Kinetics of microbial cell growth

As a rate equation of microbial cell growth, the Monod equation is widely used. However, this equation cannot fully correspond to real courses of microbial cell growth in many batch cultivations. Especially, predicted values based on this equation do not agree with observed values in many continuous cultivations. In this paper, which introduces new concepts of critical concentration and coefficient of consumption activity, the growth rate equation which corresponds to the whole period including lag period is newly derived and characteristics of microbial cell growth in batch cultivation are clarified. Further, applying the new rate equation to continuous cultivation, a general equation with which to calculate cell concentration is derived and characteristics of microbial cell growth in continuous cultivation are clarified. The calculated values of cell concentration based on the new theory showed quite good agreement with the observed values in both batch and continuous cultivation.     

广义Schumacher模型的改进及其应用

通过对前人提出的生长方程的具体分析,提出了一种改进的Schumacher生长方程．该模型包含了Gompenz函数、Schumacher方程及广义Schumacher方程,具有很强的自适应性和实用性．采用遗传算法。利用该模型对珍稀植物长苞铁杉和侧柏生长资料分别进行了拟合．结果表明,改进的Schumacher方程的拟合精度明显优于Schumache,方程和广义Schumacher方程,也优于经典的Logistic模型和李新运等自适应模型。可以在林木生长动态模拟及种群增长动态研究中广泛应用．     

Transport-limited growth in the chemostat and its competitive inhibition; A theoretical treatment

Summary An equation expressing the specific growth rate of heterotrophic cell populations in terms of yield factor and transport rate is proposed. From this equation expressions are derived for the specific growth rate when the transport of the energy source is growth0limiting. These expressions are applied to cell population growth in the chemostat limited by the transport of the energy source or of other substrates and simple mathematical tools are provided for obtaining estimates of the transport parameters. An equation is derived which predicts that at constant dilution rate in the chemostat the concentration of any substrate (whether or not the source of energy) the transport of which is growth limiting, is a linear function of the concentration of a competitive inhibitor of its transport. With this equation estimates of the Michaelis constants of competitive transport inhibitors can be obtained. The growth rate equation of Monod (1942) is discussed.     

A New Generalized Logistic Sigmoid Growth Equation Compared with the Richards Growth Equation

A new sigmoid growth equation is presented for curve-fitting,analysis and simulation of growth curves. Like the logisticgrowth equation, it increases monotonically, with both upperand lower asymptotes. Like the Richards growth equation, itcan have its maximum slope at any value between its minimumand maximum. The new sigmoid equation is unique because it alwaystends towards exponential growth at small sizes or low densities,unlike the Richards equation, which only has this characteristicin part of its range. The new sigmoid equation is thereforeuniquely suitable for circumstances in which growth at smallsizes or low densities is expected to be approximately exponential,and the maximum slope of the growth curve can be at any value.Eleven widely different sigmoid curves were constructed withan exponential form at low values, using an independent algorithm.Sets of 100 variations of sequences of 20 points along eachcurve were created by adding random errors. In general, thenew sigmoid equation fitted the sequences of points as closelyas the original curves that they were generated from. The newsigmoid equation always gave closer fits and more accurate estimatesof the characteristics of the 11 original sigmoid curves thanthe Richards equation. The Richards equation could not estimatethe maximum intrinsic rate of increase (relative growth rate)of several of the curves. Both equations tended to estimatethat points of inflexion were closer to half the maximum sizethan was actually the case; the Richards equation underestimatedasymmetry by more than the new sigmoid equation. When the twoequations were compared by fitting to the example dataset thatwas used in the original presentation of the Richards growthequation, both equations gave good fits. The Richards equationis sometimes suitable for growth processes that may or may notbe close to exponential during initial growth. The new sigmoidis more suitable when initial growth is believed to be generallyclose to exponential, when estimates of maximum relative growthrate are required, or for generic growth simulations.Copyright1999 Annals of Botany Company Asymptote,Cucumis melo,curve-fitting, exponential growth, intrinsic rate of increase, logistic equation, maximum growth rate, model, non-linear least-squares regression, numerical algorithm, point of inflexion, relative growth rate, Richards growth equation, sigmoid growth curve.     

Analytical solution for a hybrid Logistic-Monod cell growth model in batch and continuous stirred tank reactor culture

Monod and Logistic growth models have been widely used as basic equations to describe cell growth in bioprocess engineering. In the case of the Monod equation, the specific growth rate is governed by a limiting nutrient, with the mathematical form similar to the Michaelis–Menten equation. In the case of the Logistic equation, the specific growth rate is determined by the carrying capacity of the system, which could be growth-inhibiting factors (i.e., toxic chemical accumulation) other than the nutrient level. Both equations have been found valuable to guide us build unstructured kinetic models to analyze the fermentation process and understand cell physiology. In this work, we present a hybrid Logistic-Monod growth model, which accounts for multiple growth-dependent factors including both the limiting nutrient and the carrying capacity of the system. Coupled with substrate consumption and yield coefficient, we present the analytical solutions for this hybrid Logistic-Monod model in both batch and continuous stirred tank reactor (CSTR) culture. Under high biomass yield (Y_x/s) conditions, the analytical solution for this hybrid model is approaching to the Logistic equation; under low biomass yield condition, the analytical solution for this hybrid model converges to the Monod equation. This hybrid Logistic-Monod equation represents the cell growth transition from substrate-limiting condition to growth-inhibiting condition, which could be adopted to accurately describe the multi-phases of cell growth and may facilitate kinetic model construction, bioprocess optimization, and scale-up in industrial biotechnology.     

嗜酸氧化亚铁硫杆菌生长动力学方程的应用

基于Monod模型推导出了A．f的生长动力学方程模型,采用Gauss-Newton算法确定了在不同初始条件下细菌生长的动力学参数,即最大比生长速率‰、Monod常数K及R0。通过在不同初始条件下细菌生长特性的研究,得到了相应初始生长条件下以限制性底物亚铁离子浓度为表征的生长动力学方程,理论上揭示了动力学参数变化对细菌生长的影响规律,其中生长动力学方程的数值模拟与实验数据相吻合。     

Demographic analysis of continuous-time life-history models

I present a computational approach to calculate the population growth rate, its sensitivity to life-history parameters and associated statistics like the stable population distribution and the reproductive value for exponentially growing populations, in which individual life history is described as a continuous development through time. The method is generally applicable to analyse population growth and performance for a wide range of individual life-history models, including cases in which the population consists of different types of individuals or in which the environment is fluctuating periodically. It complements comparable methods developed for discrete-time dynamics modelled with matrix or integral projection models. The basic idea behind the method is to use Lotka's integral equation for the population growth rate and compute the integral occurring in that equation by integrating an ordinary differential equation, analogous to recently derived methods to compute steady-states of physiologically structured population models. I illustrate application of the method using a number of published life-history models.     

Comparison of logistic equations for population growth.

Two different forms of the logistic equation for population growth appear in the ecological literature. In the form of the logistic equation that appears in recent ecology textbooks the parameters are the instantaneous rate of natural increase per individual and the carrying capacity of the environment. In the form of the logistic equation that appears in some older literature the parameters are the instantaneous birth rate per individual and the carrying capacity. The decision whether to use one form or the other depends on which form of the equation is biologically more realistic. In this study the form of the logistic equation in which the instantaneous birth rate per individual is a parameter is shown to be more realistic in terms of the birth and death processes of population growth. Application of the logistic equation to calculate yield from an exploited fish population also shows that the parameters must be the instantaneous birth rate per individual and the carrying capacity.     

A NOTE ON THE SIMILARITIES BETWEEN THE CURVES OF GROWTH AND OF REGENERATION

The curves of growth and of regeneration follow the same course, and can be represented by the same exponential equation. This is taken to substantiate the theory that growth and regeneration are essentially identical processes governed by the same laws. A common peculiarity of the curves of growth and of regeneration is that during a short period in the early stages of regeneration and of growth, the apparent observed speed of these processes seems to be relatively slow. As a result, the curve of the fitted equation cuts the time axis not at zero, the beginning of growth or regeneration, but somewhat later. Data on regeneration are cited indicating that the initial slow phase of regeneration is due to the time required for the formation of a cap of embryonic cells which serves as a basis for the more active later regeneration; in other words, to qualitative growth which cannot be expressed in terms of quantitative units. It is suggested that the apparent initial slow phase of growth of the individual from the fertilized egg is due to a similar qualitative growth. It is suggested that if the initial qualitative changes could be converted into some common unit with the subsequent quantitative changes, the apparent initial lag would disappear, and the exponential equation representing the course of these processes would then be the same as the equation used to represent the course of a monomolecular chemical reaction. Certain implications of this reasoning are discussed in the text.     

Quantitation of microbial growth on surfaces

An equation describing the initial phases of microbial surface colonization is presented. Simultaneous microbial attachment and growth are considered as the primary components of colonization. A table is given that permits determination of growth rate from the density and distribution of cells present on surfaces after incubation in situ. Other methods used to calculate microbial growth rate on surfaces are evaluated. The new procedure is more accurate and less time consuming than those used previously. Published data on microbial surface colonization more closely follow the proposed colonization equation than the exponential growth equation, which overestimates the growth rate.     

Regulator or Driving Force? The Role of Turgor Pressure in Oscillatory Plant Cell Growth

Turgor generates the stress that leads to the expansion of plant cell walls during cellular growth. This has been formalized by the Lockhart equation, which can be derived from the physical laws of the deformation of viscoelastic materials. However, the experimental evidence for such a direct correlation between growth rate and turgor is inconclusive. This has led to challenges of the Lockhart model. We model the oscillatory growth of pollen tubes to investigate this relationship. We couple the Lockhart equation to the dynamical equations for the change in material properties. We find that the correct implementation of the Lockhart equation within a feedback loop leading to low amplitude oscillatory growth predicts that in this system changes in the global turgor do not influence the average growth rate in a linear manner, consistent with experimental observations. An analytic analysis of our model demonstrates in which regime the average growth rate becomes uncorrelated from the turgor pressure.     

A thermodynamic theory of microbial growth

Our ability to model the growth of microbes only relies on empirical laws, fundamentally restricting our understanding and predictive capacity in many environmental systems. In particular, the link between energy balances and growth dynamics is still not understood. Here we demonstrate a microbial growth equation relying on an explicit theoretical ground sustained by Boltzmann statistics, thus establishing a relationship between microbial growth rate and available energy. The validity of our equation was then questioned by analyzing the microbial isotopic fractionation phenomenon, which can be viewed as a kinetic consequence of the differences in energy contents of isotopic isomers used for growth. We illustrate how the associated theoretical predictions are actually consistent with recent experimental evidences. Our work links microbial population dynamics to the thermodynamic driving forces of the ecosystem, which opens the door to many biotechnological and ecological developments.     

Modelling the growth of Trichoderma virens with limited sampling of digital images

AIMS: Using limited digital image sampling, a model of fungal growth in soil that considers both hyphal production and lysis was constructed for two strains of Trichoderma virens over a range of four temperatures. MATERIALS AND METHODS: A growth model was developed by fitting the radial cross sectional data with a modified form of the Ratkowsky equation to determine maximum growth rate and a modified Arrhenius equation to determine maximal rate of decrease in area covered by mycelia. The parameters obtained from a combined equation were then verified by using the data obtained from the whole colony to determine the appropriateness of the model. CONCLUSIONS: Using a limited data set and a combination of the Ratkowsky and Arrhenius equations, the mycelial coverage of the T. virens colony was determined, relating microscopic hyphal growth to macroscopic colony growth. This model was sufficiently robust to predict growth across four temperatures for a genetically modified and wild-type strain of T. virens. SIGNIFICANCE AND IMPACT OF STUDY: By using simple assumptions for the increase and eventual decline in fungal growth on a resource-limited medium, this model constructs an initial framework onto which additional parameters such as nutrient consumption could be incorporated for prediction of fungal growth.     

A quantitative method for the analysis of mammalian cell proliferation in culture in terms of dividing and non-dividing cells

The application of the exponential growth equation is the standard method employed in the quantitative analyses of mammalian cell proliferation in culture. This method is based on the implicit assumption that, within a cell population under study, all division events give rise to daughter cells that always divide. When a cell population does not adhere to this assumption, use of the exponential growth equation leads to errors in the determination of both population doubling time and cell generation time. We have derived a more general growth equation that defines cell growth in terms of the dividing fraction of daughter cells. This equation can account for population growth kinetics that derive from the generation of both dividing and non-dividing cells. As such, it provides a sensitive method for detecting non-exponential division dynamics. In addition, this equation can be used to determine when it is appropriate to use the standard exponential growth equation for the estimation of doubling time and generation time.     

Modelling the ontogeny of ectotherms exhibiting indeterminate growth

Numerous growth functions exist to describe the ontogeny of animals. Such functions (e.g., von Bertalanffy's equation, thermal-unit growth coefficient) are currently applied to ectotherms even though they fail to provide analytical expressions that adapt to a wide range of fluctuating temperatures. The underlying mechanisms responsible for the ontogeny of ectotherms exhibiting indeterminate growth have not yet been summarised in terms of a simple but meaningful mathematical equation. Here, a growth function is developed, with parameters having physical or biological interpretation that accommodates indeterminate growth under fluctuating temperatures assuming the latter vary seasonally. The equation is derived as a special case of von Bertalanffy's equation providing realistic growth trajectories throughout the ontogeny of several groups of ectotherms (R²>0.90). The results suggest that the effect of temperature on growth trajectory supersedes that of reproduction in an environment with fluctuating temperature. Furthermore, values of the allometric weight exponent (0<b<0.75) indicate that the rules of body surface and body weight do not apply under certain circumstances. Finally, the growth function circumvents problems associated with models based on thermodynamic and chemical kinetic principles (e.g., inability to predict growth of organisms in which ontogeny exceeds 3 months) and on rule of thermal summation (e.g., reliable only in a certain range of temperature). The growth function can handle a wide range of temperature fluctuations, encompass life stages and apply to key organisms in ecology, fisheries and agriculture.     

副溶血性弧菌温度-盐度双因素预测模型的建立

本文以副溶血性弧菌VP BJ1.1997为研究对象, 采用均匀设计试验方法, 建立并验证了温度范围为7°C~43°C, 盐度范围为0.5%~9.5%NaCl的生长动力学模型。结果表明, 所选一级模型的拟合效果优劣依次为Logistic方程>Gompertz方程>Linear方程, 以Logistic方程为一级模型计算生长参数; 二级模型采用平方根模型进行拟合, 得到模型相关系数r为0.9863, 最低生长温度T min为9.0506°C, 最高生长盐度为5.93%NaCl(对应最低生长水分活度Aw min     

Cancer chemotherapy: Optimal control using the Verhulst-Pearl equation

General (deterministic) ordinary differential equations for the representation of cancer growth are presented when the growth is perturbed due to the action of a chemotherapeutic agent. The Verhulst-Pearl equation is introduced as a particular example of a growth equation applicable to human tumors. An optimal control problem with general performance criterion and state equation is formulated and shown to possess a novel feedback control relationship. This relationship is used in two continuous drug delivery problems involving the Verhulst-Pearl equation.     

Bifurcation of periodic oscillations due to delays in single species growth models

Summary Theorems are given which guarantee the bifurcation of non-constant, periodic solutions (of fixed period) of a scalar functional equation with two independent parameters. These results are applied to a single, isolated species growth model of general form with a general Volterra (Stieltjes) delay using the ‘magnitudes’ of the instantaneous and delayed growth rate responses as the independent bifurcation parameters. The case of linear growth rate responses (i.e. delay logistic models) is considered in more detail, particularly the often studied single lag logistic equation.     

The ferrous iron oxidation kinetics of Thiobacillus ferrooxidans in batch cultures

The ferrous iron oxidation kinetics of Thiobacillus ferrooxidans in batch cultures was examined, using on-line off-gas analyses to measure the oxygen and carbon dioxide consumption rates continuously. A cell suspension from continuous cultures at steady state was used as the inoculum. It was observed that a dynamic phase occurred in the initial phase of the experiment. In this phase the bacterial ferrous iron oxidation and growth were uncoupled. After about 16 h the bacteria were adapted and achieved a pseudo-steady state, in which the specific growth rate and oxygen consumption rate were coupled and their relationship was described by the Pirt equation. In pseudo-steady state, the growth and oxidation kinetics were accurately described by the rate equation for competitive product inhibition. Bacterial substrate consumption is regarded as the primary process, which is described by the equation for competitive product inhibition. Subsequently the kinetic equation for the specific growth rate, μ, is derived by applying the Pirt equation for bacterial substrate consumption and growth. The maximum specific growth rate, μ _max, measured in the batch culture agrees with the dilution rate at which washout occurs in continuous cultures. The maximum oxygen consumption rate, q _O2,max, of the cell suspension in the batch culture was determined by respiration measurements in a biological oxygen monitor at excess ferrous iron, and showed changes of up to 20% during the course of the experiment. The kinetic constants determined in the batch culture slightly differ from those in continuous cultures, such that, at equal ferric to ferrous iron concentration ratios, biomass-specific rates are up to 1.3 times higher in continuous cultures. Received: 8 February 1999 / Accepted: 17 February 1999