A mathematical model of the food-consumer system I. A case without food replenishment

A simple mathematical model was proposed to describe the dynamics of a food-consumer system. The model was based on the Logistic Theory and consisted of Eqs. (4), (5) and (6). The model was divided into the following three cases for further analyss; i) without food supply except at the initial time, ii) with continuous food supply at a constant rate, and iii) with food supply at varying rates. Only the first model was dealth with in this paper. The assumptions of the model 1 are that a definite amount of food is given only once at the initial time and only the feeding by animals is responsible for the decrease of food, and that the rate of decrease is proportional to the amount of animals. It is also assumed that the growth of animal population is represented by the logistic curve, and that the upper limit of the population is proportional to the amount of food at that time. For simplicity the parameters of basic differential equations are assumed to be constant throughout the time course. Analytical solutions of this non-linear model were given by Eqs. (8), (9), (10) and (11). The properties of time course of the food amount and consumer population were discussed from the mathematical and biological points of view. The method of the estimation of the three constants λ,b, and c from the experimental data was also suggested. Since we had no available data for animal populations, we applied the model, regarding reserve substance as x and new plant body as y, to the data of the initial growth of Azuki bean plant in the dark. This model is very simple, but it may be useful for analyzing the behavior of food-consumer system. And it may give some clue to the analysis of the more complex systems.     

Linking genes to communities and ecosystems: Daphnia as an ecogenomic model

How do genetic variation and evolutionary change in critical species affect the composition and functioning of populations, communities and ecosystems? Illuminating the links in the causal chain from genes up to ecosystems is a particularly exciting prospect now that the feedbacks between ecological and evolutionary changes are known to be bidirectional. Yet to fully explore phenomena that span multiple levels of the biological hierarchy requires model organisms and systems that feature a comprehensive triad of strong ecological interactions in nature, experimental tractability in diverse contexts and accessibility to modern genomic tools. The water flea Daphnia satisfies these criteria, and genomic approaches capitalizing on the pivotal role Daphnia plays in the functioning of pelagic freshwater food webs will enable investigations of eco-evolutionary dynamics in unprecedented detail. Because its ecology is profoundly influenced by both genetic polymorphism and phenotypic plasticity, Daphnia represents a model system with tremendous potential for developing a mechanistic understanding of the relationship between traits at the genetic, organismal and population levels, and consequences for community and ecosystem dynamics. Here, we highlight the combination of traits and ecological interactions that make Daphnia a definitive model system, focusing on the additional power and capabilities enabled by recent molecular and genomic advances.     

鸭绿江口及邻近海域生物群落的胁迫响应

利用ABC曲线法结合粒径谱理论对鸭绿江口近岸海域的生物群落特征、优势种演替、多样性水平、稳定性状况等进行了分析,并尝试构建了鸭绿江口近岸海域生物群落稳定性的评估模型.研究结果表明,鸭绿江口滨海湿地生物群落结构相对简单,优势种演替明显,受外界干扰较大.春、夏季浮游植物以r-对策者为主,群落数量偏离平衡点;秋季以r-对策者为主的微小型浮游植物逐渐被中大型k-对策者演替,群落完成反馈调节,恢复到平衡点.浮游动物由于较高的能量供应其幼体全年一直处于较高优势水平,但生态效率转换相对较慢.潮间带底栖生物群落结构相对简单,优势种演替明显,系统具有低多样性低密度的特点,群落在夏季受外界扰动较大.游泳生物群落主要以小型个体为主,已处于极不稳定状态.构建的群落稳定性评估模型测算结果与上述结论基本一致,这也验证了该评估模型具有一定的有效性与实用性.综合以上分析结果,说明鸭绿江口近岸海域生物群落稳定性较差,受外界干扰比较严重.     

Mathematical Models of the Kinetics of the Cultivation of Microorganisms

Various equations of mathematical models for the kinetics of the development of various biological processes were obtained on the basis of the generalized differential equation of biomass growth. Aerobic periodic cultivation of the yeast Saccharomyces cerevisiae was carried out to provide a comparative evaluation of advantages and disadvantages of four types of mathematical models. It is shown that the exponential model is a particular solution to the generalized differential equation. The developed mathematical model can be used to predict the course of biological processes in time and can serve as a tool for a computational experiment in order to clarify the dependence of the rate of a biological process on changes in certain parameters that affect the development of cells.     

A dynamic model of the receptive field of L-cells in the carp retina

A dynamic model of the receptive field of L2-cells in the carp retina is developed by using our experimental results on the basis of physiological and morphological evidences. Linear spatial summation is assumed in the model for the interactions among L2-cells. Linear forward and feedback loops are also assumed for the interactions between L2-cells and cones. The model has dynamic properties similar to the ones of the receptive field of L2-cells: L2-cells respond faster as the size of a light spot is enlarged and the L2-cells nearer to the center of the light spot respond faster. It is suggested that the faster responding properties of L2-cells are due to the feedback action.     

A Two-Dimensional Simulation Model of the Bicoid Gradient in Drosophila

Background

Bicoid (Bcd) is a Drosophila morphogenetic protein responsible for patterning the anterior structures in embryos. Recent experimental studies have revealed important insights into the behavior of this morphogen gradient, making it necessary to develop a model that can recapitulate the biological features of the system, including its dynamic and scaling properties.

Methodology/Principal Findings

We present a biologically realistic 2-D model of the dynamics of the Bcd gradient in Drosophila embryos. This model is based on equilibrium binding of Bcd molecules to non-specific, low affinity DNA sites throughout the Drosophila genome. It considers both the diffusion media within which the Bcd gradient is formed and the dynamic and other relevant properties of bcd mRNA from which Bcd protein is produced. Our model recapitulates key features of the Bcd protein gradient observed experimentally, including its scaling properties and the stability of its nuclear concentrations during development. Our simulation model also allows us to evaluate the effects of other biological activities on Bcd gradient formation, including the dynamic redistribution of bcd mRNA in early embryos. Our simulation results suggest that, in our model, Bcd protein diffusion is important for the formation of an exponential gradient in embryos.

Conclusions/Significance

The 2-D model described in this report is a simple and versatile simulation procedure, providing a quantitative evaluation of the Bcd gradient system. Our results suggest an important role of Bcd binding to non-specific, low-affinity DNA sites in proper formation of the Bcd gradient in our model. They demonstrate that highly complex biological systems can be effectively modeled with relatively few parameters.     

Capturing Intracellular pH Dynamics by Coupling Its Molecular Mechanisms within a Fully Tractable Mathematical Model

We describe the construction of a fully tractable mathematical model for intracellular pH. This work is based on coupling the kinetic equations depicting the molecular mechanisms for pumps, transporters and chemical reactions, which determine this parameter in eukaryotic cells. Thus, our system also calculates the membrane potential and the cytosolic ionic composition. Such a model required the development of a novel algebraic method that couples differential equations for slow relaxation processes to steady-state equations for fast chemical reactions. Compared to classical heuristic approaches based on fitted curves and ad hoc constants, this yields significant improvements. This model is mathematically self-consistent and allows for the first time to establish analytical solutions for steady-state pH and a reduced differential equation for pH regulation. Because of its modular structure, it can integrate any additional mechanism that will directly or indirectly affect pH. In addition, it provides mathematical clarifications for widely observed biological phenomena such as overshooting in regulatory loops. Finally, instead of including a limited set of experimental results to fit our model, we show examples of numerical calculations that are extremely consistent with the wide body of intracellular pH experimental measurements gathered by different groups in many different cellular systems.     

An investigation into the statistical properties of TB episodes in a South African community with high HIV prevalence

Continuous differential equations are often applied to small populations with little time spent on understanding uncertainty brought about by small-population effects. Despite large numbers of individuals being latently infected with Mycobacterium tuberculosis (TB), progression from latent infection to observable disease is a relatively rare event. For small communities, this means case counts are subject to stochasticity, and deterministic models may not be appropriate tools for interpreting transmission trends. Furthermore, the nonlinear nature of the underlying dynamics means that fluctuations are autocorrelated, which can invalidate standard statistical analyses which assume independent fluctuations.Here we extend recent work using a system of differential equations to study the HIV-TB epidemic in Masiphumelele, a community near Cape Town in South Africa [Bacaër, et al., J. Mol. Biol. 57(4), 557-593] by studying the statistical properties of active TB events. We apply van Kampen's system-size (or population-size) expansion technique to obtain an approximation to a master equation describing the dynamics. We use the resulting Fokker-Planck equation and point-process theory to derive two-time correlation functions for active TB events. This method can be used to gain insight into the temporal aspect of cluster identification, which currently relies on DNA classification only.     

An integro-partial differential equation for modeling biofluids flow in fractured biomaterials

A novel mathematical model in the framework of a nonlinear integro-partial differential equation governing biofluids flow in fractured biomaterials is proposed, solved, verified, and evaluated. A semi-analytical solution is derived for the equation, verified by a mass-lumped Galerkin finite element method (FEM), and calibrated with two in vitro experimental datasets. The solution process uses separation of variables and results in explicit expression involving complete and incomplete beta functions. The proposed semi-analytical model shows reasonable agreements with the finite element simulator as well as with two in vitro experimental time series and can be successfully used to simulate biofluids (e.g. water, blood, oil, etc.) flow in natural and synthetic porous biomaterials.     

Construction of a large scale integrated map of macrophage pathogen recognition and effector systems

Background

Mathematical models provide abstract representations of the information gained from experimental observations on the structure and function of a particular biological system. Conferring a predictive character on a given mathematical formulation often relies on determining a number of non-measurable parameters that largely condition the model's response. These parameters can be identified by fitting the model to experimental data. However, this fit can only be accomplished when identifiability can be guaranteed.

Results

We propose a novel iterative identification procedure for detecting and dealing with the lack of identifiability. The procedure involves the following steps: 1) performing a structural identifiability analysis to detect identifiable parameters; 2) globally ranking the parameters to assist in the selection of the most relevant parameters; 3) calibrating the model using global optimization methods; 4) conducting a practical identifiability analysis consisting of two (a priori and a posteriori) phases aimed at evaluating the quality of given experimental designs and of the parameter estimates, respectively and 5) optimal experimental design so as to compute the scheme of experiments that maximizes the quality and quantity of information for fitting the model.

Conclusions

The presented procedure was used to iteratively identify a mathematical model that describes the NF-κB regulatory module involving several unknown parameters. We demonstrated the lack of identifiability of the model under typical experimental conditions and computed optimal dynamic experiments that largely improved identifiability properties.     

Effects of Charge on Osmotic Reflection Coefficients of Macromolecules in Fibrous Membranes

A model based on continuum hydrodynamics and electrostatics was developed to predict the combined effects of molecular charge and size on the osmotic reflection coefficient (σ_o) of a macromolecule in a fibrous membrane, such as a biological hydrogel. The macromolecule was represented as a sphere with a constant surface charge density, and the membrane was assumed to consist of an array of parallel fibers of like charge, also with a constant surface charge density. The flow was assumed to be parallel to the fiber axes. The effects of charge were included by computing the electrostatic free energy for a sphere interacting with an array of fibers. It was shown that this energy could be approximated using a pairwise additivity assumption. Results for σ_o were obtained for two types of negatively charged fibers, one with properties like those of glycosaminoglycan chains, and the other for thicker fibers having a range of charge densities. Using physiologically reasonable fiber spacings and charge densities, σ_o for bovine serum albumin in either type of fiber array was shown to be much larger than that for an uncharged system. Given the close correspondence between σ_o and the reflection coefficient for filtration, the results suggest that the negative charge of structures such as the endothelial surface glycocalyx is important in minimizing albumin loss from the circulation.     

Food availability affects the strength of mutualistic host–microbiota interactions in Daphnia magna

The symbiotic gut microbial community is generally known to have a strong impact on the fitness of its host. Nevertheless, it is less clear how the impact of symbiotic interactions on the hosts'' fitness varies according to environmental circumstances such as changes in the diet. This study aims to get a better understanding of host–microbiota interactions under different levels of food availability. We conducted experiments with the invertebrate, experimental model organism Daphnia magna and compared growth, survival and reproduction of conventionalized symbiotic Daphnia with germ-free individuals given varying quantities of food. Our experiments revealed that the relative importance of the microbiota for the hosts'' fitness varied according to dietary conditions. The presence of the microbiota had strong positive effects on Daphnia when food was sufficient or abundant, but had weaker effects under food limitation. Our results indicate that the microbiota can be a potentially important factor in determining host responses to changes in dietary conditions. Characterization of the host-associated microbiota further showed that Aeromonas sp. was the most prevalent taxon in the digestive tract of Daphnia.     

Differential freshwater flagellate community response to bacterial food quality with a focus on Limnohabitans bacteria

Different bacterial strains can have different value as food for heterotrophic nanoflagellates (HNF), thus modulating HNF growth and community composition. We examined the influence of prey food quality using four Limnohabitans strains, one Polynucleobacter strain and one freshwater actinobacterial strain on growth (growth rate, length of lag phase and growth efficiency) and community composition of a natural HNF community from a freshwater reservoir. Pyrosequencing of eukaryotic small subunit rRNA amplicons was used to assess time-course changes in HNF community composition. All four Limnohabitans strains and the Polynucleobacter strain yielded significant HNF community growth while the actinobacterial strain did not although it was detected in HNF food vacuoles. Notably, even within the Limnohabitans strains we found significant prey-related differences in HNF growth parameters, which could not be related only to size of the bacterial prey. Sequence data characterizing the HNF communities showed also that different bacterial prey items induced highly significant differences in community composition of flagellates. Generally, Stramenopiles dominated the communities and phylotypes closely related to Pedospumella (Chrysophyceae) were most abundant bacterivorous flagellates rapidly reacting to addition of the bacterial prey of high food quality.     

Responses of Bacterial Communities in Arable Soils in a Rice-Wheat Cropping System to Different Fertilizer Regimes and Sampling Times

Soil physicochemical properties, soil microbial biomass and bacterial community structures in a rice-wheat cropping system subjected to different fertilizer regimes were investigated in two seasons (June and October). All fertilizer regimes increased the soil microbial biomass carbon and nitrogen. Both fertilizer regime and time had a significant effect on soil physicochemical properties and bacterial community structure. The combined application of inorganic fertilizer and manure organic-inorganic fertilizer significantly enhanced the bacterial diversity in both seasons. The bacterial communities across all samples were dominated by Proteobacteria, Acidobacteria and Chloroflexi at the phylum level. Permutational multivariate analysis confirmed that both fertilizer treatment and season were significant factors in the variation of the composition of the bacterial community. Hierarchical cluster analysis based on Bray-Curtis distances further revealed that bacterial communities were separated primarily by season. The effect of fertilizer treatment is significant (P = 0.005) and accounts for 7.43% of the total variation in bacterial community. Soil nutrients (e.g., available K, total N, total P and organic matter) rather than pH showed significant correlation with the majority of abundant taxa. In conclusion, both fertilizer treatment and seasonal changes affect soil properties, microbial biomass and bacterial community structure. The application of NPK plus manure organic-inorganic fertilizer may be a sound fertilizer practice for sustainable food production.     

Biased three-dimensional cell migration and collagen matrix modification

Various tumours can be resected even for cure with complete removal. Surgical excision with a wide margin to ensure complete removal has therefore been suggested as the primary treatment for such lesions. The histological examination of the three-dimensional (3D) excision margins (3D histology) constitutes an important part of the quality control mechanisms in tumour surgery. Understanding histologically relevant properties of the constituents of the microenvironment in tumours and the circumferential portion of non-lesional tissue is therefore critical.Accompanied by the increasing availability of high performance computers in recent decades, there has been a strong movement aiming at the development of mathematical models whose implementations allow in silico simulations of biological reaction networks. Due to its relevance in numerous biological and pathological processes, there have been various attempts to model biased migration of single cells. The model introduced in this paper plays a prominent role insofar as it covers the under-represented 3D case. Moreover, it is uniformly formulated for both two and three dimensions. The velocity of each cell is characterised by a generalised Langevin equation, a stochastic differential equation, where chemotaxis as well as contact guidance are considered to simulate selected aspects of interactions between carcinoma cell groups and fibroblast-like cells.Algorithmic and numeric aspects of the developed equations are detailed in this paper and the results of the simulations are illustrated in different manners to emphasise specific features. A simple test scenario as well as a geometry based on segmentation data of a real histological slide has served for verification of the software. The resulting morphologies closely resemble that of desmoplastic stromal reaction readily identifiable in histological slides of infiltrating carcinoma, and the images can be interpreted in the context of 3D histology.     

Lake Chapo: a baseline study of a deep, oligotrophic North Patagonian lake prior to its use for hydroelectricity generation: II. Biological properties

The present paper focuses on the biological properties of Lake Chapo prior to the installation of a hydroelectric power station. Most of the biological parameters indicate that the lake is oligotrophic, i.e., it has very low values of chlorophyll a and primary production. The phytoplankton community of 73 species shows a predominance of only a few species in terms of density and biomass. The zooplankton community is poor in species, with one of the lowest densities measured in the North Patagonian lakes. The fish community is dominated by two introduced salmonid species. Comparison with other North Patagonian lakes reveals that Lake Chapo is the most oligotrophic.