J. appl. Chem. Biotechnol. 1974, 24,453-455

Derivation of a Generalised Monod Equation and its Application

A. R. Konak

Department of Chemical Engineering, The University of Birmingham, P.O. Box 363, Birmingham BI.5 2TT (Paper received 3 ApriI 1974, and accepted 23 April 1974)

A generalised growth rate equation is derived which relates the specific growth rate of bacteria to the concentration of a limiting nutrient in a bacterial culture. Monod’s simple hyperbolic and Teissier’s exponential equations are shown to be two special cases of this general equation. It is also revealed that the so-called Michaelis-Menten constant in Monod’s simple equation is inversely proportional to the maximum specific growth rate.

1. Introduction

The well known Monod equation’ describes the relationship between the specific growth rate, p, of bacteria and concentration, S, of the limiting nutrient (substrate) in a microbial culture. It is given by

In Equation (l), M is the concentration of bacterial cells, or of biological solids, p,,, the maximum specific growth rate and K may be termed the substrate concentration when p = p J 2 or simply the Michaelis-Menten c o n ~ t a n t . ~ . ~ It is thus seen that for a true logarithmic growth phase to occur in a bacterial culture, the assumption of K = 0 must be realised.

Several criticisms of the Monod equation have been made and some modifications4 as well as other relationships have been put f ~ r w a r d . ~ ~ ~ Although the Monod equation is basically empirical, it is still the most commonly used relationship between the specific growth rate and substrate concentration, especially for pure culture systems.

In this paper, a general equation will be developed between p and S, and it will be shown that in one special case, this equation reduces to that of Monod and can, therefore, be used whenever the latter fails to fit experimental data.

2. Derivation of a general growth equation and its discussion

Experimental observations show that the specific growth rate increases with con- centration of limiting substrate and reaches a maximum value at large values of S, thus giving an exponential type of curve. At any substrate concentration S, therefore,

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454 A. R. Konak

the “driving force” for change of p with respect to S may be thought to be given by (prn Thus,

= f ( t . m - p ) *

Assuming a power law for the function f, we have

dp/dS = Wprn - (2) where k is a system constant a n d p may be termed the kinetic order of the process. Dividing equation (2) by pmp, we obtain

or - dX = kpmP-’ (I - X ) ” d S

where X = p / p m . Equation (3) can be readily integrated from X = 0 to X and S = 0 to S to give

(1-X)l-P--l = k ( p - 1 ) pmp-l S for p # 1, (4)

-In (1 - X ) = kS for p = 1. ( 5 )

If equation (5 j is rewritten in the form

p = pm(l -e-kS)

it is seen to be the one proposed by Teissier and applied by Schulze with Equation (4), however, is more general and it may be called the general Monod equation because it reduces to the simple Monod relationship for p = 2:

(6)

Comparison of equations (1) and (6) show that the so-called Michaelis-Menten constant K, is inversely proportional to maximum specific growth rate, pm. This is an interesting result which has not been revealed before and it clearly indicates that the larger the maximum specific growth rate, the steeper is the growth curve and hence the nearer the growth approaches to logarithmic phase.

Expanding the left hand sides of equations (4) and (5)

(1 --X)-X-l = X + X Z + X 3 + , , , -In (1 - X ) = X + X 2 / 2 + X 3 / 3 + . . .

show that the difference between the Monod and Teissier equations is small at low values of S and that both equations can be approximated by straight line relationships

p = kpm2S and p = kp,S

respectively. Since, for example, waste water treatment plants must yield effluent at

A generalised Monod equation 455

low concentrations, a linear relationship could be assumed in this restricted range without incurring serious error. At higher values of S, however, either the general Monod equation or its simple version as modified above should be used.

3. The procedure for experimental determination of the optimum value of p

The following procedure may be employed to avoid a trial and error solution.

and S,, we have If equation (3) is integrated between X , and X , with corresponding values of S1

(1 - X ~ ) ' - n ( R ' - P - l ) = k ( p - 1 ) p m P - ' ( S 2 - S ~ ) , (7)

( 8 ) R = (1 -X,)/(l - X I ) .

The procedure for finding p is then as follows. (a) Plot (1 - X ) uersus S from experimental data. (b) Select an arbitrary value for X I and R. (c) Find corresponding values of X , and S2 from equation (8) and the plot

described in (a), respectively. (d) Plot log (1 --XI) uersus log (S,-S1) in accordance with equation (7). From

the slope of this straight line p may be estimated.

4. Conclusions

Whenever the simple Monod equation fails to fit experimental data, the generalised Monod equation derived in this paper may be used and the corresponding value of p may be determined in accordance with the procedure outlined in section 3. Since p is the order of process, its value may be a positive fractional or whole number.

Acknowledgement The author acknowledges the financial assistance from The Scientific and Technical Research Council of Turkey.

References 1. Monod, J. A . Rev. Microbiol. 1949, 3, 371. 2. Calam, C. T.; Russell, D. W. J. appl. Chem. Biotechnol. 1973, 23, 225. 3. Dawes, E. A. Quantitative Problems in Biochemidry. Churchill and Livingstone, Edinburgh and

London, 1972. 4. Gaudy, A. F., Jr.; Ramanathan, M.; Rao, B. S. Biotechnol. Bioeng. 1967, 9, 387. 5. Chiu, S. Y.; Erickson, L. E.; Fan, L. T. ; Kao, I. C. Biotechnol. Bioeng. 1972,14,207. 6. Dabes, J. N.; Fin, R. K.; Wilke, C. R. Biotechnol. Bioeng. 1973, 15, 1159. 7. Chiu, S. Y.; Fan, L. T.; Kao, I. C.; Erickson, L. E. Biotechnol. Bioeng. 1972, 14, 179. 8. Schulze, K. L. Wat. Sewage Wks 1964, 111, 526. 9. Schulze, K. L. Wat. Sewage Wks 1965, 112, 11.