The balance between pleiotropic mutation and selection,when alleles have discrete effects

The theory of pleiotropic mutation and selection is investigated and developed for a large population of asexual organisms. Members of the population are subject to stabilising selection on Omega phenotypic characters, which each independently affect fitness. Pleiotropy is incorporated into the model by allowing each mutation to simultaneously affect all characters. To expose differences with continuous-allele models, the characters are taken to originate from discrete-effect alleles and thus have discrete genotypic effects. Each character can take the values nxDelta where n=0,+/-1,+/-2, em leader, and the splitting in character effects, Delta, is a parameter of the model. When the distribution of mutant effects is normally distributed around the parental value, and Delta is large, a "stepwise" model of mutation arises, where only adjacent trait effects are accessible from a single mutation. The present work is primarily concerned with the opposite limit, where Delta is small and many different trait effects are accessible from a single mutation.In contrast to what has been established for continuous-effect models, discrete-effect models do not yield a singular equilibrium distribution of genotypic effects for any value of Omega. Instead, for different values of Omega, the equilibrium frequencies of trait values have very different dependencies on Delta. For Omega=1 and 2, decreasing Delta broadens the width of the frequency distribution and hence increases the equilibrium level of polymorphism. For all sufficiently large values of Omega, however, decreasing Delta decreases the width of the frequency distribution and the equilibrium level of polymorphism. The connection with continuous trait models follows when the limit Delta-->0 is considered, and a singular probability density of trait values is obtained for all sufficiently large Omega.     

Exploring the common dynamics of homologous proteins. Application to the globin family

We present a procedure to explore the global dynamics shared between members of the same protein family. The method allows the comparison of patterns of vibrational motion obtained by Gaussian network model analysis. After the identification of collective coordinates that were conserved during evolution, we quantify the common dynamics within a family. Representative vectors that describe these dynamics are defined using a singular value decomposition approach. As a test case, the globin heme-binding family is considered. The two lowest normal modes are shown to be conserved within this family. Our results encourage the development of models for protein evolution that take into account the conservation of dynamical features.     

The frequency-dependent Wright–Fisher model: diffusive and non-diffusive approximations

We study a class of processes that are akin to the Wright–Fisher model, with transition probabilities weighted in terms of the frequency-dependent fitness of the population types. By considering an approximate weak formulation of the discrete problem, we are able to derive a corresponding continuous weak formulation for the probability density. Therefore, we obtain a family of partial differential equations for the evolution of the probability density, and which will be an approximation of the discrete process in the joint large population, small time-steps and weak selection limit. If the fitness functions are sufficiently regular, we can recast the weak formulation in a more standard formulation, without any boundary conditions, but supplemented by a number of conservation laws. The equations in this family can be purely diffusive, purely hyperbolic or of convection–diffusion type, with frequency dependent convection. The particular outcome will depend on the assumed scalings. The diffusive equations are of the degenerate type; using a duality approach, we also obtain a frequency dependent version of the Kimura equation without any further assumptions. We also show that the convective approximation is related to the replicator dynamics and provide some estimate of how accurate is the convective approximation, with respect to the convective-diffusion approximation. In particular, we show that the mode, but not the expected value, of the probability distribution is modelled by the replicator dynamics. Some numerical simulations that illustrate the results are also presented.     

Minimal time control of fed-batch bioreactor with product inhibition

This paper is devoted to the minimal time control problem for fed-batch bioreactors, in presence of an inhibitory product, which is released by the biomass proportionally to its growth. We first consider a growth rate with substrate saturation and product inhibition, and we prove that the optimal strategy is fill and wait (bang-bang). We then investigate the case of the Jin growth rate which takes into account substrate and product inhibition. For this type of growth function, we can prove the existence of singular arc paths defining singular strategies. Several configurations are addressed depending on the parameter set. For each case, we provide an optimal feedback control of the problem (of type bang-bang or bang-singular-bang). These results are obtained gathering the initial system into a planar one by using conservation laws. Thanks to Pontryagin maximum principle, Green’s theorem, and properties of the switching function, we obtain the optimal synthesis. A methodology is also proposed in order to implement the optimal feeding strategies.     

A spatial model of germinal center reactions: cellular adhesion based sorting of B cells results in efficient affinity maturation

Affinity maturation of humoral responses to T-cell-dependent antigens occurs in germinal centers (GC). In GCs antigen-specific B cells undergo rounds of somatic mutations that alter their affinity. High-affinity mutants take over GCs very soon after they appear; the replacement rate is as high as 4 per day (Radmacher et al., Immunol. Cell Biol. 76 (1998) 373). To gain more insight into this selection process, we present a spatial model of GC reactions, where B cells compete for survival signals from follicular dendritic cells (FDC). Assuming that high-affinity B cells have increased cellular adhesion to FDCs, we obtain an affinity-based sorting of B cells on the FDC. This sorting imposes a very strong selection and therefore results in a winner-takes-all behavior. By comparing our sorting model with "affinity-proportional selection models", we show that this winner-takes-all selection is in fact required to account for the fast rates at which high affinity mutants take over GCs. Another important feature of in vivo GC reactions is that they are non-mixed, i.e. GCs contain either no high-affinity cells at all or they are dominated by high-affinity cells. We here show that this all-or-none behavior can be obtained if B cells are sorted based on their affinity on the FDC surface. Affinity-proportional selection models, in contrast, always produce mixed GCs.     

Methods for the study of cytoplasmic effects on quantitative traits

Summary The methods used to study cytoplasmic effects in quantitative traits often do not measure quantitative genetic parameters, while those that do are either complicated or do not take into account situations where the expression of cytoplasmic effects does not persist, but decreases in advanced generations. We present two simple models that take cytoplasmic effects and the quantitative genetic parameters into account. One of the models (A) is for cases where cytoplasmic effects remain constant through successive generations, and the second model (B) is for traits where cytoplasm-genotype interactions are present. This model also takes into account the decreasing persistence of cytoplasmic effects with advancing generations, which is often reported in the literature.     

Dung beetle community (Coleoptera: Scarabaeidae: Scarabaeinae) in a tropical landscape at the Lachua Region,Guatemala

Biological diversity conservation within natural reserves has been prioritized, but conservation efforts outside protected areas (where most human activities take place) have been very little considered. In this scenario, an alternative agricultural practice that may reduce the impacts of fragmentation in outer landscapes is a perforation process, which involves conservation in agricultural fields surrounded by continuous forests. Such practices enhance the positive impact of ecological services on fields. In this study we analyzed the biological diversity state in perforation fields and their surrounding forests. The analysis was done using dung beetles as biological indicators. A nested pattern in dung beetles distribution was found, which ordered the surrounding continuous forest sites as the ones with the highest species richness, followed by the perforation fields, and placed the fragmentation practice fields (continuous agricultural fields surrounding forest patches) with the lowest one. Indicator species for perforation fields and surrounding continuous forests were chosen. In general, perforation practice fields differed in composition, based upon functional groups richness and identity; it also contained a higher species richness than the fragmentation practice. Agricultural practices that enhance biological diversity conservation such as perforation, should be recommended and considered in natural resource management by local communities in order to take advantage of ecological services that otherwise may be gradually lost.     

Natural selection for earlier male arrival to breeding grounds through direct and indirect effects in a migratory songbird

For migratory birds, the earlier arrival of males to breeding grounds is often expected to have fitness benefits. However, the selection differential on male arrival time has rarely been decomposed into the direct effect of male arrival and potential indirect effects through female traits. We measured the directional selection differential on male arrival time in the pied flycatcher (Ficedula hypoleuca) using data from 6 years and annual number of fledglings as the fitness proxy. Using structural equation modeling, we were able to take into account the temporal structure of the breeding cycle and the hierarchy between the examined traits. We found directional selection differentials for earlier male arrival date and earlier female laying date, as well as strong selection differential for larger clutch size. These selection differentials were due to direct selection only as indirect selection for these traits was nonsignificant. When decomposing the direct selection for earlier male arrival into direct and indirect effects, we discovered that it was almost exclusively due to the direct effect of male arrival date on fitness and not due to its indirect effects via female traits. In other words, we showed for the first time that there is a direct effect of male arrival date on fitness while accounting for those effects that are mediated by effects of the social partner. Our study thus indicates that natural selection directly favored earlier male arrival in this flycatcher population.     

The effect of gametic-phase disequilibrium on the prediction of response to recurrent selection in plants

Selection reduces additive genetic variation by generating gametic-phase disequilibrium, a phenomenon largely ignored when predicting response in plant breeding programs. The development of gametic-phase disequilibrium is here taken into account when predicting the response to selection for various schemes of recurrent selection applicable to plant populations. A general program permits prediction of response to selection from schemes of recurrent selection in which two or more rounds of selection occur in each cycle. An example from Sugar Beet, with alternate rounds of half-sib and S1 family selection, is illustrated. It is shown that failure to take into account the effects of gametic-phase disequilibrium can result in substantial overestimation of the response to selection as well as to changes in rank of the merits of alternative breeding schemes. For a given scheme, ignoring gametic-phase disequilibrium has only small effects on defining the optimum allocation of plots and the numbers of families tested.     

Homogenization of Heterogeneously Coupled Bistable ODE's—Applied to Excitation Waves in Pancreatic Islets of Langerhans

We consider a lattice of coupled identical differential equations. The coupling is between nearest neighbors and of resistance type, but the strength of coupling varies from site to site. Such a lattice can, for example, model an islet of Langerhans, where the sites in the lattice model individual but identical -cells, and the coupling between cells is made of gap junctions.By using a homogenization technique we approximate the coupled discrete equations by a PDE, basically a nonlinear heat equation (a Fisher equation). For appropriate parameters this equation is known to have wave-solutions. Of importance is the fact, that the resulting diffusion coefficient does not only depend on the average of the coupling, but also on the variance of the strength. This means that the heterogeneity of the coupling strength influences the wave velocity—the greater the variance, the slower is the wave. This result is illustrated by simulations, both of a simple prototype equation, and for a full model of coupled beta-cells in both one and two dimensions, and implies that the natural heterogeneity in the islets of Langerhans should be taken into account.     

A generic geometric transformation that unifies a wide range of natural and abstract shapes

To study forms in plants and other living organisms, several mathematical tools are available, most of which are general tools that do not take into account valuable biological information. In this report I present a new geometrical approach for modeling and understanding various abstract, natural, and man-made shapes. Starting from the concept of the circle, I show that a large variety of shapes can be described by a single and simple geometrical equation, the Superformula. Modification of the parameters permits the generation of various natural polygons. For example, applying the equation to logarithmic or trigonometric functions modifies the metrics of these functions and all associated graphs. As a unifying framework, all these shapes are proven to be circles in their internal metrics, and the Superformula provides the precise mathematical relation between Euclidean measurements and the internal non-Euclidean metrics of shapes. Looking beyond Euclidean circles and Pythagorean measures reveals a novel and powerful way to study natural forms and phenomena.     

Resource-aware taxon selection for maximizing phylogenetic diversity

Phylogenetic diversity (PD) is a useful metric for selecting taxa in a range of biological applications, for example, bioconservation and genomics, where the selection is usually constrained by the limited availability of resources. We formalize taxon selection as a conceptually simple optimization problem, aiming to maximize PD subject to resource constraints. This allows us to take into account the different amounts of resources required by the different taxa. Although this is a computationally difficult problem, we present a dynamic programming algorithm that solves it in pseudo-polynomial time. Our algorithm can also solve many instances of the Noah's Ark Problem, a more realistic formulation of taxon selection for biodiversity conservation that allows for taxon-specific extinction risks. These instances extend the set of problems for which solutions are available beyond previously known greedy-tractable cases. Finally, we discuss the relevance of our results to real-life scenarios.     

A Stochastic Model for Coupled Enzyme System

The molecular biology of transformed cancer cells singles out key enzymes as sensitive targets of anti-cancer drugs. Here we use one substrate–one intermediate–one final product model for a coupled enzyme system. The transfer rates for the mechanism are taken as continuous but subject to random fluctuations. Explicit formulae for the first moments of the distribution of the process are obtained. These formulae allow us to take into account not only the variability between the subjects, but also the variability of the process for a single subject. The present results allow us also to build the prediction interval for a particular time period given the observations for some preceding moments.     

Stochastic search variable selection based on two mixture components and continuous‐scale weighting

Stochastic search variable selection (SSVS) is a Bayesian variable selection method that employs covariate‐specific discrete indicator variables to select which covariates (e.g., molecular markers) are included in or excluded from the model. We present a new variant of SSVS where, instead of discrete indicator variables, we use continuous‐scale weighting variables (which take also values between zero and one) to select covariates into the model. The improved model performance is shown and compared to standard SSVS using simulated and real quantitative trait locus mapping datasets. The decision making to decide phenotype‐genotype associations in our SSVS variant is based on median of posterior distribution or using Bayes factors. We also show here that by using continuous‐scale weighting variables it is possible to improve mixing properties of Markov chain Monte Carlo sampling substantially compared to standard SSVS. Also, the separation of association signals and nonsignals (control of noise level) seems to be more efficient compared to the standard SSVS. Thus, the novel method provides efficient new framework for SSVS analysis that additionally provides whole posterior distribution for pseudo‐indicators which means more information and may help in decision making.     

Glocal alignment: finding rearrangements during alignment

MOTIVATION: To compare entire genomes from different species, biologists increasingly need alignment methods that are efficient enough to handle long sequences, and accurate enough to correctly align the conserved biological features between distant species. The two main classes of pairwise alignments are global alignment, where one string is transformed into the other, and local alignment, where all locations of similarity between the two strings are returned. Global alignments are less prone to demonstrating false homology as each letter of one sequence is constrained to being aligned to only one letter of the other. Local alignments, on the other hand, can cope with rearrangements between non-syntenic, orthologous sequences by identifying similar regions in sequences; this, however, comes at the expense of a higher false positive rate due to the inability of local aligners to take into account overall conservation maps. RESULTS: In this paper we introduce the notion of glocal alignment, a combination of global and local methods, where one creates a map that transforms one sequence into the other while allowing for rearrangement events. We present Shuffle-LAGAN, a glocal alignment algorithm that is based on the CHAOS local alignment algorithm and the LAGAN global aligner, and is able to align long genomic sequences. To test Shuffle-LAGAN we split the mouse genome into BAC-sized pieces, and aligned these pieces to the human genome. We demonstrate that Shuffle-LAGAN compares favorably in terms of sensitivity and specificity with standard local and global aligners. From the alignments we conclude that about 9% of human/mouse homology may be attributed to small rearrangements, 63% of which are duplications.     

Inferring Property Selection Pressure from Positional Residue Conservation

In this study, we attempt to understand and explain positional selection pressure in terms of underlying physical and chemical properties. We propose a set of constraining assumptions about how these pressures behave, then describe a procedure for analysing and explaining the distribution of residues at a particular position in a multiple sequence alignment. In contrast to previous approaches, our model takes into account both amino acid frequencies and a large number of physical-chemical properties. By analysing each property separately, it is possible to identify positions where distinct conservation patterns are present. In addition, the model can easily incorporate sequence weights that adjust for bias in the sample sequences. Finally, a test of statistical significance is provided for our conservation measure. The applicability of this method is demonstrated on two HIV-1 proteins: Nef and Env. The tools, data and results presented in this article are available at http://flan.blm.cs.cmu.edu.     

MEASURING NATURAL SELECTION ON PHENOTYPIC PLASTICITY

To understand natural selection we need to integrate its measure across environments. We present a method for measuring phenotypic selection that combines the potential for both environmental variation and phenotypic plasticity. The method uses path analysis and a measure of selection that is analogous to selection on breeding values. For individuals growing in alternative environments, paths are created that represent potential changes in the environment. The probabilities for these changes are then multiplied by the path coefficients to calculate selection coefficients. Selection on plasticity is measured as the difference in selection within each environment. We illustrate these methods using data on selection in an experimental population of Arabidopsis thaliana. Individuals from 36 families were grown in one of four environments, a factorial combination of shaded/open and early/late shading. For final height of the inflorescence, there was positive selection in both the open and shaded environments and negative selection on plasticity of height. For bolting time, there was also positive selection in both environments, but no selection on plasticity. We show how to use this information to examine how selection would change with changes in environmental frequencies and their transition probabilities. These methods can be expanded to encompass continuous traits and continuous environments as well as other complexities of natural selection.     

Predictions of patterns of response to artificial selection in lines derived from natural populations

The pattern of response to artificial selection on quantitative traits in laboratory populations can tell us something of the genetic architecture in the natural population from which they were derived. We modeled artificial selection in samples drawn from natural populations in which variation had been maintained by recurrent mutation, with genes having an effect on the trait, which was subject to real stabilizing selection, and a pleitropic effect on fitness (the joint-effect model). Natural selection leads to an inverse correlation between effects and frequencies of genes, such that the frequency distribution of genes increasing the trait has an extreme U-shape. In contrast to the classical infinitesimal model, an early accelerated response and a larger variance of response among replicates were predicted. However, these are reduced if the base population has been maintained in the laboratory for some generations by random sampling prior to artificial selection. When multiple loci and linkage are also taken into account, the gametic disequilibria generated by the Bulmer and Hill-Robertson effects are such that little or no increase in variance and acceleration of response in early generations of artificial selection are predicted; further, the patterns of predicted responses for the joint-effect model now become close to those of the infinitesimal model. Comparison with data from laboratory selection experiments shows that, overall, the analysis did not provide clear support for the joint-effect model or a clear case for rejection.     

Strategies of aphidophagous predators: lessons for modelling insect predator–prey dynamics

Both theoretical and empirical evidence indicate that in systems where insect predators have longer developmental times than their prey the predators have little impact on the abundance of their prey. In assessing the 'effectiveness' of a predator for biological control one should take into account that selection maximizes predator fitness, not its effctiveness as a biocontrol agent. Therefore, predators that have a long developmental time relative to their prey are unlikely to be the best biocontrol agents. If these results can be generalized to other predator–prey systems, then it is clear that an understanding of predator–prey dynamics can only be achieved by studying predators.     

An overview of the multispecies VPA — theory and applications

Summary Multispecies virtual population analysis is an attempt to take species interactions into account when assessing the status of fish stocks. It was introduced primarily with the aim of lowering the uncertainty in the natural mortality rate as used in single species VPA and to take account of variability between years and ages by calculating inside the model the part of the mortality rate caused by predation. The output of the MSVPA is therefore —in addition to stock sizes and fishing mortality rates as in single species VPA —the amounts consumed of the various species by the predators included in the analysis.The MSVPA model of the predation interactions results in a set of coupled non-linear equations which must be solved for each time step in the analysis. Key parameters in the model are the so-called suitability coefficients, measuring the relative suitability of one species as prey for another species. These parameters must be estimated inside the model and this estimation requires data on the stomach contents of the predators in the model. The MSVPA makes two key assumptions: constant ration size (i.e. independent of time for each species-age combination) and hence fixed weights-at-age and a model of prey selection which leads to a type ii functional feeding response. These assumptions do not hold for all areas and therefore limit the applicability of the MSVPA in its present form.The MSVPA is probably one of the more successful multispecies models in fisheries. Its main application to date has been to the North Sea, and although it has not been used directly as a management tool it has provided input values of parameters used in assessment models as well as valuable insights into the system. For example, it has demonstrated that an increase in mesh size can result in lower long-term yields, an effect opposite to what is predicted if species interactions are ignored. Such insights into the dynamics of the system are useful and MSVPA may therefore have an indirect role to play in management. Nevertheless, due to many uncertainties involved in multispecies modelling in general and MSVPA in particular, it seems doubtful that the use of MSVPA in fisheries management will be much greater in the immediate future than it is at present.