weighting factors, i.e. the more the thermodynamic equilibrium lies on the right-hand side of the reaction, the larger the weighting factor for the backward reaction. A linear programming technique is applied to minimize the total flux at fixed values of the target fluxes and under the constraint of flux balance (= steady-state conditions) with respect to all metabolites. The theoretical concept is applied to two metabolic schemes: the energy and redox metabolism of erythrocytes, and the central metabolism of Methylobacterium extorquens AM1. The flux rates predicted by the flux-minimization method exhibit significant correlations with flux rates obtained by either kinetic modelling or direct experimental determination. Larger deviations occur for segments of the network composed of redundant branches where the flux-minimization method always attributes the total flux to the thermodynamically most favourable branch. Nevertheless, compared with existing methods of structural modelling, the principle of flux minimization appears to be a promising theoretical approach to assess stationary flux rates in metabolic systems in cases where a detailed kinetic model is not yet available.     

Modeling of selection processes with age-dependent birth and death rates

Two simple models for the competition and selection in age-dependent populations are developed and analyzed mathematically. Following Eigen, competition is introduced by the condition of constant overall-number of the population. In the first model this condition is satisfied by regulation of a dilution flux and in the second case by regulation of a food density. The calculation of maximal fitness is given explicitly for both situations. It is shown that fitness depends in a complicated way on the age-dependence of the birth and death rates. Therefore species have to develop special aging strategies in order to survive in a population under selection pressure. In general, early reproduction is of advantage and increases fitness.     

Many Possible Worlds: Expanding the Ecological Scenarios in Experimental Evolution

Experimental microbial evolution has focused on the particular ecological scenario where a population is placed suddenly in an environment where its fitness is low, and then adapts while the environment remains stable. In line with this, most microbial evolution studies use fitness measures that report how evolved genotypes fare when competed directly against their own distant ancestor while other studies compare life history traits (such as growth rates) of ancestral and evolved genotypes. This standard way of measuring and reporting changes in fitness has resulted in a consistent body of literature that explains adaptation when populations evolve in this “standard ecological scenario.” Here, I suggest that for experimental evolution to investigate adaptation in other ecological scenarios, such as fluctuating or persistently changing environments, measures of fitness must be expanded such that they not only continue to be comparable between experiments, but also account for evolution and demographic effects in all environments that an evolving lineage experiences. I examine two non-standard measures of fitness—fitness flux and the total number of reproductive events—as potential ways to quantify adaptation by integrating historical information about selection over many environments. This approach could allow us to make quantitative and biologically-meaningful comparisons of adaptation across diverse ecological scenarios. I use the case study of understanding how phytoplankton communities may respond to global change, where environmental variables change continuously, to explore concrete ways of using non-standard fitness measures that consider both demographic effects and selection in changing, rather than in changed, environments.     

FITNESS CONSEQUENCES OF SEX‐SPECIFIC SELECTION

Theory suggests that sex‐specific selection can facilitate adaptation in sexually reproducing populations. However, sexual conflict theory and recent experiments indicate that sex‐specific selection is potentially costly due to sexual antagonism: alleles harmful to one sex can accumulate within a population because they are favored in the other sex. Whether sex‐specific selection provides a net fitness benefit or cost depends, in part, on the relative frequency and strength of sexually concordant versus sexually antagonistic selection throughout a species’ genome. Here, we model the net fitness consequences of sex‐specific selection while explicitly considering both sexually concordant and sexually antagonistic selection. The model shows that, even when sexual antagonism is rare, the fitness costs that it imposes will generally overwhelm fitness benefits of sexually concordant selection. Furthermore, the cost of sexual antagonism is, at best, only partially resolved by the evolution of sex‐limited gene expression. To evaluate the key parameters of the model, we analyze an extensive dataset of sex‐specific selection gradients from wild populations, along with data from the experimental evolution literature. The model and data imply that sex‐specific selection may likely impose a net cost on sexually reproducing species, although additional research will be required to confirm this conclusion.     

Dynamic modeling of lactic acid fermentation metabolism with Lactococcus lactis

A dynamic model of lactic acid fermentation using Lactococcus lactis was constructed, and a metabolic flux analysis (MFA) and metabolic control analysis (MCA) were performed to reveal an intensive metabolic understanding of lactic acid bacteria (LAB). The parameter estimation was conducted with COPASI software to construct a more accurate metabolic model. The experimental data used in the parameter estimation were obtained from an LC-MS/ MS analysis and time-course simulation study. The MFA results were a reasonable explanation of the experimental data. Through the parameter estimation, the metabolic system of lactic acid bacteria can be thoroughly understood through comparisons with the original parameters. The coefficients derived from the MCA indicated that the reaction rate of L-lactate dehydrogenase was activated by fructose 1,6-bisphosphate and pyruvate, and pyruvate appeared to be a stronger activator of L-lactate dehydrogenase than fructose 1,6-bisphosphate. Additionally, pyruvate acted as an inhibitor to pyruvate kinase and the phosphotransferase system. Glucose 6-phosphate and phosphoenolpyruvate showed activation effects on pyruvate kinase. Hexose transporter was the strongest effector on the flux through L-lactate dehydrogenase. The concentration control coefficient (CCC) showed similar results to the flux control coefficient (FCC).