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 comparison of seven strains of 2,4-dichlorophenoxyacetic acid-degrading bacteria.

Seven strains of 2,4-dichlorophenoxyacetic acid-degrading bacteria, including Pseudomonas, Alcaligenes, and Bordetella spp., were compared on the basis of growth kinetics. Estimates of maximum growth rate (mu max, k1) and half-saturation growth constant (Ks, k3) were obtained by fitting substrate depletion curves to a four-parameter version of the integrated Monod equation. Estimates of Ks ranged from 2.2 micrograms/ml (10 microM) to 33.8 micrograms/ml (154 microM), and estimates of mu max ranged from 0.20 h-1 (Td = 3.5 h) to 0.32 h-1 (Td = 2.2 h). Estimates of mu max, but not Ks, were affected by changes in initial inoculum density. Maximum growth rates (mu max) were also estimated from turbidity measurements. They ranged from 0.10 h-1 (Td = 6.9 h) to 1.0 h-1 (Td = 0.7 h). There was no correlation between estimates of mu max derived from substrate depletion curves and those derived from turbidity measurements (P = 0.20).     

Kinetic comparison of seven strains of 2,4-dichlorophenoxyacetic acid-degrading bacteria.

Seven strains of 2,4-dichlorophenoxyacetic acid-degrading bacteria, including Pseudomonas, Alcaligenes, and Bordetella spp., were compared on the basis of growth kinetics. Estimates of maximum growth rate (mu max, k1) and half-saturation growth constant (Ks, k3) were obtained by fitting substrate depletion curves to a four-parameter version of the integrated Monod equation. Estimates of Ks ranged from 2.2 micrograms/ml (10 microM) to 33.8 micrograms/ml (154 microM), and estimates of mu max ranged from 0.20 h-1 (Td = 3.5 h) to 0.32 h-1 (Td = 2.2 h). Estimates of mu max, but not Ks, were affected by changes in initial inoculum density. Maximum growth rates (mu max) were also estimated from turbidity measurements. They ranged from 0.10 h-1 (Td = 6.9 h) to 1.0 h-1 (Td = 0.7 h). There was no correlation between estimates of mu max derived from substrate depletion curves and those derived from turbidity measurements (P = 0.20).     

Determining design and scale-up parameters for degradation of atrazine with suspended Pseudomonas sp. ADP in aqueous bioreactors

This work investigated the kinetic parameters of atrazine mineralization by suspended cells of Pseudomonas sp. ADP in both shake flasks and spherical stirred tank batch reactors (SSTR). The degradation of atrazine and growth of Pseudomonas sp. ADP were studied. Experiments were performed at different temperatures and stirring speeds in both reactors at varying initial concentrations of atrazine. Cell growth and atrazine concentration were monitored over time, and a Monod model with one limiting substrate was used to characterize the kinetic behavior. Temperature, stirring speed, and reactor type were all found to significantly affect the regressed Monod parameters. At 27 degrees C and 200 rpm, for the shaker flask experiments, mu max and Ks were determined to be 0.14 (+/-0.01) h-1 and 1.88 (+/-1.80) mg/L, respectively. At 37 degrees C, mu max and Ks increased to 0.25 (+/-0.05) h-1 and 9.59 (+/-6.55) mg/L, respectively. As expected, stirrer speed was also found to significantly alter the kinetic parameters. At 27 degrees C and 125 rpm, mu max and Ks were 0.04 (+/-0.002) h-1 and 3.72 (+/-1.05) mg/L, respectively, whereas at 37 degrees C and 125 rpm, mu max and Ks were 0.07 (+/-0.008) h-1 and 1.65 (+/-2.06) mg/L. In the SSTR the kinetic parameters mu max and Ks at room temperature were determined to be 0.12 (+/-0.009) h-1 and 2.18 (+/-0.47) mg/L, respectively. Although the mu max values for both types of reactors were similar, the shaker flask experiments resulted in considerable error. Error analysis on calculated values of Ks were found to impact estimates in atrazine concentration by as much as two orders of magnitude, depending on the reactor design, illustrating the importance of these factors in reactor scale-up.     

Growth kinetics of Escherichia coli with galactose and several other sugars in carbon-limited chemostat culture

Kinetic models for microbial growth describe the specific growth rate (mu) as a function of the concentration of the growth-limiting nutrient (s) and a set of parameters. A typical example is the model proposed by Monod, where mu is related to s using substrate affinity (Ks) and the maximum specific growth rate (mu max). The preferred method to determine such parameters is to grow microorganisms in continuous culture and to measure the concentration of the growth-limiting substrate as a function of the dilution rate. However, owing to the lack of analytical methods to quantify sugars in the microgram per litre range, it has not been possible to investigate the growth kinetics of Escherichia coli in chemostat culture. Using an HPLC method able to determine steady-state concentrations of reducing sugars, we previously have shown that the Monod model adequately describes glucose-limited growth of E. coli ML30. This has not been confirmed for any other sugar. Therefore, we carried out a similar study with galactose and found steady-state concentrations between 18 and 840 micrograms.L-1 for dilution rates between 0.2 and 0.8.h-1, respectively. With these data the parameters of several models giving the specific growth rate as a function of the substrate concentration were estimated by nonlinear parameter estimation, and subsequently, the models were evaluated statistically. From all equations tested, the Monod model described the data best. The parameters for galactose utilisation were mu max = 0.75.h-1 and Ks = 67 micrograms.L-1. The results indicated that accurate Ks values can be estimated from a limited set of steady-state data when employing mu max measured during balanced growth in batch culture. This simplified procedure was applied for maltose, ribose, and fructose. For growth of E. coli with these sugars, mu max and Ks were for maltose 0.87.h-1, 100 micrograms.L-1; for ribose 0.57.h-1, 132 micrograms.L-1, and for fructose 0.70.h-1, 125 micrograms.L-1.     

Temperature-dependent growth kinetics of Escherichia coli ML 30 in glucose-limited continuous culture.

Detailed comparison of growth kinetics at temperatures below and above the optimal temperature was carried out with Escherichia coli ML 30 (DSM 1329) in continuous culture. The culture was grown with glucose as the sole limiting source of carbon and energy (100 mg liter(-1) in feed medium), and the resulting steady-state concentrations of glucose were measured as a function of the dilution rate at 17.4, 28.4, 37, and 40 degrees C. The experimental data could not be described by the conventional Monod equation over the entire temperature range, but an extended form of the Monod model [mu = mu(max) x (s - s(min))/(Ks + s - s(min))], which predicts a finite substrate concentration at 0 growth rate (s(min)), provided a good fit. The two parameters mu(max) and s(min) were temperature dependent, whereas, surprisingly, fitting the model to the experimental data yielded virtually identical Ks values (approximately 33 microg liter(-1)) at all temperatures. A model that describes steady-state glucose concentrations as a function of temperature at constant growth rates is presented. In similar experiments with mixtures of glucose and galactose (1:1 mixture), the two sugars were utilized simultaneously at all temperatures examined, and their steady-state concentrations were reduced compared with to growth with either glucose or galactose alone. The results of laboratory-scale kinetic experiments are discussed with respect to the concentrations observed in natural environments.     

Statistical analysis of nonlinear parameter estimation for Monod biodegradation kinetics using bivariate data

A nonlinear regression technique for estimating the Monod parameters describing biodegradation kinetics is presented and analyzed. Two model data sets were taken from a study of aerobic biodegradation of the polycyclic aromatic hydrocarbons (PAHs), naphthalene and 2-methylnaphthalene, as the growth-limiting substrates, where substrate and biomass concentrations were measured with time. For each PAH, the parameters estimated were: q(max), the maximum substrate utilization rate per unit biomass; K(S), the half-saturation coefficient; and Y, the stoichiometric yield coefficient. Estimating parameters when measurements have been made for two variables with different error structures requires a technique more rigorous than least squares regression. An optimization function is derived from the maximumlikelihood equation assuming an unknown, nondiagonal covariance matrix for the measured variables. Because the derivation is based on an assumption of normally distributed errors in the observations, the error structures of the regression variables were examined. Through residual analysis, the errors in the substrate concentration data were found to be distributed log-normally, demonstrating a need for log transformation of this variable. The covariance between ln C and X was found to be small but significantly nonzero at the 67% confidence level for NPH and at the 94% confidence level for 2MN. The nonlinear parameter estimation yielded unique values for q(max), K(S), and Y for naphthalene. Thus, despite the low concentrations of this sparingly soluble compound, the data contained sufficient information for parameter estimation. For 2-methylnaphthalene, the values of q(max) and K(S) could not be estimated uniquely; however, q(max)/K(S) was estimated. To assess the value of including the relatively imprecise biomass concentration data, the results from the bivariate method were compared with a univariate method using only the substrate concentration data. The results demonstrated that the bivariate data yielded a better confidence in the estimates and provided additional information about the model fit and model adequacy. The combination of the value of the bivariate data set and their nonzero covariance justifies the need for maximum likelihood estimation over the simpler nonlinear least squares regression.     

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.     

Simultaneous estimation ofV max,Km, and the rate of endogenous substrate production (R) from substrate depletion data

The nonlinear and 3 linearized forms of the integrated Michaelis-Menten equation were evaluated for their ability to provide reliable estimates of uptake kinetic parameters, when the initial substrate concentration (S₀) is not error-free. Of the 3 linearized forms, the one where t/(S₀–S) is regressed against ln(S₀/S)/(S₀–S) gave estimates ofV _max and K_m closest to the true population means of these parameters. Further, this linearization was the least sensitive of the 3 to errors (±1%) in S₀. Our results illustrate the danger of relying on r² values for choosing among the 3 linearized forms of the integrated Michaelis-Menten equation. Nonlinear regression analysis of progress curve data, when S₀ is not free of error, was superior to even the best of the 3 linearized forms. The integrated Michaelis-Menten equation should not be used to estimateV _max and K_m when substrate production occurs concomitant with consumption of added substrate. We propose the use of a new equation for estimation of these parameters along with a parameter describing endogenous substrate production (R) for kinetic studies done with samples from natural habitats, in which the substrate of interest is an intermediate. The application of this new equation was illustrated for both simulated data and previously obtained H₂ depletion data. The only means by whichV _max, K_m, and R may be evaluated from progress curve data using this new equation is via nonlinear regression, since a linearized form of this equation could not be derived. Mathematical components of computer programs written for fitting data to either of the above nonlinear models using nonlinear least squares analysis are presented.     

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.     

Kinetics of p-nitrophenol mineralization by a Pseudomonas sp.: effects of second substrates.

The kinetics of simultaneous mineralization of p-nitrophenol (PNP) and glucose by Pseudomonas sp. were evaluated by nonlinear regression analysis. Pseudomonas sp. did not mineralize PNP at a concentration of 10 ng/ml but metabolized it at concentrations of 50 ng/ml or higher. The Ks value for PNP mineralization by Pseudomonas sp. was 1.1 micrograms/ml, whereas the Ks values for phenol and glucose mineralization were 0.10 and 0.25 micrograms/ml, respectively. The addition of glucose to the media did not enable Pseudomonas sp. to mineralize 10 ng of PNP per ml but did enhance the degradation of higher concentrations of PNP. This enhanced degradation resulted from the simultaneous use of glucose and PNP and the increased rate of growth of Pseudomonas sp. on glucose. The Monod equation and a dual-substrate model fit these data equally well. The dual-substrate model was used to analyze the data because the theoretical assumptions of the Monod equation were not met. Phenol inhibited PNP mineralization and changed the kinetics of PNP mineralization so that the pattern appeared to reflect growth, when in fact growth was not occurring. Thus, the fitting of models to substrate depletion curves may lead to erroneous interpretations of data if the effects of second substrates on population dynamics are not considered.     

Kinetics of p-nitrophenol mineralization by a Pseudomonas sp.: effects of second substrates

The kinetics of simultaneous mineralization of p-nitrophenol (PNP) and glucose by Pseudomonas sp. were evaluated by nonlinear regression analysis. Pseudomonas sp. did not mineralize PNP at a concentration of 10 ng/ml but metabolized it at concentrations of 50 ng/ml or higher. The Ks value for PNP mineralization by Pseudomonas sp. was 1.1 micrograms/ml, whereas the Ks values for phenol and glucose mineralization were 0.10 and 0.25 micrograms/ml, respectively. The addition of glucose to the media did not enable Pseudomonas sp. to mineralize 10 ng of PNP per ml but did enhance the degradation of higher concentrations of PNP. This enhanced degradation resulted from the simultaneous use of glucose and PNP and the increased rate of growth of Pseudomonas sp. on glucose. The Monod equation and a dual-substrate model fit these data equally well. The dual-substrate model was used to analyze the data because the theoretical assumptions of the Monod equation were not met. Phenol inhibited PNP mineralization and changed the kinetics of PNP mineralization so that the pattern appeared to reflect growth, when in fact growth was not occurring. Thus, the fitting of models to substrate depletion curves may lead to erroneous interpretations of data if the effects of second substrates on population dynamics are not considered.     

Progress curve analysis for enzyme and microbial kinetic reactions using explicit solutions based on the Lambert W function

We present a simple method for estimating kinetic parameters from progress curve analysis of biologically catalyzed reactions that reduce to forms analogous to the Michaelis-Menten equation. Specifically, the Lambert W function is used to obtain explicit, closed-form solutions to differential rate expressions that describe the dynamics of substrate depletion. The explicit nature of the new solutions greatly simplifies nonlinear estimation of the kinetic parameters since numerical techniques such as the Runge-Kutta and Newton-Raphson methods used to solve the differential and integral forms of the kinetic equations, respectively, are replaced with a simple algebraic expression. The applicability of this approach for estimating Vmax and Km in the Michaelis-Menten equation was verified using a combination of simulated and experimental progress curve data. For simulated data, final estimates of Vmax and Km were close to the actual values of 1 microM/h and 1 microM, respectively, while the standard errors for these parameter estimates were proportional to the error level in the simulated data sets. The method was also applied to hydrogen depletion experiments by mixed cultures of bacteria in activated sludge resulting in Vmax and Km estimates of 6.531 microM/h and 2.136 microM, respectively. The algebraic nature of this solution, coupled with its relatively high accuracy, makes it an attractive candidate for kinetic parameter estimation from progress curve data.     

Models for mineralization kinetics with the variables of substrate concentration and population density.

The rates of mineralization of [14C]benzoate by an induced population of Pseudomonas sp. were measured at initial substrate concentrations ranging from 10 ng/ml to 100 micrograms/ml. Plots of the radioactivity remaining in the culture were fit by nonlinear regression to six kinetic models derived from the Monod equation. These models incorporate only the variables of substrate concentration and cell density. Plots of the mineralization kinetics in cultures containing low, intermediate, and high initial substrate concentrations were well fit by first-order, integrated Monod, and logarithmic kinetics, respectively. Parameters such as maximum specific growth rate, half-saturation constant, and initial population density divided by yield agreed between cultures to within a factor of 3.4. Benzoate mineralization by microorganisms in acclimated sewage was shown to fit logistic (sigmoidal), Monod, and logarithmic kinetics when the compound was added at initial concentrations of 0.1, 1.0, and 10 micrograms/ml, respectively. The mineralization of 10 micrograms of benzoate per ml in sewage also followed logarithmic kinetics in the absence of protozoa. It is concluded that much of the diversity in shapes of mineralization curves is a result of the interactions of substrate concentration and population density. Nonlinear regression with models incorporating these variables is a valuable means for analysis of microbial mineralization kinetics.     

Measurement of biodegradability parameters for single unsubstituted and methylated polycyclic aromatic hydrocarbons in liquid bacterial suspensions

Substrate depletion experiments were conducted to characterize aerobic biodegradation of 20 single polycyclic aromatic hydrocarbons (PAHs) by induced Sphingomonas paucimobilis strain EPA505 in liquid suspensions. PAHs consisted of low molecular weight, unsubstituted, and methyl-substituted homologs. A material balance equation containing the Andrews kinetic model, an extension of the Monod model accounting for substrate inhibition, was numerically fitted to batch depletion data to estimate extant kinetic parameters including the maximal specific uptake rates, q(max), the affinity coefficients, K(S), and the substrate inhibition coefficients, K(I). Strain EPA505 degraded all PAHs tested. Applied kinetic models adequately simulated experimental data. A cell proliferation assay involving reduction of the tetrazolium dye WST-1 was used to evaluate the ability of strain EPA505 to utilize individual PAHs as sole energy and carbon sources. Of the 22 PAHs tested, 9 supported bacterial growth. Evaluation of the biokinetic data showed that q(max) correlated highly with transmembrane flux as theoretically estimated by a diffusion model, pointing to transmembrane transport as a potential rate-determining process. The biodegradability data generated in this study is essential for the development of quantitative structure-activity relationships (QSARs) for biodegradability and for modeling biodegradation of simple PAH mixtures.     

Spreadsheet Method for Evaluation of Biochemical Reaction Rate Coefficients and Their Uncertainties by Weighted Nonlinear Least-Squares Analysis of the Integrated Monod Equation

A convenient method for evaluation of biochemical reaction rate coefficients and their uncertainties is described. The motivation for developing this method was the complexity of existing statistical methods for analysis of biochemical rate equations, as well as the shortcomings of linear approaches, such as Lineweaver-Burk plots. The nonlinear least-squares method provides accurate estimates of the rate coefficients and their uncertainties from experimental data. Linearized methods that involve inversion of data are unreliable since several important assumptions of linear regression are violated. Furthermore, when linearized methods are used, there is no basis for calculation of the uncertainties in the rate coefficients. Uncertainty estimates are crucial to studies involving comparisons of rates for different organisms or environmental conditions. The spreadsheet method uses weighted least-squares analysis to determine the best-fit values of the rate coefficients for the integrated Monod equation. Although the integrated Monod equation is an implicit expression of substrate concentration, weighted least-squares analysis can be employed to calculate approximate differences in substrate concentration between model predictions and data. An iterative search routine in a spreadsheet program is utilized to search for the best-fit values of the coefficients by minimizing the sum of squared weighted errors. The uncertainties in the best-fit values of the rate coefficients are calculated by an approximate method that can also be implemented in a spreadsheet. The uncertainty method can be used to calculate single-parameter (coefficient) confidence intervals, degrees of correlation between parameters, and joint confidence regions for two or more parameters. Example sets of calculations are presented for acetate utilization by a methanogenic mixed culture and trichloroethylene cometabolism by a methane-oxidizing mixed culture. An additional advantage of application of this method to the integrated Monod equation compared with application of linearized methods is the economy of obtaining rate coefficients from a single batch experiment or a few batch experiments rather than having to obtain large numbers of initial rate measurements. However, when initial rate measurements are used, this method can still be used with greater reliability than linearized approaches.     

Comparative physiology of two protozoan parasites, Leishmania donovani and Trypanosoma brucei, grown in chemostats.

Cultures of the insect stage of the protozoan parasites Leishmania donovani and Trypanosoma brucei were grown in chemostats with glucose as the growth rate-limiting substrate. L. donovani has a maximum specific growth rate (mu max) of 1.96 day-1 and a Ks for glucose of 0.1 mM; the mu max of T. brucei is 1.06 day-1 and the Ks is 0.06 mM. At each steady state (specific growth rate, mu, equals D, the dilution rate), the following parameters were measured: external glucose concentration (Glcout), cell density, dry weight, protein, internal glucose concentration (Glcin), cellular ATP level, and hexokinase activity. L. donovani shows a relationship between mu and yield that allows an estimation of the maintenance requirement (ms) and the yield per mole of ATP (YATP). Both the ms and the YATP are on the higher margin of the range found for prokaryotes grown on glucose in a complex medium. L. donovani maintains the Glcin at a constant level of about 50 mM as long as it is not energy depleted. T. brucei has a decreasing yield with increasing mu, suggesting that it oxidizes its substrate to a lesser extent at higher growth rates. Glucose is not concentrated internally but is taken up by facilitated diffusion, while phosphorylation by hexokinase is probably the rate-limiting step for glucose metabolism. The Ks is constant as long as glucose is the rate-limiting substrate. The results of this study demonstrate that L. donovani and T. brucei have widely different metabolic strategies for dealing with varying external conditions, which reflect the conditions they are likely to encounter in their respective insect hosts.     

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.