Martin S. Ridout

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.

An empirical model for underdispersed count data

We present a novel distribution for modelling count data that are underdispersed relative to the Poisson distribution. The distribution is a form of weighted Poisson distribution and is shown to have advantages over other weighted Poisson distributions that have been proposed to model underdispersion. One key difference is that the weights in our distribution are centred on the mean of the underlying Poisson distribution. Several illustrative examples are presented that illustrate the consistently good performance of the distribution.

The number and transmission of [PSI+] prion seeds (Propagons) in the yeast Saccharomyces cerevisiae

Background Yeast (Saccharomyces cerevisiae) prions are efficiently propagated and the on-going generation and transmission of prion seeds (propagons) to daughter cells during cell division ensures a high degree of mitotic stability. The reversible inhibition of the molecular chaperone Hsp104p by guanidine hydrochloride (GdnHCl) results in cell division-dependent elimination of yeast prions due to a block in propagon generation and the subsequent dilution out of propagons by cell division. Principal Findings Analysing the kinetics of the GdnHCl-induced elimination of the yeast [PSI+] prion has allowed us to develop novel statistical models that aid our understanding of prion propagation in yeast cells. Here we describe the application of a new stochastic model that allows us to estimate more accurately the mean number of propagons in a [PSI+] cell. To achieve this accuracy we also experimentally determine key cell reproduction parameters and show that the presence of the [PSI+] prion has no impact on these key processes. Additionally, we experimentally determine the proportion of propagons transmitted to a daughter cell and show this reflects the relative cell volume of mother and daughter cells at cell division. Conclusions While propagon generation is an ATP-driven process, the partition of propagons to daughter cells occurs by passive transfer via the distribution of cytoplasm. Furthermore, our new estimates of n0, the number of propagons per cell (500–1000), are some five times higher than our previous estimates and this has important implications for our understanding of the inheritance of the [PSI +] and the spontaneous formation of prion-free cells.

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.

Cell division is essential for elimination of the yeast [PSI+] prion by guanidine hydrochloride

Guanidine hydrochloride (Gdn·HCl) blocks the propagation of yeast prions by inhibiting Hsp104, a molecular chaperone that is absolutely required for yeast prion propagation. We had previously proposed that ongoing cell division is required for Gdn·HCl-induced loss of the [PSI+] prion. Subsequently, Wu et al.[Wu Y, Greene LE, Masison DC, Eisenberg E (2005) Proc Natl Acad Sci USA 102:12789–12794] claimed to show that Gdn·HCl can eliminate the [PSI+] prion from α-factor-arrested cells leading them to propose that in Gdn·HCl-treated cells the prion aggregates are degraded by an Hsp104-independent mechanism. Here we demonstrate that the results of Wu et al. can be explained by an unusually high rate of α-factor-induced cell death in the [PSI+] strain (780-1D) used in their studies. What appeared to be no growth in their experiments was actually no increase in total cell number in a dividing culture through a counterbalancing level of cell death. Using media-exchange experiments, we provide further support for our original proposal that elimination of the [PSI+] prion by Gdn·HCl requires ongoing cell division and that prions are not destroyed during or after the evident curing phase.

Generating random numbers from a distribution specified by its Laplace transform

This paper discusses simulation from an absolutely continuous distribution on the positive real line when the Laplace transform of the distribution is known but its density and distribution functions may not be available. We advocate simulation by the inversion method using a modified Newton-Raphson method, with values of the distribution and density functions obtained by numerical transform inversion. We show that this algorithm performs well in a series of increasingly complex examples. Caution is needed in some situations when the numerical Laplace transform inversion becomes unreliable. In particular the algorithm should not be used for distributions with finite range. But otherwise, except for rather pathological distributions, the approach offers a rapid way of generating random samples with minimal user effort. We contrast our approach with an alternative algorithm due to Devroye (Comput. Math. Appl. 7, 547–552, 1981).

Cation Permeability and Selectivity of a Root Plasma Membrane Calcium Channel

Abstract. 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.

Modelling digit preference in fecundability studies

Retrospective studies of fecundability, in which women are asked how many cycles they required to become pregnant, are often affected by problems of digit preference. A probability model for such digit preference is proposed in which misreporting favours 6 or 12 (and possibly also 3) cycles. It is assumed that in the absence of misreporting the number of cycles follows a beta-geometric distribution. The model is applied to two data sets, with clear-cut results: The inclusion of additional parameters to model the misreporting can lead to substantial improvements in fit, but causes little change to the estimated parameters of the underlying beta-geometric distribution. In some cases misreporting parameters may be regarded as nuisance parameters, while in others they may be of interest. We have found estimates of these parameters to vary between different categories of women in an interpretable manner. The models may also be used to estimate the percentage of couples in any study that misreport their conception waiting time.

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.

Computational aspects of N-mixture models

The N-mixture model is widely used to estimate the abundance of a population in the presence of unknown detection probability from only a set of counts subject to spatial and temporal replication (Royle, 2004, Biometrics 60, 105–115). We explain and exploit the equivalence of N-mixture and multivariate Poisson and negative-binomial models, which provides powerful new approaches for fitting these models. We show that particularly when detection probability and the number of sampling occasions are small, infinite estimates of abundance can arise. We propose a sample covariance as a diagnostic for this event, and demonstrate its good performance in the Poisson case. Infinite estimates may be missed in practice, due to numerical optimization procedures terminating at arbitrarily large values. It is shown that the use of a bound, K, for an infinite summation in the N-mixture likelihood can result in underestimation of abundance, so that default values of K in computer packages should be avoided. Instead we propose a simple automatic way to choose K. The methods are illustrated by analysis of data on Hermanns tortoise Testudo hermanni.