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Biochemical Engineering Journal 31 (2006) 102–105

Short communication

A simple thermodynamic approach for derivation of a generalMonod equation for microbial growth

Yu LiuDivision of Environmental and Water Resource Engineering, School of Civil and Environmental Engineering,

Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore

Received 29 November 2005; received in revised form 25 April 2006; accepted 30 May 2006

bstract

The Monod equation has been widely applied to describe microbial growth, but it has no any mechanistic basis. Based on the thermodynamicsf microbial growth process, a general model for microbial growth was developed. The constants involved in the present model were defined with

lear physical meanings. The model derived can be reduced to the Monod equation, Grau equation and Hill or Moser equation. Compared to theichaelis–Menten constant with the equilibrium thermodynamic characteristics, it was shown that the Monod constant (Ks) has non-equilibrium

hermodynamic characteristics.2006 Elsevier B.V. All rights reserved.

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eywords: Specific growth rate; Substrate concentration; Thermodynamics; M

. Introduction

The Monod equation is one of the best-known kinetic modelsescribing microbial growth, which shows a functional relation-hip between the specific growth rate and an essential substrateoncentration. It should be realized that the Michaelis–Mentenquation was derived from the mechanism of enzyme reaction,hile the Monod equation was developed from a curve fitting

xercise, which is an example of an empirical correlation [2,16].he Michaelis–Menten kinetics for enzymatic reactions givesechanistic meanings to the constants involved, but none of

hose meanings can be applied readily to a substrate-cell systems described by the Monod equation, even though, the Monodelationship can provide the most generally satisfactory curve fit-ing of the growth data [7]. Other models for microbial growthad also been used, such as Grau equation, Hill or Moser equa-ion in the environmental engineering and applied microbiologyelds. However, the Monod equation and Hill or Moser equationre purely empirical [8,15,16,18], and theoretical derivation ofhese models has not been readily available in the literature. Theresent study, thus attempted to derive a general equation for

icrobial growth according to the thermodynamics of a micro-

ial growth process.

E-mail address: cyliu@ntu.edu.sg.

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369-703X/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.bej.2006.05.022

equation

. Model development

According to the collision frequency theory for microbialrowth [4], it is a reasonable consideration that if the sub-trate concentration in bulk solution increases, microbial growthould be more favourable [2,16,18], i.e. the effective free energy

hange (�G◦′) would decrease with the increase of the sub-trate concentration. In the conversion of a limiting substrateo biomass without substrate inhibition effect, �G◦′ could bexpressed as

G◦′ = �G◦ − nRT ln[S] (1)

n which [S] is the molar concentration of substrate, n is the pos-tive coefficient and �G◦ is the change of standard free energy.

It is believed that cells have only a limited number of sitesor take up of substrate, e.g. macromolecules on cell surface,uch as proteins, have multiple ligand binding sites responsi-le for transferring solutes into cell [1,10]. In fact, the similarssumption about active sites has been put forward in the liter-ture [4,18]. When all sites are taken up by substrate, the ratef uptake will reach its maximum value and the specific growth

ate will also be equal to the maximum specific growth rate.n this case, the driving force of microbial growth is the num-er of reactive sites on cells, which can be described by theifference between the maximum specific growth rate (µmax)

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Y. Liu / Biochemical Enginee

nd the observed specific growth rate (µobs) under given condi-ions, and this driving force is disappearing when the microbialrocess gradually approaches its maximum. Thus, the overallhange of free energy of the microbial growth process shoulde formulated as the function of the present growth state androwth potential in a way such that

G = �G◦′ + RT lnpresent growth state

growth potential(2)

In a theoretical sense, Eq. (2) is indeed consistent with thexpression for free energy change of an ideal gas and solution asell as the collision frequency theory for microbial growth [3,4],hile this equation is also similar to the Maxell–Boltzmannistribution law. As pointed out earlier, the microbial growthecomes less favourable as the reactive sites of cells are takenp. The number of the reactive sites of cells taken up along withhe growth is directly correlated to µobs, i.e. µobs would reflecthe present growth state, while the difference between µmax and

obs may represent the growth potential of bacteria. Therefore,q. (2) can be translated to

G = �G◦′ + RT lnµobs

µmax − µobs(3)

q. (3) shows that when µobs = 0.5 µmax, �G◦′ is equal to �G.his implies that �G◦′ can be defined as the overall free energyhange at µobs = 0.5 µmax. Substitution of Eq. (1) into Eq. (3)ives

G = �G◦ − nRT ln[S] + RT lnµobs

µmax − µobs(4)

olving Eq. (4) for µobs leads to

obs = µmax[S]n

Kn + [S]n(5)

n which

n = e(�G◦−�G)/RT (6)

q. (5) indeed shows the same formulation as the Moser equa-ion. As Roels [16] noted, the Moser model is a homologue of theurely empirical Hill model and the constants in the Moser andill models have not clearly defined physical meanings. Thus,q. (5) seems to offer a theoretical basis for the empirical Moserodel.

. Discussion

A simple least-square method was developed to evaluate theonstants in Eq. (5), and literature data were used to verify Eq. (5)n this study. Zhuang et al. [20] determined the specific growthates of Bacillus naphthovorans sp. nov. at different naphtha-ene concentrations (Fig. 1). The excellent agreement betweenhe experimental data and Eq. (5) prediction is observed, and

has a value of 1.14. Koch and Schaechter [13] studied the

ffect of glucose concentration on the specific growth rate ofscherichia coli in a pure culture, and comparison of the experi-ental data with Eq. (5) prediction is shown in Fig. 2. Obviously,q. (5) provides a satisfactory description for the data, indicated

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ig. 1. Relationship between µobs and [S]. Data from Ref. [20], and Eq.6) prediction is shown by solid line with a correlation coefficient of 0.994,

max = 0.28 h−1, n = 1.14 and Kn = 3.50 × 10−5 (mol/L)1.14.

y a correlation coefficient of 0.999, while the value of n wasstimated as 2.38. There is no reason to believe that n for aicrobial growth or enzymatic reaction must be restricted to 1

s the Monod or Michaelis–Menten equation shows, e.g. in thease of phosphofructokinase, the dependence of the rate on theructose-6-phosphate concentration can be described well by thempirical Hill equation with n ≈ 3.8 [21]. As noted by Hammes10], the value of n depends on specific experimental condi-ions. The exponent n in Eq. (5) could provide a useful measuref microbial cooperativity. It should be emphasized that Eq. (5)s valid only for the growth phase of a microbial culture.

When n equals 1, Eq. (5) becomes the well-known Monodquation:

obs = µmax[S]

Kn + [S](7)

he Monod equation has been considered to be mathematicallynalogous to the Michaelis and Menten equation describingnzyme kinetics, but the meanings of the Monod constant andhe Michaelis–Menten constant are completely different. As

ig. 2. Relationship between µobs and [S]. Data from Ref. [13], and Eq.6) prediction is shown by solid line with a correlation coefficient of 0.999,

max = 0.78 h−1, n = 2.38 and Kn 2.12 × 10−6 (mol/L)2.38.

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cal. In addition, it should be pointed out that [S] in Eq. (7) isxpressed in molar concentrations (mol/L). However, in the liter-ture of microbial growth research, volumetric concentration ofubstrate (mg/L) has been commonly used in the Monod equa-ion, i.e. the Monod equation might be inadequately applied.o convert molar concentration of substrate to its volumetriconcentration, both numerator and denominator of Eq. (7) areultiplied by the molar weight of substrate, Ms:

obs = µmaxMs[S]

MsKn + Ms[S]= µmax

S

Ks + S(8)

n which S = [S]Ms, which is volumetric concentration of sub-trate and Ks = MsKn, which is so-called the Monod constant. Ksn the Monod equation is often referred to as the affinity constantf substrate–cell pair. However, it appears from Eq. (5) that Kslone cannot describe such affinity because the magnitude of nlso determines the reaction rate of substrate and subsequentlyhe affinity of substrate to cell. In this case, n in the field of

icrobiology is termed cooperativity constant [10,21].To look into the physical meaning of Kn, Eq. (6) can be rear-

anged to

G = �G◦ + RT ln1

Kn

(9)

his equation may indicate that 1/Kn is a measure of cell affinityo substrate, which is determine by energy generated for growth.lickinger and Drew [6] noted that any other definitions of Ksre speculative, e.g. Ks interpretation as dissociation constant ofnzyme–substrate complex of the cellular enzyme involved inhe first step of substrate conversion. It should be realized that

n in Eq. (5) or Ks in Eq. (8) has the non-equilibrium thermo-ynamic characteristics, while the Michaelis–Menten constantas the equilibrium thermodynamic characteristics [2,21]. Asointed out by Westerhoff et al. [19], microbial growth shoulde analyzed in terms of non-equilibrium thermodynamics ratherhan equilibrium thermodynamics. It turns out that the majorifference between the Monod equation for microbial growthnd the Michaelis–Menten equation for enzymatic reaction liesn the process state, i.e. equilibrium for enzymatic reaction andon-equilibrium for microbial growth process.

As shown by Eq. (6), Kn is a function of change in free energyf microbial growth process. It is clear now that Kn in Eq. (6) ors in the Monod equation has defined physical meaning rather

han the black box characteristics as stated in the literature [2,18].bviously, �G of microbial growth process is determined byoth bacterial species and substrate under given culture condi-ions. Thus, the value of Kn in Eq. (6) or the Monod constant

s should also be related to microbial species–substrate pairs.o date, the extremely large variation in the values of Ks in theonod equation has been reported in the literature [4,5,11,17],

.g. the data available for E. coli growing with glucose showedhat the Ks values in the Monod equation varied over more thanhree orders of magnitude for different E. coli strain-glucose

airs, and similar results were also found for Cytophaga john-onae and Klebsiella pneumoniae [12]. It had been proposedhat the large variation in Ks was simply due to changes in massransfer [14], while Ferenci [5] noted that genotype, inoculum

Journal 31 (2006) 102–105

istory, length of exposure to substrate and bacterial density inultures would lead to variations in determination of the Monodonstant. Probably, for the first time, Eq. (6) clearly shows thatn or Ks is indeed governed by change in free energy generated

n microbial growth process, which is closely related to bacte-ial species–substrate pairs present in a microbial culture. Asresult, any factor influencing the interaction between bacte-

ia and substrate would also alter the estimate of Kn or Ks. Iteems that Eq. (6) offers a new thermodynamic explanation forhe variation in the values of the Monod constant.

When [S]n in Eq. (5) is much less than the value of Kn, Eq.5) reduces to

obs = µmax

Kn

[S]n (10)

his equation is similar to the model proposed by Grau et al. [9]or the growth of activated sludge microorganisms. On the otherand, Eq. (5) can be arranged to the well-known Hill equationy letting Kn be (kn)n, i.e.

obs = µmax[S]n

(kn)n + [S]n(11)

n which kn is the Hill constant.Microbial growth process involves a series of complex bio-

hemical reactions, and the exponent n in Eq. (5) indeed couldrovide a useful measure of microbial cooperativity. It is truehat the proposed model with three constants (µmax, Kn and n)eems to be more complex than the Monod equation having twoonstants. In view of the progress of numerical methods, bothhe Monod model and Eq. (5) can be much easily solved withoutny technical difficulty. In the past half century, a tremendousuantity of experimental data of microbial growth had beennterpreted and modeled by using the Monod equation which istrictly empirical. As Grady et al. [8] noted, many people haverroneously concluded that Monod proposed the equation onheoretical base. In fact, to formulate a mathematical descrip-ion of microbial growth, one needs to seek the models withtrong theoretical characteristics rather than the simplicity ofhe models.

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