Optimizing selection for function-valued traits

We consider a function-valued trait z(t) whose pre-selection distribution is Gaussian, and a fitness function W that models optimizing selection, subject to certain natural assumptions. We show that the post-selection distribution of z(t) is also Gaussian, compute the selection differential, and derive an equation that expresses the selection gradient in terms of the parameters of W and of the pre-selection distribution. We make no assumptions on the nature of the “time” parameter t.      

Comparison between marker-assisted selection and phenotypical selection in a set of Arabidopsis thaliana recombinant inbred lines

 Parents were selected from a well-characterised Arabidopsis recombinant inbred line (RIL) population based on (1) their phenotype for flowering time or (2) marker and QTL information that had been assessed previously. The F₂ offspring obtained from pairs of selected RILs was analysed for these traits, and the results obtained with these two methods of selection were compared. Selection based on marker and QTL information gave approximately the same result as selection based on phenotype. The relative high heritability of flowering time in Arabidopsis facilitated successful phenotypical selection. The difference in selection result that was anticipated to be in favour of the marker-assisted approach was therefore not observed. Received: 29 November 1997 / Accepted: 8 June 1998     

Some additional results about polymorphisms in models of an additive quantitative trait under stabilizing selection

 The existence of two stable, symmetric (allelic frequency 0.5 in each locus) polymorphic states is demonstrated for a two-locus model of an additive quantitative trait under strong Gaussian selection. Linkage disequilibrium at one of the states is negative whereas it is positive at the other state. For a three-locus model, it is shown that in order to maintain a stable polymorphism in all three loci, selection must be sufficiently but not exces- sively strong relative to recombination. Also, positive linkage disequilibrium can be maintained in a three-locus model under stabilizing selection that is not very strong. Received 15 July 1995     

Trait mean depression for second-generation inbred strawberry populations with and without parent selection

 Strawberry genotypes selected for superior fruit yield or chosen at random from first-generation self, full-sib, and half-sib populations were crossed to provide second-generation inbred progenies and composite cross-fertilized control populations. Mean yields for inbred offspring from crosses among selected parents exceeded those from the offspring of unselected parents by 87%, 23%, and 37% for self, full-sib, and half-sib populations, respectively; yields for offspring from unrelated crosses among selected parents were 54% larger than those for crosses among unselected parents. Selection for yield also resulted in significant correlated response for fruit number and plant diameter. Mean yields for second-generation half-sib and full-sib offspring from selected parents were greater than those for offspring from the unselected but non-inbred control population. This suggests that selection can be a powerful force in counteracting most of the inbreeding depression expected in cross-fertilized strawberry breeding programs. Selection treatment× inbreeding rate interactions were non-significant for all traits; thus, selection among partially inbred offspring did not have a large effect on the rate of genetic progress. Differential realized selection intensity among individuals with differing levels of homozygosity accumulated due to inbreeding is suggested as the most likely explanation for the absence of association between pedigree inbreeding coefficients and cross performance detected previously in strawberry. Received: 21 July 1996 / Accepted: 7 March 1997     

The diffusion model for migration and selection in a plant population

 The diffusion approximation is derived for migration and selection at a multiallelic locus in a partially selfing plant population subdivided into a lattice of colonies. Generations are discrete and nonoverlapping; both pollen and seeds disperse. In the diffusion limit, the genotypic frequencies at each point are those determined at equilibrium by the local rate of selfing and allelic frequencies. If the drift and diffusion coefficients are taken as the appropriate linear combination of the corresponding coefficients for pollen and seeds, then the migration terms in the partial differential equation for the allelic frequencies have the standard form for a monoecious animal population. The selection term describes selection on the local genotypic frequencies. The boundary conditions and the unidimensional transition conditions for a geographical barrier and for coincident discontinuities in the carrying capacity and migration rate have the standard form. In the diallelic case, reparametrization renders the entire theory of clines and of the wave of advance of favorable alleles directly applicable to plant populations. Received 30 August 1995; received in revised form 23 February 1996     

Recognition and treatment of response fluctuations in Parkinson's disease: review article

Summary.  Patients with Parkinson's disease (PD) by definition benefit from treatment with the dopamine precursor levodopa. However, after 5 years of therapy 50% of patients experience motor response complications (MRC's): the benefit from each dose becomes shorter (wearing-off fluctuations), more unpredictable (on-off fluctuations) and associated with involuntary movements (dyskinesias). In addition these patients suffer from fluctuations in motor function that are inherent to the disease itself. Recent findings have lead to the suggestion that hyperfunction of NMDA receptors on striatal efferent neurons, as a consequence of chronic non-physiologic dopaminergic stimulation, contributes to the pathogenesis of MRC's. In PD patients blockade of striatal glutamate receptors with several NMDA-antagonists improve MRC's. With progression of PD the severity and complexity of MRC's magnify, obfuscating their pattern and their relation to the medication cycle. Only through detailed history taking and patient education will the physician be able to clarify the situation and establish a rational, targeted approach to the treatment of patients with advanced PD complicated by motor fluctuations and dyskinesias. Received July 7, 2001 Accepted August 6, 2001 Published online September 10, 2002     

Quantifying the effects of ecological constraints on trait expression using novel trait‐gradient analysis parameters

Complex processes related to biotic and abiotic forces can impose limitations to assembly and composition of plant communities. Quantifying the effects of these constraints on plant functional traits across environmental gradients, and among communities, remains challenging. We define ecological constraint (C_i) as the combined, limiting effect of biotic interactions and environmental filtering on trait expression (i.e., the mean value and range of functional traits). Here, we propose a set of novel parameters to quantify this constraint by extending the trait‐gradient analysis (TGA) methodology. The key parameter is ecological constraint, which is dimensionless and can be measured at various scales, for example, on population and community levels. It facilitates comparing the effects of ecological constraints on trait expressions across environmental gradients, as well as within and among communities. We illustrate the implementation of the proposed parameters using the bark thickness of 14 woody species along an aridity gradient on granite outcrops in southwestern Australia. We found a positive correlation between increasing environmental stress and strength of ecological constraint on bark thickness expression. Also, plants from more stressful habitats (shrublands on shallow soils and in sun‐exposed locations) displayed higher ecological constraint for bark thickness than plants in more benign habitats (woodlands on deep soils and in sheltered locations). The relative ease of calculation and dimensionless nature of C_i allow it to be readily implemented at various scales and make it widely applicable. It therefore has the potential to advance the mechanistic understanding of the ecological processes shaping trait expression. Some future applications of the new parameters could be investigating the patterns of ecological constraints (1) among communities from different regions, (2) on different traits across similar environmental gradients, and (3) for the same trait across different gradient types.     

Stability analysis of the partial selfing selection model

We undertake a detailed study of the one-locus two-allele partial selfing selection model. We show that a polymorphic equilibrium can exist only in the cases of overdominance and underdominance and only for a certain range of selfing rates. Furthermore, when it exists, we show that the polymorphic equilibrium is unique. The local stability of the polymorphic equilibrium is investigated and exact analytical conditions are presented. We also carry out an analysis of local stability of the fixation states and then conclude that only overdominance can maintain polymorphism in the population. When the linear local analysis is inconclusive, a quadratic analysis is performed. For some sets of selective values, we demonstrate global convergence. Finally, we compare and discuss results under the partial selfing model and the random mating model.     

A model for the evolutionary dynamics of cross-feeding polymorphisms in microorganisms

 Understanding mechanisms of evolutionary diversification is central to evolutionary biology. Microbes constitute promising model systems for observing processes of diversification directly in the laboratory. One of the main existing paradigms for microbial diversification is the evolution of cross-feeding polymorphisms, in which a strain specializing on a primary resource coexists with a cross-feeding strain that specializes on a waste product resulting from consumption of the primary resource. Here I propose a theoretical model for the evolutionary dynamics through which cross-feeding polymorphisms can gradually emerge from a single ancestral strain. The model is based on the framework of adaptive dynamics, which has proved to be very useful for studying adaptive processes of divergence under sympatric conditions. In particular, the phenomenon of evolutionary branching serves as a general paradigm for diversification. I show that evolutionary branching naturally occurs in evolutionary models of cross-feeding if (1) there is a trade-off between uptake efficiencies on the primary and secondary resources, and (2) this trade-off has positive curvature. The model also suggests that the evolution of cross-feeding should be more likely in chemostat cultures than in serial batch cultures, which conforms with empirical observations. Overall, the model provides a theoretical metaphor for the evolution of cross-feeding polymorphisms. Received: February 19, 2002 / Accepted: May 8, 2002     

Bifurcations in haploid and diploid sequence space models

 Deterministic models of mutation and selection in the space of (binary) nucleotide-type sequences have been investigated for haploid populations during the past 25 years, and, recently, for diploid populations as well. These models, in particular their ‘error thresholds’, have mainly been analyzed by numerical methods and perturbation techniques. We consider them here by means of bifurcation theory, which improves our understanding of both equilibrium and dynamical properties. In a caricature obtained from the original model by neglecting back mutation to the favourable allele, the familiar error threshold of the haploid two-class model turns out to be a simple transcritical bifurcation, whereas its diploid counterpart exhibits an additional saddle node. This corresponds to a second error threshold. Three-class models with neutral spaces of unequal size introduce further features. Such are a global bifurcation in haploid populations, and simple examples of Hopf bifurcations (as predicted by Akin’s theorem) in the diploid case. Received 13 June 1995; received in revised form 26 July 1996     

Quantitative trait locus mapping using sets of near-isogenic lines: relative power comparisons and technical considerations

 The power to detect QTL using near-isogenic line (NIL) comparisons versus recombinant inbred (RI) populations was assessed. The power to detect QTL was found to be generally greater when using RI populations than when using NIL contrasts. Power to detect QTL with NIL contrasts never exceeded that of RI populations when the number of RI lines is maximized relative to replication of lines for a given number of experimental units. The relative power of NIL contrasts is highest for traits with high heritability and when a gain in precision is realized due to increased replication of entries. Although NIL populations are generally less powerful than RI populations of similar size, some practical considerations may enhance the value of these materials. Availability of NILs allows the genetic effect of a specific chromosome region to be determined by comparing two lines; all RI lines in a population need to be rescored for each new trait even if the effect of a specific chromosome region is suspected. NIL comparisons may allow genetic differences to be detected by visual inspection; genetic effects can only be expressed as means and variances with recombinant inbred populations. In summary, RI populations generally, and in some cases, substantially, provide better power for QTL detection than NIL comparisons. Practical considerations, however, indicate that many factors need to be considered when choosing a population structure to meet an experimental objective. Received: 12 December 1996 / Accepted: 21 March 1997     

Expression and evolutionary conservation of nanos-related genes in Hydra

The Drosophila gene nanos encodes two particular zinc finger motifs which are also found in germline-associated factors from nematodes to vertebrates. We cloned two nanos (nos)-related genes, Cnnos1 and Cnnos2 from Hydra magnipapillata. Using whole-mount in situ hybridization, the expression of Cnnos1 and Cnnos2 was examined. Cnnos1 was specifically expressed in multipotent stem cells and germline cells, but not in somatic cells. Cnnos2 was weakly expressed in germline cells and more specifically in the endoderm of the hypostome where it appears to be involved in head morphogenesis. In addition to structural conservation in the zinc finger domain of nanos-related genes, functional conservation of Cnnos1 was also demonstrated by the finding that a Cnnos1 transgene can partially rescue the nos ^RC phenotype that is defective in the egg production of Drosophila. Thus, the function of nanos-related genes in the germline appears to be well conserved from primitive to highly evolved metazoans. Received: 28 April 2000 / Accepted: 1 July 2000     

The effects of the functional response on the bifurcation behavior of a mite predator–prey interaction model

 A predator–prey interaction model based on a system of differential equations with temperature-dependent parameters chosen appropriately for a mite interaction on apple trees is analyzed to determine how the type of functional response influences bifurcation and stability behavior. Instances of type I, II, III, and IV functional responses are considered, the last of which incorporates prey interference with predation. It is shown that the model systems with the type I, II, and III functional responses exhibit qualitatively similar bifurcation and stability behavior over the interval of definition of the temperature parameter. Similar behavior is found in the system with the type IV functional response at low levels of prey interference. Higher levels of interference are destabilizing, as illustrated by the prevalence of bistability and by the presence of three attractors for some values of the model parameters. All four systems are capable of modeling population oscillations and outbreaks. Received 12 March 1996; received in revised form 25 October 1996     

Convergence of multilocus systems under weak epistasis or weak selection

 The convergence of multilocus systems under viability selection with constant fitnesses is investigated. Generations are discrete and nonoverlapping; the monoecious population mates at random. The number of multiallelic loci, the linkage map, dominance, and epistasis are arbitrary. It is proved that if epistasis or selection is sufficiently weak (and satisfies a certain nondegeneracy assumption whose genericity we establish), then there is always convergence to some equilibrium point. In particular, cycling cannot occur. The behavior of the mean fitness and some other aspects of the dynamics are also analyzed. Received: 15 November 1997 / Revised version: 25 May 1998     

A theoretical method for evaluating the relative importance of positive selection and neutral drift from observed base changes

To evaluate the relative importance of positive selection and neutral drift from the nucleotide base changes observed in the homologous alignment of genes, a theoretical equation of base changes is formulated by including both the influence of selection and the base substitutions due to mutations. Under the assumption that the average rate of base substitutions estimated from synonymous changes is the ``true' mutation rate applicable at all positions, this method is applied to the vertebrate globin gene family, and evaluates the departures of base change rates from the ``true' mutation rate at the first and second codon positions as a consequence of preferential selection for the conservation of important function. In addition to the strong effect of selection on the amino acid residues in the internal region mostly common to myoglobin and hemoglobin chains, the distinctive directions of selective parameter values are seen at sites on the globin surface, distinguishing the subunit contact residues of hemoglobins from the polar residues on the surface of myoglobins. Moreover, this effect of selection distinguishing between the myoglobin and hemoglobin chain genes becomes weaker in cold-blooded vertebrates, especially in fish, strongly suggesting the possibility that the clear distinction between these globins is a result of selection out of the changes regarded as neutral ones in an ancestor of vertebrates. Thus, the present method may also serve to investigate the homology of many other proteins from the aspect of molecular evolution, mainly focusing on the evolution of their biological functions. Received: 2 January 1996 / Accepted: 20 February 1997     

Identification of quantitative trait loci influencing annual height- and diameter-increment growth in loblolly pine (Pinus taeda L.)

 Quantitative trait loci (QTLs) for annual height- and diameter-increment growth were estimated in each of two three-generation loblolly pine pedigrees. Annual height-increment was measured in three consecutive years in the BASE pedigree and in four consecutive years in the QTL pedigree. Annual diameter-increment was measured only in the QTL pedigree for two consecutive years. An interval mapping-approach was used to estimate the number of QTLs, the magnitude of QTL effects, and their position on genetic linkage maps. Thirteen different height-increment and eight different diameter-increment QTLs were detected, suggesting that the these traits are, at least in part, controlled by a few genes of large effect. Little evidence was found for the expression of individual QTLs in multiple years or in multiple genetic backgrounds. These results were discussed in terms of the power of the experiment and their consequences for marker-assisted breeding. Received: 12 July 1998/Accepted: 5 August 1998     

Covariances of relatives and selection response in generations of selfing from an outcrossed base population

Summary The covariances of relatives arising under selfing from a general outbred base population in linkage equilibrium and without epistasis given by Cockerham (1983) are expressed in an alternative form which is an extension of the treatment by Mather and Jinks (1982) of the more restricted population descended from a single F₁ family. Whereas no more than two quadratic components are required to describe any covariance in the case of F₁, descendants, this more general case calls for a total of four, three of which are needed for any particular covariance. The estimation of covariances and their use for the prediction of selection response is described for breeding programs initiated by one or more cycles of intermating among a number of parental lines, as advocated by Hansel (1964) and Jensen (1970). It is pointed out that the homozygous lines descended from such a population will have up to twice as much variance as those from an F₁ between a randomly chosen pair from the same population of parents. The selection method is especially recommended for undeveloped species in which the parental lines are not well characterized and large selection responses are needed.     

QTL analysis of an advanced backcross of Lycopersicon peruvianum to the cultivated tomato and comparisons with QTLs found in other wild species

 A BC₃ population previously developed from a backcross of Lycopersicon peruvianum, a wild relative of tomato, into the cultivated variety L. esculentum was analyzed for QTLs. Approximately 200 BC₄ families were scored for 35 traits in four locations worldwide. One hundred and sixty-six QTLs were detected for 29 of those traits. For more than half of those 29 traits at least 1 QTL was detected for which the presence of the wild allele was associated with an agronomically beneficial effect despite the inferior phenotype of the wild parent. Eight QTLs for fruit weight could be followed through the BC₂, BC₃, and BC₄, generations, supporting the authenticity of these QTLs. Comparisons were made between the QTLs found in this study and those found in studies involving two other wild species; the results showed that while some of these QTLs can be presumed to be allelic, most of the QTLs detected in this study are ones not previously discovered. Received: 9 April 1997 / Accepted: 20 May 1997     

On the non-existence of an optimal migration rate

 We show that an optimal migration rate may not exist in a population distributed over an infinite number of individual living sites if empty sites occur. This is the case when the mean number of offspring per individual μ is finite. We make the assumption of uniform migration to other sites whose rate is determined by the parent’s genotype or the offspring’s genotype at a single locus in a diploid hermaphrodite population undergoing random mating. In both cases, for μ small enough, any population at fixation would go to extinction. Moreover, in the latter case, for intermediate values of μ, the only fixation state that could resist the invasion of any mutant would lead the population to extinction. These are the two conditions for the non-existence of an optimal migration rate. They become less stringent as the cost for migration expressed by a coefficient of selection 1−β becomes larger, that is, closer to 1. The results are obtained assuming that the allele at fixation is either nondominant or dominant. Although the optimal migration rate is the same in both cases when it exists, the optimality properties may differ. Received 14 December 1995; received in revised form 5 April 1996