Kinetic models of guanidine hydrochloride-induced curing of the yeast [PSI+] prion.

A population of [PSI(+)] Saccharomyces cerevisiae cells can be cured of the [PSI(+)] prion by the addition of guanidine hydrochloride (GdnHCl). In this paper we extend existing nucleated polymerisation simulation models to investigate the mechanisms that might underlie curing. Our results are consistent with the belief that prions are dispersed through the cells at division following GdnHCl addition. A key feature of the simulation model is that the probability that a polymer is transmitted from mother to daughter during cell division is dependent upon the length of the polymer. The model is able to reproduce the essential features of data from several different experimental protocols involving addition and removal of GdnHCl.

Approximations for expected generation number.

A deterministic formula is commonly used to approximate the expected generation number of a population of growing cells. However, this can give misleading results because it does not allow for natural variation in the times that individual cells take to reproduce. Here we present more accurate approximations for both symmetric and asymmetric cell division. Based on the first two moments of the generation time distribution, these approximations are also robust. We illustrate the improved approximations using data that arise from monitoring individual yeast cells under a microscope and also demonstrate how the approximations can be used when such detailed data are not available.

New approximations to the Malthusian parameter.

Approximations to the Malthusian parameter of an age-dependent branching process are obtained in terms of the moments of the lifetime distribution, by exploiting a link with renewal theory. In several examples, the new approximations are more accurate than those currently in use, even when based on only the first two moments. The new approximations are extended to include a form of asymmetric cell division that occurs in some species of yeast. When used for inference, the new approximations are shown to have high efficiency.

Single-locus gametophytic incompatibility in autotetraploids.

It is known that a single-locus gametophytic self-incompatibility (GSI) system can persist with just two distinct alleles in an autotetraploid population, in contrast to diploid GSI systems, assuming "competitive interaction" in which heteroallelic pollen is universally compatible. The steady-state population structure of a GSI system in autotetraploids was investigated in an undivided population assuming "competitive interaction." A deterministic model was developed to predict the frequencies of genotypes with two, three, or four distinct S alleles, assuming no mutation or population subdivision. The model showed that unlike in diploid GSI systems, the limiting values of the frequencies of genotype classes do not minimize pollen wastage.

Estimating the number of prions in yeast cells.

Certain yeast cells contain proteins that behave like the mammalian prion PrP and are called yeast prions. The yeast prion protein Sup35p can exist in one of two stable forms, giving rise to phenotypes [PSI(+)] and [psi(-)]. If the chemical guanidine hydrochloride (GdnHCl) is added to a culture of growing [PSI(+)] cells, the proportion of [PSI(+)] cells decreases over time. This process is called curing and is due to a failure to propagate the prion form of Sup35p. We describe how curing can be modelled, and improve upon previous models for the underlying processes of cell division and prion segregation; the new model allows for asymmetric cell division and unequal prion segregation. We conclude by outlining plans for future experimentation and modelling.

Modelling the establishment and spread of autotetraploid plants in a spatially heterogeneous environment.

The establishment and spread of autotetraploids from an original diploid population in a heterogeneous environment were studied using a stochastic simulation model. Specifically, we investigated the effects of heterogeneous habitats and nonrandom pollen/seed dispersal on the critical value (micro) of unreduced 2n gamete production necessary for the establishment of autotetraploids as predicted by deterministic models. Introduction of a heterogeneous environment with random pollen/seed dispersal had little effect on the micro value. In contrast, incorporating nonrandom pollen/seed dispersal into a homogeneous environment considerably reduced the micro value. Incorporating both heterogeneous habitats and nonrandom pollen/seed dispersal may lead either to an increase or to a decrease in the micro value compared to that with random dispersal, indicating that the two factors interact in a complex way.

Models for yeast prions.

The cytoplasmic heritable determinant [PSI+] of the yeast Saccharomyces cerevisiae exhibits prion-like properties. The properties of yeast prions are studied in the hope that this will enhance the understanding of mammalian prions, which cause mad-cow, Creutzfeldt-Jakob, and related neurodegenerative diseases. When host cells divide, the yeast prions distribute themselves without loss over the daughter cells. Experimental data provide information on how the proportion of cells with prions decreases over time when priori replication is inhibited. One feature of scientific interest is the unknown mean number, n0, of prions assumed to be present in the cells at the start of the experiment. We develop several stochastic models and by fitting them to the data, we obtain substantially larger estimates of n0 compared with a previous analysis. An interesting feature of a model with constant cell generation times is that the predicted proportion of cells with prions varies over time as a sequence of linked hyperbolic curves. Avenues for future research are outlined, which relax simplifying assumptions made in the models. We make several recommendations for the design of future experiments.

Polylink: to support two-point linkage analysis in autotetraploids.

SUMMARY: Polylink runs under Microsoft Windows (95 or later). It performs various calculations that are useful for investigating two-point linkage analysis for autopolyploids, based on the random chromosome pairing model. These include calculation of offspring phenotypic probabilities as functions of the recombination fraction, calculation of theoretical standard errors for the maximum likelihood estimator of and numerical computation of maximum likelihood estimates. It also includes simulation facilities. AVAILABILITY: Polylink is free and available from Xiangming Xu via email

Analysis of segregation data from selfed progeny of allopolyploids.

This paper discusses the inference of parental genotype based on segregation data from selfed progeny of allopolyploids when there is incomplete information about genotypes and when alleles are codominant or null. The distinct alleles that are present in a genotype are assumed to be known, but not the frequency with which they occur. These assumptions may be appropriate when genotypes are deduced from DNA or protein banding patterns on electrophoretic gels. A computer program, SELF, is described that can generate all possible parental genotypes and rank them on the basis of their agreement with the progeny data. The program caters for tetraploids, hexaploids and octoploids. The methods are illustrated using data from a study of the inheritance of isoenzymes in selfed progeny of octoploid strawberry cultivars.

Cation permeability and selectivity of a root plasma membrane calcium channel.

Calcium channels in the plasma membrane of root cells fulfill both nutritional and signaling roles. The permeability of these channels to different cations determines the magnitude of their cation conductances, their effects on cell membrane potential and their contribution to cation toxicities. The selectivity of the rca channel, a Ca2+-permeable channel from the plasma membrane of wheat (Triticum aestivum L.) roots, was studied following its incorporation into planar lipid bilayers. The permeation of K+, Na+, Ca2+ and Mg2+ through the pore of the rca channel was modeled. It was assumed that cations permeated in single file through a pore with three energy barriers and two ion-binding sites. Differences in permeation between divalent and monovalent cations were attributed largely to the affinity of the ion binding sites. The model suggested that significant negative surface charge was present in the vestibules to the pore and that the pore could accommodate two cations simultaneously, which repelled each other strongly. The pore structure of the rca channel appeared to differ from that of L-type calcium channels from animal cell membranes since its ion binding sites had a lower affinity for divalent cations. The model adequately accounted for the diverse permeation phenomena observed for the rca channel. It described the apparent submillimolar Km for the relationship between unitary conductance and Ca2+ activity, the differences in selectivity sequences obtained from measurements of conductance and permeability ratios, the changes in relative cation permeabilities with solution ionic composition, and the complex effects of Ca2+ on K+ and Na+ currents through the channel. Having established the adequacy of the model, it was used to predict the unitary currents that would be observed under the ionic conditions employed in patch-clamp experiments and to demonstrate the high selectivity of the rca channel for Ca2+ influx under physiological conditions.

Improved estimation of the proportion of triploids in populations with diploid and triploid individuals.

We consider the estimation of the proportion of triploids in populations of plants or animals in which diploid and triploid individuals coexist, using data from electrophoretic analysis of isozyme or microsatellite markers. Individuals that have three distinct alleles at a locus are unambiguously triploid. However, other individuals cannot be classified with certainty as diploid or triploid, unless allelic dosage can be determined reliably. This is impossible for microsatellite markers, and for many isozyme markers. We therefore present a maximum likelihood method of estimating the proportion of triploids based only on the presence or absence of different alleles.

Effects of quadrat size and shape, initial epidemic conditions, and spore dispersal gradient on spatial statistics of plant disease epidemics.

ABSTRACT The spatiotemporal spread of plant diseases was simulated using a stochastic model to study the effects of initial conditions (number of plants initially infected and their spatial pattern), spore dispersal gradient, and size and shape of sampling quadrats on statistics describing the spatiotemporal dynamics of epidemics. The spatial spread of disease was simulated using a half-Cauchy distribution with median dispersal distance mu (units of distance). A total of 54 different quadrat types, including 23 distinct sizes ranging from 4 to 144 plants, were used to sample the simulated epidemics. A symmetric form of the binary power law with two parameters (alpha, beta) was fitted to the sampled epidemic data using each of the 54 quadrats for each replicate simulation run. The alpha and beta estimates were highly correlated positively with each other, and their estimates were comparable to those estimated from observed epidemics. Intraclass correlation (kappa) was calculated for each quadrat type; kappa decreased exponentially with increasing quadrat size. An asymmetric form of the binary power law with three parameters (alpha (1), beta(1), beta(2)) was used to relate kappa to the disease incidence (p); beta1 was highly correlated to beta: beta1 approximately beta - 1. In general, initial conditions and quadrat size affected alpha, beta, alpha(1), beta(1), and beta(2) greatly. The parameter estimates increased as quadrat size increased, and the relationships were described well by a linear regression model on the logarithm of quadrat size with the slope or intercept parameters dependent on initial conditions and mu. Compared with initial conditions and quadrat size, the overall effects of mu and quadrat shape were generally small, although within each quadrat size and initial condition they could be substantial. Quadrat shape had the greatest effect when the quadrat was long and thin. The relationship of the index of dispersion (D) to p and quadrat size was determined from the alpha and beta estimates. D was greatest when p was 0.5 and decreased when p approached 0 or 1. It increased with quadrat size and the rate of the increase was maximum when p was 0.5 and decreased when p approached 0 or 1.

Relationships between several quadrat-based statistical measures used to characterize spatial aspects of disease incidence data.

ABSTRACT This article investigates the relationships between various statistical measures that are used to summarize spatial aspects of disease incidence data. The focus is on quadrat data in which each plant in a quadrat is classified as diseased or healthy. We show that spatial autocorrelation plays a central role via the mean intraclass correlation, rho, which is defined as the average correlation of the disease status of all pairs of plants within the quadrat. The value of rho determines the variance of the number of infected plants in the quadrat and, if this variable follows a beta-binomial distribution, the heterogeneity parameter of the beta-binomial distribution is directly related to the mean intraclass correlation. We consider in detail a model in which the spatial autocorrelation depends only on the distance between the plants. For illustration, we consider a specific autocorrelation model that was derived from simulated data. We show that this model leads, approximately, to the binary form of the power law relating the variance of the number of infected plants per quadrat to the mean. Using an approximation technique, we then show how the index of dispersion is related to quadrat size and shape. The index of dispersion increases with quadrat size. The rate of increase is dependent on quadrat shape, but the effect of quadrat shape is small in comparison to the effect of quadrat size. Finally, we note that if the spatial autocorrelation depends on the relative orientation of the plants, as well as the distance between them, there are connections with distance class methods.

An energy-barrier model for the permeation of monovalent and divalent cations through the maxi cation channel in the plasma membrane of rye roots.

The depolarization-activated, high-conductance "maxi" cation channel in the plasma membrane of rye (Secale cereale L.) roots is permeable to a wide variety of monovalent and divalent cations. The permeation of K+, Na+, Ca2+ and Ba2+ through the pore could be simulated using a model composed of three energy barriers and two ion binding sites (a 3B2S model), which assumed single-file permeation and the possibility of double cation occupancy. The model had an asymmetrical free energy profile. Differences in permeation between cations were attributed primarily to differences in their free energy profiles in the regions of the pore adjacent to the extracellular solution. In particular, the height of the central free energy peak differed between cations, and cations differed in their affinities for ion binding sites. Significant ion repulsion occurred within the pore, and the mouths of the pore had considerable surface charge. The model adequately described the diverse current vs. voltage (I/V) relationships obtained over a wide variety of experimental conditions. It described the phenomena of non-Michaelian unitary conductance vs. activity relationships for K+, Na+ and Ca2+, differences in selectivity sequences obtained from measurements of conductance and permeability ratios, changes in relative cation permeabilities with solution composition, and the complex effects of Ba2+ and Ca2+ on K+ currents through the channel. The model enabled the prediction of unitary currents and ion fluxes through the maxi cation channel under physiological conditions. It could be used, in combination with data on the kinetics of the channel, as input to electrocoupling models allowing the relationships between membrane voltage, Ca2+ influx and Ca2+ signaling to be studied theoretically.

Estimating intraclass correlation for binary data.

This paper reviews many different estimators of intraclass correlation that have been proposed for binary data and compares them in an extensive simulation study. Some of the estimators are very specific, while others result from general methods such as pseudo-likelihood and extended quasi-likelihood estimation. The simulation study identifies several useful estimators, one of which does not seem to have been considered previously for binary data. Estimators based on extended quasi-likelihood are found to have a substantial bias in some circumstances.

Memory in coal tits: an alternative model.

Jolliffe and Jolliffe (1997, Biometrics 53, 1136-1142) proposed various models for data from an experiment on memory in coal tits. This article describes an alternative model, which fits equally well and which may be simpler to interpret.

The estimation of rapid rate constants from current-amplitude frequency distributions of single-channel recordings.

A method is described for estimating rapid rate constants from the distributions of current amplitude observed in single-channel electrical recordings. It has the advantages over previous, similar approaches that it can accommodate both multistate kinetic models and adjustable filtering of the data using an 8-pole Bessel filter. The method is conceptually straightforward: the observed distributions of current amplitude are compared with theoretical distributions derived by combining several simplifying assumptions about the underlying stochastic process with a model of the filter and electrical noise. Parameters are estimated by approximate maximum likelihood. The method was used successfully to estimate rate constants for both a simple two-state kinetic model (the transitions between open and closed states during the rapid gating of an outward-rectifying K(+)-selective channel in the plasma membrane of Acetabularia) and a complex multistate kinetic model (the blockade of the maxi cation channel in the plasma membrane of rye roots by verapamil). For the two-state model, parameters were estimated well, provided that they were not too fast or too slow in relation to the sampling rate. In the three-state model the precision of estimates depended in a complex way on the values of all rate parameters in the model.

Effects of initial epidemic conditions, sporulation rate, and spore dispersal gradient on the spatio-temporal dynamics of plant disease epidemics.

ABSTRACT A stochastic model that simulates the spread of disease over space and time was developed to study the effects of initial epidemic conditions (number of initial inocula and their spatial pattern), sporulation rate, and spore dispersal gradient on the spatio-temporal dynamics of plant disease epidemics. The spatial spread of disease was simulated using a half-Cauchy distribution with median dispersal distance mu (units of distance). The rate of temporal increase in disease incidence (beta(I), per day) was influenced jointly by mu and by the sporulation rate lambda (spores per lesion per day). The relationship between beta(I) and mu was nonlinear: the increase in beta(I) with increasing mu was greatest when mu was small (i.e., when the dispersal gradient was steep). The rate of temporal increase in disease severity of diseased plants (beta(S)) was affected mainly by lambda: beta(S) increased directly with increasing lambda. Intraclass correlation (kappa(t)), the correlation of disease status of plants within quadrats, increased initially with disease incidence, reached a peak, and then declined as disease incidence approached 1.0. This relationship was well described by a power-law model that is consistent with the binary form of the variance power law. The amplitude of the model relating kappa(t) to disease incidence was affected mainly by mu: kappa(t) decreased with increasing mu. The shape of the curve was affected mainly by initial conditions, especially the spatial pattern of the initial inocula. Generally, the relationship of spatial autocorrelation (rho(t,k)), the correlation of disease status of plants at various distances apart, to disease incidence and distance was well described by a four-parameter power-law model. rho(t,k) increased with disease incidence to a maximum and then declined at higher values of disease incidence, in agreement with a power-law relationship. The amplitude of rho(t,k) was determined mainly by initial conditions and by mu: rho(t,k) decreased with increasing mu and was lower for regular patterns of initial inocula. The shape of the rho(t,k) curve was affected mainly by initial conditions, especially the spatial pattern of the initial inocula. At any level of disease incidence, autocorrelation declined exponentially with spatial lag; the degree of this decline was determined mainly by mu: it was steeper with decreasing mu.

Finite mixture models for proportions.

Six data sets recording fetal control mortality in mouse litters are presented. The data are clearly overdispersed, and a standard approach would be to describe the data by means of a beta-binomial model or to use quasi-likelihood methods. For five of the examples, we show that beta-binomial model provides a reasonable description but that the fit can be significantly improved by using a mixture of a beta-binomial model with a binomial distribution. This mixture provides two alternative solutions, in one of which the binomial component indicates a high probability of death but is selected infrequently; this accounts for outlying litters with high mortality. The influence of the outliers on the beta-binomial fits is also demonstrated. The location and nature of the two main maxima to the likelihood are investigated through profile log-likelihoods. Comparisons are made with the performance of finite mixtures of binomial distributions.

Micropropagation of adult birch trees: production and field performance.

We compared the growth of trees produced by micropropagation from nodal stem sections or callus tissue of a 20-year-old silver birch (Betula pendula Roth) tree with that of seedlings; growth was monitored for 17 months in pots followed by six years in the field. Micropropagated trees from both nodal stem sections and callus tissue grew at a similar rate to seedling trees and no obvious mutant types were observed. However, micropropagated trees were more uniform in height and trunk girth than seedling trees and more than 80% flowered within three years of field planting, whereas only 39% of seedling trees flowered within this time. Micropropagated trees had less bark fissuring, a mature characteristic, than seedling trees.