A mathematical model for the growth of mycelial pellet populations

In liquid culture, filamentous organisms often grow in the form of pellets. Growth result in an increase in radius, whereas shear forces result in release of hyphal fragments which act as centers for further pellet growth and development. A previously published model for pellet growth of filamentous microorganisms has been examined and is found to be unstable for certain parameter values. This instability has been identified as being due to inaccuracies in estimating the numbers of fragments which seed the pellet population. A revised model has been formulated, based on similar premises, but adopting a finite element approach. This considers the population of pellets to be distributed in a range of size classes. Growth results in movement to classes of increasing pellet size, while fragments enter the smallest size class, from which they grow to form further pellets. The revised model is stable and predicts changes in the distribution of pellet sizes within a population growing in liquid batch culture. It considers pellet growth and death, with fragmentation providing new centers of growth within the pellet population, and predicts the effects of shear forces on pellet growth and size distribution. Predictions of pellet size distributions are tested using previously published data on the growth of fungal pellets and further predictions are generated which are suitable for experimental testing using cultures of filamentous fungi or actinomycetes. (c) 1995 John Wiley & Sons, Inc.     

Experimental design of a simple medium for the bioluminescence of Aliivibrio fischeri and mathematical modelling for growth estimation

Increasing numbers of studies are using Aliivibrio fischeri (A. fischeri), a marine bioluminescent bacterium as a model, however the culture medium used for its growth are complex and expensive. The objectives of this study were: (1) to evaluate the effect of yeast extract, tryptone, and NaCl to select a simple and inexpensive culture medium suitable for A. fischeri growth and bioluminescence induction; and (2) to compare the performance of mathematical models to predict the growth of A. fischeri. A fractional factorial design was performed to evaluate the effect of yeast extract, tryptone, and sodium chloride on the luminescence of A. fischeri. The result showed that sodium chloride is the most important factor, congruent with its inducer role in bioluminescence. The best medium for bioluminescence induction was selected through an optimization plot, this medium is inexpensive, and generates the same luminescence as commercial formulations. The estimation of A. fischeri growth at OD₆₀₀ measurement was statistically analyzed. All evaluated models fitted the data adequately (r²  > 0.96). The nonlinear models Gompertz, Richards and logistic provided a lower variation and a better fit of the growth estimation (r² >0.99), showing that these mathematical models can be used for the accurate growth prediction of A. fischeri.     

Mortality as a mathematical function of organic growth and diameter structure

ABSTRACT

An indirect method to determine the exponential mortality coefficient (λ) as well as the annual mortality (m) is proposed. In this method, both parameters are deduced mathematically from a model of organic growth (von Bertalanffy) and from a model of uneven-aged population structure (De Liocourt and Meyer) used with forest trees, which relates the frequency of individuals by size classes. The resultant equations allow the determination of either the instantaneous or mean value for λ or m between any two ages t ₁ and t ₂. Both concepts of mortality were used to determine the half-life (t _0.5) as a function of λ and m . The method is applied to determine the curves of mortality and the half-life of Otoba gracilipes (Myristicaceae), a tropical tree that grows in the forested wetlands of the Colombian Pacific coast.     

A mathematical model of epiphyseal development: hypothesis on the cartilage canals growth

The role of cartilage canals is to transport nutrients and biological factors that cause the appearance of the secondary ossification centre (SOC). The SOC appears in the centre of the epiphysis of long bones. The canal development is a complex interaction between mechanical and biological factors that guide its expansion into the centre of the epiphysis. This article introduces the ‘Hypothesis on the growth of cartilage canals’. Here, we have considered that the development of these canals is an essential event for the appearance of SOC. Moreover, it is also considered to be important for the transport of molecular factors (RUNX2 and MMP9) at the ends of such canals. Once the canals are merged in the centre of the epiphysis, these factors are released causing hypertrophy of adjacent cells. This RUNX2 and MMP9 release occurs due to the action of mechanical loads that supports the epiphysis. In order to test this hypothesis, we use a hybrid approach using the finite element method to simulate the mechanical stresses present in the epiphysis and the cellular automata to simulate the expansion of the canals and the hypertrophy factors pathway. By using this hybrid approach, we have obtained as a result the spatial–temporal patterns for the growth of cartilage canals and hypertrophy factors within the epiphysis. The model is in qualitative agreement with experimental results previously reported by other authors. Thus, we conclude that this model may be used as a methodological basis to present a complete mathematical model of the processes involved in epiphyseal development.     

A mathematical model of epiphyseal development: hypothesis of growth pattern of the secondary ossification centre

This paper introduces a ‘hypothesis about the growth pattern of the secondary ossification centre (SOC)’, whereby two phases are assumed. First, the formation of cartilage canals as an event essential for the development of the SOC. Second, once the canals are merged in the central zone of the epiphysis, molecular factors are released (primarily Runx2 and MMP9) spreading and causing hypertrophy of adjacent cells. In addition, there are two important molecular factors in the epiphysis: PTHrP and Ihh. The first one inhibits chondrocyte hypertrophy and the second helps the cell proliferation. Between these factors, there is negative feedback, which generates a highly localised and stable pattern over time. From a mathematical point of view, this pattern is similar to the patterns of Turing. The spread of Runx2 hypertrophies the cells from the centre to the periphery of the epiphysis until found with high levels of PTHrP to inhibit hypertrophy. This mechanism produces the epiphyseal bone-plate. Moreover, the hypertrophy is inhibited when the cells sense low shear stress and high pressure levels that maintain the articular cartilage structure. To test this hypothesis, we solve a system of coupled partial differential equations using the finite element method and we have obtained spatio-temporal patterns of the growth process of the SOC. The model is in qualitative agreement with experimental results previously reported by other authors. Thus, we conclude that this model can be used as a methodological basis to present a complete mathematical model of the whole epiphyseal development.     

A mathematical model for pH patterns in the rhizospheres of growth zones

In the classical model by Nye (1981), the main process for the change in pH across the rhizosphere is assumed to be diffusion. The classical model focuses on the non-growing part of the root and assumes that the distribution of ion fluxes along the root is spatially uniform. We consider the rhizosphere of the growth zone and take into account the root growth rate and spatially varying flux along the root surface. We present both analytical (dimensional analysis) and experimental (computational) evidence of the importance of taking into account the root growth rate. We describe a conceptual and mathematical model to analyse the pH field around the root tip over time. The model is used with published data to show that, for typical growth rates in sandy soil, the pH field becomes steady (independent of time) after 6 h. Dimensional analysis reveals that a version of the Péclet number, related to the quotient of root elongation rate and proton diffusivity, can be used to predict the extent of the rhizosphere and the time required for it to become steady. For Péclet numbers much greater than 1 (soils), the root influences soil pH for distances on the millimetre scale. In contrast, for Péclet numbers much less than one (agar, aqueous solution), the root influences substrate pH for radial distances on the scale of centimetres. We also present some evidence that agar-contact techniques to measure the soil pH may not be appropriate for measuring the millimetre-scale gradients in soil pH.     

A mathematical model to study the effects of drugs administration on tumor growth dynamics

A mathematical model for describing the cancer growth dynamics in response to anticancer agents administration in xenograft models is discussed. The model consists of a system of ordinary differential equations involving five parameters (three for describing the untreated growth and two for describing the drug action). Tumor growth in untreated animals is modelled by an exponential growth followed by a linear growth. In treated animals, tumor growth rate is decreased by an additional factor proportional to both drug concentration and proliferating cells. The mathematical analysis conducted in this paper highlights several interesting properties of this tumor growth model. It suggests also effective strategies to design in vivo experiments in animals with potential saving of time and resources. For example, the drug concentration threshold for the tumor eradication, the delay between drug administration and tumor regression, and a time index that measures the efficacy of a treatment are derived and discussed. The model has already been employed in several drug discovery projects. Its application on a data set coming from one of these projects is discussed in this paper.     

Insight into the basis of root growth in Arabidopsis thaliana provided by a simple mathematical model

Plant organ growth changes under genetic and environmental influences can be observed as altered cell proliferation and volume growth. The two aspects are mutually dependent and intricately related. For comprehensive growth analysis, it is necessary to specify the relationship quantitatively. Here, we develop a simple mathematical model for this purpose. Our model assumes that the biological activity of a given organ is proportional to the cell number of the organ and is allocated into three aspects: cell proliferation, volume growth, and organ maintenance. We analyzed the growth of primary roots of Arabidopsis thaliana (L.) Heynh. in one tetraploid and four diploid strains using this model. The analysis determined various growth parameters, such as specific cost coefficients of cell proliferation and volume growth for each strain. The results provide insight into the basis of interstrain variations and ploidy effects in root growth.     

展望数学生态学与生态模型的未来

首先简要回顾了２０世纪数学生态学发展的历史,特别是半２个世纪以来在中国的发展。然后指出了生物学的进步为数学生态学的发展提供了机遇。作者列出了当前数学生态学和生态模型研究的几个热点：⑴非线性动力学;⑵种群的时空动态：包括异质种群动态,源－汇理论以及种群对时、空变化的响应等;⑶多样性和稳定性的关系;⑷行为的动态模型;⑸基于个体的模型。最后指出,生态学中混沌现象,可能表明多年来理论生态学家寻找的种群动态     

A mathematical model of the regulation system of the secretion of glucocorticoids

We propose a mathematical model for the regulation system of the secretion of glucocorticoids and determined the coefficients in the system of ordinary differential equations. Some results are calculated which agree with the experimental results.     

A mathematical model for the leukocyte filtration process

Leukocyte filters are applied clinically to remove leukocytes from blood. In order to optimize leukocyte filters, a mathematical model to describe the leukocyte filtration process was developed by modification of a general theoretical model for depth filtration. The model presented here can be used to predict the time-dependent leukocyte filtration as a function of cell-cell interaction in the filter, filter efficiency, filter capacity, filter dimensions, and leukocyte concentration in the suspension applied to the filter. The results of different leukocyte filtration experiments previously reported in the literature could be well described by the present model. (c) 1995 John Wiley & Sons, Inc.     

Use of a simple mathematical model to predict the behavior of Escherichia coli overproducing beta-lactamase within continuous single- and two-stage reactor systems

A simple mathematical model is developed to help explain the complex population dynamics of an Escherichia coli host-plasmid expression/excretion system for beta-lactamase within single- and two-stage reactors. The model successfully integrates the individual regulatory (tac promoter induction), genetic (runaway plasmid replication), and population dynamics (culture instability) aspects of the system. The model predicts, and experiment confirms, that high-level beta-lactamase production and excretion cannot be easily maintained in single-stage reactors using the current plasmid construction. Stable target protein production and excretion is mathematically predicted, and experimentally confirmed, within two-stage reactors. The model is used to provide insight into engineering a more stable host-vector expression/excretion system for use in single-stage reactors. (c) 1993 John Wiley & Sons, Inc.     

Analysis of a time-delayed mathematical model for tumour growth with an almost periodic supply of external nutrients

In this paper, the existence, uniqueness and exponential stability of almost periodic solutions for a mathematical model of tumour growth are studied. The establishment of the model is based on the reaction–diffusion dynamics and mass conservation law and is considered with a delay in the cell proliferation process. Using a fixed-point theorem in cones, the existence and uniqueness of almost periodic solutions for different parameter values of the model is proved. Moreover, by the Gronwall inequality, sufficient conditions are established for the exponential stability of the unique almost periodic solution. Results are illustrated by computer simulations.     

A nonlinear structured population model of tumor growth with quiescence

A nonlinear structured cell population model of tumor growth is considered. The model distinguishes between two types of cells within the tumor: proliferating and quiescent. Within each class the behavior of individual cells depends on cell size, whereas the probabilities of becoming quiescent and returning to the proliferative cycle are in addition controlled by total tumor size. The asymptotic behavior of solutions of the full nonlinear model, as well as some linear special cases, is investigated using spectral theory of positive simigroup of operators. Supported in part by the National Science Foundation under Grant No. DMS-8722947     

Neural network modeling of batch cell growth pattern

The capability of neural networks in modeling batch cell growth by providing initial conditions only is tested in this study. The neural network tested is of the back-propagation-type including a newly discovered saturation-type transfer function. The simulation and prediction results of this neural network modeling will be demonstrated. (c) 1993 John Wiley & Sons, Inc.     

A simple model for periodic cyclostat growth of algae

Summary This paper provides a simple model of nutrient limited periodic cyclostat growth for algae.The basic growth function is assumed to be a time dependent variation of the empirical growth equation developed by Droop (1968). The authors also present the relations for n species' cyclostat coexistence and a stability analysis for the model growth equation.The model, although limited in some respects, agrees very well with available experimental data on Euglena gracilis. The significance of the time dependent amplitude functions developed in this study is also discussed.The term used to describe this system is the cyclostat (see Chisholm et al., 1975).