A general model to explore complex dominance patterns in plant sporophytic self-incompatibility systems

We developed a general model of sporophytic self-incompatibility under negative frequency-dependent selection allowing complex patterns of dominance among alleles. We used this model deterministically to investigate the effects on equilibrium allelic frequencies of the number of dominance classes, the number of alleles per dominance class, the asymmetry in dominance expression between pollen and pistil, and whether selection acts on male fitness only or both on male and on female fitnesses. We show that the so-called "recessive effect" occurs under a wide variety of situations. We found emerging properties of finite population models with several alleles per dominance class such as that higher numbers of alleles are maintained in more dominant classes and that the number of dominance classes can evolve. We also investigated the occurrence of homozygous genotypes and found that substantial proportions of those can occur for the most recessive alleles. We used the model for two species with complex dominance patterns to test whether allelic frequencies in natural populations are in agreement with the distribution predicted by our model. We suggest that the model can be used to test explicitly for additional, allele-specific, selective forces.     

The different mechanisms of sporophytic self-incompatibility

Flowering plants have evolved a multitude of mechanisms to avoid self-fertilization and promote outbreeding. Self-incompatibility (SI) is by far the most common of these, and is found in ca. 60% of flowering plants. SI is a genetically controlled pollen-pistil recognition system that provides a barrier to fertilization by self and self-related pollen in hermaphrodite (usually co-sexual) flowering plants. Two genetically distinct forms of SI can be recognized: gametophytic SI (GSI) and sporophytic SI (SSI), distinguished by how the incompatibility phenotype of the pollen is determined. GSI appears to be the most common mode of SI and can operate through at least three different mechanisms, two of which have been characterized extensively at a molecular level in the Solanaceae and Papaveraceae. Because molecular studies of SSI have been largely confined to species from the Brassicaceae, predominantly Brassica species, it is not yet known whether SSI, like GSI, can operate through different molecular mechanisms. Molecular studies of SSI are now being carried out on Ipomoea trifida (Convolvulaceae) and Senecio squalidus (Asteraceae) and are providing important preliminary data suggesting that SSI in these two families does not share the same molecular mechanism as that of the Brassicaceae. Here, what is currently known about the molecular regulation of SSI in the Brassicaceae is briefly reviewed, and the emerging data on SSI in I. trifida, and more especially in S. squalidus, are discussed.     

Allelic genealogies in sporophytic self-incompatibility systems in plants.

Expectations for the time scale and structure of allelic genealogies in finite populations are formed under three models of sporophytic self-incompatibility. The models differ in the dominance interactions among the alleles that determine the self-incompatibility phenotype: In the SSIcod model, alleles act codominantly in both pollen and style, in the SSIdom model, alleles form a dominance hierarchy, and in SSIdomcod, alleles are codominant in the style and show a dominance hierarchy in the pollen. Coalescence times of alleles rarely differ more than threefold from those under gametophytic self-incompatibility, and transspecific polymorphism is therefore expected to be equally common. The previously reported directional turnover process of alleles in the SSIdomcod model results in coalescence times lower and substitution rates higher than those in the other models. The SSIdom model assumes strong asymmetries in allelic action, and the most recessive extant allele is likely to be the most recent common ancestor. Despite these asymmetries, the expected shape of the allele genealogies does not deviate markedly from the shape of a neutral gene genealogy. The application of the results to sequence surveys of alleles, including interspecific comparisons, is discussed.     

Uneven segregation of sporophytic self-incompatibility alleles in Arabidopsis lyrata

Self-incompatibility in Arabidopsis lyrata is sporophytically controlled by the multi-allelic S-locus. Self-incompatibility alleles (S-alleles) are under strong negative frequency dependent selection because pollen carrying common S-alleles have fewer mating opportunities. Population genetics theory predicts that deleterious alleles can accumulate if linked to the S-locus. This was tested by studying segregation of S-alleles in 11 large full sib families in A. lyrata. Significant segregation distortion leading to an up to fourfold difference in transmission rates was found in six families. Differences in transmission rates were not significantly different in reciprocal crosses and the distortions observed were compatible with selection acting at the gametic stage alone. The S-allele with the largest segregation advantage is also the most recessive, and is very common in natural populations concordant with its apparent segregation advantage. These results imply that frequencies of S-alleles in populations of A. lyrata cannot be predicted based on simple models of frequency-dependent selection alone.     

Numbers of sporophytic self-incompatibility alleles in populations of wild radish

Summary To estimate the numbers of sporophytic S-alleles in two adjacent populations of wild radish, we performed 701 reciprocal crosses among 50 individuals. Each cross was replicated five times in each direction. Sixteen plants were fully intercompatible, indicating the presence of at least 32 S-alleles in the two populations. A minimum of 22 S-alleles occur in a single population. The frequency of incompatibility was significantly higher for within-population crosses (14.5%) than for between-population crosses (7.8%). This suggests that the two populations differ in the composition and frequency of alleles at the S-locus.     

A general model of stochastic neural processing

A model of neural processing is proposed which is able to incorporate a great deal of neurophysiological detail, including effects associated with the mechanics of postsynaptic summation and cell surface geometry and is capable of hardware realisation as a probabilistic random access memory (pRAM). The model is an extension of earlier work by the authors, which by operating at much shorter time scales (of the order of the lifetime of a quantum of neurotransmitter in the synaptic cleft) allows a greater amount of information to be retrieved from the simulated spike train. The mathematical framework for the model appears to be that of an extended Markov process (involving the firing histories of the N neurons); simulation work has yielded results in excellent agreement with theoretical predictions. The extended neural model is expected to be particularly applicable in situations where timing constraints are of special importance (such as the auditory cortex) or where firing thresholds are high, as in the case for the granule and pyramidal cells of the hippocampus.     

COMPUTER PROGRAMS: nessi: a program for numerical estimations in sporophytic self-incompatibility genetic systems

nessi is a computer program generating predictions about allelic and genotypic frequencies at the S-locus in sporophytic self-incompatibility systems under finite and infinite populations. For any pattern of dominance relationships among self-incompatibility alleles, nessi computes deterministic equilibrium frequencies and estimates distributions in samples from finite populations of the number of alleles at equilibrium, allelic and genotypic frequencies at equilibrium and allelic and genotypic frequency changes in a single generation. These predictions can be used to rigorously test the impact of negative frequency-dependent selection on diversity patterns in natural populations.     

A general model for stochastic SIR epidemics with two levels of mixing

This paper is concerned with a general stochastic model for susceptible→infective→removed epidemics, among a closed finite population, in which during its infectious period a typical infective makes both local and global contacts. Each local contact of a given infective is with an individual chosen independently according to a contact distribution ‘centred’ on that infective, and each global contact is with an individual chosen independently and uniformly from the whole population. The asymptotic situation in which the local contact distribution remains fixed as the population becomes large is considered. The concepts of local infectious clump and local susceptibility set are used to develop a unified approach to the threshold behaviour of this class of epidemic models. In particular, a threshold parameter R_* governing whether or not global epidemics can occur, the probability that a global epidemic occurs and the mean proportion of initial susceptibles ultimately infected by a global epidemic are all determined. The theory is specialised to (i) the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective’s household; (ii) the overlapping groups model, in which the population is partitioned in several ways, with local uniform mixing within the elements of the partitions; and (iii) the great circle model, in which individuals are equally spaced on a circle and local contacts are nearest-neighbour.     

New hypothesis elucidates self-incompatibility in the olive tree regarding S-alleles dominance relationships as in the sporophytic model

Most olive varieties are not strictly self-incompatible, nevertheless, they request foreign pollen to enhance fruit yield, and consequently orchards should contain pollinisers to ensure fruit set of the main variety. The best way to choose pollinisers is to experiment numerous crosses in a diallel design. Here, the genetic mode of inheritance of SI in the olive is deciphered and it does not correspond to the GSI type, but to the SSI type. It leaves S-allele dominance relationship expression in the male (pollen and pollen tube), but not in the female (stigma and style). Thus, a pair-wise combination of varieties may be inter-compatible in one direction (male to female, or female to male) and inter-incompatible in the other direction. Dominance relationships also explain different levels of self-pollination observed in varieties. Little efficient pollinisers were found and predicted in varieties; nevertheless, some new efficient pair-wise allele combinations are predicted and could be created. This model enables one to forecast compatibility without waiting for several years of yield records and to choose pollinisers in silico.     

A stochastic model for cancer risk

We propose a simple stochastic model based on the two successive mutations hypothesis to compute cancer risks. Assume that only stem cells are susceptible to the first mutation and that there are a total of D stem cell divisions over the lifetime of the tissue with a first mutation probability mu(1) per division. Our model predicts that cancer risk will be low if m = mu(1)D is low even in the case of very advantageous mutations. Moreover, if mu(1)D is low the mutation probability of the second mutation is practically irrelevant to the cancer risk. These results are in contrast with existing models but in agreement with a conjecture of Cairns. In the case where m is large our model predicts that the cancer risk depends crucially on whether the first mutation is advantageous or not. A disadvantageous or neutral mutation makes the risk of cancer drop dramatically.     

A stochastic model for head lice infections

We investigate the dynamics of head lice infections in schools, by consideringa model for endemic infection based on a stochastic SIS (susceptible-infected-susceptible) epidemic model, with the addition of an external source of infection. We deduce a range of properties of our model, including the length of a single outbreak of infection. We use the stationary distribution of the number of infected individuals, in conjunction with data from a recent study carried out in Welsh schools on the prevalence of head lice infections, and employ maximum likelihood methods to obtain estimates of the model parameters. A complication is that, for each school, only a sample of the pupils was checked for infection. Our likelihood function takes account of the missing data by incorporating a hypergeometric sampling element. We arrive at estimates of the ratios of the “within school” and “external source” transmission rates to the recovery rate and use these to obtain estimates for various quantities of interest.      

A stochastic bioburden model for spacecraft sterilization

An important factor in the calculation of sterilization cycles for spacecraft is the bioburden on the surface of the spacecraft at the start of the cycle. This bioburden must be predicted by the use of models. This paper presents a stochastic model for the prediction of the bioburden on the surfaces of spacecraft. This model has many desirable properties as well as being consistent with observations.This work was conducted under Contract No. W-12,853, Planetary Programs, Office, of Space Science and Applications, NASA Headquarters, Washington, D.C.     

A stochastic model for then-compartment irreversible system

This paper deals with a stochasticn-compartment irreversible system with a non-homogeneous Poisson input and arbitrary residence time for each of the compartments. Results relating to the number of particles present in each of the compartments as well as the total number of particles present in the system at any time are derived. Further, explicit expressions for the auto covariance function for each compartment and the cross-covariance function between any two compartments with a given time lag are obtained. As a particular case, then-compartment irreversible system is analyzed with homogeneous Poisson input and exponential residence time distribution for each of the compartments. The possible applications of the model are discussed.     

A stochastic model for prostate-specific antigen levels

We introduce a continuous stochastic model for the prostate-specific antigen (PSA) levels following radiotherapy and derive solutions for the associated partial differential (Kolmogorov-Chapman) equation. The solutions describe the evolution of the time-dependent density for PSA levels which take into account an absorbing condition along the boundary and various initial conditions. We include implications for single-dose and multi-dose radiation treatment regimens and discuss parameter estimation and sensitivity issues.     

A stochastic model for a progressive chronic disease

For many progressive chronic diseases, there exist useful prognostic indicators for the course of the disease and the survival of the patient. The evolution of such an indicator is modelled as a monotone transformation of a pure birth process with killing. Explicit formulas are derived for the probability distribution of this process at an arbitrary time, the distribution of the first-passage times, the joint distribution of the survival time and the maximum of the process, and the marginals of this joint distribution. In two examples, the general formulas are evaluated in closed form.     

A stochastic metapopulation model accounting for habitat dynamics

A stochastic metapopulation model accounting for habitat dynamics is presented. This is the stochastic SIS logistic model with the novel aspect that it incorporates varying carrying capacity. We present results of Kurtz and Barbour, that provide deterministic and diffusion approximations for a wide class of stochastic models, in a form that most easily allows their direct application to population models. These results are used to show that a suitably scaled version of the metapopulation model converges, uniformly in probability over finite time intervals, to a deterministic model previously studied in the ecological literature. Additionally, they allow us to establish a bivariate normal approximation to the quasi-stationary distribution of the process. This allows us to consider the effects of habitat dynamics on metapopulation modelling through a comparison with the stochastic SIS logistic model and provides an effective means for modelling metapopulations inhabiting dynamic landscapes.     

A spatiotemporal stochastic process model for species spread

We use a spatiotemporal Markov process to model the spread of an ecological population through its environment over time. Available habitat is divided into sites, and a parametric function of spatial variables is used to model the probability that one site is colonized from another. This allows us both to make predictions about the future spread of a population, and to determine which are the important factors governing colonizations. The model evolves in discrete time, allowing the population distribution to change seasonally in accordance with breeding patterns. Discrete time formulations are natural for ecological populations, but are problematic due to difficulties of fitting and predicting over irregular time intervals. The model described here can accommodate years of missing data and can therefore fit and predict at irregular intervals. Two methods of approximating the likelihood are described and applied to ornithological survey data for the woodlark, Lullula arborea, from Thetford Forest in the U.K.