Vol. 45, No. 5APPLIED AND ENVIRONMENTAL MICROBIOLOGY, May 1983, p. 1453-14580099-2240/83/051453-06$02.00/0Copyright C 1983, American Society for Microbiology

Nonlinear Estimation of Monod Growth Kinetic Parametersfrom a Single Substrate Depletion Curvet

JOSEPH A. ROBINSONt AND JAMES M. TIEDJE*

Department of Microbiology and Public Health, Michigan State University, East Lansing, Michigan 48824

Received 27 September 1982/Accepted 25 February 1983

Monod growth kinetic parameters were estimated by fitting sigmoidal substratedepletion data to the integrated Monod equation, using nonlinear least-squaresanalysis. 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 Px. Ks, and Y values used to create thesedata. Nonlinear regression analysis of sigmoidal substrate depletion data was alsoevaluated for H2-limited batch growth of Desulfovibrio sp. strain Gil. Theintegrated Monod equation can be more convenient for the estimation of growthkinetic parameters, particularly for gaseous substrates, but it must be recognizedthat the estimates of Rma,x K5, and Y obtained may be influenced by the growthrate history of the inoculum.

Both derivative and integrated forms of equa-tions derived for enzyme-catalyzed reactionshave been used to estimate kinetic parametersfor microbially mediated processes. In particu-lar, Vma and Km estimates have been calculatedby fitting data to either integrated (2, 5, 6, 8, 14,15) or derivative (3, 14, 15) forms of the Michae-lis-Menten equation. However, Michaelis-Men-ten kinetics only describe bacterial substrateconsumption either (i) when this process is un-linked to growth, such as under resting condi-tions, or (ii) when the amount of growth occur-ring is less than that which gives sigmoidalsubstrate depletion. Vm., is no longer a parame-ter (i.e., a constant) if its value changes duringsubstrate consumption.When substrate consumption is linked to

growth, the number of catalytic units, or activi-ty, increases with time. Assuming that the initialsubstrate concentration is greater than thatwhich gives one-half of the maximum growthrate, an increase in activity concomitant withsubstrate consumption yields an S-shaped sub-strate depletion curve, or sigmoidal kinetics(12). As mentioned above, sigmoidal kinetics isinconsistent with the Michaelis-Menten model,but is predicted by Monod kinetics.

In this paper, we demonstrate the fitting ofsubstrate depletion data (i.e., progress curvedata) to the integrated Monod equation by usingnonlinear regression, which is advantageous

t Journal article no. 10579 of the Michigan AgriculturalExperiment Station.

t Present address: Department of Civil Engineering, Mon-tana State University, Bozeman, MT 59717.

since estimates of growth kinetic parametersmay be calculated from a single progress curve.We show how initial estimates of Uma, K, andY, which are required for nonlinear least-squares analysis, can be calculated by usinglinearized, discretized forms of the Monod equa-tions describing the rates of change of biomassand growth-limiting substrate concentrationsduring batch growth. Additionally, we discussthe use of sensitivity coefficients for the optimalestimation of growth kinetic parameters and theadvantages of using the integrated Monod equa-tion for estimation of Lmax, K, and Y. Finally,we give attention to the limitations of fittingtransient-growth data to the Monod model,which describes balanced bacterial growth only(13).

THEORY AND METHODSThe rate of change of substrate consumption by a

bacterium growing in batch may be described bydS/dt = -[ m,,.,,S/(K5 + S)]X/Y (1)

where ,u, is the maximum specific growth rate, K5 isthe half-saturation constant for growth, and Y is theyield coefficient. The variable X is the biomass con-centration and may be eliminated from equation 1 byusing

X = Y(SO- S) + XO (2)which relates X at time t to S. After elimination of X,equation 1 becomes

dS/dt = -[pL,,S/(K, + S)][Y(So- S) + Xo)]/Y (3)which may be integrated to give

Cjln{[Y(SO - S) + XO]/XO} - C21n(S/So} = Lmaxt (4)

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1454 ROBINSON AND TIEDJE

where C1 = (KSY + SOY + XO)/(YSO + XO) and C2 =K5Y/(YSo + XO). Equation 4 gives the familiar S-shaped curve for substrate depletion during batchgrowth. A relation similar to equation 4 describing theincrease of biomass during batch growth has beenderived by integrating the expression obtained aftereliminating S from equation 1 by using the massbalance relation (equation 2) (11, 12).

Equation 4 cannot be explicitly solved for S, andhence its solution must be numerically approximated.Solution curves (S versus t) may be estimated bysolving equation 3 by numerical integration, or, alter-nately, solution curves for both S and X may begenerated by solving equation 1 simultaneously with

dX/dt = [~z,u.S/(K, + S)]X (5)again by numerical integration (4).

Nonlinear regression analysis requires that the sen-sitivity of the dependent variable to changes in each ofthe parameters be calculable. The first derivatives of Swith respect to p. ,s K5, and Y satisfy this require-ment (although these may be numerically approximat-ed), and these expressions can be derived from theintegrated Monod equation by using implicit differenti-ation (16). The sensitivity equations for tmax, KS, andY are given below.

dS/dR,max = t/C4 (6)

dS/dK, = {(Y/C3}Rln(X/XO} - ln(S/So}]}/C4 (7)dS/dY = {C1(S0 - S)/X + ln(X/XO}/C3[K, + (1 -

C1)SO] - ln(S/SO)/C3(K, - C2SO)}/C4 (8)In the above three equations the terms C3 and C4 equalYSO + XO and C3Y/X + C4/S, respectively. Note thatequations 6, 7, and 8 are all functions of KS and Y. Bydefinition, then, the integrated Monod equation isnonlinear, since its sensitivity equations are not inde-pendent of the parameters (1).

In addition to being required for nonlinear regres-sion, the sensitivity equations predict (i) whether theparameters may be uniquely estimated, (ii) the relativeprecision of the estimated parameters, and (iii) therange of the independent variable over which themodel is most sensitive to changes in the parameters.The last item is useful in designing optimal experi-ments for parameter estimation (1). If the sensitivityequations are proportional (i.e., if they are multiples ofone another), then it is impossible to obtain uniqueestimates of the parameters from the data by least-squares analysis (1). Unique parameter estimates maybe obtained when the sensitivity equations are verysimilar, but they are highly correlated. This situation isundesirable since it implies that several combinationsof parameter estimates may describe the same dataset, and this is true for equation 4, depending on SO.The sensitivity equations for Imax, KS, and Y yieldsimilar curves (Fig. 1), with the sensitivity equationsfor m.Lmax and Y numerically dominating the one for KS.

In the first-order region it is not possible to obtainunique estimates of m KS, and Y because thesensitivity coefficients for Lmax and Y are proportional(Fig. 1A). This is intuitively clear since an increase inIL,,I is equivalent to a decrease in Y in the first-orderregion. The sensitivity equations also predict thatrelatively poor estimates of .,max, KS, and Y will beobtained when So is saturating. This results from the

parameters having nearly proportional sensitivitiesover most of the progress curve (Fig. 1C). The value ofSo for the optimal estimation of Lmax, KS, and Y is inthe mixed-order region (Fig. 1B).A number of techniques may be used to fit data to

nonlinear models, but we chose the Gaussian methodbecause of its relative simplicity (1). The Gaussianmethod uses the equation

S - ST = AlXraxdS/dp.max + AK5dS/dK, + AYdS/dY(9)

where ST iS the theoretically predicted substrate con-centration at time t given current parameter estimates,and AILmax, AK,, and AY are correction terms for theseparameters. The application of equation 9 proceeds byevaluating the sensitivity equations 6, 7, and 8 andcalculating the residual errors (ST - S) at the mea-sured substrate concentrations for the initial parame-ter estimates. The correction terms are then calculatedvia multiple linear regression and added (they can bepositive or negative) to the initial parameter estimates.These corrected parameter estimates then serve as theinitial estimates for the next iteration, and this processcontinues until the correction terms are less than somesmall value (e.g., 0.01). Once the solution has con-verged, estimates of the standard errors of ILmax, KS,and Y may be calculated from the variances of theirrespective correction terms (8).

Nonlinear regression analysis for any model that isnonlinear in its parameters requires initial estimates or"'guesses" of the parameters (1, 7). This is usuallydone by fitting transformed data to a linear version ofthe model, but the integrated Monod equation isintrinsically nonlinear (7) and cannot be transformedinto a linear expression for the purpose of estimatingP'max, KS, and Y. However, provisional estimates ofthese parameters may be obtained by using

-AS/At = [(LmaxS/(Ks + S)]X/YAX/At = [j±maxS/(Ks + S)]X

(10)(11)

Equations 10 and 11 are derived by replacing infinitesi-mal time (dt) with finite time (At) and are approximate-ly correct if At is relatively small (16). These finite-difference equations are nonlinear, but may beconverted into the following linearized forms:

-AtX/AS = (KsY4lumax)(1/S) + Y/Imax (12)and

AtX/AX = (Ks4Imax)(1/S) + l4lmax (13)which yield straight lines when -AtX/AS and AtX/AXare plotted against 1/S.

RESULTS AND DISCUSSIONTo evaluate nonlinear regression analysis for

equation 4, we fitted simulated data containingknown errors to this equation, using the Gauss-ian method. Theoretically, estimates of Imax,K, and Y should be close to those parametervalues used to generate the simulated data. Byanalyzing simulated data it is possible to checkwhether the sensitivity coefficients have been

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NONLINEAR REGRESSION ANALYSIS OF MONOD DATA 1455

A

0

0la'a

B

0a0

'0-X

10-

C

0

0

o0'0

Time

FIG. 1. Sensitivity coefficients for Prax(dS/diL,,), Ks (dS/dK5), and Y (dS/dY). For all threepanels, A,,, = 0.1, Ks = 5, Y = 0.2, and XO = 1. SO =0.1, 20, and 250 for panels A, B, and C, respectively.

correctly derived, since incorrect sensitivity ex-pressions can (i) prevent fitting data to a givenmodel altogether or (ii) lead to dramatic errors inthe estimated parameters. Twelve simulateddata sets were created by numerically integrat-ing equation 2 for the following parameter valuesand initial conditions: m = 0.1, Ks = 5, Y =

0.2, So = 20, and Xo = 1. Measurement errors ofthe relative type (constant coefficient of varia-tion) were added to six of the error-free datasets, using a pseudo-random number generatoraccording to the procedure described by Har-baugh and Bonham-Carter (10). Similarly, sim-ple errors (constant standard deviation) wereintroduced into the remaining six data sets.

Initial , K5, and Y estimates were ob-tained from linear regression analysis of thesimulated S-t and X-t data pairs transformedaccording to equations 12 and 13. (Actually,only initial and final estimates of the biomassconcentration are required; a provisional esti-mate of Y can be obtained from the difference

between these values, and provisional pm. andKs estimates can be calculated by using a linear-ized, discretized version of equation 3.) To fitdata to equations 12 and 13, estimates of -dS/dtand dX/dt are required. These were obtained byfitting the S-t and X-t data pairs to cubic splines(4) and then evaluating the first derivatives ofthe fitted cubic polynomials. Two examples offitting transformed data having either relative orsimple measurement errors are depicted in Fig.2. For both cases (A and B), Ks and pmae werecalculated from the y intercepts and slopes of thestraight lines fitted to the transformed biomassdata. The two Y estimates were calculated bydividing the y intercepts of the best-fit lines fortransformed substrate data by the y intercepts ofthe linear equations fitted to the transformedbiomass data.The initial estimates of ium., K5, and Y were

entered into a computer program (MON-ODCRV) that fits S data to equation 4 by theGaussian method. In addition to estimating thegrowth kinetic parameters, MONODCRV ap-proximates (i) the standard errors of FLm., K5,and Y, assuming no correlation among measure-ment errors, and (ii) the absolute residual sum-of-squares at each iteration. The pattern of pa-

A An

so60

la 40

20

0

B 12

9IGo10

x

3

0

400

300

x

200x

100

o 2 4 6 a1/8

I ist~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

0 0.2 0.4118

0.6 0.8

Uvv

40

30Xx

20 0

10

n

FIG. 2. Linearized, discretized Monod data. Simu-lated data containing errors having a constant coeffi-cient of variation (CV) of 0.5% (A) and a constantstandard deviation (SD) of 0.01 (B) were transformedaccording to equations 12 and 13. The values of lmax,

K5, and Y are the same as those used in Fig. 1.

0

SD-=0.01

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1456 ROBINSON AND TIEDJE

rameter updating for data fitted to equation 4 isillustrated in Table 1. In both cases, five itera-tions were required before the correction termswere less than 0.01 and the residual sum-of-squares was minimized. The updating of initialparameter estimates was usually complete afterfour or five iterations when a convergence crite-rion of 0.01 was used.

Nonlinear regression analysis of the simulatedMonod data generally provided better estimatesof FLmax, Ks, and Y than did least-squares analy-sis of the linearized data (Table 2, Fig. 3). Thiswas always true for Ks, but in a few cases thefinal Rmax (simulations 4 and 9) and Y (simula-tion 5) estimates were slightly more in error thanthe initial estimates of these two parameters.Nonlinear regression analysis always increasedthe overall goodness-of-fit by reducing the resid-ual sum-of-squares, in some cases by as much as104-fold (simulation 9). In addition to the resultspresented above, we have found that fitting datato equation 4 by using nonlinear regressionanalysis is far superior to linear analysis oftransformed data for several combinations ofparameter values and error levels. However,when the relative error level is greater than orequal to 5%, then estimates of iJmax, Ks, and Ymay be in substantial error (e.g., 500%), al-though the residual sum-of-squares has beenminimized.To demonstrate the fitting of biological data to

equation 4 by using nonlinear regression, weobtained H2 depletion and biomass formationdata for the sulfate reducer Desulfovibrio sp.strain Gl1 growing on H2 as the sole electrondonor. The culture conditions and experimentaldetails will be described elsewhere (Robinson

and Tiedje, manuscript in preparation). Thegrowth of this organism was monitored by fol-lowing the protein content.

Initial estimates of the growth kinetic parame-ters were calculated according to the above-described procedure, and these were updated byMONODCRV. The goodness-of-fit for the H2concentration data versus the theoretical curvecalculated from the best parameter estimates isshown in Fig. 4; for these data, estimates oflLmax, Ks, and Y, respectively, were 5.6 x 10-2h-1, 2.4 jxM dissolved H2, and 0.99 g of proteinper mol of H2.

Others have determined bacterial growth pa-rameters from batch data, but generally theyhave used computationally inefficient proce-dures or paid little attention to the limitations ofestimating these parameters under transient ornonsteady-state conditions. Graham and Canale(9), in a recent investigation on the kinetics of afour-trophic level predator-prey system, fittedbatch data to Monod growth models withoutconsidering the influence that the growth ratehistories of their predator and prey culturesmight have had on the optimal parameter esti-mates. Before this, investigators typically fittedbiomass formation data to the integrated versionof the Monod equation describing biomass pro-duction, using a systematic procedure (11)which involves changing Ptmax, Ks, and Y untilthe theoretical biomass formation curve agreeswith the experimental points. Systematic (orrandom) search techniques are inefficient andshould be avoided if possible (1). The largerdanger, however, is that since sensitivity coeffi-cients are not required for systematic fittingprocedures, the researcher may believe that it is

TABLE 1. Course of parameter updating during nonlinear regression analysis of simulated Monod dataaRelative error' Simple error'

Estimate b _____________________ ____________________IJLmax K5 Y RSS' ILmax Ks Y RSS

Initialf 0.14 10.5 0.31 80.9 0.11 5.99 0.18 46.70Iteration 1 0.12 7.26 0.21 8.9 0.12 7.40 0.21 0.83Iteration 2 0.10 4.58 0.20 1.1 0.10 4.49 0.20 0.16Iteration 3 0.10 5.02 0.21 0.09 0.10 5.00 0.20 0.004Iteration 4 0.10 4.95 0.21 0.08 0.10 5.04 0.20 0.003Final (iteration 5) 0.10 4.95 0.21 0.08 0.10 5.04 0.20 0.003

True value 0.10 5.00 0.20 0.10 5.00 0.20a Untransformed data were directly fitted to equation 4 by using nonlinear regression analysis, given the initial

parameter estimates.b Generally, four or five passes were required before the solution converged. For these simulated data, a

convergence criterion of 0.01 was used.c The relative errors have a constant coefficient of variation (0.5%).d The simple errors have a constant standard deviation (0.01).e RSS, Residual sum-of-squares.f Initial estimates of fLmax, Ks, and Y were calculated by fitting transformed S-t and X-t data pairs to equations

12 and 13.

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NONLINEAR REGRESSION ANALYSIS OF MONOD DATA 1457

TABLE 2. Comparison of errors in parameter estimates and residual sums-of-squares for linear versusnonlinear regression analysis of simulated Monod dataa

% Error in:Simulation Initial estimates Final estimates RSSj/RSSf'

i-Mx K,nKY > K, Y

1 12.2 29.8 14 2 1.2 3.5 71.82 29 80.4 37 9 12.8 9.5 3373 41 110 56.5 1 1 2.5 9744 1.2 10.4 8.5 3 2.4 4 1445 7.1 10.8 5.5 5 5.2 6.5 4.016 22.5 52.4 20 1 0.4 2.5 69.57 7 12.8 5.5 1 1.2 1 15.88 16 26.6 22 0 0.2 0 1,1509 0 5.6 23.5 1 1.6 1 22,22010 21 33 10 0.5 1 0 3,43011 5.2 7.2 2 2 2.6 1.5 46.112 4.8 4.2 5 0 0 0 75.4

a True parameter values were identical for all 12 simulations: eLmax = 0.1, K, = 5, and Y = 0.2.b Simulations 1 through 6 contain errors having a constant coefficient of variation (0.5%) (relative errors);

simulations 7 through 12 contain errors with a constant standard deviation (0.01) (simple errors).I Ratio of residual sum-of-squares for initial parameter estimates to residual sum-of-squares for final parameter

estimates.

possible to uniquely estimate parameters for agiven nonlinear model when the sensitivity coef-ficients predict otherwise.The integrated Monod equation is advanta-

geous for the estimation of Umax, K,, and Y

Ac

0

-

O-Cec0C

0dUeU-S1

0o:

(particularly for gaseous substrates) but suffersfrom some limitations. It is advantageous sinceestimates of growth kinetic parameters may beobtained from a single substrate depletion curve.Further, estimating growth kinetic parametersby using the integrated Monod equation offersan alternative to estimating these parameters forgaseous substrates by using chemostat culturesin which mass transport limitations may yieldunrealistically high K, estimates (14). The use ofthe integrated Monod equation is limited be-cause the derivative form, from which the inte-grated version is derived, describes balanced(steady-state) bacterial growth and may fail todescribe transient (batch) bacterial growth (13).Powell has shown that apparent Monod parame-

Time

Bc

0

a4-

C0e0C0Ue

0-aa,.

20C

16 Final estimates

12 cvo.o12~~~CV= 0.005

8

4 . Initial estimates ',

O 1 . . *i-m5 10 15 20 2S5 3Time

FIG. 3. Comparison of theoretical curves calculat-

ed by using initial versus final (best) , K,, and Yestimates with simulated data points. The data are

those that were transformed and plotted in Fig. 2.

cOI

co00am

._

0 8 16Hour

24 32 40

FIG. 4. Comparison of theoretical curve with mea-sured H2 concentrations (O) for growth of Desulfovib-rio sp. strain Gll on H2 as the sole electron donor. Theinitial substrate concentration (SO) was estimated byextrapolating back to t = 0 on the H2 concentrationaxis.

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1458 ROBINSON AND TIEDJE

ters determined by using batch growth data maydepend on the growth rate history of the orga-nism (13). Lastly, maintenance requirements,which we neglected for strain Gil, may produceerroneous parameter estimates (particularly ofKs) if ignored. In summary, the integratedMonod equation offers an alternative to chemo-stat studies for estimating growth kinetic param-eters, but it should be used only after its limita-tions have been considered.

ACKNOWLEDGMENTSWe thank Dave Myrold for use of a computer program that

fits cubic splines to data and Dan Shelton for many helpfuldiscussions on bacterial growth kinetics. We gratefully ac-

knowledge the expert technical assistance of Walter Smo-lenski and thank James Beck for a critical review of themanuscript. A copy of MONODCRV is available from usupon request.

This work was supported by National Science Foundationgrants DEB 78-05321 and DEB 81-09994.

LITERATURE CITED1. Beck, J. V., and K. J. Arnold. 1976. Parameter estimation

in engineering and science, p. 334-350. John Wiley &Sons, Inc., New York.

2. Betlach, M. R., and J. M. Tiedje. 1981. Kinetic explana-tion for accumulation of nitrite, nitric oxide, and nitrousoxide during bacterial denitrification. Appl. Environ. Mi-crobiol. 42:1074-1084.

3. Betlach, M. R., J. M. Tiedje, and R. B. Firestone. 1981.Assimilatory nitrate uptake in Pseudomonas fluorescensstudied using nitrogen-13. Arch. Microbiol. 129:135-140.

4. Burden, R. L., J. D. Faires, and A. C. Reynolds. 1978.Numerical analysis, p. 116-128 and 239-245. Prindle,Weber and Schmidt, Boston, Mass.

5. Cornish-Bowden, A. 1979. Fundamentals of enzyme kinet-ics, p. 200-210. Butterworth, Inc., Boston, Mass.

6. Counotte, G. H. M., and R. A. Prins. 1979. Calculation ofKm and V,,,0, from substrate concentration versus timeplot. Appl. Environ. Microbiol. 38:758-760.

7. Draper, N. R., and H. Smith. 1981. Applied regressionanalysis, p. 459. John Wiley & Sons, Inc., New York.

8. Duggleby, R. G., and J. F. Morrison. 1977. The analysisof progress curves for enzyme-catalyzed reactions bynonlinear regression. Biochim. Biophys. Acta 481:297-312.

9. Graham, J. M., and R. P. Canale. 1982. Experimental andmodeling studies of a four-trophic level predator-preysystem. Microb. Ecol. 8:217-232.

10. Harbaugh, J., and G. Bonham-Carter. 1970. Computersimulation in geology, p. 61-97. John Wiley & Sons, Inc.,New York.

11. Knowles, G., A. L. Downing, and M. J. Barre"t. 1965.Determination of kinetic constants for nitrifying bacteriain mixed culture, with the aid of an electronic computer.J. Gen. Microbiol. 38:263-278.

12. Pirt, S. J. 1975. Principles of microbe and cell cultivation,p. 22-28. John Wiley & Sons, Inc., New York.

13. Powell, E. 0. 1967. The growth rate of microorganisms asa function of substrate concentration, p. 34-56. In E. 0.Powell, C. G. T. Evans, R. E. Strange, and D. W. Tem-pest (ed.), Microbial physiology and continuous culture.Her Majesty's Stationery Office, London, United King-dom.

14. Robinson, J. A., and J. M. Tiedje. 1982. Kinetics ofhydrogen consumption by rumen fluid, anaerobic digestorsludge, and sediment. Appl. Environ. Microbiol. 44:1374-1384.

15. Strayer, R. F., and J. M. TiedUe. 1978. Kinetic parametersof the conversion of methane precursors to methane in ahypereutrophic lake sediment. Appl. Environ. Microbiol.36:330-340.

16. Thomas, G. B., Jr. 1972. Calculus and analytical geome-try. p. 76-81. Addison-Wesley, Inc., Reading, Mass.

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