Stochastic description of the three-state model of the Na channel

The three-state model of the Na channel is formulated stochastically and various observable quantities calculated from this model. This model assumes that the Na channel can occupy one of three states--resting, open and inactivated--and that each channel is independent. This is a simplified representation of the model proposed by Aldrich et al. (1983) mainly with respect to the neuroblastoma. When the system contains many channels, the probability distribution of the numbers of the channels that occupy each of these states is a time-dependent multinomial distribution and the distribution of the first passage time from resting state to open state becomes an exponential decay function with higher components. The average and variance of the channel current calculated by the distribution show the time course with a single peak and its ratio converges to the current of a single channel. Stochastic treatment of the transient process developed here gives a method of estimating all of the transition rates and the number of channels in the system.     

The null distribution of the heterogeneity lod score does depend on the assumed genetic model for the trait.

It is well known that the asymptotic null distribution of the homogeneity lod score (LOD) does not depend on the genetic model specified in the analysis. When appropriately rescaled, the LOD is asymptotically distributed as 0.5 chi(2)(0) + 0.5 chi(2)(1), regardless of the assumed trait model. However, because locus heterogeneity is a common phenomenon, the heterogeneity lod score (HLOD), rather than the LOD itself, is often used in gene mapping studies. We show here that, in contrast with the LOD, the asymptotic null distribution of the HLOD does depend upon the genetic model assumed in the analysis. In affected sib pair (ASP) data, this distribution can be worked out explicitly as (0.5 - c)chi(2)(0) + 0.5chi(2)(1) + cchi(2)(2), where c depends on the assumed trait model. E.g., for a simple dominant model (HLOD/D), c is a function of the disease allele frequency p: for p = 0.01, c = 0.0006; while for p = 0.1, c = 0.059. For a simple recessive model (HLOD/R), c = 0.098 independently of p. This latter (recessive) distribution turns out to be the same as the asymptotic distribution of the MLS statistic under the possible triangle constraint, which is asymptotically equivalent to the HLOD/R. The null distribution of the HLOD/D is close to that of the LOD, because the weight c on the chi(2)(2) component is small. These results mean that the cutoff value for a test of size alpha will tend to be smaller for the HLOD/D than the HLOD/R. For example, the alpha = 0.0001 cutoff (on the lod scale) for the HLOD/D with p = 0.05 is 3.01, while for the LOD it is 3.00, and for the HLOD/R it is 3.27. For general pedigrees, explicit analytical expression of the null HLOD distribution does not appear possible, but it will still depend on the assumed genetic model.     

Modelling the known and unknown plant biodiversity of the Amazon Basin

Aims The overall aim of this study is to provide the data needed for Amazonian conservation and the sustainable management of the region. To this end I model the hypothetical distribution of plant species richness across the Amazon Basin, the distribution of the proportion of this species richness that can be accounted for by described species, and hence the distribution of the biodiversity which remains unknown. Location Amazonia, Neotropics. Methods Species richness across the Amazon Basin is estimated by comparing the occurrences of 1584 species of Magnoliophyta whose taxonomy and geographical distributions are relatively well known. These data are used to collate checklists for squares of 1° latitude by 1° longitude. Comparison of the checklists allows estimation of the relative expected diversity in the vicinity of each degree square. Summing the distributions of the hypothetical real ranges gives the proportion of the biodiversity that can be accounted for by described species. Subtraction of the second distribution from the first gives a distribution of the contribution to the overall biodiversity that the model predicts, potentially, results from as yet undescribed species. Results Collections documented in recent botanical monographs show an extremely biased distribution with the best knowledge being found in a very few relatively well‐collected areas. At the degree square level, this model predicts that gamma biodiversity in the Amazon Basin is uniformly high across most of the basin. The model predicts that four large areas of the basin are particularly poorly known, and that they should contain large numbers of uncollected species. Main conclusions The model presented here highlights the difficulties of quantifying Amazonian plant diversity and its distribution. The low density of collections, and especially their extremely clumped distribution, undermines confidence in theories that seek to explain the apparent distribution of biodiversity. The model's prediction is substantially different from published predictions of the distribution of alpha diversity. Testing of this model in the areas identified as lacunae would require collecting programmes designed to collect fertile material of rare species. If the model's predictions are approximately accurate, the plant biodiversity of the Amazon Basin is considerably underestimated.     

Modelling the distribution of diameter growth along the stem in Scots pine

This paper presents an empirical model for the distribution of diameter growth along the stem in Scots pine (Pinus sylvestris L.) and for the consequent stem form over time. First, the distribution of annual mass growth in the stem is determined as a function of the total annual growth in stem mass, current stem mass and the distribution of the latter along the stem. Second, the distribution of diameter growth is obtained by converting the fraction of annual growth in the stem mass at a given height in the stem into the thickness of the annual ring at the same height. Application of the model to Scots pine data sets including both young and mature trees not used in parameter estimation showed that the model was capable of reconstructing the distribution of diameter growth from the stem butt to the apex and from the pith to the stem surface at any height in the stem in both young and mature trees. The resulting empirical model was also linked to a physiological, process-based model in order to study its performance in a simulated stand. Simulations representing trees grown in unthinned and thinned Scots pine stands with trees of different status (from dominant to suppressed) showed that the response in tree growth to thinning in terms of the distribution of diameter growth along the stem was quite realistic relative to measured data.     

State vector models of the cell cycle II: The first three moments of the transit time distribution.

A discrete time state vector model (the Hahn model) has been used to simulate many experiments in cell kinetics. In the first paper in this series the authors described a new method to define the parameters of the Hahn model suitable for use in automatic fitting of fraction of labelled mitoses (FLM) experiments. In this paper it is shown how to compute the first three moments of the transit time distribution which arises from a Hahn model. These moments are compared analytically and numerically to the corresponding moments of the distribution the authors used to define the Hahn model. Finally, the problems involved in estimating the moments of the transit time distribution observed in fitting FLM data using a Hahn model are discussed.     

Mathematical model of auxin distribution in the plant root

Auxin regulation of plant growth and development is mediated by controlled distribution of this hormone and dose-dependent mechanisms of its action. A mathematical model is proposed, which describes auxin distribution in the cell array along the root longitudinal axis in Arabidopsis thaliana. The model qualitatively simulates auxin distribution over the longitudinal axis in intact roots, changes in this distribution at decreased auxin transport rates, and restoration of the auxin distribution pattern with subsequent establishment of new root meristem in the course of root regeneration after the ablation of its tip. The model shows the presence of different auxin distribution patterns over the longitudinal root axis and suggests possible scenarios for root growth and lateral root formation. Biological interpretation of different regimes of model behavior is presented.     

A statistical model of the VA/Q distribution

A "blocks model" is proposed to model the distribution of the ventilation-perfusion ratio (VA/Q distribution). This model is developed from statistical principles and enables the estimated VA/Q distribution to be interpreted in a straightforward and intuitive manner. Estimation of parameters of the blocks model uses a constrained weighted least-squares procedure. Although developed initially to estimate VA/Q distributions from data generated by the multiple inert gas elimination technique (P. D. Wagner, H. A. Saltzman, and J. B. West, J. Appl. Physiol. 36: 588-599, 1974), the blocks method is applicable to any problem in which the unknown distribution is related to the data through an ill-posed integral equation and is particularly suited for problems in which the data are scarce. The method is illustrated with several examples--hypothetical data representing a wide range of VA/Q distributions as well as some real data.     

A stochastic model for the sizes of detectable metastases

A stochastic entirely mechanistic model of metastatic progression of cancer is developed. Based on this model the joint conditional distribution of the ordered sizes of detectable metastases given their number, n, is computed. It is shown that this distribution coincides with the joint distribution of order statistics for a random sample of size n derived from some probability distribution, and a formula for the latter is obtained. This formula is specialized for the case of exponentially growing primary and secondary tumors and exponentially distributed metastasis promotion times, and identifiability of model parameters is ascertained. These results allow for estimation of the natural history of cancer. As an example, it is estimated for a breast cancer patient with 31 bone metastases of known sizes. The proposed model for the sizes of detectable metastases provided an excellent fit to these data.     

Potential Distribution of the Australian Native Chloris truncata Based on Modelling Both the Successful and Failed Global Introductions

Our aim was to model the current and future potential global distribution of Chloris truncata (windmill grass) based on the plant's biology, soil requirements and colonisation success. The growth response of C. truncata to constant temperatures and soil moisture levels were measured and estimated respectively, to develop parameters for a CLIMEX bioclimatic model of potential distribution. The native distribution in eastern Australia and naturalised distribution in Western Australia was also used to inform the model. Associations with soil types were assessed within the suitable bioclimatic region in Australia. The global projection of the model was tested against the distribution of soil types and the known successful and failed global introductions. The verified model was then projected to future conditions due to climate change. Optimal temperature for plant development was 28°C and the plant required 970 degree-days above a threshold of 10°C. Early collection records indicate that the species is native to Queensland, New South Wales and Victoria. The plant has been introduced elsewhere in Australia and throughout the world as a wool contaminant and as a potential pasture species, but some of the recorded establishments have failed to persist. The CLIMEX model projected to the world reflected effectively both the successful and failed distributions. The inclusion of soil associations improved the explanation of the observed distribution in Australia, but did not improve the ability to determine the potential distribution elsewhere, due to lack of similarity of soil types between continents. The addition of a climate change projection showed decreased suitability for this species in Australia, but increased suitability for other parts of the world, including regions where the plant previously failed to establish.     

Use of the Multinomial Dirichlet Model for Analysis of Subdivided Genetic Populations

The distribution found by compounding the multinomial distribution with the Dirichlet distribution has been suggested as a basis for the estimation of parameters in subdivided populations, in particular of the "correlation between genotypes" within subpopulations. It is shown that the estimators deriving from these procedures perform poorly when the data are generated by the classical Wright drift model of subdivided populations. This conclusion suggests that the compound distribution estimation approach does not provide a good estimation procedure for real populations which are reasonably described by the Wright model.     

塔里木荒漠河岸林不同生境群落物种多度分布格局

为解释塔里木荒漠河岸林群落构建和物种多度分布格局形成的机理, 本文以塔里木荒漠河岸林2个不同生境(沙地、河漫滩) 4 ha固定监测样地为研究对象, 基于两样地物种调查数据, 采用统计模型(对数级数模型、对数正态模型、泊松对数正态分布模型、Weibull分布模型)、生态位模型(生态位优先占领模型、断棍模型)和中性理论模型(复合群落零和多项式模型、Volkov模型)拟合荒漠河岸林群落物种多度分布, 并用K-S检验与赤池信息准则(AIC)筛选最优拟合模型。结果表明: (1)随生境恶化(土壤水分降低), 植物物种多度分布曲线变化减小, 群落物种多样性、多度和群落盖度降低, 常见种数减少。(2)选用的3类模型均可拟合荒漠河岸林不同生境群落物种多度分布格局, 统计模型和中性理论模型拟合效果均优于生态位模型。复合群落零和多项式模型对远离河岸的干旱沙地生境拟合效果最好; 对数正态模型和泊松对数正态模型对洪水漫溢的河漫滩生境拟合效果最优; 中性理论模型与统计模型无显著差异。初步推断中性过程在荒漠河岸林群落构建中发挥着主导作用, 但模型拟合结果只能作为推断群落构建过程的必要非充分条件, 不能排除生态位过程的潜在作用。     

A physiological analogy of the niche for projecting the potential distribution of plants

Aim  To develop a physiologically based model of the plant niche for use in species distribution modelling. Location  Europe. Methods  We link the Thornley transport resistance (TTR) model with functions which describe how the TTR’s model parameters are influenced by abiotic environmental factors. The TTR model considers how carbon and nutrient uptake, and the allocation of these assimilates, influence growth. We use indirect statistical methods to estimate the model parameters from a high resolution data set on tree distribution for 22 European tree species. Results  We infer, from distribution data and abiotic forcing data, the physiological niche dimensions of 22 European tree species. We found that the model fits were reasonable (AUC: 0.79–0.964). The projected distributions were characterized by a false positive rate of 0.19 and a false negative rate 0.12. The fitted models are used to generate projections of the environmental factors that limit the range boundaries of the study species. Main conclusions  We show that physiological models can be used to derive physiological niche dimensions from species distribution data. Future work should focus on including prior information on physiological rates into the parameter estimation process. Application of the TTR model to species distribution modelling suggests new avenues for establishing explicit links between distribution and physiology, and for generating hypotheses about how ecophysiological processes influence the distribution of plants.     

Random effects modeling of multiple binomial responses using the multivariate binomial logit-normal distribution

The multivariate binomial logit-normal distribution is a mixture distribution for which, (i) conditional on a set of success probabilities and sample size indices, a vector of counts is independent binomial variates, and (ii) the vector of logits of the parameters has a multivariate normal distribution. We use this distribution to model multivariate binomial-type responses using a vector of random effects. The vector of logits of parameters has a mean that is a linear function of explanatory variables and has an unspecified or partly specified covariance matrix. The model generalizes and provides greater flexibility than the univariate model that uses a normal random effect to account for positive correlations in clustered data. The multivariate model is useful when different elements of the response vector refer to different characteristics, each of which may naturally have its own random effect. It is also useful for repeated binary measurement of a single response when there is a nonexchangeable association structure, such as one often expects with longitudinal data or when negative association exists for at least one pair of responses. We apply the model to an influenza study with repeated responses in which some pairs are negatively associated and to a developmental toxicity study with continuation-ratio logits applied to an ordinal response with clustered observations.     

Stereotactic model of the electrical distribution within the internal globus pallidus during deep brain stimulation

Deep brain stimulation (DBS) of the internal globus pallidus (GPi) is an established surgical technique for the treatment of movement disorders. The objective of this study was to propose a computational stereotactic model of the electrical distribution around the electrode within the targeted GPi in order to optimize parameter adjustment in clinical practice. The outline of the GPi can be defined precisely by using stereotactic magnetic resonance imaging (MRI) and from this it is possible to model its three-dimensional structure. The electrode and the distribution of the patient-specific parameters can then be co-registered with the GPi volume. By using this methodology, it is possible to visualize and measure the relationship between the electrical distribution of patient-specific parameters and the morphology of the GPi. The model could be applied in clinical practice to help determine the threshold for achieving a therapeutic effect and consequently may aid in optimizing parameter settings for individual patients.     

黑河上游祁连山区青海云杉生长状况及其潜在分布区的模拟

根据Hutchinson的n维超体积概念以及物种与资源利用之间的关系, 构建了青海云杉(Picea crassifolia)在三维环境资源空间中的生物-地理模型, 并利用该模型模拟了青海云杉的潜在分布及其对环境资源的利用状况。结果表明: 青海云杉在生长季平均气温、多年平均降水量及太阳直接辐射三维环境资源空间上的最佳配置为9 ℃、360 mm和1.9 × 10³ kW·h·m^-2; 用三元方程式的拟合结果在大范围上预测了青海云杉的潜在分布区, 并给出了其在对应地理位置上的生长状况。     

A frequency distribution for the number of hematogenous organ metastases

The number of metastases associated with a primary tumor can be a major determinant for the chance of cure. A model is proposed here where the frequency distribution of the number of experimental organ metastases is affected by Poisson statistics, and by stochastic variations in regional blood flow. The model predicts that the mean E (X) and variance var (X) of the number of metastases per organ should relate according to a power function,var (X)= E (X)(p)+E (X), where and p are the constants, and that the actual distribution has a Poisson-negative binomial form. This model was found consistent with the data derived from a meta-analysis of over 47 000 murine experimental metastases. This frequency distribution (together with knowledge of the size distribution for metastases and the fraction of clonogenic cells) could permit more accurate assessments for tumor control probabilities with adjuvant therapies.     

On the size distribution of private microsatellite alleles

Private microsatellite alleles tend to be found in the tails rather than in the interior of the allele size distribution. To explain this phenomenon, we have investigated the size distribution of private alleles in a coalescent model of two populations, assuming the symmetric stepwise mutation model as the mode of microsatellite mutation. For the case in which four alleles are sampled, two from each population, we condition on the configuration in which three distinct allele sizes are present, one of which is common to both populations, one of which is private to one population, and the third of which is private to the other population. Conditional on this configuration, we calculate the probability that the two private alleles occupy the two tails of the size distribution. This probability, which increases as a function of mutation rate and divergence time between the two populations, is seen to be greater than the value that would be predicted if there was no relationship between privacy and location in the allele size distribution. In accordance with the prediction of the model, we find that in pairs of human populations, the frequency with which private microsatellite alleles occur in the tails of the allele size distribution increases as a function of genetic differentiation between populations.     

Stochastic constancy,variability and adaptation of spike generation: Performance of a giant neuron in the visual system of the fly

The stochastic structure of the spike activity generated by a movement processing wide-field element in the visual system of the fly has been studied over the whole performance area of the neuron. The structure of this discharge is described in terms of an Adaptive Integrate-to-Threshold model for a wide variety of spatio-temporal stimuli as well as steady-state stimuli. In order to reproduce the experimental results it is shown that the source of randomness in the model (e.g. the threshold) behaves like a random variable which is distributed according to a two-state Markov renewal process. In the case of stationary discharges generated by moving sinewave patterns the shape of the interspike interval distribution (which, in the Integrate-to-Threshold model, reflects the shape of the threshold distribution) changes continuously from a two-state distribution at low firing rates to a one-state distribution at high firing rates. In dynamic conditions of the discharge, generated by temporal dynamic stimuli, the experimental results show that the shape of the (demodulated) interval distribution of the discharge is determined by the highest instantaneous firing rate with an adaptation time constant of a few seconds. The physioligical origin of this intriguing behaviour remains — up till now — out of the picture.