Re-interpretation of the logistic equation for batch microbial growth in relation to Monod kinetics

Aims:  To determine the underlying substrate utilization mechanism in the logistic equation for batch microbial growth by revealing the relationship between the logistic and Monod kinetics. Also, to determine the logistic rate constant in terms of Monod kinetic constants.
Methods and Results:  The logistic equation used to describe batch microbial growth was related to the Monod kinetics and found to be first-order in terms of the substrate and biomass concentrations. The logistic equation constant was also related to the Monod kinetic constants. Similarly, the substrate utilization kinetic equations were derived by using the logistic growth equation and related to the Monod kinetics.
Conclusion:  It is revaled that the logistic growth equation is a special form of the Monod growth kinetics when substrate limitation is first-order with respect to the substrate concentration. The logistic rate constant ( k ) is directly proportional to the maximum specific growth rate constant ( μ _m) and initial substrate concentration ( S ₀) and also inversely related to the saturation constant ( K _s).
Significance and Impact of the Study:  The semi-empirical logistic equation can be used instead of Monod kinetics at low substrate concentrations to describe batch microbial growth using the relationship between the logistic rate constant and the Monod kinetic constants.     

A dynamic mathematical model of the chemostat

A number of experimental studies on the dynamic, behavior of the chemostat have shown that the specific growth rate does not, instantaneously adjust to changes in the concentration of limiting substrate in the chemostat following disturbances in the steady state input limiting substrate concentration or in the steady state dilution rate. Instead of an instantaneous response, as would be predicted by the Monod equation, experimental studies have shown that the specific growth rate experiences a dynamic lag in responding to the changes in the concentration of limiting substrate in the culture vessel. The observed dynamic lag has been recognized by researchers in such terms as an inertial phenomenon and as a hysteresis effect, but as yet a systems engineering approach has not been applied to the observed data. The present paper criticizes the use of the Monod equation as a dynamic relationship and offers as an alternative a dynamic equation relating specific growth rate to the limiting substrate concentration in the chemostat. Following the development of equations, experimental methods of evaluating parameters are discussed. Dynamic responses of analog simulations (incorporating the newly derived equations) are compared with the dynamic responses predicted by the Monod equation and with the dynamic responses of experimental chemostats.     

A theoretical derivation of the Contois equation for kinetic modeling of the microbial degradation of insoluble substrates

Although developed as an empirical model to describe microbial growth on soluble substrates, the Contois equation has been widely accepted for kinetic modeling of insoluble substrate degradation. Yet, the mechanistic basis underlining these successful applications remains unanswered. Unlike soluble substrates that mainly cultivate suspended cultures, microbes cultivated on insoluble substrates have the populations that attach to the substrate surface or remain suspended in the bulk solution, while those attached usually grow faster than those suspended due to their proximity to food resources. This imbalanced growth provides a plausible explanation to the inverse relationship between microbial concentration and their specific growth rate as conveyed in the Contois equation. Based on a theoretical derivation, this study revealed that the Contois equation holds true only when attached microbes substantially obstruct the access of food to their suspended counterparts. On the other hand, when plentiful insoluble substrate surfaces are exposed for cell attachment, the Contois equation will be reduced back to the classic Monod equation.     

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.     

Extended monod kinetics for substrate, product, and cell inhibition

A generalized form of Monod kinetics is proposed to account for all kinds of product, cell, and substrate inhibition. This model assumes that there exists a critical inhibitor concentration above which cells cannot grow, and that the constants of the Monod equation are functions of this limiting inhibitor concentration. Methods for evaluating the constants of this rate form are presented. Finally the proposed kinetic form is compared with the available data in the literature, which unfortunately is very sparse. In all cases, this equation form fitted the data very well.     

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.     

初始底物浓度对序批式培养光合细菌产氢动力学影响

实验研究了初始底物浓度对序批式培养光合细菌生长、降解及产氢过程的影响,根据最大比生长速率实验数据拟合得到其关于初始底物浓度影响的关联式,并在建立的修正Monod模型基础上建立了光合细菌比生长速率、基质比消耗速率和比产氢速率关于底物初始浓度影响的数学模型,模型预测值与实验结果在光合细菌生长期和稳定期内得到较好吻合,反映了光合细菌生长、降解和产氢过程中受底物初始浓度限制性和抑制性影响的基本规律。分析发现光合细菌生长、降解基质和产氢过程中最适底物浓度为50 mmol/L,初始底物浓度低于或高于该浓度时,光合细菌生长、降解及产氢过程都受到限制性或抑制性影响,且抑制性影响较限制性影响效果更明显;底物比消耗速率受初始底物浓度影响较小。     

Effect of Nutrient Concentration on the Growth of Escherichia coli

The relationship between specific growth rate of Escherichia coli and the concentration of limiting nutrient (glucose or phosphate or tryptophan) has been determined for populations in a steady state. At high concentrations the specific growth rate is independent of the concentration of nutrient, but at low concentrations the specific growth rate is a strong function of the nutrient concentration. Such a relationship was predicted by Monod; however, Monod's equation does not predict the relationship over the entire range of nutrient concentration. If parameters of the equation are estimated from the results obtained at low concentrations, then at high concentrations of nutrient, the specific growth rate is significantly higher than that predicted by Monod's equation. These results were interpreted on the basis that the rate of growth is controlled by at least two parallel reactions and that the affinities of the enzymes catalyzing these reactions are different. The relationship between specific growth rate and mean cell volume was also measured, and the results indicate that mean cell volume depends not only on the specific growth rate but also on the nature of the limiting nutrient. There are different mean cell volumes at the same specific growth rate established by different limiting nutrients. Therefore, the mean cell volume is not uniquely determined by the specific growth rate.     

Influence of some physicochemical parameters on bacterial activity of biofilm: Ferrous iron oxidation by Thiobacillus ferrooxidans

The influence of temperature, pH, and substrate and product concentrations on the oxidation rate of ferrous iron by biofilm of Thiobacillus ferrooxidans was determined. The experiments were performed in an inverse fluidized-bed biofilm reactor in which the biofilm thickness was kept constant at 80 mum. Oxygen concentration and diffusion through the biofilm did not limit the oxidation rate. The oxidation rate was almost unaffected by temperature between 13 and 38 degrees C, pH between 1.3 and 2.2, ferric iron concentration up to 14 g/L, or ferrous iron concentration from 4 to 13 g/L. The kinetics of the process was described by the Monod equation with respect to the mass of the biofilm and with ferrous ions as the limiting substrate.     

Application of bioenergetics to modelling the microbial conversion of D-xylose to 2,3-butanediol

During the oxygen limiting growth of Klebsiella oxytoca, the xylose metabolism may be considered as consisting of three components: conversion to 2,3-butanediol by "fermentation," oxidation to carbon dioxide by respiration, and assimilation to cell mass. The amount of energy required for the assimilation of cell mass is assumed to determine the extent to which the two energy producing reactions occur. The activity of each energy producing pathway is also determined by the availability of oxygen and by the energy yield of each pathway. These relationships can be quantified by equating the ATP required for growth and maintenance to the ATP produced by the energy producing reactions. The resulting equation for butanediol production appears similar to the Luedeking and Piret model where the parameters alpha and beta are related to the maximum cell yield from ATP and the maintenance energy requirement. These parameters were estimated from 14 batch fermentations, and the resulting simulation was used to describe the effects of the oxygen transfer rate and the initial xylose concentration on the yields and rates of the 2,3-butanediol fermentation.     

Models for the continuous culture of microorganisms under both oxygen and carbon limiting conditions

Two models for predicting the behavior of cultures of microorganisms under both oxygen and carbon limiting conditions have been evlauated on a chemostat growing Candida utilis on a glycolysis suppressing glycerol medium. The work indicated that parameter values obtained under wholly oxygen limiting or wholly carbon limiting conditions successfully predict the behavior of the chemostat under the wide range of flow and substrate concentration conditions tested. Both models are satisfactory and hence it is deduced that the simpler one may be used with confidence. It was found that Monod kinetics were applicable to the growth rate dependence on oxygen concentration but that Contois kinetics were superior for the corresponding dependence on carbon substrate concentration.     

Kinetic Constants for Aerobic Growth of Microbial Populations Selected with Various Single Compounds and with Municipal Wastes as Substrates

The general applicability of the Monod relationship between the logarithmic growth rate constant and substrate concentration was studied for heterogeneous populations metabolizing a variety of substrates including concentrated municipal sewage. It was found that growth could be described by the Monod equation, mu = mu(m)/k(s) + s. The kinetic "constants" for heterogeneous populations growing on concentrated sewage were comparable to those found with glucose as substrate.     

Effects of dissolved oxygen concentration on biodegradation of 2,4-dichlorophenoxyacetic acid.

Batch experiments were conducted to examine the effects of dissolved oxygen concentration on the degradation of 2,4-dichlorophenoxyacetic acid (2,4-D) by an enrichment culture of 2,4-D-utilizing bacteria. A modified Monod equation was found to describe the relationship between the specific growth rate and the concentrations of both the organic substrate and dissolved oxygen. Values for the maximum specific growth rate, yield, and Monod coefficient for growth on 2,4-D were 0.09 h-1, 0.14 g/g, and 0.6 mg/liter, respectively. The half-saturation constant for dissolved oxygen was estimated to be 1.2 mg/liter. These results suggest that dissolved oxygen concentrations below 1 mg/liter may be rate limiting for the biodegradation of chlorinated aromatic compounds such as 2,4-D, which have a requirement for molecular oxygen as a cosubstrate for metabolism.     

Effects of dissolved oxygen concentration on biodegradation of 2,4-dichlorophenoxyacetic acid

Batch experiments were conducted to examine the effects of dissolved oxygen concentration on the degradation of 2,4-dichlorophenoxyacetic acid (2,4-D) by an enrichment culture of 2,4-D-utilizing bacteria. A modified Monod equation was found to describe the relationship between the specific growth rate and the concentrations of both the organic substrate and dissolved oxygen. Values for the maximum specific growth rate, yield, and Monod coefficient for growth on 2,4-D were 0.09 h-1, 0.14 g/g, and 0.6 mg/liter, respectively. The half-saturation constant for dissolved oxygen was estimated to be 1.2 mg/liter. These results suggest that dissolved oxygen concentrations below 1 mg/liter may be rate limiting for the biodegradation of chlorinated aromatic compounds such as 2,4-D, which have a requirement for molecular oxygen as a cosubstrate for metabolism.     

Kinetics of microbial growth on pentachlorophenol

Batch and fed-batch experiments were conducted to examine the kinetics of pentachlorophenol utilization by an enrichment culture of pentachlorophenol-degrading bacteria. The Haldane modification of the Monod equation was found to describe the relationship between the specific growth rate and substrate concentration. Analysis of the kinetic parameters indicated that the maximum specific growth rate and yield coefficients are low, with values of 0.074 h-1 and 0.136 g/g, respectively. The Monod constant (Ks) was estimated to be 60 micrograms/liter, indicating a high affinity of the microorganisms for the substrate. However, high concentrations (KI = 1,375 micrograms/liter) were shown to be inhibitory for metabolism and growth. These kinetic parameters can be used to define the optimal conditions for the removal of pentachlorophenol in biological treatment systems.     

Kinetics of microbial growth on pentachlorophenol.

Batch and fed-batch experiments were conducted to examine the kinetics of pentachlorophenol utilization by an enrichment culture of pentachlorophenol-degrading bacteria. The Haldane modification of the Monod equation was found to describe the relationship between the specific growth rate and substrate concentration. Analysis of the kinetic parameters indicated that the maximum specific growth rate and yield coefficients are low, with values of 0.074 h-1 and 0.136 g/g, respectively. The Monod constant (Ks) was estimated to be 60 micrograms/liter, indicating a high affinity of the microorganisms for the substrate. However, high concentrations (KI = 1,375 micrograms/liter) were shown to be inhibitory for metabolism and growth. These kinetic parameters can be used to define the optimal conditions for the removal of pentachlorophenol in biological treatment systems.     

Control of Growth Rate by Initial Substrate Concentration at Values Below Maximum Rate

The hyperbolic relationship between specific growth rate, mu, and substrate concentration, proposed by Monod and used since as the basis for the theory of steady-state growth in continuous-flow systems, was tested experimentally in batch cultures. Use of a Flavobacterium sp. exhibiting a high saturation constant for growth in glucose minimal medium allowed direct measurement of growth rate and substrate concentration throughout the growth cycle in medium containing a rate-limiting initial concentration of glucose. Specific growth rates were also measured for a wide range of initial glucose concentrations. A plot of specific growth rate versus initial substrate concentration was found to fit the hyperbolic equation. However, the instantaneous relationship between specific growth rate and substrate concentration during growth, which is stated by the equation, was not observed. Well defined exponential growth phases were developed at initial substrate concentrations below that required for support of the maximum exponential growth rate and a constant doubling time was maintained until 50% of the substrate had been used. It is suggested that the external substrate concentration initially present "sets" the specific growth rate by establishing a steady-state internal concentration of substrate, possibly through control of the number of permeation sites.     

Biokinetic models for representing the complete inhibition of microbial activity at high substrate concentrations.

This paper reintroduces the Wayman and Tseng model for representing substrate inhibition effects on specific growth rate by further documenting its potential predictive capabilities. It also introduces a modification to this model in which an Andrews inhibition function is used in place of the Monod noninhibitory substrate function. This modification better represents the relationship between specific growth rate and substrate concentration for those substrates that show Andrews type inhibition at lower substrate concentrations, rather than the Monod type noninhibitory behavior described in the model of Wayman and Tseng. Results from nonlinear, least squares regression analysis are used to evaluate the ability of these models to empirically represent experimental data (both new and from the literature). The statistical goodness of fit is evaluated by comparing the regression results against those obtained using other empirical models. Finally, possible mechanisms of toxicity responsible for the observed inhibition trends are used to further justify use of these empirical models. The dominant mechanism considered to be relevant for conceptually explaining complete inhibition at high concentrations of solvents is the deterioration of cell membrane integrity. Literature citations are used to support this argument. This work should lead to improvements in the mathematical modeling of contaminant fate and transport in the environment and in the simulation of microbial growth and organic compound biodegradation in engineered systems.     

A Thermodynamic Interpretation of the Monod Equation

The Monod equation for microbial growth is purely empirical, and the theoretical basis of this model stays unclear. Similar to any chemical reactions, overall microbial growth process is dependent upon the changes in free energy. This study showed that the Monod equation could be interpreted in a thermodynamic sense very well. It was probably for the first time demonstrated that the Monod constant (K _s ) was inversely related to the equilibrium constant of the overall microbial growth process. Received: 2 August 2002 / Accepted: 23 September 2002     

An integrated mathematical model for co-composting of agricultural solid wastes with industrial wastewater

An integrated model for the composting process was developed. The structure of the model is such that it can be implemented in any mixture of different substrates, even in the case of co-composting of a solid waste with industrial wastewater. This paper presents a mathematical formulation of the physicochemical and biological principles that govern the composting process. The model of the co-composting ecosystem included mass transfer, heat transfer and biological processes. The biological processes included in the model were hydrolysis of particulate substrates, microbial growth and death. Two microbial populations (bacteria and fungi) were selected using Monod kinetics. Growth limiting functions of inhibitory factors, moisture and dissolved oxygen were added in the Monod kinetics. The bacteria were considered to utilise the easy biodegradable carbon hydrolysis product, fungi the difficult one, while both could degrade the carbon of wastewater. The mass balances of the most important nutrients, nitrogen and phosphorous, were also included in this approach. Model computer simulations provided results that fitted satisfactory the experimental data. Conclusively, the model could be a useful tool for the prediction of the co-composting process performance in the future and could be used to assist in the operation of co-composting plants.