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.
The rates of methane utilization and trichloroethylene (TCE) cometabolism by a methanotrophic mixed culture were characterized in batch and pseudo‐steady‐state studies. Procedures for determination of the rate coefficients and their uncertainties by fitting a numerical model to experimental data are described. The model consisted of a system of differential equations for the rates of Monod kinetics, cell growth on methane and inactivation due to TCE transformation product toxicity, gas/liquid mass transfer of methane and TCE, and the rate of passive losses of TCE. The maximum specific rate of methane utilization (k CH 4) was determined by fitting the numerical model to batch experimental data, with the initial concentration of active methane‐oxidizing cells (X0a) also used as a model fitting parameter. The best estimate of k CH 4 was 2.2 g CH4/g cells‐d with excess copper available, with a single‐parameter 95% confidence interval of 2.0–2.4 mg/mg‐d. The joint 95% confidence region for k CH 4 and X0a is presented graphically. The half‐velocity coefficient (K italicS,CH 4) was 0.07 mg CH4/L with excess copper available and 0.47 mg CH4/L under copper limitation, with 95% confidence intervals of 0.02–0.11 and 0.35–0.59 mg/L, respectively. Unique values of the TCE rate coefficients kTCE and KS,TCE could not be determined because they were found to be highly correlated in the model fitting analysis. However, the ratio kTCE/KS,TCE and the TCE transformation capacity (TC) were well defined, with values of 0.35 L/mg‐day and 0.21 g TCE/g active cells, respectively, for cells transforming TCE in the absence of methane or supplemental formate. The single‐parameter 95% confidence intervals for kTCE/KS,TCE and TC were 0.27–0.43 L/mg‐d and 0.18–0.24 g TCE/g active cells, respectively. The joint 95% confidence regions for kTCE/KS,TCE and TC are presented graphically. © 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 53: 320–331, 1997.
Two technologies in combination, cometabolic bioremediation and in-well vapor stripping, were applied to reduce trichloroethylene (TCE) concentrations in groundwater at a contaminant source area without the need to pump contaminated groundwater to the surface for treatment. The vapor-stripping well reduced source TCE concentrations (as high as 6-9 mg/L) by over 95%. Effluent from the well then flowed to two bioremediation wells, where additional reductions of approximately 60% were achieved. TCE removal was extensively monitored (for research and not regulatory purposes) using an automated system that collected samples about every 45 min at 55 locations over an area of approximately 50 x 60 m2. During 4.5 months of system operation, total TCE mass removal was 8.1 kg, 7.1 kg of which resulted from in-well vapor stripping and 1.0 kg from biotreatment. The system reduced the average TCE concentration of about 3000 microg/L in the source-zone groundwater to about 250 microg/L in water leaving the treatment zone, effecting greater than 92% TCE removal. A 6 month rebound study after system operation ceased found TCE concentrations then increased significantly in the treatment zone due to diffusion from the fractured rock below and perhaps other processes, with mass increases of about 1.5 kg in the lower aquifer and 0.3 kg in the upper aquifer.
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