A Thermodynamic Interpretation of the Monod Equation

Yu Liu, Yue-Mei Lin, Shu-Fang Yang

Environmental Engineering Research Centre, School of Civil and Environmental Engineering, Nanyang Technological University, 50 NanyangAvenue, Singapore 639798

Received: 2 August 2002 / Accepted: 23 September 2002

Abstract. The Monod equation for microbial growth is purely empirical, and the theoretical basis of thismodel stays unclear. Similar to any chemical reactions, overall microbial growth process is dependentupon the changes in free energy. This study showed that the Monod equation could be interpreted in athermodynamic sense very well. It was probably for the first time demonstrated that the Monod constant(Ks) was inversely related to the equilibrium constant of the overall microbial growth process.

Acording to Monod [5], the relationship of the specificgrowth rate (�) to a growth-limiting substrate concen-tration (S) in bulk solution is subject to

� � �max

S

Ks � S(1)

in which �max is the maximum specific growth rate and Ks

is the Monod constant. As Grady and Lim [4] noted, manypeople have erroneously concluded that Monod proposedthe equation on theoretical base. In fact, this equation has noany mechanistic basis [1, 3, 4]. So far, little information isavailable on theoretical derivation of the Monod equation.This study therefore attempted to provide a thermodynamicinterpretation for the Monod equation.

Model Development

For a substrate-limiting microbial culture, the microbialgrowth reaction can be regarded as a simple change inthe nature of the substrate, that is,

S 3 Sb �G0� (2)

in which Sb is apparent substrate concentration that is in-corporated into new biomass. Consider the evidence that ifS increases, microbial growth becomes more favorable [1,3], the free energy change, �G0�, would decrease with theincrease of S. Hence, it can be assumed that

�G0� � �G0 � RT ln S (3)

in which �G0 is change of standard free energy. Analogto a chemical reaction, the overall change in free energyof the microbial growth reaction (�G) can be expressedas a function of S and Sb as follows:

�G � �G0� � RT lnSb

S(4)

Assume that substrate going to carbon dioxide throughcatabolism is negligible, we have

Sb � S0 � S (5)

in which S0 is initial substrate concentration. Thus,

�G � �G0� � RT lnS0 � S

S(6)

For easily biodegradable organic compounds, Ecken-felder [2] proposed that the specific substrate utilizationrate (q) could be expressed as

q �S0 � S

Xt(7)

in which X is biomass concentration, and t is culturetime. In this case, Eq. 6 can be further rearranged in thefollowing form:

�G � �G0� � RT lnq

�S0/Xt� � q(8)

The specific growth rate (�) is related to q through thegrowth yield coefficient, Yxs, that is,

� � qYxs (9)Correspondence to: Y. Liu; email: cyliu@ntu.edu.sg

CURRENT MICROBIOLOGY Vol. 46 (2003), pp. 233–234DOI: 10.1007/s00284-002-3934-z Current

MicrobiologyAn International Journal© Springer-Verlag New York Inc. 2003

Substitution of Eq. 9 into Eq. 8 leads to

�G � �G0� � RT ln�

Yxs�S0/Xt� � � (10)

It appears from Eqs. 7 and 9 that the Yxs(S0/Xt) term inEq. 10 can be approximately replaced by the theoreticalmaximum specific growth rate (�max). In this case, Eq.10 becomes

�G � �G0� � RT ln�

�max � �(11)

Substituting Eq. 3 into Eq. 11 gives

�G � �G0 � RT ln S � RT ln�

�max � �(12)

When microbial growth reaches its equilibrium, an en-ergy balance must be maintained around the energycarrier at the steady state, i.e., �G � 0. Therefore, Eq. 12is further reduced to

�

�max � �� S e��G0/RT (13)

Solving Eq. 13 for � gives

� � �max

S

Ks � S(14)

and

Ks � e�G0/RT (15)

It can be seen that Eq. 14 and the Monod equation (Eq.1) have the same form. Analog to a chemical reaction,the equilibrium constant (K) of a balanced microbialgrowth reaction can be defined as

K � e��G0/RT (16)

Comparison of Eqs. 15 and 16 yields

Ks �1

K(17)

Equation 17 probably for the first time reveals the realphysical meaning of Ks, i.e., Ks is a function of theequilibrium constant of overall microbial growth pro-cess.

Discussion

Since Monod [5] proposed Eq. 1 on experimental base,the Monod equation has been considered strictly empir-ical. This study provides a thermodynamic interpretationof the Monod equation. In the fields of applied microbi-ology and environmental engineering, the question of thereal physical meaning of Ks in the Monod equation hasbeen the subject of much debate. As Roels [6] noted, theMichaelis-Menten constant has a direct physical mean-ing, it is a ratio of kinetics constants, but the Monodconstant Ks has black box characteristics. However, Eq.17 shows that Ks is related to the inverse of the equilib-rium constant of a balanced microbial growth reaction.When the equilibrium constant is very large, i.e., Ks isvery small, the microbial growth process proceeds fartowards completion, and the position of equilibrium liesfar toward the growth of biomass. In this case, � wouldapproach �max. On the other hand, when the equilibriumconstant in Eq. 16 is very small, i.e., Ks is very large,extremely small amount of biomass is formed, and theposition of equilibrium lies far towards the substrate. It ismost likely that the value of Ks is determined by thenatures of both microbial species and limiting substrateemployed in culture. Consequently, the magnitude of Ks

value represents the equilibrium position of a microbialgrowth process, while �max reflects the limit of microbialgrowth. It is expected that this study is helpful for betterunderstanding large variations observed in the values ofKs as commonly reported in the literature.

Literature Cited1. Bailey JE, Ollis DF (1986) Biochemical engineering fundamentals.

Singapore: McGraw-Hill2. Eckenfelder WW (1989) Industrial water pollution control. New

York: McGraw-Hill3. Gaudy AF, Gaudy ET (1980) Microbiology for environmental sci-

entists and engineers. Singapore: McGraw-Hill4. Grady CPL, Lim HC (1980) Biological wastewater treatment. New

York: Marcel Dekker5. Monod J (1942) Recherche sur la croissance des cultures bacteri-

ennes. Paris: Harmann et Cie6. Roels JA (1983) Energetics and kinetics in biotechnology. New

York: Elsevier

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