Spreadsheet Method for Evaluation of Biochemical Reaction Rate Coefficients and Their Uncertainties...

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Appl Environ Microbiol. 1998 June; 64(6): 20442050.

PMCID: PMC106276

Spreadsheet Method for Evaluation of BiochemicalReaction Rate Coefficients and Their Uncertainties byWeighted Nonlinear Least-Squares Analysis of theIntegrated Monod Equation

Laurence H. Smith,1,*Perry L. McCarty,2 and Peter K. Kitanidis2

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ABSTRACT

A convenient method for evaluation of biochemical reaction rate coefficientsand their uncertainties is described. The motivation for developing thismethod 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-squaresmethod provides accurate estimates of the rate coefficients and theiruncertainties from experimental data. Linearized methods that involveinversion of data are unreliable since several important assumptions oflinear regression are violated. Furthermore, when linearized methods areused, there is no basis for calculation of the uncertainties in the ratecoefficients. Uncertainty estimates are crucial to studies involvingcomparisons of rates for different organisms or environmental conditions.The spreadsheet method uses weighted least-squares analysis to determinethe best-fit values of the rate coefficients for the integrated Monod equation.

Although the integrated Monod equation is an implicit expression ofsubstrate concentration, weighted least-squares analysis can be employed tocalculate approximate differences in substrate concentration between modelpredictions and data. An iterative search routine in a spreadsheet programis utilized to search for the best-fit values of the coefficients by minimizingthe sum of squared weighted errors. The uncertainties in the best-fit valuesof the rate coefficients are calculated by an approximate method that canalso be implemented in a spreadsheet. The uncertainty method can be usedto calculate single-parameter (coefficient) confidence intervals, degrees of

correlation between parameters, and joint confidence regions for two ormore parameters. Example sets of calculations are presented for acetate
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utilization by a methanogenic mixed culture and trichloroethylenecometabolism by a methane-oxidizing mixed culture. An additionaladvantage of application of this method to the integrated Monod equationcompared with application of linearized methods is the economy of

obtaining rate coefficients from a single batch experiment or a few batchexperiments rather than having to obtain large numbers of initial ratemeasurements. However, when initial rate measurements are used, thismethod can still be used with greater reliability than linearized approaches.

The evaluation of bacterial and enzymatic reaction rates requiresrepresentative rate data and a valid method for fitting appropriate rateequations to the data. In addition, estimation of uncertainties in ratecoefficients is crucial for informed comparisons between cultures orenvironmental conditions. Nonlinear least-squares analysis of nonlinear

equations, such as the Monod and Michaelis-Menten equations, can provideaccurate estimates of rate coefficients and reliable estimates of theuncertainties in the coefficients.

Transformations of the nonlinear rate equations to linear forms, such asLineweaver-Burk and Eadie-Hofstee plots, are undesirable for numerousreasons that have been discussed repeatedly (3, 5, 9, 10). The deficiencies inthe use of linearized forms have been recognized for many years (6) buthave often been overlooked due to the time-consuming calculations andcomplexity of nonlinear least-squares analysis.

The integrated Monod equation is useful in many applications forevaluation of bacterial transformation rate coefficients. Coefficients can beevaluated from progress curves from a few batch experiments or even one

batch experiment of a reaction. This fact can be very important when dataare costly to obtain, such as in animal studies or human studies.

However, the integrated Monod equation is somewhat cumbersome to usebecause it is a nonlinear implicit expression for substrate and organismconcentrations. Weighted least-squares analysis is an approach that can be

used to minimize differences between experimental data and modelpredictions when it is necessary to use an implicit expression in the model.This paper describes a simple method for determining the best-fit values forrate coefficients in the Monod equation and their uncertainties by using

weighted least-squares analysis. The method is straightforward and isdesigned for easy implementation in a computer spreadsheet program. Asexamples, results from two rate studies were used together with anintegrated Monod equation weighted least-squares analysis to determinerate coefficients and their uncertainties. A simple example involving a dataset for acetate utilization by a methanogenic mixed culture is described. Asecond, more complex data set for trichloroethylene (TCE) cometabolism by
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a methane-oxidizing mixed culture is used to illustrate application of thismethod to cometabolism and verification of the method by comparison witha more rigorous numerical model. The experimental techniques used aredescribed elsewhere (7, 12).

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MATERIALS AND METHODSIntegrated Monod equation.

An integrated form of the Monod equation for utilization or cometabolismof a substrate in a batch reactor can be obtained. The Monod equation forthe substrate reaction rate in a bacterial culture is

1

where CL is the liquid-phase concentration of the rate-limiting substrate (inmilligrams per liter), tis time (in days), k is the maximum specific rate ofsubstrate utilization (in milligrams of substrate per milligram of active cellsper day),Xa is the concentration of active cells (in milligrams per liter),andKS is the half-velocity coefficient (in milligrams of substrate per liter).

The active cell concentration with growth substrate utilization can bedescribed by

2where Yis the cell yield coefficient (in grams of active cells per gram ofsubstrate) and the subscript 0 denotes time zero.Equations 1 and 2 can be combined to eliminateXa and can be integrated toobtain the integrated Monod equation for a growth substrate

3

Alternatively, when a volatile nongrowth substrate, such as TCE, iscometabolized in the absence of growth substrate, it is necessary to accountfor the gas-liquid partitioning of the substrate and the effect of TCEtransformation product toxicity. For gaseous substrates, a mass balance ona closed system gives

4whereMis the mass of substrate present (in milligrams), CG is the gas-

phase substrate concentration (in milligrams per liter), VL is the liquidvolume (in liters), and VG is the gas volume (in liters).
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Henrys Law (11) can be used to account for the distribution of the substratebetween the liquid and gas phases, if an assumption of gas-liquidequilibrium is valid

5whereHC is the Henrys constant of the substrate ().

Adequate mass transfer conditions can be verified by various means, suchas calculation of the Damkohler number (Da) or comparison of sampleshaving different biomass concentrations at a high substrate concentration.

An increase in biomass concentration results in a proportional increase inthe substrate reaction rate if the system is not mass transfer limited. Da isdefined as follows

6where kLa is the mass transfer rate coefficient (per hour). If Da is 1, then the system is reaction rate limited, and if Da is 1, the s ystem is masstransfer limited (2).The biomass concentration decreases due to TCE transformation producttoxicity (1, 8), and the mass of cells inactivated is proportional to the massof TCE transformed

7where TC is the TCE transformation capacity (in grams of TCE per gram ofactive cells) (1).Equations 1, 4, 5, and 7 can be combined to eliminate CG, CL, andXa and can

be integrated (when Da is 1) to obtain the integrated Monod equation for cometabolism of TCE (a volatile nongrowth substrate with transformationproduct toxicity)

8Equation 8 can also be applied to the simpler case of nonvolatile substrates

by settingHC equal to zero.The best estimates of the rate coefficients, such as k andKS, or otherconstants, such as CL0, can be determined by comparing model predictions

to observed values ofCL (orM) and tby using the known valuesfor Y, VL, VG,HC, etc., together with initial estimates of the coefficients that
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are sought in equation 3 or 8. The rate coefficients and other constantsestimated by fitting the integrated Monod equation to experimental data arereferred to as model fitting parameters. The estimates of the parameters aresuccessively revised through trial and error or other more sophisticated

searching techniques (9) to minimize the sum of the squared weightederrors (SSWE)

9where wi is an appropriate weighting factor, tiobs is the time of the ithobservation, and tipred is the tvalue predicted by the model for themeasured CL value, ofCLiobs.Ideally, it would be desirable to minimize the differences between themeasured and predicted CL values (CLiobs CLipred) because the errors in the

measurement ofCL are generally much larger than the errors in themeasurement oft. However, for implicit equations, such as equations 3 and8, only the differences between predicted and observed tvalues (tiobs tipred)can be calculated explicitly. The differences between predicted andobserved CL values can be estimated by multiplying tiobs tipred by the localslope of the substrate disappearance curve, CL/t. Therefore, we proposethat the logical weighting factor is the local slope of the substratedisappearance curve

10Given this weighting factor, the quantitywi(tiobs tipred) in equation 9 is

11which is the error in the predicted CLiobs. This approach provides an explicitapproximate method to minimize CLiobs CLipred in lieu of an explicitexpression for CL as a function oft.Uncertainty in fitted parameters.

The uncertainties in the best estimates of the model parameters can be

evaluated by an approximate method similar to that described elsewhere fornumerical modeling applications (4, 12).

The mean square error (MSE) of a fitting parameter, which is used tocalculate the 95% confidence interval of a given parameter estimate, iscalculated from the mean square fitting error and the sensitivity of themodel to the parameter. In general, the parameter uncertainty increases

with the mean square fitting error and decreases with increasing sensitivityto the parameter.

The mean square fitting error is
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12where n is the number of observations andp is the number of parameters

being determined.The model sensitivity to the parameters is evaluated by calculatingapproximate first derivatives of the model predictions with respect to theparameters. Two sets of model predictions are compared, in which oneparameter is varied by a small step. The sensitivity coefficient for the fittingparameter is evaluated at each observation as follows

13

where is the best estimate of , + is a nearby value of , and witipred( )and witipred( + ) are the weighted predictions for the times correspondingto an observed substrate concentration for the values and + ,respectively. Note that although the model predictions are not compared tothe observations in these calculations, the two sets of model predictions areevaluated at the points corresponding to the observations. Therefore, if theexperimental observations are made at substrate concentrations that aresensitive to the model parameters, high accuracy of parameter estimatescan be ensured.

In cases with only one fitting parameter, the MSE of the parameter is

14

where the denominator contains the sensitivity coefficients, squared andsummed over all observations.

The square root of the MSE is the standard deviation, and the approximate95% confidence interval for [()95%] is

15For cases involving more than one fitting parameter, the sensitivity of themodel predictions to the parameters is represented by ap p matrix,A,

wherep is the number of fitting parameters. Such a matrix is given inequation 16 for the case involving two parameters, such as 1 and 2. Thediagonal elements of the matrix (e.g.,A11 andA22) are the sensitivitycoefficients for the individual parameters, squared and summed over allobservation times. The off-diagonal elements (i.e.,A12 andA21) are products


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of the sensitivity coefficients for pairs of the parameters, summed over allobservation times.

16The single-parameter standard error for each parameter can be calculatedfrom the diagonal elements of the MSE matrix, V, which is related to thesensitivity matrix as follows

17

where 2

is given by equation 12 andA1

is the inverse ofA. The inverse of amatrix can be calculated by using the MINVERSE function in MicrosoftExcel (Microsoft Corp., Redmond, Wash.). The standard error of 1, for

example, is . The off-diagonal elements ofVcan be used to computethe correlation coefficients of the estimation error. The correlation

coefficient of 1 and 2 is V12/ .The single-parameter 95% confidence interval of 1 [(1)95%] is

18

The single-parameter confidence intervals should be used with cautionbecause they do not reflect the joint variability of all of the fittingparameters. Joint confidence regions are more informative.

The joint confidence region, which contains the set of parameters with aprobability of 95%, is the interior of an ellipse for the case with twoparameters. It is the region where values of the variables 1 and 2satisfy theinequality

19where 1 and 2 are the best estimates of the parameters. The value ofZisbased on theFdistribution for finite sample sizes and is a function of thenumbers of observations and parameters and the selected confidenceinterval probability as follows

20whereF(p, n-p, 1-) is taken from anF-distribution table (4) and 1- is thefractional probability (0.95 for the 95% probability case) (4). The joint 95%

confidence region can be plotted with the Goal Seek function in Excel byentering 1 values and using Goal Seek to find 2 values that satisfy equation
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19. Alternatively, a graphics program with a contour plotting function, suchas Mathematica, can be used.

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RESULTSApplication. (i) Acetate utilization.

Experimental data for acetate utilization by a methanogenic mixed cultureat 25C (7) were analyzed to determine the rate coefficients k andKS and theinitial substrate concentration CL0 by fitting equation 3 to acetate utilizationrate data by using a computer spreadsheet and weighted nonlinear least-squares analysis (equations 9 and 10). CL0 was used as a fitting parameter

because it was not known with greater certainty than the other data pointsand it would not be appropriate to force the best-fit curve through themeasured value ofCL0. The input value for Ywas determined in separate

experiments, andXa0 was estimated based on the combined results ofseveral experiments. All of the calculations were done on an AppleMacintosh computer (Apple Computer, Cupertino, Calif.) by using aspreadsheet program (Microsoft Excel 4.0). The joint 95% confidenceregions were plotted with Mathematica 2.2 for Macintosh (WolframResearch Inc., Champaign, Ill.).

An example of a spreadsheet for fitting the integrated Monod equation tothe data is shown in Table Table1. 1. The values of the rate coefficients andother constants, some of which were used as fitting parameters, are given in

rows 1 through 5. The values of the fitting parameters shown were initialguesses that were subsequently changed by the program as the model wasfitted to the experimental data. The experimental data are listed in columns

A and B, in order of decreasing CL. Column C contains the calculated tvaluefor each observed CL value (equation 3). The calculated values are shown inTable Table1 1 rather than the equations. Column D contains the CL valuesused to calculate the values in column C. These CL values are the same as the

values in column B except for the first value, which is a variable fittingparameter. The biomass concentrations were calculated (column E) but

were not required for the purpose of curve fitting. Additional modelpredictions used to calculate the weighting factors are shown in columns Fand G; the CL values in column G are slightly lower (0.1 mg/liter lower) thanthe values in column D, and the tvalues in column F were calculated fromthe values in column G by using equation 3. Column H shows the local slopeof the model curve, which was calculated for each observation by using thetwo sets of model predictions; for example, the value at H11 (column H, row11) equals (D11G11)/(C11F11). The differences between model andpredicted tvalues (column A column C) were calculated (column I), andthe results were multiplied by the weighting factors (column H) to give the

weighted errors (column J). The weighted errors were squared (column K),and the squared weighted errors were summed (cell K36).
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The model was fitted to the data by using the Solver function under theFormula menu in Excel to adjust the parameter estimates to minimize theSSWE (cell K36). The best fit was obtained with Solver, which uses aniterative search for the parameter values that yielded the minimum SSWE,

and the initial estimates of the parameters were automatically replaced bythe best estimates in rows 2, 3, and 5. Solver can search quickly for amaximum, minimum, or specified value for any selected cell by varying the

values for one or more other selected cells in a spreadsheet. It is alsopossible to define limits for the values of the variables, such asKS > 0, ifSolver gives unrealistic results.The best estimates of the rate coefficients k (7.4 day1) andKS (23.2mg/liter), obtained from fitting equation 3 to the data set in whichthe CL0 value was 50 mg/liter, yielded a model curve that is an excellent fitto all three data sets shown in Fig. Fig.1. 1. Since the data were obtained

from mixed-culture experiments, these rate coefficients represent theoverall values for the different organisms present and may not be directlycomparable to other previously published values.

FIG. 1Results of fitting the integrated Monod equation to acetate utilization rate

data by using the weighted nonlinear least-squares spreadsheet method.The model (heavy solid line) was fitted to one data set (), which resulted inthe following values: ...Only one data set was used to estimate the rate coefficients. The resulting

values for the coefficients were used to generate the two additional curvesfor comparison with the other two data sets (CL0 was used as the first datapoint for the two additional curves). The two data sets obtained with lowerinitial acetate concentrations (Fig. (Fig.1) 1) were not very sensitive to theparameters because CL was less thanKS throughout the experiments, and

therefore, these data did not add much information about theparameters k andKS. If all three data sets were used in the parameter fittingprocedure with five fitting parameters (k,KS, and a CL0 value for eachexperiment), the estimation accuracy would not improve. Robinson (9)gives a very good explanation of experimental design and substrateconcentrations that yield data that are sensitive to the parameters ofinterest.The uncertainty calculations were performed by using additionalspreadsheet calculations not shown in Table Table1. 1. The mean squarefitting error was calculated from the SSWE value in cell K36 andthe n andpvalues (25 data points and 3 parameters, respectively) by using
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equation 12. The sensitivity coefficients for each parameter weredetermined by calculating a new set of model points by using the bestestimates of all fitting parameters except the parameter of interest, which

was varied by a small increment. The sensitivity coefficient was evaluated at

each point by using equation 13, and the individual values were squared andsummed for use in equation 14 or for the diagonal elements of the matrix inequation 16. The off-diagonal elements of equation 16 were calculated bysumming the products of the individual sensitivity values. The resultingsensitivity coefficients obtained in equation 16 were used in the equation forthe 95% confidence region, equation 19.Since three fitting parameters were used in this case, the joint 95%confidence region was a three-dimensional ellipsoid which represented theregion containing k,KS, and CL0 with a probability of 95%. Three slices ofthis ellipsoid are shown in Fig. Fig.2. 2. The slices were taken in the

plane CL0 = L0 and CL0 =L0 CL0, where L0 is the best estimateofCL0 and CL0 is the standard error ofCL0, as defined in Materials andMethods. There were no points in the plane CL0 = L0 2 CL0 due to the lowcorrelation of uncertainty between CL0 and the other two parameters.

FIG. 2Joint 95% confidence region for k andKS for the data in Fig. Fig.1. 1. Slices ofthe three-dimensional ellipsoid were taken along the CL0 axis atL0 (largeellipse) and L0 CL0 (small ellipse; there ...(ii) TCE cometabolism.

TCE cometabolism rate data were obtained from batch experimentsperformed with resting cells (no methane or other carbon or energy source

was present) in a mixed culture that were transforming TCE. The apparatuswas a closed 2-liter vessel with a headspace present to provide oxygen fromair and for the convenience of headspace analysis of TCE. The methods are

described in detail elsewhere (12). Gas-liquid equilibrium was verified bymass transfer rate measurements. After the reaction was started by addingcells, the disappearance of TCE was monitored by periodically measuringthe TCE in the headspace and calculating the TCE mass present (Miobs) (Fig.(Fig.3). 3). The experimental methods used are described elsewhere (12).

FIG. 3

Comparison of model fits for TCE cometabolism data. Parameter estimateswere obtained by using the integrated Monod equation (), the numerical
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model ( ), and the numerical model with simplifying assumptions(). ...The integrated Monod equation for a nongrowth substrate with producttoxicity (equation 8) was fitted to the data in Fig. Fig.3. 3. For this data set, it

was not possible to obtain unique estimates for kTCE andKS,TCE by using thespreadsheet model because the coefficients were highly correlated. Manypairs ofkTCE andKS,TCE values that yielded a kTCE/KS,TCE ratio of 0.31liter/mg/day were found that had SSWE values very near the sameminimum value. Therefore, the kTCE/KS,TCE ratio was used as a fittingparameter as an alternative approach. The kTCE/KS,TCE and TC values were

varied by Solver until the minimum SSWE was found. The kTCE/KS,TCE valuewas varied by leaving the kTCE value fixed and allowing Solver to varytheKS,TCE value. As shown in Fig. Fig.3, 3, the spreadsheet fitting methodgave a model curve that is an excellent fit to the data. As stated above for the

acetate utilization data, these results for mixed cultures represent overallrates for several organisms and might not be directly comparable to otherpreviously published values.In order to test the reliability of this method, the best estimates and joint95% confidence regions obtained from the integrated Monod spreadsheetanalysis were compared to estimates and confidence regions obtained byfitting a more rigorous numerical model to the data (12). The numericalmodel used included a fourth-order Runge-Kutta solution for the system ofdifferential equations describing the experimental conditions, including

Monod kinetics, changes in active organism concentration due to producttoxicity and endogenous decay, gas-liquid mass transfer of TCE, and passivelosses of TCE from the apparatus. The numerical model calculated fittingerrors (Miobs Mipred)2 at the times corresponding to each data point.The estimates ofkTCE/KS,TCE and TC values obtained by using the integratedMonod spreadsheet analysis method were not significantly different thanthose obtained by using the more rigorous numerical model, as shown inTable Table2 2 and Fig Fig3 3 and and4. 4. For both models, it was assumedthat the initial concentration of active organisms was equal to 20% of the

initial total suspended-solids concentration, based on the results ofmethane utilization rate experiments and modeling results obtained withthe mixed culture, as described previously (12).

FIG. 4Joint 95% confidence regions for kTCE/KS,TCE and TC for the TCE

cometabolism results shown in Fig. Fig.3, 3, using the numerical model (
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/figure/F3/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/figure/F3/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/#B12http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/table/T2/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/figure/F3/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/figure/F4/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/#B12http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/figure/F4/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/figure/F3/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/figure/F4/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/figure/F3/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/figure/F3/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/#B12http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/table/T2/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/figure/F3/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/figure/F4/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/#B12http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/figure/F4/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/figure/F3/


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significant, and the integrated Monod equation weighted least-squaresapproach might then not provide reliable estimates of the rate coefficients.

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DISCUSSION

The integrated Monod equation method is a simple procedure for obtainingreliable estimates of microbiological reaction rate coefficients as long as therequirements of the underlying assumptions are met. The fraction of cellslost over the course of the experiment due to endogenous decay of cellsmust be small. Changes in cell concentration due to growth or producttoxicity do not limit the applicability of this method because they can beincluded in the integrated form of the equation. For volatile substrates, theexperimental system must be at gas-liquid equilibrium and the rate of

passive loss of substrates, such as TCE, must be low relative to thetransformation rate.

The weighting method described in this paper has a rational statisticalbasis. The difference between the observed and predicted substrateconcentrations is estimated by multiplying the slope of the substratedisappearance curve by the difference between measured and predictedsample times. This approach is appropriate because it approximates theerror in the measurement of substrate concentration and such errors areusually larger than errors in time measurement in biochemical rate

experiments.The use of linearized forms of the Monod equation, such as Lineweaver-Burk and Eadie-Hofstee plots, should be avoided because several criticalassumptions of linear regression are violated by these methods, which leadsto inaccurate parameter estimates and a lack of any way to evaluate theuncertainties in the parameter estimates (9). Because of the availability ofcomputers and software that are easy to use, the use of linearizedapproaches is no longer justified. The weighted nonlinear least-squaresmethod described in this paper is very straightforward and easy to apply

with a computer spreadsheet program.Another important advantage of the integrated Monod equation methodover linearized methods is the economy of obtaining rate coefficients from asingle batch experiment or a few batch experiments rather than having toobtain large numbers of initial rate measurements. However, theuncertainty calculations from a single experiment do not reflect the run-to-run variability in parameter estimates. When initial rate measurements areused, the rate coefficients and their uncertainties can be determined moreprecisely from nonlinear least-squares analysis (e.g., applied to a plot of

dCL/dtversus CL) than from linearized plots of inverted data.
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Statistical software packages for personal computers may also allowconvenient application of the integrated Monod equation if they provide for

weighting in the nonlinear least-squares fitting and uncertainty analysis. Inthe absence of a weighting factor, a nonlinear least-squares approach would

minimize the differences between measured and predicted tvalues ratherthan CL values, which would be inappropriate for an implicit expression,such as the integrated Monod equation. Users should be careful not to usenonlinear least-squares fitting software without appropriate provisions,such as a weighting factor, when they use implicit equations, such as theintegrated Monod equation.

Comparison with a rigorous numerical model validated the results obtainedby the integrated Monod equation spreadsheet method. One importantadvantage of the integrated Monod equation spreadsheet method over a

numerical model is the flexibility that it offers to use the initial substrateconcentration as a fitting parameter, which is desirable when the initialsubstrate concentration data point is known with certainty equal to that ofall other data points.

Rate experiments can be designed to maximize the informative value of theresults. Coefficients, such asY, that are measured separately and used asconstants in the parameter fitting process should be measured as accuratelyas possible. Accurate measurements of other constants, such as the initialactive biomass concentration, are crucial for minimizing the uncertainties in

the fitted parameters. The number of fitting parameters should beminimized, and the concentrations of biomass and substrate should beselected to yield the most information about the rate parameters of interest.The use of sensitivity analysis to design experiments has been described byRobinson (9).

The method described above for calculating the uncertainties in ratecoefficients makes great use of the information available from the data

without requiring the use of sophisticated mathematics. When the initialsubstrate concentration is not used as a fitting parameter, this method can

be used to describe a confidence region precisely based on numerous datasets if desired. However, when the initial substrate concentration is used asa fitting parameter, each data set included in the analysis adds one fittingparameter, and both the fitting calculations and the graphicalrepresentation of the confidence region become difficult with three or moredata sets. In this situation, it may be preferable to use the fitting techniquedescribed in this paper to determine the best-fit value of each ratecoefficient for each experiment and use the mean and standard deviation ofthe best-fit values from all of the experiments to evaluate the confidence

interval for each rate coefficient.
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Copies of the spreadsheet are available on diskette from the correspondingauthor.

TABLE 1

Example spreadsheet for fitting the integrated Monod equation to acetateutilization ratedataa

ad, day; L, liter.

TABLE 2

Best estimates, approximate 95% confidence intervals, and correlationcoefficients for TCE cometabolism rate data shown in Fig. 3
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TCE cometabolism

rate coefficient

Best estimate of

previous

numerical modela

Present study

Best estimate of

numerical model

assuming no cell decay

and no TCE losses

Integrated Monod equation

weighted least-squares

analysis

Best estimate

Correlation

coefficients

k/KS TC

k/KS(liter/mg/day) 0.35 0.08 0.35 0.08 0.31 0.06 1

TC (g/g) 0.21 0.03 0.23 0.02 0.23 0.02 0.63 1aData from reference 12.

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ACKNOWLEDGMENTS

This study was supported by the Gas Research Institute and theWestinghouse Savannah River Corporation through the U.S. Environmental

Protection Agency-sponsored Western Region Hazardous SubstanceResearch Center.

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REFERENCES1. Alvarez-Cohen L, McCarty P L. A cometabolic biotransformation modelfor halogenated aliphatic compounds exhibiting product toxicity. EnvironSci Technol. 1991;25:13811387.2. Bailey J E, Ollis D F. Biochemical engineering fundamentals. New York,

N.Y: McGraw-Hill Book Company; 1986.3. Dowd J E, Riggs D S. A comparison of estimates of Michaelis-Mentenkinetic constants from various linear transformations. J BiolChem. 1965;240:863869. [PubMed]4. Draper N R, Smith H. Applied regression analysis. New York, N.Y: John

Wiley & Sons; 1981.5. Eisenthal R, Cornish-Bowden A. The direct linear plot: a new graphicalprocedure for estimating enzyme kinetic parameters. BiochemJ. 1974;139:715720. [PMC free article] [PubMed]
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/#B12http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/#ui-ncbiinpagenav-2http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/#ui-ncbiinpagenav-2http://www.ncbi.nlm.nih.gov/pubmed/14275146http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1166335/http://www.ncbi.nlm.nih.gov/pubmed/4854723http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/#B12http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/#ui-ncbiinpagenav-2http://www.ncbi.nlm.nih.gov/pmc/articles/PMC106276/#ui-ncbiinpagenav-2http://www.ncbi.nlm.nih.gov/pubmed/14275146http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1166335/http://www.ncbi.nlm.nih.gov/pubmed/4854723


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6. Hanes C S. Studies on plant amylases. I. The effect of starchconcentration upon the velocity of hydrolysis by the amylase of germinated

barley. Biochem J. 1932;26:14061421.7. Montgomery M S. Kinetics of methane fermentation in anaerobic

biofilms. Ph.D. thesis. Stanford, Calif: Stanford University; 1983.8. Oldenhuis R, Oedzes J Y, van der Waarde J J, Janssen D B. Kinetics ofchlorinated hydrocarbon degradation byMethylosinus trichosporium OB3band toxicity of trichloroethylene. Appl Environ Microbiol. 1991;57:714. [PMC free article] [PubMed]9. Robinson J A. Determining microbial kinetic parameters using nonlinearregression analysis. Adv Microb Ecol. 1985;8:61114.10. Robinson J A, Charaklis W G. Simultaneous estimation of Vmax, Km, andthe rate of endogenous substrate production (R) from substrate depletiondata. Microb Ecol. 1984;10:165178.

11. Sawyer C N, McCarty P L, Parkin G F. Chemistry for environmentalengineering. New York, N.Y: McGraw-Hill, Inc.; 1994.12. Smith L H, Kitanidis P K, McCarty P L. Numerical modeling anduncertainties in rate coefficients for methane utilization and TCEcometabolism by a methane-oxidizing mixed culture. BiotechnolBioeng.1997;53:320331.

Articles from Applied and Environmental Microbiology are provided here courtesy ofAmerican

Society for Microbiology (ASM)
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC182657/http://www.ncbi.nlm.nih.gov/pubmed/2036023http://www.ncbi.nlm.nih.gov/pmc/articles/PMC182657/http://www.ncbi.nlm.nih.gov/pubmed/2036023

Spreadsheet Method for Evaluation of Biochemical Reaction Rate Coefficients and Their Uncertainties...

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