Perturbation expansions of multilocus fixation probabilities for frequency-dependent selection with applications to the Hill-Robertson effect and to the joint evolution of helping and punishment.

Natural populations are of finite size and organisms carry multilocus genotypes. There are, nevertheless, few results on multilocus models when both random genetic drift and natural selection affect the evolutionary dynamics. In this paper we describe a formalism to calculate systematic perturbation expansions of moments of allelic states around neutrality in populations of constant size. This allows us to evaluate multilocus fixation probabilities (long-term limits of the moments) under arbitrary strength of selection and gene action. We show that such fixation probabilities can be expressed in terms of selection coefficients weighted by mean first passages times of ancestral gene lineages within a single ancestor. These passage times extend the coalescence times that weight selection coefficients in one-locus perturbation formulas for fixation probabilities. We then apply these results to investigate the Hill-Robertson effect and the coevolution of helping and punishment. Finally, we discuss limitations and strengths of the perturbation approach. In particular, it provides accurate approximations for fixation probabilities for weak selection regimes only (Ns < or = 1), but it provides generally good prediction for the direction of selection under frequency-dependent selection.

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.

Modeling morphogen gradient formation from arbitrary realistically shaped sources.

Much of the analytical modeling of morphogen profiles is based on simplistic scenarios, where the source is abstracted to be point-like and fixed in time, and where only the steady state solution of the morphogen gradient in one dimension is considered. Here we develop a general formalism allowing to model diffusive gradient formation from an arbitrary source. This mathematical framework, based on the Green's function method, applies to various diffusion problems. In this paper, we illustrate our theory with the explicit example of the Bicoid gradient establishment in Drosophila embryos. The gradient formation arises by protein translation from a mRNA distribution followed by morphogen diffusion with linear degradation. We investigate quantitatively the influence of spatial extension and time evolution of the source on the morphogen profile. For different biologically meaningful cases, we obtain explicit analytical expressions for both the steady state and time-dependent 1D problems. We show that extended sources, whether of finite size or normally distributed, give rise to more realistic gradients compared to a single point-source at the origin. Furthermore, the steady state solutions are fully compatible with a decreasing exponential behavior of the profile. We also consider the case of a dynamic source (e.g. bicoid mRNA diffusion) for which a protein profile similar to the ones obtained from static sources can be achieved.

On dome-shaped norms of reaction for size-to-age at maturity in fishes

The optimal size-to-age at maturity depends on growth and mortality rates, which vary with environment. Therefore, organisms in spatially or temporaly changing environments should develop adaptative phenotypic plasticity for this trait. Experimental work by Alm (1959) on several fish species shows a dome-shape norm of reaction for size-to-age at maturity: size at maturity is smaller in both fast-growing and slow-growing fishes, than it is in fish with a medium growth rate. Using computer simulations, we show that such a dome-shaped norm of reaction is optimal when assuming a finite life span and a negative relationship between production and survival rates. This latter assumption is supported by empirical data, as well as by physiological and emographic arguments.

Random mating with a finite number of matings.

Random mating is the null model central to population genetics. One assumption behind random mating is that individuals mate an infinite number of times. This is obviously unrealistic. Here we show that when each female mates a finite number of times, the effective size of the population is substantially decreased.

Optimal growth strategies when mortality and production rates are size-dependent

Pontryagin's maximum principle from optimal control theory is used to find the optimal allocation of energy between growth and reproduction when lifespan may be finite and the trade-off between growth and reproduction is linear. Analyses of the optimal allocation problem to date have generally yielded bang-bang solutions, i.e. determinate growth: life-histories in which growth is followed by reproduction, with no intermediate phase of simultaneous reproduction and growth. Here we show that an intermediate strategy (indeterminate growth) can be selected for if the rates of production and mortality either both increase or both decrease with increasing body size, this arises as a singular solution to the problem. Our conclusion is that indeterminate growth is optimal in more cases than was previously realized. The relevance of our results to natural situations is discussed.

Influence of the implanted pulse generator as reference electrode in finite element model of monopolar deep brain stimulation.

Electrical deep brain stimulation (DBS) is an efficient method to treat movement disorders. Many models of DBS, based mostly on finite elements, have recently been proposed to better understand the interaction between the electrical stimulation and the brain tissues. In monopolar DBS, clinically widely used, the implanted pulse generator (IPG) is used as reference electrode (RE). In this paper, the influence of the RE model of monopolar DBS is investigated. For that purpose, a finite element model of the full electric loop including the head, the neck and the superior chest is used. Head, neck and superior chest are made of simple structures such as parallelepipeds and cylinders. The tissues surrounding the electrode are accurately modelled from data provided by the diffusion tensor magnetic resonance imaging (DT-MRI). Three different configurations of RE are compared with a commonly used model of reduced size. The electrical impedance seen by the DBS system and the potential distribution are computed for each model. Moreover, axons are modelled to compute the area of tissue activated by stimulation. Results show that these indicators are influenced by the surface and position of the RE. The use of a RE model corresponding to the implanted device rather than the usually simplified model leads to an increase of the system impedance (+48%) and a reduction of the area of activated tissue (-15%).

Why are offspring born larger when it is colder? Phenotypic plasticity for offspring size in the cladoceran Simocephalus vetulus (Müller)

It has been predicted on theorerical grounds (Sibly & Calow, 1983; Taylor & Williams, 1984) that optimal offspring size should be highly sensitive to juvenile growth and survival rates. To test such models, genetically-identical individuals of Simicephalus vetulus were reared at different temperatures and monitored for offspring size and juvenile growth rate. As adult size correlates negatively with temperature, an analysis of covariance was performed to separate the effects of temperature and maternal size. The result is that offspring size indeed correlates negatively with juvenile growth rate. Comparisons are made with field observation of several authors on seasonal variation of offspring size and alternative explanations are discussed. It is concluded that present experiments support the prediction of the theoretical models.

Sexual dimorphism in resource acquisition and deployment: both size and timing matter.

Background and Aims The males and females of many dioecious plant species differ from one another in important life-history traits, such as their size. If male and female reproductive functions draw on different resources, for example, one should expect males and females to display different allocation strategies as they grow. Importantly, these strategies may differ not only between the two sexes, but also between plants of different age and therefore size. Results are presented from an experiment that asks whether males and females of Mercurialis annua, an annual plant with indeterminate growth, differ over time in their allocation of two potentially limiting resources (carbon and nitrogen) to vegetative (below-and above-ground) and reproductive tissues.Methods Comparisons were made of the temporal patterns of biomass allocation to shoots, roots and reproduction and the nitrogen content in the leaves between the sexes of M. annua by harvesting plants of each sex after growth over different periods of time.Key Results and Conclusions Males and females differed in their temporal patterns of allocation. Males allocated more to reproduction than females at early stages, but this trend was reversed at later stages. Importantly, males allocated proportionally more of their biomass towards roots at later stages, but the roots of females were larger in absolute terms. The study points to the important role played by both the timing of resource deployment and the relative versus absolute sizes of the sinks and sources in sexual dimorphism of an annual plant.

Entrance pupil size predicts retinal illumination in darkly pigmented eyes, but not lightly pigmented eyes.

PURPOSE: We determined the effect of entrance pupil size on retinal illumination. The influence of unilateral miosis on the magnitude of the pupil light reflex was studied to ascertain how a clinically significant anisocoria influences the relative afferent pupil defect (RAPD). METHODS: Miosis was induced by topical 1% pilocarpine in the right eye of 14 healthy subjects with normal eyes. The interocular difference in retinal illumination was assessed by computerized pupillometry from the stimulus response curve of the right and left eyes. The main outcome measure was the RAPD, determined by computerized pupillography, at baseline and after pilocarpine-induced anisocoria. RESULTS: Induced anisocoria produced a significant change in RAPD from baseline (mean = 1.60 dB in the miotic eye, P = 0.007). However, anisocoria correlated with RAPD only in subjects with darkly pigmented irides (Pearson correlation coefficient 0.793, P = 0.05). CONCLUSIONS: In darkly pigmented eyes, entrance pupil size significantly influenced the retinal illumination. However, retinal illumination of lightly pigmented eyes is relatively independent of entrance pupil size, presumably due to extrapupillary transmission of light through the iris and sclera. This has important implications in understanding the potential influence of anisocoria on the RAPD and also greater susceptibility of lightly pigmented eyes to light toxicity.

EEG-based functional brain networks: does the network size matter?

Functional connectivity in human brain can be represented as a network using electroencephalography (EEG) signals. These networks--whose nodes can vary from tens to hundreds--are characterized by neurobiologically meaningful graph theory metrics. This study investigates the degree to which various graph metrics depend upon the network size. To this end, EEGs from 32 normal subjects were recorded and functional networks of three different sizes were extracted. A state-space based method was used to calculate cross-correlation matrices between different brain regions. These correlation matrices were used to construct binary adjacency connectomes, which were assessed with regards to a number of graph metrics such as clustering coefficient, modularity, efficiency, economic efficiency, and assortativity. We showed that the estimates of these metrics significantly differ depending on the network size. Larger networks had higher efficiency, higher assortativity and lower modularity compared to those with smaller size and the same density. These findings indicate that the network size should be considered in any comparison of networks across studies.

Effect of training-sample size and classification difficulty on the accuracy of genomic predictors.

Introduction: As part of the MicroArray Quality Control (MAQC)-II project, this analysis examines how the choice of univariate feature-selection methods and classification algorithms may influence the performance of genomic predictors under varying degrees of prediction difficulty represented by three clinically relevant endpoints. Methods: We used gene-expression data from 230 breast cancers (grouped into training and independent validation sets), and we examined 40 predictors (five univariate feature-selection methods combined with eight different classifiers) for each of the three endpoints. Their classification performance was estimated on the training set by using two different resampling methods and compared with the accuracy observed in the independent validation set. Results: A ranking of the three classification problems was obtained, and the performance of 120 models was estimated and assessed on an independent validation set. The bootstrapping estimates were closer to the validation performance than were the cross-validation estimates. The required sample size for each endpoint was estimated, and both gene-level and pathway-level analyses were performed on the obtained models. Conclusions: We showed that genomic predictor accuracy is determined largely by an interplay between sample size and classification difficulty. Variations on univariate feature-selection methods and choice of classification algorithm have only a modest impact on predictor performance, and several statistically equally good predictors can be developed for any given classification problem.

Male dominance linked to size and age, but not to 'good genes' in brown trout (Salmo trutta).

BACKGROUND: Males that are successful in intra-sexual competition are often assumed to be of superior quality. In the mating system of most salmonid species, intensive dominance fights are common and the winners monopolise most mates and sire most offspring. We drew a random sample of mature male brown trout (Salmo trutta) from two wild populations and determined their dominance hierarchy or traits linked to dominance. The fish were then stripped and their sperm was used for in vitro fertilisations in two full-factorial breeding designs. We recorded embryo viability until hatching in both experiments, and juvenile survival during 20 months after release into a natural streamlet in the second experiment. Since offspring of brown trout get only genes from their fathers, we used offspring survival as a quality measure to test (i) whether males differ in their genetic quality, and if so, (ii) whether dominance or traits linked to dominance reveal 'good genes'. RESULTS: We found significant additive genetic variance on embryo survival, i.e. males differed in their genetic quality. Older, heavier and larger males were more successful in intra-sexual selection. However, neither dominance nor dominance indicators like body length, weight or age were significantly linked to genetic quality measured as embryo or juvenile survival. CONCLUSION: We found no evidence that females can improve their offspring's genetic viability by mating with large and dominant males. If there still were advantages of mating with dominant males, they may be linked to non-genetic benefits or to genetic advantages that are context dependent and therefore possibly not revealed under our experimental conditions - even if we found significant additive genetic variation for embryo viability under such conditions.