Deterministic Three-Half-Order Kinetic Model for Microbial Degradation of Added Carbon Substrates in Soil

The kinetics of mineralization of carbonaceous substrates has been explained by a deterministic model which is applicable to either growth or nongrowth conditions in soil. The mixed-order nature of the model does not require a priori decisions about reaction order, discontinuity period of lag or stationary phase, or correction for endogenous mineralization rates. The integrated equation is simpler than the integrated form of the Monod equation because of the following: (i) only two, rather than four, interdependent constants have to be determined by nonlinear regression analysis, (ii) substrate or product formation can be expressed explicitly as a function of time, (iii) biomass concentration does not have to be known, and (iv) the required initial estimate for the nonlinear regression analysis can be easily obtained from a linearized form rather than from an interval estimate of a differential equation. ¹⁴CO₂ evolution data from soil have been fitted to the model equation. All data except those from irradiated soil gave better fits by residual sum of squares (RSS) by assuming growth in soil was linear (RSS = 0.71) as opposed to exponential (RSS = 2.87). The underlying reasons for growth (exponential versus linear), no growth, and relative degradation rates of substrates are consistent with the basic mechanisms from which the model is derived.     

Modeling of heat and mass transfer for solid state fermentation process in tray bioreactor

A mathematical model is developed to simulate oxygen consumption, heat generation and cell growth in solid state fermentation (SSF). The fungal growth on the solid substrate particles results in the increase of the cell film thickness around the particles. The model incorporates this increase in the biofilm size which leads to decrease in the porosity of the substrate bed and diffusivity of oxygen in the bed. The model also takes into account the effect of steric hindrance limitations in SSF. The growth of cells around single particle and resulting expansion of biofilm around the particle is analyzed for simplified zero and first order oxygen consumption kinetics. Under conditions of zero order kinetics, the model predicts upper limit on cell density. The model simulations for packed bed of solid particles in tray bioreactor show distinct limitations on growth due to simultaneous heat and mass transport phenomena accompanying solid state fermentation process. The extent of limitation due to heat and/or mass transport phenomena is analyzed during different stages of fermentation. It is expected that the model will lead to better understanding of the transport processes in SSF, and therefore, will assist in optimal design of bioreactors for SSF.     

Biosorption Processes for Natural and Waste Water Treatment – Part II: Experimental Studies and Theoretical Model of a Biosorption Fixed Bed

A model has been developed for fixed‐bed biosorption performance, i.e. combined action of adsorption of organic water contaminants and their biological destruction in a column. The model contains an adsorption isotherm of the Freundlich type, adsorption kinetics by an overall film mass transfer (Glueckauf equation), maximum bacterial growth,and biological aerobic destruction (Monod model) of the organics by exoenzymes. Bacteria can not penetrate into the pores of the adsorbent. The model was tested using the system aqueous solution of aniline/Pseudomonas putida/Polysorb 40/100. Breakthrough curves in shorter columns have been measured and a velocity‐dependent steady‐state exit concentration was achieved. These curves could be simulated with sufficient accuracy on the basis of isotherm data, mass transfer coefficients and values of biological growth and destruction activity estimated from independent measurements.     

Modelling inert gas exchange in tissue and mixed-venous blood return to the lungs

Inert gas exchange in tissue has been almost exclusively modelled by using an ordinary differential equation. The mathematical model that is used to derive this ordinary differential equation assumes that the partial pressure of an inert gas (which is proportional to the content of that gas) is a function only of time. This mathematical model does not allow for spatial variations in inert gas partial pressure. This model is also dependent only on the ratio of blood flow to tissue volume, and so does not take account of the shape of the body compartment or of the density of the capillaries that supply blood to this tissue. The partial pressure of a given inert gas in mixed-venous blood flowing back to the lungs is calculated from this ordinary differential equation. In this study, we write down the partial differential equations that allow for spatial as well as temporal variations in inert gas partial pressure in tissue. We then solve these partial differential equations and compare them to the solution of the ordinary differential equations described above. It is found that the solution of the ordinary differential equation is very different from the solution of the partial differential equation, and so the ordinary differential equation should not be used if an accurate calculation of inert gas transport to tissue is required. Further, the solution of the PDE is dependent on the shape of the body compartment and on the density of the capillaries that supply blood to this tissue. As a result, techniques that are based on the ordinary differential equation to calculate the mixed-venous blood partial pressure may be in error.     

Importance of scaling exponents and other parameters in growth mechanism: an analytical approach

The growth process of a living organism is studied with the help of a mathematical model where a part of the surplus power is assumed to be used for growth. In the present study, the basic mathematical framework of the growth process is based on a pioneering theory proposed by von Bertalanffy and his work is the main intellectual driving force behind the present analysis. Considering the existence of an optimum size for which the surplus power becomes maximum, it has been found that the scaling exponent for the intake rate must be smaller than the exponent for the metabolic cost. A relationship among the empirical constants in allometric scaling has also been established on the basis of the fact that an organism never ceases to generate surplus energy. The growth process is found to continue forever, although with a decreasing rate. Beyond the optimum point the percentage of shortfall in energy has been calculated and its dependence on scaling exponents has been determined. The dependence of optimum mass on the empirical constants has been shown graphically. The functional dependence of mass variation on time has been obtained by solving a differential equation based on the concept of surplus energy. The dependence of the growth process on scaling exponent and empirical constants has been shown graphically.     

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。     

Laccase production from oil palm industry solid waste: Statistical optimization of selected process parameters

Oil palm frond parenchyma tissue was used as a solid substrate for the production of laccase via solid‐state fermentation using the white rot fungus Pycnoporus sanguineus. With a rectangular aluminium tray as solid‐state fermentation bioreactor, process parameters such as bed height, moisture and supplemented nitrogen (as urea solution) levels were studied and optimized using a statistical design of experiment. The moisture level exerted a significant effect on the process. The interaction effect observed between bed height and supplemented nitrogen level suggested that uniform distribution of supplemented nitrogen into the substrate bed was important. The proposed regression model sufficiently predicted the process response over the experimental range tested. The optimum parameter combination for laccase production was a 3‐cm bed height, 72% w/w moisture and 0.21% w/v supplemented nitrogen. Laccase productivity remained constant when the tray size was increased from 1.4 to 3.4‐fold.     

Advances in modeling microbial growth

Summary A mathematical model for bacterial growth, survival and death has been developed. This equation has been applied to a large set of data obtained withYersinia enterocolitica to produce a predictive model. Favorable comparisons were obtained between predictions from the model, primarily as estimates of the population size as affected by local conditions, and data from an experiment with an inoculated food.     

pH gradients in immobilized amidases and their influence on rates and yields of beta-lactam hydrolysis

The pH gradients developing within immobilized biocatalysts during hydrolysis of penicillin G and glutaryl-7-aminocephalosporanic acid have been estimated both theoretically and experimentally. For the latter a fluorimetric method for the direct measurement of the average pH value within the carrier during reaction has been developed using the pH-dependent fluorescence intensity of an enzyme-bound fluorophore determined with a fiber bundle. The theoretical calculations were based on a model for the hydrolysis with immobilized enzymes using a kinetic expression with five pH-dependent, measurable kinetic and equilibrium constants. The transport reaction differential equation which considers the laminar boundary layer has been solved numerically for the key component. The calculated values agreed well with the experimental data. Under the typical reaction conditions of penicillin G hydrolysis the average pH value in the carrier was 1 and 2.5 pH units below the bulk pH (=8) with and without buffer, respectively. The corresponding changes for the hydrolysis of glutaryl-7-aminocephalosporanic acid at bulk pH 8 in the presence of buffer was 0.5. This demonstrates the existence of considerable pH gradients in carriers during hydrolytic reactions, even in buffered systems with negligible mass transfer resistance. The low pH value causes suboptimal reaction rates, reduced equilibrium conversion, and reduced enzyme stability. These pH gradients can be minimised by using buffers with pK values approximately equal to the bulk pH used for the hydrolysis. The prediction quality of the model has been tested applying it to fixed bed reactor design. The reduction in rate and yield due to concentration and pH gradients can be overcome with simple measures such as high initial pH value and pH adjustments in segmented or recycling fixed bed reactors. Thus, enzymatic conversions with high yield and high operational effectiveness are achieved.     

Hydrodynamic study of biogranules obtained from an anaerobic hybrid reactor

The bed expansion characteristics of a fluidized bed containing bacterial granules have been studied. These biogranules were obtained from an anaerobic hybrid reactor, which uses biogranules (without carrier particle) in fluidized condition. The settling velocity study of biogranules has shown that the drag coefficient of biogranule is greater than that of the rigid particle at the same Reynolds number. A new correlation based on this finding has been developed. The bed expansion study has demonstrated that a linear relationship exists between the natural logarithm of bed porosity and the natural logarithm of upflow superficial liquid velocity for the bed containing either a particular fraction of biogranule size or biogranules with wide size distribution. For a fluidized bed having a particular granule size, the bed porosity, and liquid superficial velocity could be related by the classic equation suggested by Richardson and Zaki (1954). The characteristic parameter of this correlation, the slope of the line n, has been related with Reynolds number. The intercept of the line gave a smaller value than the unhindered settling velocity of the particle. For fluidized bed having wide size distribution, the characteristic parameter n could not be related to Reynolds number. But the correlation suggested for single biogranule size has been found to predict n value with an average error of 2.3%.     

A mathematical model for the motion of mechanoreceptor hairs in fluid environments.

A differential equation has been derived for the motion of the mechanosensory hairs of animals when they are stimulated by the motion of their fluid environment. Specific solutions of the equation are obtained for three states of fluid flow including steady-state sinusoidal oscillations. The model is specifically applied to crayfish sensilla in an aqueous medium, but the assumptions of the model are also shown to be valid in air for the sensory hairs of insects. The calculations are consistent with available experimental data.     

Diffusion along intercellular spaces: An analysis of a transient situation

We present an analysis of the diffusion of a tracer in a model of a cell-intercellular space system. The problem reduces to the resolution of a system of a linear partial differential equation and of a linear integral differential equation. The mathematical results have been obtained in terms of their Laplace transforms, which have been inverted by a numerical procedure for some parameter values. The importance of considering gradients of concentrations in intercellular spaces instead of lumping them with the external mediums has been discussed together with the possibility of extending Ussing's relation to transient cases, in order to detect active transports. Some possible implementations of the model to take into account more general situations have been considered.     

An alternative formulation for a delayed logistic equation

We derive an alternative expression for a delayed logistic equation, assuming that the rate of change of the population depends on three components: growth, death, and intraspecific competition, with the delay in the growth component. In our formulation, we incorporate the delay in the growth term in a manner consistent with the rate of instantaneous decline in the population given by the model. We provide a complete global analysis, showing that, unlike the dynamics of the classical logistic delay differential equation (DDE) model, no sustained oscillations are possible. Just as for the classical logistic ordinary differential equation (ODE) growth model, all solutions approach a globally asymptotically stable equilibrium. However, unlike both the logistic ODE and DDE growth models, the value of this equilibrium depends on all of the parameters, including the delay, and there is a threshold that determines whether the population survives or dies out. In particular, if the delay is too long, the population dies out. When the population survives, i.e., the attracting equilibrium has a positive value, we explore how this value depends on the parameters. When this value is positive, solutions of our DDE model seem to be well approximated by solutions of the logistic ODE growth model with this carrying capacity and an appropriate choice for the intrinsic growth rate that is independent of the initial conditions.     

Game dynamics in mendelian populations

A game theoretical model for the evolution of strategies in animal conflicts is considered, using methods from dynamical systems and population genetics. It is shown that the Hardy-Weinbergequilibrium is readily approached. The differential equation for the gene frequencies is more complicated than that which has been studied previously in the corresponding asexual case.This work has been supported financially by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, Projekt No. 3502     

A Model for Growth and Self-thinning in Even-aged Monocultures of Plants

A theoretical model is derived from simple postulates to describethe rates of growth and mortality of plants in populations ofdifferent densities. The growth rate is described by a modificationof the logistic growth differential equation in which the increasein weight of an individual plant depends on its area, s_i ratherthan on its weight. The effective area for growth of a plantis reduced by an empirical function, f(s_i) with two terms: oneterm expresses the constraint imposed upon the increasing totalarea of plants by the limited physical area of the plot; theother term allows for a competitive advantage or disadvantagefor plants of varying sizes. Depending on the value of the parametercontrolling the relative competitive advantage term, intrinsicvariability between plants can be amplified or suppressed. Anindividual plant dies if the f(s_i) results in a negative growthrate for that plant. Computer simulations of the growth andsurvival of plants at different population densities were run.The results exhibit characteristics that appear realistic uponcomparison with published data: a survival of the fittest occurringduring thinning; a line of slope close to –3/2 boundingthe graphs of log weight versus log density; and the occurrenceof bimodality, associated with subsequent mortality, on frequencydistribution of log weight. computer logistic model, growth differential equation, density-effect, competition, mortality, self-thinning     

A nonlinear,second-order population model

Delay differential, difference, and partial differential equation models are being used more extensively to explain single-species population oscillations and limit cycle behavior. Ordinary differential equation (ODE) models have been largely ignored. This is because first-order ODE models are inherently monotonic. Certainly this is not usual population behavior in the real world. If it is assumed that the per capita growth rate of a population changes over time as a result of regulating factors impinging on it, then a more realistic biological model results. The model translates into a second-order nonlinear ODE. Such a model can exhibit oscillatory and limit cycle as well as monotonic solutions, i.e., behavior for which non-ODE models have been used to explain. Although first-order ODE models are gross simplifications of real phenomena, ODE models in general should not be disregarded as important analytical tools.     

Bionomics dynamic model of a class of competition systems ☆

Differential equation problem is an important research topic in the international academia. In accordance with certain ecological phenomena, previous research was conducted based on simple observational and statistical data. But this approach does not effectively study the essence of the ecological phenomena. Recently, one dynamic approach has been proposed for the study of ecology in the international academia. According to this approach, first of all, the ecology is reduced to the differential equation model which represents the essential phenomenon, and then the dynamic law and rules of mathematics and biology will be studied. Currently, an extensive research is conducted on the differential equation problem. This paper primarily explores a type of competitive ecological model, which is a system of differential equation with infinite integral. we first study the existence of positive periodic solution to this model, and then present sufficient conditions for the global attractivity of positive periodic solutions.     

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.     

Curvature-Driven Pore Growth in Charged Membranes during Charge-Pulse and Voltage-Clamp Experiments

We find that curvature-driven growth of pores in electrically charged membranes correctly reproduces charge-pulse experiments. Our model, consisting of a Langevin equation for the time dependence of the pore radius coupled to an ordinary differential equation for the number of pores, captures the statistics of the pore population and its effect on the membrane conductance. The calculated pore radius is a linear, and not an exponential, function of time, as observed experimentally. Two other important features of charge-pulse experiments are recovered: pores reseal for low and high voltages but grow irreversibly for intermediate values of the voltage. Our set of coupled ordinary differential equations is equivalent to the partial differential equation used previously to study pore dynamics, but permits the study of longer timescales necessary for the simulations of voltage-clamp experiments. An effective phase diagram for such experiments is obtained.