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 study of some diffusion models of population growth

The stochastic differential equations of many diffusion processes which arise in studies of population growth in random environments can be transformed, if the Stratonovich stochastic calculus is employed, to the equation of the Wiener process. If the transformation function has certain properties then the transition probability density function and quantities relating to the time to first attain a given population size can be obtained from the known results for the Wiener process. Some other random growth processes can be derived from the Ornstein-Uhlenbeck process. These transformation methods are applied to the random processes of Malthusian growth, Pearl-Verhulst logistic growth and a recent model of density independent growth due to Levins.     

On relationships between various distribution functions in balanced unicellular growth

The relationships between various size distributions in balanced exponential growth of a batch culture of microorganisms are presented. Starting from the partial differential integral equations (Eakmanet al., 1966; Fredricksonet al., 1967) derived for the growth of a microbial culture expressions are obtained for the growth rate of organisms of specific size and size range. These expressions were first obtained by Collins and Richmond (1962) by an entirely different method. Also derived are equations which link probability functions, which are basic to the growth of a microbial culture, with other size distributions that can be estimated experimentally.     

Mathematical models for optimizing tumour radiotherapy. I: A simple two compartment cell-kinetic model for the unperturbed growth of transplantable tumours

Tumour growth kinetics has been analysed on the basis of interactions between two compartments comprising the proliferating and non-proliferating cells. Starting from the differential equations of growth of the cell-populations in the two compartments and assuming the process of intercompartmental cell transfers to be linear, an analytic expression on the variation of growth-fraction with the age of the tumour is obtained. The restricted conditions on the cell-cycle time and cell-loss-rate, under which these differential equations lead to a Gompertzian growth of the tumour, are critically analysed. The formalism permits the estimation of some important cell-kinetic parameters, like growth-fraction or cell-loss-factor, from a knowledge of the tumour-growth curve, cell-cycle-time and a single measurement of the cell-loss-rate of the matured tumour, provided the tumour follows a Gompertzian growth. The validity of the model has been verified with the experimental data on 4 different transplantable murine tumour systems. Usefulness of the model has been demonstrated by making some interesting predictions regarding the radiation response of the tumours from the cell-kinetic parameters.     

Mathematical description of modifying synapses of hippocampal pyramidal neurons

The system of differential equations describing the plasticity of the hippocampal pyramidal neuron CA3, developed before, was analyzed. The system was divided into two groups according to magnitudes of the biochemical reaction constants. The first group with large values of the constants was transformed into quasi stationary algebraic equations. This allowed one to transform the system of 32 differential equations to a system containing only 4 differential equations, which can be used for modeling of learning processes in various parts of the brain.     

Supercomputer analysis of purine and pyrimidine metabolism leading to DNA synthesis

A model-system is established to analyze purine and pyrimidine metabolism leading to DNA synthesis. The principal aim is to explore the flow and regulation of terminal deoxynucleoside triophosphates (dNTPs) in various input and parametric conditions. A series of flow equations are established, which are subsequently converted to differential equations. These are programmed (Fortran) and analyzed on a Cray X-MP/48 supercomputer. The pool concentrations are presented as a function of time in conditions in which various pertinent parameters of the system are modified. The system is formulated by 100 differential equations.     

模拟青霉素发酵过程中菌体生长动态的细胞自动机模型

在青霉素发酵生产机理及其动力学微分方程模型的基础上,建立了模拟青霉素分批发酵过程中菌体生长动态的细胞自动机模型(CABGM)。CABGM采用三维细胞自动机作为菌体生长空间,采用Moore型邻域作为细胞邻域,其演化规则根据青霉素分批发酵过程中菌体生长机理和动力学微分方程模型设计。CABGM中的每一个细胞既可代表单个的青霉素产生菌,又可代表特定数量的青霉素产生菌,它具有不同的状态。对CABGM进行了统计特性的理论分析和仿真实验,理论分析和仿真实验结果均证明了CABGM能一致地复现动力学微分方程模型所描述的青霉素分批发酵菌体生长过程。最后,对所建模型在实际生产过程中的应用问题进行了分析,指出了需要进一步研究的问题。     

Supercomputer analysis of purine and pyrimidine metabolism leading to DNA synthesis

A model-system is established to analyze purine and pyrimidine metabolism leading to DNA synthesis. The principal aim is to explore the flow and regulation of terminal deoxynucleoside triophosphates (dNTPs) in various input and parametric conditions. A series of flow equations are established, which are subsequently converted to differential equations. These are programmed (Fortran) and analyzed on a Cray chi-MP/48 supercomputer. The pool concentrations are presented as a function of time in conditions in which various pertinent parameters of the system are modified. The system is formulated by 100 differential equations.     

On the Distinction Between Lag and Delay in Population Growth

The analysis and results presented in this paper provide conclusive evidence to distinguish between the delay effect and the lag as two biologically distinct phenomena. It therefore dispels the incorrect notion that delay effects represented by delay differential equations are the biological reason behind the lag phase in microorganism growth. The resulting consequence so far is that the only other reason for the lag phase is the existence of unstable stationary states. The latter are a result of accounting for the microbial metabolic mass transfer in the population growth process.     

Age-dependent models of population growth

P. H. Leslie (1945, Biometrika, 33, 183–212) (and others) introduced vector-matrix growth difference equations to model populations in which birth and death rates are age-dependent. We develop differential versions of these equations in both the unrestricted growth and logistic cases. We find that the vector logistic equations (difference and differential) are explicitly solvable in terms of solutions of the unrestricted equations, even when vital rates vary with time. These explicit solution formulas make it easy to determine the behavior of solutions as time goes on.     

The attractiveness of the Droop equations.

The Droop equations are a system of three coupled, nonlinear ordinary differential equations describing the growth of a microorganism in a chemostat. The growth rate of the organism is limited by the availability of a single nutrient. In contrast to the better known Monod equations, the nutrient is divided into external and internal cellular pools. Only the internal pool can catalyze growth. This paper proves that the Droop equations are globally stable. Based on a single combination of parameters, either the chemostat organism goes extinct or it tends to a fixed, positive concentration.     

A model of hematopoietic bone marrow apoptosis during growth factor deprivation in combination with a cytokine

Background

The process by which blood cells are formed is referred to as hematopoiesis. This process involves a complex sequence of phases that blood cells must complete. During hematopoiesis, a small fraction of cells undergo cell death. Causes of cell death are dependent upon various factors; one such factor being growth factor deprivation.

Methods

In this paper, a mathematical model of hematopoiesis during growth factor deprivation is presented. The model consists of a set of three coupled differential delay equations. Phase plane and linear stability analysis are performed in order to locate and determine stability of fixed points. Numerical simulations of the governing equations are run and provide a visual display of the behavior of the stem cell population undergoing growth factor deprivation. In addition, the effect of cytokine administration is incorporated in the model in an effort to understand how cytokine administration can offset the negative effects of apoptosis caused by growth factor deprivation.

Conclusions

The model produces qualitatively similar results to that observed during serum deprivation. The model captures apoptosis levels of cells at different time points. Additionally, it is shown that cytokine administration stabilizes the stem cell count.

     

Development of a biologically-based controlled growth and differentiation model for developmental toxicology

 A mathematical model is developed with a highly controlled birth and death process for precursor cells. This model is both biologically- and statistically-based. The controlled growth and differentiation (CGD) model limits the number of replications allowed in the development of a tissue or organ and thus, more closely reflects the presence of a true stem cell population. Leroux et al. (1996) presented a biologically-based dose-response model for developmental toxicology that was derived from a partial differential equation for the generating function. This formulation limits further expansion into more realistic models of mammalian development. The same formulae for the probability of a defect (a system of ordinary differential equations) can be derived through the Kolmogorov forward equations due to the nature of this Markov process. This modified approach is easily amenable to the expansion of more complicated models of the developmental process such as the one presented here. Comparisons between the Leroux et al. (1996) model and the controlled growth and differentiation (CGD) model as developed in this paper are also discussed. Received: 8 June 2001 / Revised version: 15 June 2002 / Published online: 26 September 2002 Keywords or phrases: Teratology – Multistate process – Cellular kinetics – Numerical simulation     

Process-based simulation library SALMO-OO for lake ecosystems. Part 2: Multi-objective parameter optimization by evolutionary algorithms

SALMO-OO represents an object-oriented simulation library for lake ecosystems that allows to determine generic model structures for certain lake categories. It is based on complex ordinary differential equations that can be assembled by alternative process equations for algal growth and grazing as well as zooplankton growth and mortality. It requires 128 constant parameters that are causally related to the metabolic, chemical and transport processes in lakes either estimated from laboratory and field experiments or adopted from the literature.An evolutionary algorithm (EA) was integrated into SALMO-OO in order to facilitate multi-objective optimization for selected parameters and to substitute them by optimum temperature and phosphate functions. The parameters were related to photosynthesis, respiration and grazing of the three algal groups diatoms, green algae and blue-green algae. The EA determined specific temperature and phosphate functions for same parameters for 3 lake categories that were validated by ecological data of six lakes from Germany and South Africa.The results of this study have demonstrated that: (1) the hybridization of ordinary differential equations by EA provide a sophisticated approach to fine-tune crucial parameters of complex ecological models, and (2) the multi-objective parameter optimization of SALMO-OO by EA has significantly improved the accuracy of simulation results for three different lake categories.     

Transport of bacteria in porous media: II. A model for convective Transport and growth

A model is presented for the coupled processes of bacterial growth and convective transport of bacteria has been modeled using a fractional flow approach. The various mechanisms of bacteria retention can be incorporated into the model through selection of an appropriate shape of the fractional flow curve. Permeability reduction due to pore plugging by bacteria was simulated using the effective medium theory. In porous media, the rates of transport and growth of bacteria, the generation of metabolic products, and the consumption of nutrients are strongly coupled processes. Consequently, the set of governing conservation equations form a set of coupled, nonlinear partial differential equations that were solved numerically. Reasonably good agreement between the model and experimental data has been obtained indicating that the physical processes incorporated in the model are adequate. The model has been used to predict the in situ transport and growth of bacteria, nutrient consumption, and metabolite production. It can be particularly useful in simulating laboratory experiments and in scaling microbial-enhanced oil recovery or bioremediation processes to the field. (c) 1994 John Wiley & Sons, Inc.     

A stochastic model of a cell population with quiescence

A cell population in which cells are allowed to enter a quiescent (nonproliferating) phase is analyzed using a stochastic approach. A general branching process is used to model the population which, under very mild conditions, exhibits balanced exponential growth. A formula is given for the asymptotic fraction of quiescent cells, and a numerical example illustrates how convergence toward the asymptotic fraction exhibits a typical oscillatory pattern. The model is compared with deterministic models based on semigroup analysis of systems of differential equations.     

A stochastic model of a cell population with quiescence

A cell population in which cells are allowed to enter a quiescent (nonproliferating) phase is analyzed using a stochastic approach. A general branching process is used to model the population which, under very mild conditions, exhibits balanced exponential growth. A formula is given for the asymptotic fraction of quiescent cells, and a numerical example illustrates how convergence toward the asymptotic fraction exhibits a typical oscillatory pattern. The model is compared with deterministic models based on semigroup analysis of systems of differential equations.     

Sex chromosomes and human growth

Studies on tooth crown size and structure of individuals with various sex chromosome anomalies and their normal male and female relatives have demonstrated differential direct effects on growth of genes on the human X and Y chromosomes. The Y chromosome promotes growth of both tooth enamel and dentin, whereas the effect of the X chromosome on tooth growth seems to be restricted to enamel formation. Enamel growth is decisively influenced by cell secretory function and dentin growth by cell proliferation. It is suggested that these differential effects of the X and Y chromosomes on growth explain the expression of sexual dimorphism in various somatic features, such as the size, shape and number of teeth, and, under the assumption of genetic pleiotropy, torus mandibularis, statural growth, and sex ratio. Future questions concern, among other topics, the Y chromosome and the mineralization process, concentric control of enamel and dentin growth, and gene expression. Received: 11 March 1997 / Accepted: 10 June 1997     

Transient growth- and product-formation kinetics of acetic acid bacteria

Rates of growth and product formation under non-stationary conditions were measured in fermentations of industrial acetic acid bacteria. A repeated-batch process, where conditions change rapidly, and a slower shift experiment in CSTR culture were examined. Significant deviations from the steady-state kinetics determined in continuous fermentations were found for cell growth as well as for the formation of acetic acid. Algebraic functions of the inhibiting acid concentration were identified to describe the rates of reaction under stationary conditions. Transient kinetics are modeled by phenomenological differential equations. The data from both the repeated-batch experiments and the CSTR shift is consistently reproduced. Measurements and simulation results are presented in phase diagrams of the reaction rates over the concentration of acetic acid. Due to the dynamic effects, which enhance the transient rates of both growth and product formation, the repeated-batch process is superior to a continuous fermentation in terms of total volumetric productivity and final acid concentration.