Modelling logistic growth by a new diffusion process: Application to biological systems

The present paper introduces a new diffusion process for the purpose of modelling logistic-type behaviour patterns. Unlike other processes in the same context, this one verifies that its mean function is a logistic curve. In addition, its transition density can be found explicitly, which allows to analyse inference from the discrete sampling of trajectories. The main features of the process will be analysed and the maximum likelihood estimation of parameters will be carried out through discrete sampling. Regarding the numerical problems found to solve the likelihood equations, several strategies are developed for obtaining initial solutions for the usual numerical procedures. Such strategies are compared by means of a simulation example. Also, another simulation study is carried out in order to compare the estimation in this process to that developed by means of continuous sampling in the logistic diffusion model considered by Giovanis and Skiadas (1999). Finally an example is given for the growth of a microorganism culture. This example illustrates the predictive possibilities of the new process, as well as its ability to study time variables formulated as first-passage-times.     

Modeling of Branching Patterns in Plants

A major determinant of plant architecture is the arrangement of branches around the stem, known as phyllotaxis. However, the specific form of branching conditions is not known. Here we discuss this question and suggest a branching model which seems to be in agreement with biological observations. Recently, a number of models connected with the genetic network or molecular biology regulation of the processes of pattern formation appeared. Most of these models consider the plant hormone, auxin, transport and distribution in the apical meristem as the main factors for pattern formation and phyllotaxis. However, all these models do not take into consideration the whole plant morphogenesis, concentrating on the events in the shoot or root apex. On the other hand, other approaches for modeling phyllotaxis, where the whole plant is considered, usually are mostly phenomenological, and due to it, do not describe the details of plant growth and branching mechanism. In this work, we develop a mathematical model and study pattern formation of the whole, though simplified, plant organism where the main physiological factors of plant growth and development are taken into consideration. We model a growing plant as a system of intervals, which we will consider as branches. We assume that the number and location of the branches are not given a priori, but appear and grow according to certain rules, elucidated by the application of mathematical modeling. Four variables are included in our model: concentrations of the plant hormones auxin and cytokinin, proliferation and growth factor, and nutrients—we observe a wide variety of plant forms and study more specifically the involvement of each variable in the branching process. Analysis of the numerical simulations shows that the process of pattern formation in plants depends on the interaction of all these variables. While concentrations of auxin and cytokinin determine the appearance of a new bud, its growth is determined by the concentrations of nutrients and proliferation factors. Possible mechanisms of apical domination in the frame of our model are discussed.     

Limiting genotype frequencies of Y-linked genes through bisexual branching processes with blind choice

The limiting genotype growth rates and the limiting genotype frequencies of Y-linked genes are studied in a two-sex monogamous population. To this end, the evolution of the numbers of females, males, and mating units of each genotype is modelled by a multitype bisexual branching process in which it assumed that the gene has no influence on the mating process. It is deduced from this model that the average numbers of female and male descendants per mating unit of a genotype determine its growth rate, which does not depend on the behaviour of the other genotypes. Hence, the dominant genotype is found. Conditions for the simultaneous survival of genotypes to have positive probability are also investigated. Finally, the main results are illustrated by means of examples.     

An experimental study of craniofacial growth in a heterotopic rat head transplant

In order to examine whether or not facial bones are themselves able to regulate their own growth, we devised a new experimental model in which we transplanted the whole head of an infant rat to the body of an isohistogeneic adult rat by means of microvascular anastomoses. The advantage of this model is that the transplanted head has neither scars nor any moving soft tissues that could modify growth around facial bones. Using this model, we conducted a study of the nasomaxillary region that has led us to conclude that facial bones do in fact regulate their own growth. The results also suggested that facial bone sutures play a more active role in the growth process than presently suspected.     

A class of linear spectral models and analyses for the study of longitudinal data

Longitudinal data can always be represented by a time series with a deterministic trend and randomly correlated residuals, the latter of which do not usually form a stationary process. The class of linear spectral models is a basis for the exploratory analysis of these data. The theory and techniques of factor analysis provide a means by which one component of the residual series can be separated from an error series, and then partitioned into a sum of randomly scaled metameters that characterize the sample paths of the residuals. These metameters, together with linear modelling techniques, are then used to partition the nonrandom trend into a determined component, which is associated with the sample paths of the residuals, and an independent inherent component. Linear spectral models are assumption-free and represent both random and nonrandom trends with fewer terms than any other mixed-effects linear model. Data on body-weight growth of juvenile mice are used in this paper to illustrate the application of linear spectral models, through a relatively sophisticated exploratory analysis.     

Computational model of flow-tissue interactions in intussusceptive angiogenesis

Angiogenesis, the growth of vascular structures, is a complex biological process which has long puzzled scientists. Better physiological understanding of this phenomenon could result in many useful medical applications such as the development of new methods for cancer therapy. We report on the development of a simple computational model of micro-vascular structure formation in intussusceptive angiogenesis observed in vivo. The tissue is represented by a discrete set of basic structural entities and flow conditions within the resulting domain are obtained by solving the Navier-Stokes equations. The tissue is then remodelled according to the tangential shear stress while approximating advection by means of simple non-diffusive heuristics. The updated tissue geometry then becomes the input for the next remodelling step. The model, consisting of steady-state flow and a simple mechanistic tissue response, successfully predicts bifurcation formation and micro-vessel separation in a porous cellular medium. This opens new modelling possibilities in computational studies of the cellular transport involved in micro-vascular growth.     

Acclimation in dynamic models based on structural relationships

1. A number of recent tree growth models have been based on the assumption that tree structure follows certain empirical rules and carbon allocation is performed so as to maintain these rules when growth and senescence occur simultaneously. This paper introduces a method of combining structural rules with adaptive regulation in a carbon balance model.
2. The method is based on regulating gross growth but situations are analysed where increased senescence of tissue functions as an alternative means of re-establishing the balance.
3. The model does not aim to be mechanistic but merely to describe the process of regulation, or acclimation to changing environments and situations, as constrained by the carbon balance. The method incorporates parameters that describe the rate of returning to the balance after disturbance or after a change in the goal, which can be determined empirically.
4. Two models are given as examples of the use of the method, and the requirements and limitations of control in a carbon balance framework are discussed.
5. The method is best applied as a technical tool to describe variations and disturbance in balanced growth models when the variations are present but not very large. In addition, it can be used as a theoretical framework for the analysis of regulation as constrained by the conservation of mass.     

Confidence intervals for population growth rate of organisms with two-stage life histories

Although life histories can be modelled with great generality using projection matrices, for organisms with life histories that can be accurately described by a simplified set of parameters, e.g. when adult fecundity and mortality are independent of age, more accurate estimates of life table parameters and of population growth rate and its standard error can be readily obtained. Here an analytic method for calculating approximate confidence intervals for population growth rate is given for two-stage life histories that can be described by four variables representing age at first breeding, fecundity per unit time, and juvenile and adult survivorships per unit time. The method is applied to experimental data on Capitella sp. I obtained by Hansen et al., and quite good agreement is found between the analytic and bootstrap estimates of the standard error of Λ. The analytic estimates were a little conservative, probably because of the way the action of mortality was modelled. Alternative life-history models are briefly discussed, and the desirability of formulating life-history models so that the variables involved are independent of each other is stressed. Analytic estimates of Λ may be biassed if an inappropriate model is chosen or if variables are not independent and the correlations between them are not measured. To allow for these possibilities, where necessary a conservative approach should be taken to significance testing using the analytic method.     

New stochastic carcinogenesis model with covariates: an approach involving intracellular barrier mechanisms

In this paper we present a new multiple-pathway stochastic model of carcinogenesis with potential of predicting individual incidence risks on the basis of biomedical measurements. The model incorporates the concept of intracellular barrier mechanisms in which cell malignization occurs due to an inefficient operation of barrier cell mechanisms, such as antioxidant defense, repair systems, and apoptosis. Mathematical formalism combines methodological innovations of mechanistic carcinogenesis models and stochastic process models widely used in studying biodemography of aging and longevity. An advantage of the modeling approach is in the natural combining of two types of measures expressed in terms of model parameters: age-specific hazard rate and means of barrier states. Results of simulation studies allow us to conclude that the model parameters can be estimated in joint analyses of epidemiological data and newly collected data on individual biomolecular measurements of barrier states. Respective experimental designs for such measurements are suggested and discussed. An analytical solution is obtained for the simplest design when only age-specific incidence rates are observed. Detailed comparison with TSCE model reveals advantages of the approach such as the possibility to describe decline in risk at advanced ages, possibilities to describe heterogeneous system of intermediate cells, and perspectives for individual prognoses of cancer risks. Application of the results to fit the SEER data on cancer risks demonstrates a strong predictive power of the model. Further generalizations of the model, opportunities to measure barrier systems, biomedical and mathematical aspects of the new model are discussed.     

Modelling of continuous microbial cultivation taking into account the memory effects

The problem of chemostat dynamics modelling for the purpose of control is considered. The "memory" of the culture is explicitly taken into account. Two possibilities for improving the quality of the proposed modelling approaches are discussed. A general model that accounts for the culture `memory' by means of different `memory' functions in the expressions of the specific growth rate and of the specific consumption rate and a polynomial function of the substrate concentration for the yield factor is proposed. The case where the maintenance energy is taken into account is also discussed. Two modifications of the general model (w-type and S-type) are presented. A zero-order `memory' function and a i-function with delay are applied in order to describe the `memory' effects. Continuous growth of the strain Saccharomyces cerevisiae on a glucose limited medium is considered as a case study. Detailed investigations of the variety of models, derived from the general model by applying different `memory' functions and different assumptions are carried out. The results are compared with those previously reported for the same process. It is shown that a significant improvement in predicting the substrate dynamics (not accompanied by any decrease in the quality of the model with respect to the biomass concentration) could be achieved, involving a first- or second-order polynomial function for the yield factor. It is also shown that the quality of the model mainly depends on the way that `memory' function is incorporated. The detailed investigations give priority to the w-type models. In this case past values of both biomass and substrate variables are considered. The time delay models with pure (constant) delay and those which account for the culture `memory' by zero-order `memory' function (adaptability parameter) are compared with respect to their utilization for the purpose of model-based control.     

面向生态文明的超循环经济:理论、模型与实例

在全球气候变化背景下,研究适应生态文明要求的新的经济运行模式——超循环经济的基本原理、结构模型、运行机理和实际应用。首先,运用系统结构分析方法梳理不同时期经济运行模式的演进过程:由"从摇篮到产品"的粗放经济,到"从摇篮到坟墓"的末端治理经济,再到"从摇篮到摇篮"的循环经济。延续这一绿色发展趋势,根据文明演化的共生理论和艾根创立的超循环理论,提出超循环经济的理论构想,并描绘"从孕育到孕育"的超循环经济的概念结构。其次,将超循环经济思想应用于林纸拓展系统(EFPS)。在分析中国造纸工业的发展现状和瓶颈及其与林业和生态环境相互关系的基础上,按照资源链、生态链和价值链(简称"三链",或3C)逐层拓展的逻辑顺序,研究EFPS超循环经济的系列结构模型。具体来说,依次建立起各层次的超循环结构模型——制浆造纸系统的资源链核心层模型、供应链系统的资源链拓展层模型、生态环境系统的生态链拓展层模型,以及社会经济系统的价值链拓展层模型。然后将各层次的结构模型综合起来,形成EFPS超循环经济的多重拓展-嵌套整体模型。该模型既能展示EFPS超循环经济系统的全貌,又包含其各层次的系统结构。因此既能为各级政府在制定国家和地区的循环经济总体发展规划时提供参考,又能为制造企业、营林组织、供应链、行业协会等各类经济主体的绿色经营决策提供支撑。最后,在上述实证研究的基础上进行理论提升。一方面提出超循环经济的5R原则:减量化、再循环、再利用、再分配和再培育。另一方面基于5R原则和3C循环链,构建5R-3C理论模型,并研究其5R-3C共生运行机理。研究表明,在超循环经济模式下,产业与生态系统可以实现互利共生的良性循环。以上研究成果的主要创新之处在于:在原理上,揭示了面向生态文明的超循环经济的概念结构、本质属性和5R原则;在机理上,创立了超循环经济的5R-3C模型及其共生运行机理;在实施上,以林纸拓展系统为例,为超循环经济理论的"落地生根"和推广应用构建了具体的多重拓展-嵌套模型。     

Evaluation of kinetic models for industrial acetic fermentation: proposal of a new model optimized by genetic algorithms

The most important kinetic models developed for acetic fermentation were evaluated to study their ability to explain the behavior of the industrial process of acetification. Each model was introduced into a simulation environment capable of replicating the conditions of the industrial plant. In this paper, it is proven that these models are not suitable to predict the evolution of the industrial fermentation by the comparison of the simulation results with an average sequence calculated from the industrial data. Therefore, a new kinetic model for the industrial acetic fermentation was developed. The kinetic parameters of the model were optimized by a specifically designed genetic algorithm. Only the representative sequence of industrial concentrations of acetic acid was required. The main novelty of the algorithm is the four-composed desirability function that works properly as the response to maximize. The new model developed is capable of explaining the behavior of the industrial process. The predictive ability of the model has been compared with that of the other models studied.     

Dimensional rearrangement of rod-shaped bacteria following nutritional shift-up. II. Experiments with Escherichia coliBr

The dimensions of Escherichia coli$\text{B}\text{r}$ (strain H266) in transition between two states of balanced growth, were determined from electron micrographs of fixed cells by sampling the culture at various times following nutritional shift-up from a doubling time of 72 min to one of 24 min. Mean cell length rises immediately and overshoots its final steady-state value, cell diameter increases monotonically; both approach their asymptotic levels only after several hours.The results are compared with the dimensions predicted by each of two models of cell growth and morphogenesis in rod-shaped bacteria. The first attributes cell elongation to circular zones that double in number at a particular time during the cell cycle and which act at rates proportional to the growth rate; the second is similar, except that it considers surface growth rather than length extension as the active process, length being determined passively. Two possibilities are examined, that the zonal growth rate adjusts immediately to the new growth conditions, and that it does so gradually.The experimental data appear consistent with the gradual response version of the surface growth model.     

Exclusion processes on a growing domain

A discrete model provides a useful framework for experimentalists to understand the interactions between growing tissues and other biological mechanisms. A cellular automata (CA) model with domain growth, cell motility and cell proliferation, based on cellular exclusion processes, is developed here. Average densities can be defined from the CA model and a continuum representation can be determined. The domain growth mechanism in the CA model gives rise to a Fokker-Planck equation in the corresponding continuum model, with a diffusive and a convective term. Deterministic continuum models derived from conservation laws do not include this diffusive term. The new diffusive term arises because of the stochasticity inherited from the CA mechanism for domain growth. We extend the models to multiple species and investigate the influence of the flux terms arising from the exclusion processes. The averaged CA agent densities are well approximated by the solution of nonlinear advection-diffusion equations, provided that the relative size of the proliferation processes to the diffusion processes is sufficiently small. This dual approach provides an understanding of the microscopic and macroscopic scales in a developmental process.     

Variational learning for Generalized Associative Functional Networks in modeling dynamic process of plant growth

This paper presents a new statistical techniques — Bayesian Generalized Associative Functional Networks (GAFN), to model the dynamical plant growth process of greenhouse crops. GAFNs are able to incorporate the domain knowledge and data to model complex ecosystem. By use of the functional networks and Bayesian framework, the prior knowledge can be naturally embedded into the model, and the functional relationship between inputs and outputs can be learned during the training process. Our main interest is focused on the Generalized Associative Functional Networks (GAFNs), which are appropriate to model multiple variable processes. Three main advantages are obtained through the applications of Bayesian GAFN methods to modeling dynamic process of plant growth. Firstly, this approach provides a powerful tool for revealing some useful relationships between the greenhouse environmental factors and the plant growth parameters. Secondly, Bayesian GAFN can model Multiple-Input Multiple-Output (MIMO) systems from the given data, and presents a good generalization capability from the final single model for successfully fitting all 12 data sets over 5-year field experiments. Thirdly, the Bayesian GAFN method can also play as an optimization tool to estimate the interested parameter in the agro-ecosystem. In this work, two algorithms are proposed for the statistical inference of parameters in GAFNs. Both of them are based on the variational inference, also called variational Bayes (VB) techniques, which may provide probabilistic interpretations for the built models. VB-based learning methods are able to yield estimations of the full posterior probability of model parameters. Synthetic and real-world examples are implemented to confirm the validity of the proposed methods.     

Plant Growth Analysis: a Program for the Fitting of Lengthy Series of Data by the Method of B-splines*

A method is presented for fitting curves to lengthy and/or complicatedseries of data on plant growth. A computer program which derivesthe usual plant growth analytical quantities, and their errors,from these fitted curves is also described and offered for circulation.The fitted curves are splined cubic polynomial exponentials.Examples of their application are given, employing both realand artificial data. In any set of data the number of splines,and the position of the knots joining them, may be determinedeither by objective statistical tests or by the experimenterhimself, who thus retains a considerable degree of control overthe process but can call on the assistance of the program ifrequired. The value of this method is considered in relationto other curve-fitting approaches to plant growth analysis andit is concluded that, provided sufficient primary data are available,the method is free from many of the problems which beset earlierwork in this field, and also provides new possibilities of itsown. growth curves, approximating functions, empirical models, regression analysis, growth analysis     

Inferring Metabolic States in Uncharacterized Environments Using Gene-Expression Measurements

The large size of metabolic networks entails an overwhelming multiplicity in the possible steady-state flux distributions that are compatible with stoichiometric constraints. This space of possibilities is largest in the frequent situation where the nutrients available to the cells are unknown. These two factors: network size and lack of knowledge of nutrient availability, challenge the identification of the actual metabolic state of living cells among the myriad possibilities. Here we address this challenge by developing a method that integrates gene-expression measurements with genome-scale models of metabolism as a means of inferring metabolic states. Our method explores the space of alternative flux distributions that maximize the agreement between gene expression and metabolic fluxes, and thereby identifies reactions that are likely to be active in the culture from which the gene-expression measurements were taken. These active reactions are used to build environment-specific metabolic models and to predict actual metabolic states. We applied our method to model the metabolic states of Saccharomyces cerevisiae growing in rich media supplemented with either glucose or ethanol as the main energy source. The resulting models comprise about 50% of the reactions in the original model, and predict environment-specific essential genes with high sensitivity. By minimizing the sum of fluxes while forcing our predicted active reactions to carry flux, we predicted the metabolic states of these yeast cultures that are in large agreement with what is known about yeast physiology. Most notably, our method predicts the Crabtree effect in yeast cells growing in excess glucose, a long-known phenomenon that could not have been predicted by traditional constraint-based modeling approaches. Our method is of immediate practical relevance for medical and industrial applications, such as the identification of novel drug targets, and the development of biotechnological processes that use complex, largely uncharacterized media, such as biofuel production.