Bayesian inference for stationary points in Gaussian process regression models for event-related potentials analysis

Stationary points embedded in the derivatives are often critical for a model to be interpretable and may be considered as key features of interest in many applications. We propose a semiparametric Bayesian model to efficiently infer the locations of stationary points of a nonparametric function, which also produces an estimate of the function. We use Gaussian processes as a flexible prior for the underlying function and impose derivative constraints to control the function's shape via conditioning. We develop an inferential strategy that intentionally restricts estimation to the case of at least one stationary point, bypassing possible mis-specifications in the number of stationary points and avoiding the varying dimension problem that often brings in computational complexity. We illustrate the proposed methods using simulations and then apply the method to the estimation of event-related potentials derived from electroencephalography (EEG) signals. We show how the proposed method automatically identifies characteristic components and their latencies at the individual level, which avoids the excessive averaging across subjects that is routinely done in the field to obtain smooth curves. By applying this approach to EEG data collected from younger and older adults during a speech perception task, we are able to demonstrate how the time course of speech perception processes changes with age.     

Phenotypic selection in natural populations: what limits directional selection?

Studies of phenotypic selection document directional selection in many natural populations. What factors reduce total directional selection and the cumulative evolutionary responses to selection? We combine two data sets for phenotypic selection, representing more than 4,600 distinct estimates of selection from 143 studies, to evaluate the potential roles of fitness trade-offs, indirect (correlated) selection, temporally varying selection, and stabilizing selection for reducing net directional selection and cumulative responses to selection. We detected little evidence that trade-offs among different fitness components reduced total directional selection in most study systems. Comparisons of selection gradients and selection differentials suggest that correlated selection frequently reduced total selection on size but not on other types of traits. The direction of selection on a trait often changes over time in many temporally replicated studies, but these fluctuations have limited impact in reducing cumulative directional selection in most study systems. Analyses of quadratic selection gradients indicated stabilizing selection on body size in at least some studies but provided little evidence that stabilizing selection is more common than disruptive selection for most traits or study systems. Our analyses provide little evidence that fitness trade-offs, correlated selection, or stabilizing selection strongly constrains the directional selection reported for most quantitative traits.     

Stripes,spots, or reversed spots in two-dimensional Turing systems

Two-dimensional Turing models can generate stationary striped patterns or spotted patterns, and are used to explain the body pattern formation of animals. We studied the effects of the choice of reaction terms on pattern selection, i.e., which pattern is likely to be formed. We examined in detail a model with linear reaction terms and additional constraint terms that confine two variables within a finite range. In the one-dimensional model, a periodic stationary pattern can be formed only when the activator level is constrained both from below and from above. In the two-dimensional model, the relative distance of the equilibrium level of the activator between the upper and lower limitations determines the pattern selection. Striped patterns are produced when the equilibrium is equally distant from the upper and the lower limitations, but spotted patterns are produced when the equilibrium is clearly closer to one than to the other of two limitations. We then examined models with nonlinear reaction terms, including both activator-inhibitor and activator-depletion substrate type models; we attempted to explain the pattern selection of these nonlinear models based on the results of linear models with constraints. The distribution of the activator level is skewed positively and negatively for spotted patterns and reversed spotted patterns, respectively. In contrast, the skew of the distribution of the activator level was close to zero in the case of striped patterns. This observation provides a heuristic argument of how the location of the equilibrium between the constraints leads to pattern selection.     

Cost-benefit theory and optimal design of gene regulation functions

Cells respond to the environment by regulating the expression of genes according to environmental signals. The relation between the input signal level and the expression of the gene is called the gene regulation function. It is of interest to understand the shape of a gene regulation function in terms of the environment in which it has evolved and the basic constraints of biological systems. Here we address this by presenting a cost-benefit theory for gene regulation functions that takes into account temporally varying inputs in the environment and stochastic noise in the biological components. We apply this theory to the well-studied lac operon of E. coli. The present theory explains the shape of this regulation function in terms of temporal variation of the input signals, and of minimizing the deleterious effect of cell-cell variability in regulatory protein levels. We also apply the theory to understand the evolutionary tradeoffs in setting the number of regulatory proteins and for selection of feed-forward loops in genetic circuits. The present cost-benefit theory can be used to understand the shape of other gene regulatory functions in terms of environment and noise constraints.     

Using semidefinite programming to optimize unequal deployment of genotypes to a clonal seed orchard

Tree breeders must often consider the conservation of genetic diversity, while at the same time, maximizing response to selection. In the case of seed orchards, the buyer of seed wants maximum performance, while satisfying a restriction, sometimes legislated, on the diversity deployed to the forest. Optimal selection will not completely avoid kinship but rather maximize gain while imposing a constraint on average relatedness. Here, we present the application of semidefinite programming (SDP) as a flexible approach to optimize the deployment of genotypes to a clonal seed orchard. We formulate the selection problem as an SDP, where average breeding value is to be maximized, while imposing constraints on relatedness, as well as maximum and minimum contributions from each candidate. An open-source solver, SDPA, was embedded into a tool designed to make the optimization of seed orchards by SDP simple and flexible. Case studies optimizing seed orchards for Scots pine and loblolly pine illustrate how this flexibility can be used to impose additional constraints on the scion material available from some candidate genotypes and optimize selection even when related candidates have varying degrees of coancestry among them. Additional situations where SDP can be employed are discussed.     

The stationary distribution of a continuously varying strategy in a class-structured population under mutation-selection-drift balance

Many traits and/or strategies expressed by organisms are quantitative phenotypes. Because populations are of finite size and genomes are subject to mutations, these continuously varying phenotypes are under the joint pressure of mutation, natural selection and random genetic drift. This article derives the stationary distribution for such a phenotype under a mutation-selection-drift balance in a class-structured population allowing for demographically varying class sizes and/or changing environmental conditions. The salient feature of the stationary distribution is that it can be entirely characterized in terms of the average size of the gene pool and Hamilton's inclusive fitness effect. The exploration of the phenotypic space varies exponentially with the cumulative inclusive fitness effect over state space, which determines an adaptive landscape. The peaks of the landscapes are those phenotypes that are candidate evolutionary stable strategies and can be determined by standard phenotypic selection gradient methods (e.g. evolutionary game theory, kin selection theory, adaptive dynamics). The curvature of the stationary distribution provides a measure of the stability by convergence of candidate evolutionary stable strategies, and it is evaluated explicitly for two biological scenarios: first, a coordination game, which illustrates that, for a multipeaked adaptive landscape, stochastically stable strategies can be singled out by letting the size of the gene pool grow large; second, a sex-allocation game for diploids and haplo-diploids, which suggests that the equilibrium sex ratio follows a Beta distribution with parameters depending on the features of the genetic system.     

Engineering gene overlaps to sustain genetic constructs in vivo

Evolution is often an obstacle to the engineering of stable biological systems due to the selection of mutations inactivating costly gene circuits. Gene overlaps induce important constraints on sequences and their evolution. We show that these constraints can be harnessed to increase the stability of costly genes by purging loss-of-function mutations. We combine computational and synthetic biology approaches to rationally design an overlapping reading frame expressing an essential gene within an existing gene to protect. Our algorithm succeeded in creating overlapping reading frames in 80% of E. coli genes. Experimentally, scoring mutations in both genes of such overlapping construct, we found that a significant fraction of mutations impacting the gene to protect have a deleterious effect on the essential gene. Such an overlap thus protects a costly gene from removal by natural selection by associating the benefit of this removal with a larger or even lethal cost. In our synthetic constructs, the overlap converts many of the possible mutants into evolutionary dead-ends, reducing the evolutionary potential of the system and thus increasing its stability over time.     

Modularity, noise, and natural selection

Most biological systems are formed by component parts that are to some degree interrelated. Groups of parts that are more associated among themselves and are relatively autonomous from others are called modules. One of the consequences of modularity is that biological systems usually present an unequal distribution of the genetic variation among traits. Estimating the covariance matrix that describes these systems is a difficult problem due to a number of factors such as poor sample sizes and measurement errors. We show that this problem will be exacerbated whenever matrix inversion is required, as in directional selection reconstruction analysis. We explore the consequences of varying degrees of modularity and signal-to-noise ratio on selection reconstruction. We then present and test the efficiency of available methods for controlling noise in matrix estimates. In our simulations, controlling matrices for noise vastly improves the reconstruction of selection gradients. We also perform an analysis of selection gradients reconstruction over a New World Monkeys skull database to illustrate the impact of noise on such analyses. Noise-controlled estimates render far more plausible interpretations that are in full agreement with previous results.     

Thermodynamic stability of ecosystems

The stability of ecosystems during periods of stasis in their macro-evolutionary trajectory is studied from a non-equilibrium thermodynamic perspective. Individuals of the species are considered as units of entropy production and entropy exchange in an open thermodynamic system. Within the framework of the classical theory of irreversible thermodynamics, and under the condition of constant external constraints, such a system will naturally evolve toward a globally stable thermodynamic stationary state. It is thus suggested that the ecological steady state, or stasis, is a particular case of the thermodynamic stationary state, and that the evolution of community stability through natural selection is a manifestation of non-equilibrium thermodynamic directives. Furthermore, it is argued that punctuation of stasis leading to ecosystem succession, may be a manifestation of non-equilibrium "phase transitions" brought on by a change of external constraints through a thermodynamic critical point.     

The stationary distribution of allele frequencies when selection acts at unlinked loci

We consider population genetics models where selection acts at a set of unlinked loci. It is known that if the fitness of an individual is multiplicative across loci, then these loci are independent. We consider general selection models, but assume parent-independent mutation at each locus. For such a model, the joint stationary distribution of allele frequencies is proportional to the stationary distribution under neutrality multiplied by a known function of the mean fitness of the population. We further show how knowledge of this stationary distribution enables direct simulation of the genealogy of a sample at a single-locus. For a specific selection model appropriate for complex disease genes, we use simulation to determine what features of the genealogy differ between our general selection model and a multiplicative model.     

Brain evolution and development: adaptation,allometry and constraint

Phenotypic traits are products of two processes: evolution and development. But how do these processes combine to produce integrated phenotypes? Comparative studies identify consistent patterns of covariation, or allometries, between brain and body size, and between brain components, indicating the presence of significant constraints limiting independent evolution of separate parts. These constraints are poorly understood, but in principle could be either developmental or functional. The developmental constraints hypothesis suggests that individual components (brain and body size, or individual brain components) tend to evolve together because natural selection operates on relatively simple developmental mechanisms that affect the growth of all parts in a concerted manner. The functional constraints hypothesis suggests that correlated change reflects the action of selection on distributed functional systems connecting the different sub-components, predicting more complex patterns of mosaic change at the level of the functional systems and more complex genetic and developmental mechanisms. These hypotheses are not mutually exclusive but make different predictions. We review recent genetic and neurodevelopmental evidence, concluding that functional rather than developmental constraints are the main cause of the observed patterns.     

Selection, Hitchhiking and Disequilibrium Analysis at Three Linked Loci with Application to Hla Data

The HLA system has been extensively studied from an evolutionary perspective. Although it is clear that selection has acted on the genes in the HLA complex, the nature of this selection has yet to be fully clarified. A study of constrained disequilibrium values is presented that is applicable to HLA and other less polymorphic systems with three or more linked loci, with the purpose of identifying selection events. The method uses the fact that three locus systems impose additional constraints on the range of possible disequilibrium values for any pair of loci. We have thus examined the behavior of the normalized pairwise disequilibrium measures using two locus (D'), and also three locus (D"), constraints on pairwise disequilibria in a three locus system when one of the three loci is under positive selection. The difference between these measures, delta = magnitude of D' - magnitude of D", has a distribution for the two unselected loci differing from that for the selected locus with either of the unselected loci (the hallmark is a high positive value of delta for the two unselected loci). An examination of genetic drift indicates that positive delta values are unlikely to be found in human populations in the absence of selection when recombination is greater than about 0.1%. This measure can thus provide insight into which allele of several linked loci might have been subject to selection. Application of this method to HLA haplotypes from a large French population study (Provinces Francaise) identifies selected alleles on particular haplotypes. Application of a complementary method, disequilibrium pattern analysis also confirms the action of selection on these haplotypes.     

Trends, stasis, and drift in the evolution of nematode vulva development

BACKGROUND: A surprising amount of developmental variation has been observed for otherwise highly conserved features, a phenomenon known as developmental system drift. Either stochastic processes (e.g., drift and absence of selection-independent constraints) or deterministic processes (e.g., selection or constraints) could be the predominate mechanism for the evolution of such variation. We tested whether evolutionary patterns of change were unbiased or biased, as predicted by the stochastic or deterministic hypotheses, respectively. As a model, we used the nematode vulva, a highly conserved, essential organ, the development of which has been intensively studied in the model systems Caenorhabditis elegans and Pristionchus pacificus. RESULTS: For 51 rhabditid species, we analyzed more than 40 characteristics of vulva development, including cell fates, fate induction, cell competence, division patterns, morphogenesis, and related aspects of gonad development. We then defined individual characters and plotted their evolution on a phylogeny inferred for 65 species from three nuclear gene sequences. This taxon-dense phylogeny provides for the first time a highly resolved picture of rhabditid evolution and allows the reconstruction of the number and directionality of changes in the vulva development characters. We found an astonishing amount of variation and an even larger number of evolutionary changes, suggesting a high degree of homoplasy (convergences and reversals). Surprisingly, only two characters showed unbiased evolution. Evolution of all other characters was biased. CONCLUSIONS: We propose that developmental evolution is primarily governed by selection and/or selection-independent constraints, not stochastic processes such as drift in unconstrained phenotypic space.     

THE EVOLUTION OF PHENOTYPIC CORRELATIONS AND “DEVELOPMENTAL MEMORY”

Development introduces structured correlations among traits that may constrain or bias the distribution of phenotypes produced. Moreover, when suitable heritable variation exists, natural selection may alter such constraints and correlations, affecting the phenotypic variation available to subsequent selection. However, exactly how the distribution of phenotypes produced by complex developmental systems can be shaped by past selective environments is poorly understood. Here we investigate the evolution of a network of recurrent nonlinear ontogenetic interactions, such as a gene regulation network, in various selective scenarios. We find that evolved networks of this type can exhibit several phenomena that are familiar in cognitive learning systems. These include formation of a distributed associative memory that can “store” and “recall” multiple phenotypes that have been selected in the past, recreate complete adult phenotypic patterns accurately from partial or corrupted embryonic phenotypes, and “generalize” (by exploiting evolved developmental modules) to produce new combinations of phenotypic features. We show that these surprising behaviors follow from an equivalence between the action of natural selection on phenotypic correlations and associative learning, well‐understood in the context of neural networks. This helps to explain how development facilitates the evolution of high‐fitness phenotypes and how this ability changes over evolutionary time.     

Predicting evolutionary potential: A numerical test of evolvability measures

Despite sophisticated mathematical models, the theory of microevolution is mostly treated as a qualitative rather than a quantitative tool. Numerical measures of selection, constraints, and evolutionary potential are often too loosely connected to theory to provide operational predictions of the response to selection. In this paper, we study the ability of a set of operational measures of evolvability and constraint to predict short‐term selection responses generated by individual‐based simulations. We focus on the effects of selective constraints under which the response in one trait is impeded by stabilizing selection on other traits. The conditional evolvability is a measure of evolutionary potential explicitly developed for this situation. We show that the conditional evolvability successfully predicts rates of evolution in an equilibrium situation, and further that these equilibria are reached with characteristic times that are inversely proportional to the fitness load generated by the constraining characters. Overall, we find that evolvabilities and conditional evolvabilities bracket responses to selection, and that they together can be used to quantify evolutionary potential on time scales where the G‐matrix remains relatively constant.     

Annotation of selection strengths in viral genomes

MOTIVATION: Viral genomes tend to code in overlapping reading frames to maximize informational content. This may result in atypical codon bias and particular evolutionary constraints. Due to the fast mutation rate of viruses, there is additional strong evidence for varying selection between intra- and intergenomic regions. The presence of multiple coding regions complicates the concept of K(a)/K(s) ratio, and thus begs for an alternative approach when investigating selection strengths. Building on the paper by McCauley and Hein, we develop a method for annotating a viral genome coding in overlapping reading frames. We introduce an evolutionary model capable of accounting for varying levels of selection along the genome, and incorporate it into our prior single sequence HMM methodology, extending it now to a phylogenetic HMM. Given an alignment of several homologous viruses to a reference sequence, we may thus achieve an annotation both of coding regions as well as selection strengths, allowing us to investigate different selection patterns and hypotheses. RESULTS: We illustrate our method by applying it to a multiple alignment of four HIV2 sequences, as well as of three Hepatitis B sequences. We obtain an annotation of the coding regions, as well as a posterior probability for each site of the strength of selection acting on it. From this we may deduce the average posterior selection acting on the different genes. Whilst we are encouraged to see in HIV2, that the known to be conserved genes gag and pol are indeed annotated as such, we also discover several sites of less stringent negative selection within the env gene. To the best of our knowledge, we are the first to subsequently provide a full selection annotation of the Hepatitis B genome by explicitly modelling the evolution within overlapping reading frames, and not relying on simple K(a)/K(s) ratios.     

Exponential Signaling Gain at the Receptor Level Enhances Signal-to-Noise Ratio in Bacterial Chemotaxis

Cellular signaling systems show astonishing precision in their response to external stimuli despite strong fluctuations in the molecular components that determine pathway activity. To control the effects of noise on signaling most efficiently, living cells employ compensatory mechanisms that reach from simple negative feedback loops to robustly designed signaling architectures. Here, we report on a novel control mechanism that allows living cells to keep precision in their signaling characteristics – stationary pathway output, response amplitude, and relaxation time – in the presence of strong intracellular perturbations. The concept relies on the surprising fact that for systems showing perfect adaptation an exponential signal amplification at the receptor level suffices to eliminate slowly varying multiplicative noise. To show this mechanism at work in living systems, we quantified the response dynamics of the E. coli chemotaxis network after genetically perturbing the information flux between upstream and downstream signaling components. We give strong evidence that this signaling system results in dynamic invariance of the activated response regulator against multiplicative intracellular noise. We further demonstrate that for environmental conditions, for which precision in chemosensing is crucial, the invariant response behavior results in highest chemotactic efficiency. Our results resolve several puzzling features of the chemotaxis pathway that are widely conserved across prokaryotes but so far could not be attributed any functional role.     

The evolution of partial reproductive isolation as an adaptive optimum

Decades of theoretical work on the evolution of adaptive prezygotic isolation have led to an interesting finding—namely that stable partial reproductive isolation is a relatively common outcome. This conclusion is generally lost, however, in the desire to pinpoint when exactly speciation occurs. Here, we argue that the evolution of partial reproductive isolation is of great interest in its own right and matches empirical findings that ongoing hybridization is taxonomically widespread. We present the mechanisms by which partial reproductive isolation can be a stable evolutionary endpoint, concentrating on insights from theoretical studies. We focus not on cases in which hybridization results from constraints imposed by ongoing migration or mutation, but on the intriguing idea that partial reproductive isolation may instead be an adaptive optimum. We identify three general categories of selective mechanisms that can lead to partial reproductive isolation: context-dependent hybrid advantage, indirect selection due to the varying actions of sexual selection in different geographic contexts, and a balance of costs of choosiness with indirect selection for stronger mating preferences. By any of these mechanisms, stable partial reproductive isolation can potentially provide a robust evolutionary alternative to either complete speciation or population fusion.     

Traits and selection strategies to improve root systems and water uptake in water-limited wheat crops

Wheat yields globally will depend increasingly on good management to conserve rainfall and new varieties that use water efficiently for grain production. Here we propose an approach for developing new varieties to make better use of deep stored water. We focus on water-limited wheat production in the summer-dominant rainfall regions of India and Australia, but the approach is generally applicable to other environments and root-based constraints. Use of stored deep water is valuable because it is more predictable than variable in-season rainfall and can be measured prior to sowing. Further, this moisture is converted into grain with twice the efficiently of in-season rainfall since it is taken up later in crop growth during the grain-filling period when the roots reach deeper layers. We propose that wheat varieties with a deeper root system, a redistribution of branch root density from the surface to depth, and with greater radial hydraulic conductivity at depth would have higher yields in rainfed systems where crops rely on deep water for grain fill. Developing selection systems for mature root system traits is challenging as there are limited high-throughput phenotyping methods for roots in the field, and there is a risk that traits selected in the lab on young plants will not translate into mature root system traits in the field. We give an example of a breeding programme that combines laboratory and field phenotyping with proof of concept evaluation of the trait at the beginning of the selection programme. This would greatly enhance confidence in a high-throughput laboratory or field screen, and avoid investment in screens without yield value. This approach requires careful selection of field sites and years that allow expression of deep roots and increased yield. It also requires careful selection and crossing of germplasm to allow comparison of root expression among genotypes that are similar for other traits, especially flowering time and disease and toxicity resistances. Such a programme with field and laboratory evaluation at the outset will speed up delivery of varieties with improved root systems for higher yield.     

Microgeographic life history variation in an alpine caddisfly: plasticity in response to seasonal time constraints

1. Temporally constrained environments, such as habitats with short growth seasons or short hydroperiods, cause potentially strong selection on life histories. Depending on the predictability of these events and the extent of spatial and temporal heterogeneity, local populations could become adapted either via a fixed phenotype or via life history plasticity in response to these environmental cues. 2. We used a common garden experiment to investigate microgeographic variation in life history responses to combined changes in photoperiod (ambient/late) and hydroperiod (constant/drying) time constraint cues in an alpine caddisfly (Trichoptera). We compared six populations (three permanent/three temporary streams) originating from a small, alpine floodplain and which spanned an expected gradient in growth period duration (GPD) with distance from glaciers. 3. We made two main predictions in relation to locally varying selection pressures: (i) populations nearest glaciers (shorter GPD and strongest time constraints) should have the fastest development rates and (ii) populations from permanent streams should be less able to respond to drying hydroperiods than populations from temporary streams. 4. All populations and both sexes accelerated development in response to late season photoperiod cues. However, only permanent stream populations showed an increase in development time with increasing GPD, suggesting that other factors were influencing populations in temporary streams. 5. Permanent stream populations showed countergradient variation (genetic and environmental influences were in opposition) in development time, and under‐compensation of growth rates resulted in a converse Bergmann cline in body size (smaller body size along gradients of declining season length). The extent of plasticity in response to hydroperiod, and the combined effects of both time constraints, differed between populations and sexes, but were not consistent among populations. 6. Taken together, our results suggest adaptive plasticity in response to season length. The lack of a predictable pattern in response to hydroperiod may be due to gene flow or weak selection. We conclude that spatially structured populations can strongly differ in phenotypic plasticity even at microgeographic scales.