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.     

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.     

Equations of substrate-limited growth: the case for blackman kinetics

A simplified model of cell metabolism, consisting of a series of linked reversible enzymatic reactions dependent on the concentration of a single external substrate has been developed. The general mathematical solution for this system of reactions is presented. This general solution confirms the concept of a rate-limiting step, or “master reaction”, in biological systems as first proposed by Blackman. The maximum rate of such a process is determined by, and equal to, the maximum rate of the slowest forward reaction in the series. Of practical interest in modeling the growth rate of cells are three cases developed from the general model. The simplest special case results in the Monod equation when the maximum forward rate of one enzymatic reaction in the cell is much less than the maximum forward rate of any other enzymatic reactions. More realistic is the case where the maximum forward rates of more than one enzymatic reaction are slow. When two slow enzymatic reactions are separated from each other by any number of fast reactions that overall can be described by a large equilibrium constant, the Blackman form results: and A third case is that in which two slow enzymatic steps are separated by an equilibrium constant that is not large. Unlike the Monod and Blackman forms, which contain only two arbitrary constants, this model contains three arbitrary constants: The Monod and Blackman forms are special cases of this three constant form. In comparing equations with two arbitrary constants the Monod equation gave poorer fit of the data in most cases than the Blackman form. It is concluded that workers modeling the growth of microorganisms should give a t least as much consideration to the Blackman form as is given to the Monod equation.     

Enzyme reaction kinetics in organic solvents: A theoretical kinetic model and comparison with experimental observations

A theoretical kinetic model has been developed in order to describe the enzyme reaction in organic solvents. In this model the hydration of the enzyme molecule was examined and the equilibrium kinetic constants expressed in terms of thermodynamic activity. Analysis of a proposed kinetic model shows that the enzyme reaction rate in organic solvents is determined by two factors: substrate solvation and enzyme hydration, which are determined by the activity coefficient of the substrate and the water activity of the reaction media, respectively. The activity coefficient of the substrate and the water activity have been calculated using the UNIFAC equation to analyze the effects of organic solvents on the rate of enzyme reaction, and the results were compared with experimental data. Predictions of the proposed model were found to be in good agreement with previous experimental observations.     

Nonlinear estimation of Monod growth kinetic parameters from a single substrate depletion curve

Monod growth kinetic parameters were estimated by fitting sigmoidal substrate depletion data to the integrated Monod equation, using nonlinear least-squares analysis. When the initial substrate concentration was in the mixed-order region, nonlinear estimation of simulated data sets containing known measurement errors provided accurate estimates of the mu max, Ks, and Y values used to create these data. Nonlinear regression analysis of sigmoidal substrate depletion data was also evaluated for H2-limited batch growth of Desulfovibrio sp. strain G11. The integrated Monod equation can be more convenient for the estimation of growth kinetic parameters, particularly for gaseous substrates, but it must be recognized that the estimates of mu max, Ks, and Y obtained may be influenced by the growth rate history of the inoculum.     

Nonlinear estimation of Monod growth kinetic parameters from a single substrate depletion curve.

Monod growth kinetic parameters were estimated by fitting sigmoidal substrate depletion data to the integrated Monod equation, using nonlinear least-squares analysis. When the initial substrate concentration was in the mixed-order region, nonlinear estimation of simulated data sets containing known measurement errors provided accurate estimates of the mu max, Ks, and Y values used to create these data. Nonlinear regression analysis of sigmoidal substrate depletion data was also evaluated for H2-limited batch growth of Desulfovibrio sp. strain G11. The integrated Monod equation can be more convenient for the estimation of growth kinetic parameters, particularly for gaseous substrates, but it must be recognized that the estimates of mu max, Ks, and Y obtained may be influenced by the growth rate history of the inoculum.     

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.     

A fast kinetic model discrimination applied to the continuous two-phase anaerobic digestion of fruit and vegetable wastes

Summary A simple method to fit the constants of first order, Monod and Chen and Hashimoto models using only one single linear regression, is presented. The method can be applied to continuous operated systems. As an example, the kinetic behaviour of the anaerobic digestion of a mixture of fruit and vegetable wastes is studied. Chen and Hashimoto model yields the best fit.     

STEADY STATE GROWTH OF PHYTOPLANKTON IN CONTINUOUS CULTURE: COMPARISON OF INTERNAL AND EXTERNAL NUTRIENT EQUATIONS1,2

Previously, there have been conflicts over whether external or internal nutrient concentrations control phytoplankton growth rates at steady state in continuous culture. To experimentally demonstrate that both equations equally describe steady state growth, continuous culture studies with phosphorus-limited growth of the chrysophyte Monochrysis lutheri Droop were carried out over the entire growth rate region up to biomass washout. Data were examined using both the Monod and Droop equations, and, even though there were significant variations in the yield coefficient with growth rate, the data fit both equations reasonably well. Because of their relative simplicity, the Droop equation and an equation combining both the Monod and Droop equations are better suited for expressing kinetic data than the Monod equation. It is crucial, though, that the criteria necessary to achieve steady state be fulfilled.     

Process model comparison and transferability across bioreactor scales and modes of operation for a mammalian cell bioprocess

A Monod kinetic model, logistic equation model, and statistical regression model were developed for a Chinese hamster ovary cell bioprocess operated under three different modes of operation (batch, bolus fed‐batch, and continuous fed‐batch) and grown on two different bioreactor scales (3 L bench‐top and 15 L pilot‐scale). The Monod kinetic model was developed for all modes of operation under study and predicted cell density, glucose glutamine, lactate, and ammonia concentrations well for the bioprocess. However, it was computationally demanding due to the large number of parameters necessary to produce a good model fit. The transferability of the Monod kinetic model structure and parameter set across bioreactor scales and modes of operation was investigated and a parameter sensitivity analysis performed. The experimentally determined parameters had the greatest influence on model performance. They changed with scale and mode of operation, but were easily calculated. The remaining parameters, which were fitted using a differential evolutionary algorithm, were not as crucial. Logistic equation and statistical regression models were investigated as alternatives to the Monod kinetic model. They were less computationally intensive to develop due to the absence of a large parameter set. However, modeling of the nutrient and metabolite concentrations proved to be troublesome due to the logistic equation model structure and the inability of both models to incorporate a feed. The complexity, computational load, and effort required for model development has to be balanced with the necessary level of model sophistication when choosing which model type to develop for a particular application. © 2012 American Institute of Chemical Engineers Biotechnol. Prog., 2013     

Kinetics of sequential nitrification and denitrification processes

Kinetics of nitrification and denitrification of synthetic wastewater was investigated by using two reactors in series. An activated sludge unit was used for nitrification followed by a downflow biofilter for denitrification. Glucose solution was fed to the denitrification column to supply carbon source. Reactors were operated at different operating conditions and data were collected for determination of kinetic constants. Experimental data indicated that nitrification and denitrification kinetics followed Monod kinetics. By using the experimental data, kinetic constants for nitrification were determined as k = 1.15 d(-1), K(N) = 5.14 mg/l, Y = 0.34 mgX/mgN and b = -0.021 d(-1). Similarly, kinetic constants for denitrification were determined as k = 0.23 d(-1) and K(DN) = 0.27 mg/l. Rates of nitrification and denitrification increased with increasing nitrogen loading rate.     

The removal of nitrogen and organics in vertical flow wetland reactors: predictive models

Three kinetic models, for predicting the removal of nitrogen and organics in vertical flow wetlands, have been developed and evaluated. These models were established by combining first-order, Monod and multiple Monod kinetics with continuous stirred-tank reactor (CSTR) flow pattern. Critical evaluations of these models using three statistical parameters, coefficient of determination, relative root mean square error and model efficiency, indicated that when the Monod/multiple Monod kinetics was combined with CSTR flow pattern it allowed close match between theoretical prediction and experiment data of nitrogen and organics removal. The kinetic coefficients (derived from Monod/multiple Monod kinetics) was found to increase with pollutant loading, indicating that the coefficients may vary based on different factors, such as influent pollutant concentration, hydraulic loading, and water depth. Overall, this study demonstrated the validity of combining Monod and multiple Monod kinetics with CSTR flow pattern for the modelling and design of vertical flow wetland systems.     

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。     

Allosteric kinetics of pyruvate kinase of Saccharomyces carlsbergensis

The allosteric model of Monod et al. (1965) has been used to analyse the steadystate kinetics of pyruvate kinase from Saccharomyces carlsbergensis. The dissociation constants for the substrate phosphoenolpyruvate, the inhibitor ATP as well as the activator fructose-1, 6-diphosphate from the R and T state were calculated using a series of computer programs. On the basis of a crucial relation (derived in the Appendix), which correlates the Hill coefficient and the half-saturating concentration of substrate saturation curves with the parameters of the model of Monod et al., it is possible to differentiate between exclusive and non-exclusive ligand binding. On the other hand, this relation makes it possible to fit the experimental data to an extended model assuming only partially concerted transitions in each enzyme molecule.The physical data of yeast pyruvate kinase point to a tetrameric structure, whereas the steady-state kinetics favour a trimeric one. This discrepancy in the number of protomers can be overcome by the use of an extended model, which permits the occurrence of hybrid states R_tT_n?t. The introduction of one symmetrical hybrid state R₂T₂ into the model explains the kinetic data of yeast pyruvate kinase on the basis of four, probably identical, protomers. The equilibrium constants between the states are given.In the Appendix the derivation of the equation describing the occurrence of hybrid states is reported.     

Overview of some theoretical approaches for derivation of the Monod equation

The Monod equation has been widely applied to describe microbial growth, but it has no mechanistic basis and is purely empirical. Extensive efforts have been dedicated to develop theoretical approaches for derivation of the Monod equation, which can be classified into three major groups, i.e., kinetic, thermodynamic, and substance transport approaches. In this review, four representative approaches are thus discussed. Due to the fact that different assumptions are made in each approach, no universal physical meaning of the Monod constant (K (s)) can be revealed. However, it seems that the Monod constant would be free energy-dependent and have nonequilibrium thermodynamic characteristics.     

Kinetic modeling of aerobic biodegradation of high oil and grease rendering wastewater

Batch scale activated sludge kinetic studies were undertaken for the treatment of pet food wastewater characterized by oil and grease concentrations of up to 21,500 mg/L, COD and BOD concentrations of 75,000 and 60,000 mg/L, respectively as well as effluent from the batch dissolved air flotation (DAF) system. The conducted kinetics studies showed that Haldane Model fit the substrates and biomass data better than Monod model in DAF-pretreated wastewater, while the modified hydrolysis Monod model better fit the raw wastewater kinetic data. For the DAF pretreated batches, Haldane Model kinetic coefficients k, K(S), Y and Ki values of 1.28-5.35 g COD/g VSS-d, 17,833-23,477 mg/L, 0.13-0.41 mg VSS/mg COD and 48,168 mg/L, respectively were obtained reflecting the slow biodegradation rate. Modified hydrolysis Monod model kinetic constants for the raw wastewater i.e., k, K(S), Y, and K(H) varied from 1-1.3 g COD/g VSS-d, 5580-5600 mg COD/l, 0.08-0.85 mg VSS/mg COD, and 0.21-0.66 d(-1), respectively.     

A new multi-wavelength model-based method for determination of enzyme kinetic parameters

Lineweaver-Burk plot analysis is the most widely used method to determine enzyme kinetic parameters. In the spectrophotometric determination of enzyme activity using the Lineweaver-Burk plot, it is necessary to find a wavelength at which only the substrate or the product has absorbance without any spectroscopic interference of the other reaction components. Moreover, in this method, different initial concentrations of the substrate should be used to obtain the initial velocities required for Lineweaver-Burk plot analysis. In the present work, a multi-wavelength model-based method has been developed and validated to determine Michaelis-Menten constants for some enzyme reactions. In this method, a selective wavelength region and several experiments with different initial concentrations of the substrate are not required. The absorbance data of the kinetic assays are fitted by non-linear regression coupled to the numeric integration of the related differential equation. To indicate the applicability of the proposed method, the Michaelis-Menten constants for the oxidation of phenanthridine, 6-deoxypenciclovir and xanthine by molybdenum hydroxylases were determined using only a single initial concentration of the substrate, regardless of any spectral overlap.     

Kinetic study of the conversion of different substrates to lactic acid using Lactobacillus bulgaricus

Lactic acid fermentation includes several reactions in association with the microorganism growth. A kinetic study was performed of the conversion of multiple substrates to lactic acid using Lactobacillus bulgaricus. Batch experiments were performed to study the effect of different substrates (lactose, glucose, and galactose) on the overall bioreaction rate. During the first hours of fermentation, glucose and galactose accumulated in the medium and the rate of hydrolysis of lactose to glucose and galactose was faster than the convesion of these substrates. Once the microorganism built the necessary enzymes for the substrate conversion to lactic acid, the conversion rate was higher for glucose than for galactose. The inoculum preparation was performed in such a way that healthy young cells were obtained. By using this inoculum, shorter fermentation times with very little lag phase were observed. The consumption patterns of the different substrates converted to lactic acid were studied to determine which substrate controls the overall reaction for lactic acid production. A mathematical model (unstructured Monod type) was developed to describe microorganism growth and lactic acid production. A good fit with a simple equation was obtained. It was found experimentally that the approximate ratio of cell to substrate was 1 to 10, the growth yield coefficient (Y(XS)) was 0.10 g cell/g substrate, the product yield (Y(PS)) was 0.90 g lactic acid/g substrate, and the alpha parameter in the Luedeking-Piret equation was 9. The Monod kinetic parameters were obtained. The saturation constant (K(S)) was 3.36 g/L, and the specific growth rate (microm ) was 1.14 l/h.     

Kinetic model for dynamic response of three-phase fluidized bed biofilm reactor for wastewater treatment

Step changes in inlet concentration has been introduced into the completely mixed three-phase fluidized bed biofilm reactor treating simulated domestic wastewater to study the dynamic behavior of the system and to establish the suitable kinetic model from the response curve. Three identical reactors having different biomass volumes were operated in parallel. It was found that the response curves showed second-order characteristics, and thus at least two first-order differential equations are necessary to simulate the substrate and biomass response curves. Nonlinear regression analysis was performed using different types of rate equations and their corresponding kinetic parameters were used to simulate the theoretical response curve using the Runge–Kutta numerical integration method. As a result, although various types of conventional biokinetic models such as Monod, Haldane and Andrew types were examined, all the theoretical substrate response curves underestimated time constants compared to the actual substrate response plots. On the other hand, the theoretical curve of the kinetic model that incorporates adsorption term has best fit to the actual response in most of the cases. Thus, it was concluded that adsorption of substrate onto biofilm and carrier particles has significant effect on the dynamic response in biofilm processes.