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Appl Environ Microbiol, June 1998, p. 2044-2050, Vol. 64, No. 6
0099-2240/98/$04.00+0
Copyright © 1998, American Society for Microbiology. All rights reserved.
Spreadsheet Method for Evaluation of Biochemical Reaction Rate
Coefficients and Their Uncertainties by Weighted Nonlinear
Least-Squares Analysis of the Integrated Monod Equation
Laurence H.
Smith,1,*
Perry L.
McCarty,2 and
Peter K.
Kitanidis2
Institute of Technology and Engineering,
Massey University, Palmerston North, New
Zealand,1 and
Department of Civil
and Environmental Engineering, Stanford University, Stanford,
California 94305-40202
Received 24 September 1997/Accepted 19 March 1998
 |
ABSTRACT |
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.
| |
INTRODUCTION |
The evaluation of bacterial and
enzymatic reaction rates requires representative rate data and a valid
method for fitting appropriate rate equations to the data. In addition,
estimation of uncertainties in rate coefficients is crucial for
informed comparisons between cultures or environmental conditions.
Nonlinear least-squares analysis of nonlinear equations, such as the
Monod and Michaelis-Menten equations, can provide accurate estimates of
rate coefficients and reliable estimates of the uncertainties in the
coefficients.
Transformations of the nonlinear rate equations to linear forms, such
as Lineweaver-Burk and Eadie-Hofstee plots, are undesirable for
numerous reasons that have been discussed repeatedly (3, 5, 9,
10). The deficiencies in the use of linearized forms have been
recognized for many years (6) but have often been overlooked
due to the time-consuming calculations and complexity of nonlinear
least-squares analysis.
The integrated Monod equation is useful in many applications for
evaluation of bacterial transformation rate coefficients. Coefficients
can be evaluated from progress curves from a few batch experiments or
even one batch experiment of a reaction. This fact can be very
important when data are costly to obtain, such as in animal studies or
human studies.
However, the integrated Monod equation is somewhat cumbersome to
use because it is a nonlinear implicit expression for substrate and
organism concentrations. Weighted least-squares analysis is an
approach that can be used to minimize differences between experimental data and model predictions when it is necessary to use an implicit expression in the model. This paper describes a simple method for
determining the best-fit values for rate coefficients in the Monod
equation and their uncertainties by using weighted least-squares analysis. The method is straightforward and is designed for easy implementation in a computer spreadsheet program. As examples, results
from two rate studies were used together with an integrated Monod
equation weighted least-squares analysis to determine rate coefficients and their uncertainties. A simple example involving a data
set for acetate utilization by a methanogenic mixed culture is
described. A second, more complex data set for trichloroethylene (TCE)
cometabolism by a methane-oxidizing mixed culture is used to illustrate
application of this method to cometabolism and verification of the
method by comparison with a more rigorous numerical model. The
experimental techniques used are described elsewhere (7, 12).
| |
MATERIALS AND METHODS |
Integrated Monod equation.
An integrated form of the Monod
equation for utilization or cometabolism of a substrate in a batch
reactor can be obtained. The Monod equation for the substrate reaction
rate in a bacterial culture is
|
(1)
|
where CL is the liquid-phase
concentration of the rate-limiting substrate (in milligrams per liter),
t is time (in days), k is the maximum specific
rate of substrate utilization (in milligrams of substrate per milligram
of active cells per day), Xa is the concentration of active cells (in milligrams per liter), and
KS is the half-velocity coefficient (in
milligrams of substrate per liter).
The active cell concentration with growth substrate utilization can be
described by
|
(2)
|
where
Y is the cell yield coefficient (in grams of
active cells per gram of substrate) and the subscript 0 denotes time
zero.
Equations
1 and
2 can be combined to eliminate
Xa and can be integrated to obtain the
integrated Monod equation for a growth
substrate
|
(3)
|
Alternatively, when a volatile nongrowth substrate, such as TCE,
is cometabolized in the absence of growth substrate, it
is necessary to
account for the gas-liquid partitioning of the
substrate and the effect
of TCE transformation product toxicity.
For gaseous substrates, a mass
balance on a closed system gives
|
(4)
|
where
M is the mass of substrate present (in
milligrams),
CG is the gas-phase substrate
concentration (in milligrams per
liter),
VL is
the liquid volume (in liters), and
VG is the gas
volume (in liters).
Henry's Law (
11) can be used to account for the
distribution of the substrate between the liquid and gas phases, if an
assumption
of gas-liquid equilibrium is valid
|
(5)
|
where
HC is the Henry's constant of the
substrate (
).
Adequate mass transfer conditions can be verified by various means,
such as calculation of the Damkohler number (Da) or comparison
of
samples having different biomass concentrations at a high substrate
concentration. An increase in biomass concentration results in
a
proportional increase in the substrate reaction rate if the
system is
not mass transfer limited. Da is defined as follows
|
(6)
|
where
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 system is mass transfer limited
(
2).
The biomass concentration decreases due to TCE transformation product
toxicity (
1,
8), and the mass of cells inactivated
is
proportional to the mass of TCE transformed
|
(7)
|
where
TC is the TCE transformation
capacity (in grams of TCE per gram of active cells) (
1).
Equations
1,
4,
5, and
7 can be combined to eliminate
CG,
CL, and
Xa and can be integrated (when Da is
1) to
obtain the integrated
Monod equation for cometabolism of TCE (a
volatile nongrowth substrate
with transformation product toxicity)
|
(8)
|
Equation
8 can also be applied to the simpler case of
nonvolatile substrates by setting
HC equal to
zero.
The best estimates of the rate coefficients, such as
k and
KS, or other constants, such as
CL0, can be determined
by comparing model
predictions to observed values of
CL (or
M)
and
t by using the known values for
Y,
VL,
VG,
HC, etc., together
with initial estimates of the
coefficients that are sought in
equation 3 or 8. The rate coefficients
and other constants estimated
by fitting the integrated Monod equation
to experimental data
are referred to as model fitting parameters. The
estimates of
the parameters are successively revised through trial and
error
or other more sophisticated searching techniques (
9)
to minimize
the sum of the squared weighted errors (SSWE)
![<UP>SSWE</UP>=<LIM><OP>∑</OP><LL>i=1</LL><UL>n</UL></LIM> [w<SUB>i</SUB>(t<SUP><UP>obs</UP></SUP><SUB>i</SUB>−t<SUP><UP>pred</UP></SUP><SUB>i</SUB>)]<SUP>2</SUP>](/content/vol64/issue6/fulltext/2044/img011.gif)
|
(9)
|
where
wi is an appropriate weighting
factor,
tiobs is the time of the
ith observation, and
tipred is
the
t value predicted by the model for the measured
CL value,
of
CLiobs.
Ideally, it would be desirable to minimize the differences between the
measured and predicted
CL values
(
CLiobs CLipred) because the errors in the
measurement of
CL are generally much
larger than
the errors in the measurement of
t. However, for implicit
equations, such as equations 3 and 8, only the differences between
predicted and observed
t values
(
tiobs tipred)
can be calculated explicitly. The differences between predicted
and
observed
CL values can be estimated by
multiplying
tiobs tipred by the local slope of the substrate
disappearance curve,
CL/t.
Therefore, we
propose that the logical weighting factor is the
local slope of the
substrate disappearance curve
|
(10)
|
Given this weighting factor, the quantity
wi(
tiobs tipred) in equation 9 is
|
(11)
|
which is the error in the predicted
CLiobs. This approach provides an explicit
approximate method to minimize
CLiobs CLipred in lieu of an explicit expression
for
CL as a function of
t.
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 for numerical
modeling applications (4, 12).
The mean square error (MSE) of a fitting parameter, which is used to
calculate the 95% confidence interval of a given parameter
estimate,
is calculated from the mean square fitting error and
the sensitivity of
the model to the parameter. In general, the
parameter uncertainty
increases with the mean square fitting error
and decreases with
increasing sensitivity to the parameter.
The mean square fitting error is
![&sfgr;<SUP>2</SUP>=<FR><NU>1</NU><DE>n−p</DE></FR> <LIM><OP>∑</OP><LL>i=1</LL><UL>n</UL></LIM> [w<SUB>i</SUB>(t<SUP><UP>obs</UP></SUP><SUB>i</SUB>−t<SUP><UP>pred</UP></SUP><SUB>i</SUB>)]<SUP>2</SUP>](/content/vol64/issue6/fulltext/2044/img014.gif)
|
(12)
|
where
n is the number of observations and
p is the number of parameters being determined.
The model sensitivity to the parameters is evaluated by calculating
approximate first derivatives of the model predictions
with respect to
the parameters. Two sets of model predictions
are compared, in which
one parameter is varied by a small step.
The sensitivity coefficient
for the fitting parameter
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 corresponding
to an observed
substrate concentration for the
values
and
+
, respectively. Note that although the model predictions
are not
compared to the observations in these calculations, the
two sets of
model predictions are evaluated at the points corresponding
to the
observations. Therefore, if the experimental observations
are made at
substrate concentrations that are sensitive to the
model parameters,
high accuracy of parameter estimates can be
ensured.
In cases with only one fitting parameter, the MSE of the parameter is
|
(14)
|
where the denominator contains the sensitivity coefficients,
squared and summed over all observations.
The square root of the MSE is the standard deviation, and the
approximate 95% confidence interval for
[(
)
95%]
is
|
(15)
|
For cases involving more than one fitting parameter, the
sensitivity of the model predictions to the parameters is represented
by a
p ×
p matrix,
A, where
p is the number of fitting parameters.
Such a matrix is
given in equation 16 for the case involving two
parameters, such as
1 and
2. The diagonal elements of the
matrix
(e.g.,
A11 and
A22) are the sensitivity coefficients for the
individual
parameters, squared and summed over all observation times.
The
off-diagonal elements (i.e.,
A12 and
A21) are products of the
sensitivity
coefficients for pairs of the parameters, summed over
all observation
times.
|
(16)
|
The single-parameter standard error for each parameter can be
calculated from the diagonal elements of the MSE matrix,
V,
which is related to the sensitivity matrix as follows
|
(17)
|
where
2 is given by equation 12 and
A
1 is the inverse of
A. The inverse
of a matrix can be calculated by using the MINVERSE
function in
Microsoft Excel (Microsoft Corp., Redmond, Wash.).
The standard error
of
1, for example, is
.
The off-diagonal elements of
V can be used to compute the 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
caution because they do not reflect the joint variability of
all of the
fitting parameters. Joint confidence regions are more
informative.
The joint confidence region, which contains the set of parameters with
a probability of 95%, is the interior of an ellipse
for the case with
two parameters. It is the region where values
of the variables
1 and
2 satisfy the inequality
![<FR><NU>1</NU><DE>&sfgr;<SUP>2</SUP></DE></FR> [A<SUB>11</SUB>(&thgr;<SUB>1</SUB>−<A><AC><UP>&thgr;</UP></AC><AC>ˆ</AC></A><SUB>1</SUB>)<SUP>2</SUP>+A<SUB>22</SUB>(&thgr;<SUB>2</SUB>−<A><AC><UP>&thgr;</UP></AC><AC>ˆ</AC></A><SUB>2</SUB>)<SUP>2</SUP>+2A<SUB>12</SUB>(&thgr;<SUB>1</SUB>−<A><AC><UP>&thgr;</UP></AC><AC>ˆ</AC></A><SUB>1</SUB>)(&thgr;<SUB>2</SUB>−<A><AC><UP>&thgr;</UP></AC><AC>ˆ</AC></A><SUB>2</SUB>)]≤Z](/content/vol64/issue6/fulltext/2044/img024.gif)
|
(19)
|
where
1 and
2 are the
best estimates of the parameters. The value of
Z is based on
the
F distribution for finite sample
sizes and is a function
of the numbers of observations and parameters
and the selected
confidence interval probability as follows
|
(20)
|
where
F(
p,
n-p, 1-
) is taken
from an
F-distribution table (
4) and 1-
is the
fractional probability (0.95 for the 95%
probability case)
(
4). The joint 95% confidence region can
be plotted with
the Goal Seek function in Excel by entering
1 values and
using Goal Seek to find
2 values that satisfy equation
19. Alternatively, a graphics program with a contour plotting
function,
such as Mathematica, can be used.
| |
RESULTS |
Application. (i) Acetate utilization.
Experimental data for
acetate utilization by a methanogenic mixed culture at 25°C
(7) were analyzed to determine the rate coefficients
k and KS and the initial substrate
concentration CL0 by fitting equation 3 to
acetate utilization rate 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 points and it
would not be appropriate to force the best-fit curve through the
measured value of CL0. The input value for
Y was determined in separate experiments, and
Xa0 was estimated based on the combined results
of several experiments. All of the calculations were done on an Apple Macintosh computer (Apple Computer, Cupertino, Calif.) by using a
spreadsheet program (Microsoft Excel 4.0). The joint 95% confidence regions were plotted with Mathematica 2.2 for Macintosh (Wolfram Research Inc., Champaign, Ill.).
An example of a spreadsheet for fitting the integrated Monod equation
to the data is shown in Table
1. The
values of the
rate coefficients and other constants, some of which were
used
as fitting parameters, are given in rows 1 through 5. The values
of the fitting parameters shown were initial guesses that were
subsequently changed by the program as the model was fitted to
the
experimental data. The experimental data are listed in columns
A and B,
in order of decreasing
CL. Column C contains the
calculated
t value for each observed
CL value (equation 3). The calculated
values are
shown in Table
1 rather than the equations. Column
D contains the
CL values used 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 fitting
parameter. The
biomass concentrations were calculated (column E) but
were not
required for the purpose of curve fitting. Additional model
predictions
used to calculate the weighting factors are shown in
columns F
and G; the
CL values in column G are
slightly lower (0.1 mg/liter
lower) than the values in column D, and
the
t values in column
F were calculated from the values in
column G by using equation
3. Column H shows the local slope of the
model curve, which was
calculated for each observation by using the two
sets of model
predictions; for example, the value at H11 (column H, row
11)
equals (D11
G11)/(C11
F11). The differences between model and
predicted
t values (column A
column C) were
calculated (column
I), and the 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).
The model was fitted to the data by using the Solver function under the
Formula menu in Excel to adjust the parameter estimates
to minimize the
SSWE (cell K36). The best fit was obtained with
Solver, which uses an
iterative search for the parameter values
that yielded the minimum
SSWE, and the initial estimates of the
parameters were automatically
replaced by the best estimates in
rows 2, 3, and 5. Solver can search
quickly for a maximum, minimum,
or specified value for any selected
cell by varying the values
for one or more other selected cells in a
spreadsheet. It is also
possible to define limits for the values of the
variables, such
as
KS > 0, if Solver gives
unrealistic results.
The best estimates of the rate coefficients
k (7.4 day
1) and
KS (23.2 mg/liter),
obtained from fitting equation 3 to the data
set in which the
CL0 value was 50 mg/liter, yielded a
model curve
that is an excellent fit to all three data sets shown
in Fig.
1. Since the data were obtained from
mixed-culture experiments,
these rate coefficients represent the
overall values for the different
organisms present and may not be
directly comparable to other
previously published values.
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FIG. 1.
Results 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 in the following values:
k = 7.4 day1, KS = 23.2 mg/liter, and CL0 = 53.5 mg/liter. Model
curves obtained with the same k and
KS values are shown for other data ( and  )
for comparison. L, liter; d, day.
|
|
Only one data set was used to estimate the rate coefficients. The
resulting values for the coefficients were used to generate
the two
additional curves for comparison with the other two data
sets
(
CL0 was used as the first data point for the
two
additional curves). The two data sets obtained with lower initial
acetate concentrations (Fig.
1) were not very sensitive to the
parameters because
CL was less than
KS throughout the experiments,
and therefore,
these data did not add much information about the
parameters
k and
KS. If all three data sets were
used in the parameter
fitting procedure with five fitting parameters
(
k,
KS, and a
CL0 value for each experiment), the estimation
accuracy would not
improve. Robinson (
9) gives a very good
explanation of experimental
design and substrate concentrations that
yield data that are sensitive
to the parameters of interest.
The uncertainty calculations were performed by using additional
spreadsheet calculations not shown in Table
1. The mean square
fitting
error was calculated from the SSWE value in cell K36 and
the
n and
p values (25 data points and 3 parameters,
respectively)
by using equation 12. The sensitivity coefficients for
each parameter
were determined by calculating a new set of model points
by using
the best estimates 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 and summed for use in
equation
14 or for the diagonal elements of the matrix in equation 16.
The off-diagonal elements of equation 16 were calculated by summing
the
products of the individual sensitivity values. The resulting
sensitivity coefficients obtained in equation 16 were used in
the
equation for the 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
the region containing
k,
KS, and
CL0 with
a probability of 95%. Three slices of
this ellipsoid are shown
in Fig.
2. The
slices were taken in the plane
CL0 =
L0 and
CL0 =
L0 ±
CL0,
where
L0 is the best estimate of
CL0 and
CL0 is the standard error of
CL0, as defined in Materials
and Methods. There
were no points in the plane
CL0 =
L0 ± 2
CL0 due to the low correlation
of
uncertainty between
CL0 and the other two
parameters.
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FIG. 2.
Joint 95% confidence region for k and
KS for the data in Fig. 1. Slices of the
three-dimensional ellipsoid were taken along the
CL0 axis at
L0 (large ellipse) and
L0 ± CL0 (small ellipse; there were two
ellipses, but they had the same size and location). L, liter; d, day.
|
|
(ii) TCE cometabolism.
TCE cometabolism rate data were
obtained from batch experiments performed with resting cells (no
methane or other carbon or energy source was present) in a mixed
culture that were transforming TCE. The apparatus was a closed 2-liter
vessel with a headspace present to provide oxygen from air and for the
convenience of headspace analysis of TCE. The methods are described in
detail elsewhere (12). Gas-liquid equilibrium was verified
by mass transfer rate measurements. After the reaction was started by adding cells, the disappearance of TCE was monitored by periodically measuring the TCE in the headspace and calculating the TCE mass present
(Miobs) (Fig.
3). The experimental methods used are
described elsewhere (12).
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FIG. 3.
Comparison of model fits for TCE cometabolism data.
Parameter estimates were obtained by using the integrated Monod
equation ( ), the numerical model ( ), and the numerical model
with simplifying assumptions (---). Two of the curves are shown with
the same symbol, because they are indistinguishable at the scale of
this graph.
|
|
The integrated Monod equation for a nongrowth substrate with product
toxicity (equation 8) was fitted to the data in Fig.
3. For this data
set, it was not possible to obtain unique estimates
for
kTCE and
KS,TCE by using
the spreadsheet model because
the coefficients were highly correlated.
Many pairs of
kTCE and
KS,TCE values that yielded a
kTCE/
KS,TCE ratio
of 0.31 liter/mg/day were found that had SSWE values very near
the same minimum
value. Therefore, the
kTCE/
KS,TCE ratio
was
used as a fitting parameter 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 value was
varied by leaving the
kTCE value fixed and
allowing Solver to
vary the
KS,TCE value. As
shown in Fig.
3, the spreadsheet
fitting method gave 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 overall
rates for several
organisms and might not be directly comparable to
other previously
published values.
In order to test the reliability of this method, the best estimates and
joint 95% confidence regions obtained from the integrated
Monod
spreadsheet analysis were compared to estimates and confidence
regions
obtained by fitting a more rigorous numerical model to
the data
(
12). The numerical model used included a fourth-order
Runge-Kutta solution for the system of differential equations
describing the experimental conditions, including Monod kinetics,
changes in active organism concentration due to product toxicity
and
endogenous decay, gas-liquid mass transfer of TCE, and passive
losses
of TCE from the apparatus. The numerical model calculated
fitting
errors (
Miobs Mipred)
2 at the times
corresponding to each data point.
The estimates of
kTCE/
KS,TCE and
TC values obtained by using the integrated Monod
spreadsheet analysis method were
not significantly different than those
obtained by using the more
rigorous numerical model, as shown in Table
2 and Fig
3 and
4.
For both models, it was assumed that
the initial concentration
of active organisms was equal to 20% of the
initial total suspended-solids
concentration, based on the results of
methane utilization rate
experiments and modeling results obtained with
the mixed culture,
as described previously (
12).
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TABLE 2.
Best estimates, approximate 95% confidence intervals,
and correlation coefficients for TCE cometabolism rate data shown
in Fig. 3
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|
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FIG. 4.
Joint 95% confidence regions for
kTCE/KS,TCE and
TC for the TCE cometabolism results shown in
Fig. 3, using the numerical model ( - ), the numerical model with
simplifying assumptions (), and integrated Monod equation weighted
least-squares analysis (---). L, liter; d, day.
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|
The most significant differences between the numerical model and the
integrated Monod equation weighted least-squares approach
are the error
calculation methods and the assumptions regarding
TCE losses,
endogenous decay of the organisms, and gas-liquid
equilibrium. The TCE
losses and endogenous decay were neglected,
and gas-liquid equilibrium
was assumed in the integrated Monod
equation approach because the
equation cannot be integrated if
these processes are included. In many
cases, such assumptions
are justified. The dissolved TCE concentrations
predicted by the
numerical model were within 1% of the equilibrium
concentrations,
and therefore it was known a priori that the
equilibrium assumption
was valid for this data set. Justification for
this assumption
would be necessary in other applications of this
approach.
In order to compare the two models more directly, the numerical model
was fitted to the data assuming that there were no TCE
losses from the
reactor and there was no endogenous cell decay
(Table
2). A comparison
of the estimates in Table
2 indicates
that the assumption concerning no
TCE losses and the different
methods of error calculations had only
slight effects, while the
assumption that there was no endogenous decay
had an insignificant
effect on the analysis of this data set. The
integrated Monod
equation weighted least-squares analysis and numerical
model with
simplifying assumptions yielded a higher
TC estimate than the
complete numerical model
because all of the TCE removed was considered
biodegraded in the former
two cases, which yielded a higher
TC estimate
than the numerical model, which correctly accounted for
the TCE removal
due to passive losses. The estimated
kTCE/
KS,TCE value
obtained from the integrated Monod equation weighted least-squares
approach was somewhat lower, although the difference was not
statistically
significant. The slight difference was due to the
different methods
of calculating fitting errors and was verified by
comparing the
calculated SSWE (equation 9) for both
kTCE/
KS,TCE estimates
with the true errors,
,
calculated by trial and error application of equation 8 for
each
kTCE/
KS,TCE
estimate.
The integrated Monod equation weighted least-squares analysis method is
a good approximation of the more rigorous numerical
model for this data
set because the best estimates of each model
were within the bounds of
the joint 95% confidence region of the
other model (Fig.
4). The
differences in the estimates for the
rate coefficient ratio shown in
Table
2 and discussed above are
not statistically significant, but they
do illustrate the differences
between the modeling techniques. It is
important to recognize
that under other conditions, such as
experimental runs of more
than 1 day, passive TCE losses and endogenous
decay are likely
to be more significant, and the integrated Monod
equation weighted
least-squares approach might then not provide
reliable estimates
of the rate coefficients.
| |
DISCUSSION |
The integrated Monod equation method is a simple procedure for
obtaining reliable estimates of microbiological reaction rate coefficients as long as the requirements of the underlying assumptions are met. The fraction of cells lost over the course of the experiment due to endogenous decay of cells must be small. Changes in cell concentration due to growth or product toxicity do not limit the applicability of this method because they can be included in the integrated form of the equation. For volatile substrates, the experimental system must be at gas-liquid equilibrium and the rate of
passive loss of substrates, such as TCE, must be low relative to the
transformation rate.
The weighting method described in this paper has a rational statistical
basis. The difference between the observed and predicted substrate
concentrations is estimated by multiplying the slope of the substrate
disappearance curve by the difference between measured and predicted
sample times. This approach is appropriate because it approximates the
error in the measurement of substrate concentration and such errors are
usually 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 critical assumptions of linear regression are violated by these
methods, which leads to inaccurate parameter estimates and a lack of
any way to evaluate the uncertainties in the parameter estimates
(9). Because of the availability of computers and software
that are easy to use, the use of linearized approaches is no longer
justified. The weighted nonlinear least-squares method described in
this paper is very straightforward and easy to apply with a computer
spreadsheet program.
Another important advantage of the integrated Monod equation method
over 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,
the uncertainty calculations from a single experiment do not reflect
the run-to-run variability in parameter estimates. When initial rate
measurements are used, the rate coefficients and their uncertainties
can be determined more precisely from nonlinear least-squares analysis
(e.g., applied to a plot of dCL/dt versus
CL) than from linearized plots of inverted data.
Statistical software packages for personal computers may also allow
convenient application of the integrated Monod equation if they provide
for weighting in the nonlinear least-squares fitting and uncertainty
analysis. In the absence of a weighting factor, a nonlinear
least-squares approach would minimize the differences between measured
and predicted t values rather than CL
values, which would be inappropriate for an implicit expression, such as the integrated Monod equation. Users should be careful not to use
nonlinear least-squares fitting software without appropriate provisions, such as a weighting factor, when they use implicit equations, such as the integrated Monod equation.
Comparison with a rigorous numerical model validated the results
obtained by the integrated Monod equation spreadsheet method. One
important advantage of the integrated Monod equation spreadsheet method
over a numerical model is the flexibility that it offers to use the
initial substrate concentration as a fitting parameter, which is
desirable when the initial substrate concentration data point is known
with certainty equal to that of all other data points.
Rate experiments can be designed to maximize the informative value of
the results. Coefficients, such as Y, that are measured separately and used as constants in the parameter fitting process should be measured as accurately as possible. Accurate measurements of
other constants, such as the initial active biomass concentration, are
crucial for minimizing the uncertainties in the fitted parameters. The
number of fitting parameters should be minimized, and the concentrations of biomass and substrate should be selected to yield the
most information about the rate parameters of interest. The use of
sensitivity analysis to design experiments has been described by
Robinson (9).
The method described above for calculating the uncertainties in rate
coefficients makes great use of the information available from the data
without requiring the use of sophisticated mathematics. When the
initial substrate concentration is not used as a fitting parameter,
this method can be used to describe a confidence region precisely based
on numerous data sets if desired. However, when the initial substrate
concentration is used as a fitting parameter, each data set included in
the analysis adds one fitting parameter, and both the fitting
calculations and the graphical representation of the confidence region
become difficult with three or more data sets. In this situation, it
may be preferable to use the fitting technique described in this paper
to determine the best-fit value of each rate coefficient for each
experiment and use the mean and standard deviation of the best-fit
values from all of the experiments to evaluate the confidence interval
for each rate coefficient.
Copies of the spreadsheet are available on diskette from the
corresponding author.
| |
ACKNOWLEDGMENTS |
This study was supported by the Gas Research Institute and the
Westinghouse Savannah River Corporation through the U.S. Environmental Protection Agency-sponsored Western Region Hazardous Substance Research
Center.
| |
FOOTNOTES |
*
Corresponding author. Mailing address: Institute of
Technology and Engineering, Massey University, Palmerston North, New
Zealand. Phone: (64) 6-356-9099. Fax: (64) 6-350-5604. E-mail:
l.h.smith{at}massey.ac.nz.
| |
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Appl Environ Microbiol, June 1998, p. 2044-2050, Vol. 64, No. 6
0099-2240/98/$04.00+0
Copyright © 1998, American Society for Microbiology. All rights reserved.
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Top
Abstract
Introduction
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