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A simple thermodynamic approach for derivation of a general Monod equation for microbial growth
Biochemical Engineering Journal 31 (2006) 102–105
Short communication
A simple thermodynamic approach for derivation of a general Monod equation for microbial growth Yu Liu Division 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
Abstract The Monod equation has been widely applied to describe microbial growth, but it has no any mechanistic basis. Based on the thermodynamics of microbial growth process, a general model for microbial growth was developed. The constants involved in the present model were defined with clear physical meanings. The model derived can be reduced to the Monod equation, Grau equation and Hill or Moser equation. Compared to the Michaelis–Menten constant with the equilibrium thermodynamic characteristics, it was shown that the Monod constant (Ks ) has non-equilibrium thermodynamic characteristics. © 2006 Elsevier B.V. All rights reserved. Keywords: Specific growth rate; Substrate concentration; Thermodynamics; Monod equation
1. Introduction
2. Model development
The Monod equation is one of the best-known kinetic models describing microbial growth, which shows a functional relationship between the specific growth rate and an essential substrate concentration. It should be realized that the Michaelis–Menten equation was derived from the mechanism of enzyme reaction, while the Monod equation was developed from a curve fitting exercise, which is an example of an empirical correlation [2,16]. The Michaelis–Menten kinetics for enzymatic reactions gives mechanistic meanings to the constants involved, but none of those meanings can be applied readily to a substrate-cell system as described by the Monod equation, even though, the Monod relationship can provide the most generally satisfactory curve fitting of the growth data [7]. Other models for microbial growth had also been used, such as Grau equation, Hill or Moser equation in the environmental engineering and applied microbiology fields. However, the Monod equation and Hill or Moser equation are purely empirical [8,15,16,18], and theoretical derivation of these models has not been readily available in the literature. The present study, thus attempted to derive a general equation for microbial growth according to the thermodynamics of a microbial growth process.
According to the collision frequency theory for microbial growth [4], it is a reasonable consideration that if the substrate concentration in bulk solution increases, microbial growth would be more favourable [2,16,18], i.e. the effective free energy change (G◦ ) would decrease with the increase of the substrate concentration. In the conversion of a limiting substrate to biomass without substrate inhibition effect, G◦ could be expressed as
E-mail address: [email protected]. 1369-703X/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.bej.2006.05.022
G◦ = G◦ − nRT ln[S]
(1)
in which [S] is the molar concentration of substrate, n is the positive coefficient and G◦ is the change of standard free energy. It is believed that cells have only a limited number of sites for take up of substrate, e.g. macromolecules on cell surface, such as proteins, have multiple ligand binding sites responsible for transferring solutes into cell [1,10]. In fact, the similar assumption about active sites has been put forward in the literature [4,18]. When all sites are taken up by substrate, the rate of uptake will reach its maximum value and the specific growth rate will also be equal to the maximum specific growth rate. In this case, the driving force of microbial growth is the number of reactive sites on cells, which can be described by the difference between the maximum specific growth rate (µmax )
Y. Liu / Biochemical Engineering Journal 31 (2006) 102–105
103
and the observed specific growth rate (µobs ) under given conditions, and this driving force is disappearing when the microbial process gradually approaches its maximum. Thus, the overall change of free energy of the microbial growth process should be formulated as the function of the present growth state and growth potential in a way such that G = G◦ + RT ln
present growth state growth potential
(2)
In a theoretical sense, Eq. (2) is indeed consistent with the expression for free energy change of an ideal gas and solution as well as the collision frequency theory for microbial growth [3,4], while this equation is also similar to the Maxell–Boltzmann distribution law. As pointed out earlier, the microbial growth becomes less favourable as the reactive sites of cells are taken up. The number of the reactive sites of cells taken up along with the growth is directly correlated to µobs , i.e. µobs would reflect the present growth state, while the difference between µmax and µobs may represent the growth potential of bacteria. Therefore, Eq. (2) can be translated to µobs G = G◦ + RT ln (3) µmax − µobs Eq. (3) shows that when µobs = 0.5 µmax , G◦ is equal to G. This implies that G◦ can be defined as the overall free energy change at µobs = 0.5 µmax . Substitution of Eq. (1) into Eq. (3) gives µobs G = G◦ − nRT ln[S] + RT ln (4) µmax − µobs Solving Eq. (4) for µobs leads to µobs
(5)
in which ◦ −G)/RT
Kn = e(G
by a correlation coefficient of 0.999, while the value of n was estimated as 2.38. There is no reason to believe that n for a microbial growth or enzymatic reaction must be restricted to 1 as the Monod or Michaelis–Menten equation shows, e.g. in the case of phosphofructokinase, the dependence of the rate on the fructose-6-phosphate concentration can be described well by the empirical Hill equation with n ≈ 3.8 [21]. As noted by Hammes [10], the value of n depends on specific experimental conditions. The exponent n in Eq. (5) could provide a useful measure of microbial cooperativity. It should be emphasized that Eq. (5) is valid only for the growth phase of a microbial culture. When n equals 1, Eq. (5) becomes the well-known Monod equation: µobs = µmax
n
[S] = µmax Kn + [S]n
Fig. 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, µmax = 0.28 h−1 , n = 1.14 and Kn = 3.50 × 10−5 (mol/L)1.14 .
(6)
Eq. (5) indeed shows the same formulation as the Moser equation. As Roels [16] noted, the Moser model is a homologue of the purely empirical Hill model and the constants in the Moser and Hill models have not clearly defined physical meanings. Thus, Eq. (5) seems to offer a theoretical basis for the empirical Moser model.
[S] Kn + [S]
(7)
The Monod equation has been considered to be mathematically analogous to the Michaelis and Menten equation describing enzyme kinetics, but the meanings of the Monod constant and the Michaelis–Menten constant are completely different. As pointed out by Monod [15], there is no relationship between two constants. In the past half century, a tremendous quantity of experimental data of microbial growth had been interpreted and modeled by using the Monod equation which is strictly empir-
3. Discussion A simple least-square method was developed to evaluate the constants in Eq. (5), and literature data were used to verify Eq. (5) in this study. Zhuang et al. [20] determined the specific growth rates of Bacillus naphthovorans sp. nov. at different naphthalene concentrations (Fig. 1). The excellent agreement between the experimental data and Eq. (5) prediction is observed, and n has a value of 1.14. Koch and Schaechter [13] studied the effect of glucose concentration on the specific growth rate of Escherichia coli in a pure culture, and comparison of the experimental data with Eq. (5) prediction is shown in Fig. 2. Obviously, Eq. (5) provides a satisfactory description for the data, indicated
Fig. 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, µmax = 0.78 h−1 , n = 2.38 and Kn 2.12 × 10−6 (mol/L)2.38 .
104
Y. Liu / Biochemical Engineering Journal 31 (2006) 102–105
ical. In addition, it should be pointed out that [S] in Eq. (7) is expressed in molar concentrations (mol/L). However, in the literature of microbial growth research, volumetric concentration of substrate (mg/L) has been commonly used in the Monod equation, i.e. the Monod equation might be inadequately applied. To convert molar concentration of substrate to its volumetric concentration, both numerator and denominator of Eq. (7) are multiplied by the molar weight of substrate, Ms : µobs = µmax
Ms [S] S = µmax Ms Kn + Ms [S] Ks + S
(8)
in which S = [S]Ms , which is volumetric concentration of substrate and Ks = Ms Kn , which is so-called the Monod constant. Ks in the Monod equation is often referred to as the affinity constant of substrate–cell pair. However, it appears from Eq. (5) that Ks alone cannot describe such affinity because the magnitude of n also determines the reaction rate of substrate and subsequently the affinity of substrate to cell. In this case, n in the field of microbiology is termed cooperativity constant [10,21]. To look into the physical meaning of Kn , Eq. (6) can be rearranged to 1 G = G◦ + RT ln Kn
(9)
This equation may indicate that 1/Kn is a measure of cell affinity to substrate, which is determine by energy generated for growth. Flickinger and Drew [6] noted that any other definitions of Ks are speculative, e.g. Ks interpretation as dissociation constant of enzyme–substrate complex of the cellular enzyme involved in the first step of substrate conversion. It should be realized that Kn in Eq. (5) or Ks in Eq. (8) has the non-equilibrium thermodynamic characteristics, while the Michaelis–Menten constant has the equilibrium thermodynamic characteristics [2,21]. As pointed out by Westerhoff et al. [19], microbial growth should be analyzed in terms of non-equilibrium thermodynamics rather than equilibrium thermodynamics. It turns out that the major difference between the Monod equation for microbial growth and the Michaelis–Menten equation for enzymatic reaction lies on the process state, i.e. equilibrium for enzymatic reaction and non-equilibrium for microbial growth process. As shown by Eq. (6), Kn is a function of change in free energy of microbial growth process. It is clear now that Kn in Eq. (6) or Ks in the Monod equation has defined physical meaning rather than the black box characteristics as stated in the literature [2,18]. Obviously, G of microbial growth process is determined by both bacterial species and substrate under given culture conditions. Thus, the value of Kn in Eq. (6) or the Monod constant Ks should also be related to microbial species–substrate pairs. To date, the extremely large variation in the values of Ks in the Monod equation has been reported in the literature [4,5,11,17], e.g. the data available for E. coli growing with glucose showed that the Ks values in the Monod equation varied over more than three orders of magnitude for different E. coli strain-glucose pairs, and similar results were also found for Cytophaga johnsonae and Klebsiella pneumoniae [12]. It had been proposed that the large variation in Ks was simply due to changes in mass transfer [14], while Ferenci [5] noted that genotype, inoculum
history, length of exposure to substrate and bacterial density in cultures would lead to variations in determination of the Monod constant. Probably, for the first time, Eq. (6) clearly shows that Kn or Ks is indeed governed by change in free energy generated in microbial growth process, which is closely related to bacterial species–substrate pairs present in a microbial culture. As a result, any factor influencing the interaction between bacteria and substrate would also alter the estimate of Kn or Ks . It seems that Eq. (6) offers a new thermodynamic explanation for the 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 µobs =
µmax n [S] Kn
(10)
This equation is similar to the model proposed by Grau et al. [9] for the growth of activated sludge microorganisms. On the other hand, Eq. (5) can be arranged to the well-known Hill equation by letting Kn be (kn )n , i.e. µobs = µmax
[S]n (kn ) + [S]n n
(11)
in which kn is the Hill constant. Microbial growth process involves a series of complex biochemical reactions, and the exponent n in Eq. (5) indeed could provide a useful measure of microbial cooperativity. It is true that the proposed model with three constants (µmax , Kn and n) seems to be more complex than the Monod equation having two constants. In view of the progress of numerical methods, both the Monod model and Eq. (5) can be much easily solved without any technical difficulty. In the past half century, a tremendous quantity of experimental data of microbial growth had been interpreted and modeled by using the Monod equation which is strictly empirical. As Grady et al. [8] noted, many people have erroneously concluded that Monod proposed the equation on theoretical base. In fact, to formulate a mathematical description of microbial growth, one needs to seek the models with strong theoretical characteristics rather than the simplicity of the models. References [1] B. Alberts, A. Johnson, M. Raff, K. Roberts, P. Walter, Molecular Biology of the Cell, Garland Science, 2001. [2] J.E. Bailey, D.F. Ollis, Biochemical Engineering Fundamentals, McGraw-Hill, Singapore, 1986. [3] J.E. Brady, J.W. Russell, J.R. Holum, Chemistry: Matter and its Changes, John Wiley & Sons, Singapore, 2000. [4] D.K. Button, Nutrient uptake by microorganisms according to kinetic parameters from theory as related to cytoarchitecture, Microbiol. Mol. Biol. Rev. 62 (1998) 636–645. [5] T. Ferenci, Growth of bacterial cultures’ 50 years on: towards an uncertainty principle instead of constants in bacterial growth kinetics, Res. Microbiol. 150 (1999) 431–438. [6] M.C. Flickinger, S.W. Drew, Encyclopedia of Bioprocess Technology: Fermentation, Biocatalysis and Bioseparation, John Wiley & Sons, Inc., 1999. [7] A.F. Gaudy, E.T. Gaudy, Microbiology for Environmental Scientists and Engineers, McGraw-Hill, New York, 1980.
Y. Liu / Biochemical Engineering Journal 31 (2006) 102–105 [8] C.P.L. Grady, G.T. Daigger, H.C. Lim, Biological Wastewater Treatment, Marcel Dekker, New York, 1999. [9] P. Grau, M. Dohanyos, J. Chudoba, Kinetics of multicomponent substrate removal by activated sludge, Water Res. 9 (1975) 171–180. [10] G.G. Hammes, Thermodynamics and Kinetics for the Biological Sciences, Wiley Interscience, New York, 2000. [11] I. Kefalogianni, G. Aggelis, Modeling growth and biochemical activities of Azospirillum spp, Appl. Microbiol. Biotehcnol. 58 (2002) 352–357. [12] K. Kovarova-Kovar, T. Egli, Growth kinetics of suspended microbial cells: from single-substrate-controlled growth to mixed-substrate kinetics, Microbiol. Mol. Biol. Rev. 62 (1998) 646–666. [13] A.L. Koch, M. Schaechter, The world and ways of E. coli, in: A.L. Demain, N.A. Solomon (Eds.), Biology of Industrial Microorganisms, 1–25, Butterworths, London, 1985. [14] J.C. Merchuk, J.A. Asenjo, The Monod equation and mass transfer, Biotehcnol. Bioeng. 45 (1995) 91–94. [15] J. Monod, The growth of bacterial cultures, Ann. Rev. Microbiol. 3 (1949) 371–394.
105
[16] J.A. Roels, Energetics and Kinetics in Biotechnology, Elsevier, New York, 1983. [17] H. Senn, U. Lendenmann, M. Snozzi, G. Hamer, T. Egli, The growth of Escherichia coli in glucose-limited chemostat cultures-a reexamination of the kinetics, Biochim. Biophys. Acta-Gen. Sub. 1201 (1994) 424– 436. [18] M.L. Shulter, F. Kargi, Bioprocess Engineering: Basic Concepts, Prentice Hall PTR, 2002. [19] H.V. Westerhoff, J.S. Lolkema, R. Otto, K.J. Hellingwerf, Thermodynamics of growth. Non-equilibrium thermodynamics of bacterial growth. The phenomenological and the mosaic approach, Biochim. Biophys. Acta 683 (1982) 181–220. [20] W.Q. Zhuang, J.H. Tay, A.M. Maszenan, S.T.L. Tay, Bacillus naphthovorans sp. nov. from oil-contaminated tropical marine sediments and its role in naphthalene biodegradation, Appl. Microbial. Biotechnol. 58 (2002) 547–554. [21] G.L. Zubay, Biochemistry, Wm. C. Brown Publishers, Dubuque, 1998.
Short communication
A simple thermodynamic approach for derivation of a general Monod equation for microbial growth Yu Liu Division 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
Abstract The Monod equation has been widely applied to describe microbial growth, but it has no any mechanistic basis. Based on the thermodynamics of microbial growth process, a general model for microbial growth was developed. The constants involved in the present model were defined with clear physical meanings. The model derived can be reduced to the Monod equation, Grau equation and Hill or Moser equation. Compared to the Michaelis–Menten constant with the equilibrium thermodynamic characteristics, it was shown that the Monod constant (Ks ) has non-equilibrium thermodynamic characteristics. © 2006 Elsevier B.V. All rights reserved. Keywords: Specific growth rate; Substrate concentration; Thermodynamics; Monod equation
1. Introduction
2. Model development
The Monod equation is one of the best-known kinetic models describing microbial growth, which shows a functional relationship between the specific growth rate and an essential substrate concentration. It should be realized that the Michaelis–Menten equation was derived from the mechanism of enzyme reaction, while the Monod equation was developed from a curve fitting exercise, which is an example of an empirical correlation [2,16]. The Michaelis–Menten kinetics for enzymatic reactions gives mechanistic meanings to the constants involved, but none of those meanings can be applied readily to a substrate-cell system as described by the Monod equation, even though, the Monod relationship can provide the most generally satisfactory curve fitting of the growth data [7]. Other models for microbial growth had also been used, such as Grau equation, Hill or Moser equation in the environmental engineering and applied microbiology fields. However, the Monod equation and Hill or Moser equation are purely empirical [8,15,16,18], and theoretical derivation of these models has not been readily available in the literature. The present study, thus attempted to derive a general equation for microbial growth according to the thermodynamics of a microbial growth process.
According to the collision frequency theory for microbial growth [4], it is a reasonable consideration that if the substrate concentration in bulk solution increases, microbial growth would be more favourable [2,16,18], i.e. the effective free energy change (G◦ ) would decrease with the increase of the substrate concentration. In the conversion of a limiting substrate to biomass without substrate inhibition effect, G◦ could be expressed as
E-mail address: [email protected]. 1369-703X/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.bej.2006.05.022
G◦ = G◦ − nRT ln[S]
(1)
in which [S] is the molar concentration of substrate, n is the positive coefficient and G◦ is the change of standard free energy. It is believed that cells have only a limited number of sites for take up of substrate, e.g. macromolecules on cell surface, such as proteins, have multiple ligand binding sites responsible for transferring solutes into cell [1,10]. In fact, the similar assumption about active sites has been put forward in the literature [4,18]. When all sites are taken up by substrate, the rate of uptake will reach its maximum value and the specific growth rate will also be equal to the maximum specific growth rate. In this case, the driving force of microbial growth is the number of reactive sites on cells, which can be described by the difference between the maximum specific growth rate (µmax )
Y. Liu / Biochemical Engineering Journal 31 (2006) 102–105
103
and the observed specific growth rate (µobs ) under given conditions, and this driving force is disappearing when the microbial process gradually approaches its maximum. Thus, the overall change of free energy of the microbial growth process should be formulated as the function of the present growth state and growth potential in a way such that G = G◦ + RT ln
present growth state growth potential
(2)
In a theoretical sense, Eq. (2) is indeed consistent with the expression for free energy change of an ideal gas and solution as well as the collision frequency theory for microbial growth [3,4], while this equation is also similar to the Maxell–Boltzmann distribution law. As pointed out earlier, the microbial growth becomes less favourable as the reactive sites of cells are taken up. The number of the reactive sites of cells taken up along with the growth is directly correlated to µobs , i.e. µobs would reflect the present growth state, while the difference between µmax and µobs may represent the growth potential of bacteria. Therefore, Eq. (2) can be translated to µobs G = G◦ + RT ln (3) µmax − µobs Eq. (3) shows that when µobs = 0.5 µmax , G◦ is equal to G. This implies that G◦ can be defined as the overall free energy change at µobs = 0.5 µmax . Substitution of Eq. (1) into Eq. (3) gives µobs G = G◦ − nRT ln[S] + RT ln (4) µmax − µobs Solving Eq. (4) for µobs leads to µobs
(5)
in which ◦ −G)/RT
Kn = e(G
by a correlation coefficient of 0.999, while the value of n was estimated as 2.38. There is no reason to believe that n for a microbial growth or enzymatic reaction must be restricted to 1 as the Monod or Michaelis–Menten equation shows, e.g. in the case of phosphofructokinase, the dependence of the rate on the fructose-6-phosphate concentration can be described well by the empirical Hill equation with n ≈ 3.8 [21]. As noted by Hammes [10], the value of n depends on specific experimental conditions. The exponent n in Eq. (5) could provide a useful measure of microbial cooperativity. It should be emphasized that Eq. (5) is valid only for the growth phase of a microbial culture. When n equals 1, Eq. (5) becomes the well-known Monod equation: µobs = µmax
n
[S] = µmax Kn + [S]n
Fig. 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, µmax = 0.28 h−1 , n = 1.14 and Kn = 3.50 × 10−5 (mol/L)1.14 .
(6)
Eq. (5) indeed shows the same formulation as the Moser equation. As Roels [16] noted, the Moser model is a homologue of the purely empirical Hill model and the constants in the Moser and Hill models have not clearly defined physical meanings. Thus, Eq. (5) seems to offer a theoretical basis for the empirical Moser model.
[S] Kn + [S]
(7)
The Monod equation has been considered to be mathematically analogous to the Michaelis and Menten equation describing enzyme kinetics, but the meanings of the Monod constant and the Michaelis–Menten constant are completely different. As pointed out by Monod [15], there is no relationship between two constants. In the past half century, a tremendous quantity of experimental data of microbial growth had been interpreted and modeled by using the Monod equation which is strictly empir-
3. Discussion A simple least-square method was developed to evaluate the constants in Eq. (5), and literature data were used to verify Eq. (5) in this study. Zhuang et al. [20] determined the specific growth rates of Bacillus naphthovorans sp. nov. at different naphthalene concentrations (Fig. 1). The excellent agreement between the experimental data and Eq. (5) prediction is observed, and n has a value of 1.14. Koch and Schaechter [13] studied the effect of glucose concentration on the specific growth rate of Escherichia coli in a pure culture, and comparison of the experimental data with Eq. (5) prediction is shown in Fig. 2. Obviously, Eq. (5) provides a satisfactory description for the data, indicated
Fig. 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, µmax = 0.78 h−1 , n = 2.38 and Kn 2.12 × 10−6 (mol/L)2.38 .
104
Y. Liu / Biochemical Engineering Journal 31 (2006) 102–105
ical. In addition, it should be pointed out that [S] in Eq. (7) is expressed in molar concentrations (mol/L). However, in the literature of microbial growth research, volumetric concentration of substrate (mg/L) has been commonly used in the Monod equation, i.e. the Monod equation might be inadequately applied. To convert molar concentration of substrate to its volumetric concentration, both numerator and denominator of Eq. (7) are multiplied by the molar weight of substrate, Ms : µobs = µmax
Ms [S] S = µmax Ms Kn + Ms [S] Ks + S
(8)
in which S = [S]Ms , which is volumetric concentration of substrate and Ks = Ms Kn , which is so-called the Monod constant. Ks in the Monod equation is often referred to as the affinity constant of substrate–cell pair. However, it appears from Eq. (5) that Ks alone cannot describe such affinity because the magnitude of n also determines the reaction rate of substrate and subsequently the affinity of substrate to cell. In this case, n in the field of microbiology is termed cooperativity constant [10,21]. To look into the physical meaning of Kn , Eq. (6) can be rearranged to 1 G = G◦ + RT ln Kn
(9)
This equation may indicate that 1/Kn is a measure of cell affinity to substrate, which is determine by energy generated for growth. Flickinger and Drew [6] noted that any other definitions of Ks are speculative, e.g. Ks interpretation as dissociation constant of enzyme–substrate complex of the cellular enzyme involved in the first step of substrate conversion. It should be realized that Kn in Eq. (5) or Ks in Eq. (8) has the non-equilibrium thermodynamic characteristics, while the Michaelis–Menten constant has the equilibrium thermodynamic characteristics [2,21]. As pointed out by Westerhoff et al. [19], microbial growth should be analyzed in terms of non-equilibrium thermodynamics rather than equilibrium thermodynamics. It turns out that the major difference between the Monod equation for microbial growth and the Michaelis–Menten equation for enzymatic reaction lies on the process state, i.e. equilibrium for enzymatic reaction and non-equilibrium for microbial growth process. As shown by Eq. (6), Kn is a function of change in free energy of microbial growth process. It is clear now that Kn in Eq. (6) or Ks in the Monod equation has defined physical meaning rather than the black box characteristics as stated in the literature [2,18]. Obviously, G of microbial growth process is determined by both bacterial species and substrate under given culture conditions. Thus, the value of Kn in Eq. (6) or the Monod constant Ks should also be related to microbial species–substrate pairs. To date, the extremely large variation in the values of Ks in the Monod equation has been reported in the literature [4,5,11,17], e.g. the data available for E. coli growing with glucose showed that the Ks values in the Monod equation varied over more than three orders of magnitude for different E. coli strain-glucose pairs, and similar results were also found for Cytophaga johnsonae and Klebsiella pneumoniae [12]. It had been proposed that the large variation in Ks was simply due to changes in mass transfer [14], while Ferenci [5] noted that genotype, inoculum
history, length of exposure to substrate and bacterial density in cultures would lead to variations in determination of the Monod constant. Probably, for the first time, Eq. (6) clearly shows that Kn or Ks is indeed governed by change in free energy generated in microbial growth process, which is closely related to bacterial species–substrate pairs present in a microbial culture. As a result, any factor influencing the interaction between bacteria and substrate would also alter the estimate of Kn or Ks . It seems that Eq. (6) offers a new thermodynamic explanation for the 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 µobs =
µmax n [S] Kn
(10)
This equation is similar to the model proposed by Grau et al. [9] for the growth of activated sludge microorganisms. On the other hand, Eq. (5) can be arranged to the well-known Hill equation by letting Kn be (kn )n , i.e. µobs = µmax
[S]n (kn ) + [S]n n
(11)
in which kn is the Hill constant. Microbial growth process involves a series of complex biochemical reactions, and the exponent n in Eq. (5) indeed could provide a useful measure of microbial cooperativity. It is true that the proposed model with three constants (µmax , Kn and n) seems to be more complex than the Monod equation having two constants. In view of the progress of numerical methods, both the Monod model and Eq. (5) can be much easily solved without any technical difficulty. In the past half century, a tremendous quantity of experimental data of microbial growth had been interpreted and modeled by using the Monod equation which is strictly empirical. As Grady et al. [8] noted, many people have erroneously concluded that Monod proposed the equation on theoretical base. In fact, to formulate a mathematical description of microbial growth, one needs to seek the models with strong theoretical characteristics rather than the simplicity of the models. References [1] B. Alberts, A. Johnson, M. Raff, K. Roberts, P. Walter, Molecular Biology of the Cell, Garland Science, 2001. [2] J.E. Bailey, D.F. Ollis, Biochemical Engineering Fundamentals, McGraw-Hill, Singapore, 1986. [3] J.E. Brady, J.W. Russell, J.R. Holum, Chemistry: Matter and its Changes, John Wiley & Sons, Singapore, 2000. [4] D.K. Button, Nutrient uptake by microorganisms according to kinetic parameters from theory as related to cytoarchitecture, Microbiol. Mol. Biol. Rev. 62 (1998) 636–645. [5] T. Ferenci, Growth of bacterial cultures’ 50 years on: towards an uncertainty principle instead of constants in bacterial growth kinetics, Res. Microbiol. 150 (1999) 431–438. [6] M.C. Flickinger, S.W. Drew, Encyclopedia of Bioprocess Technology: Fermentation, Biocatalysis and Bioseparation, John Wiley & Sons, Inc., 1999. [7] A.F. Gaudy, E.T. Gaudy, Microbiology for Environmental Scientists and Engineers, McGraw-Hill, New York, 1980.
Y. Liu / Biochemical Engineering Journal 31 (2006) 102–105 [8] C.P.L. Grady, G.T. Daigger, H.C. Lim, Biological Wastewater Treatment, Marcel Dekker, New York, 1999. [9] P. Grau, M. Dohanyos, J. Chudoba, Kinetics of multicomponent substrate removal by activated sludge, Water Res. 9 (1975) 171–180. [10] G.G. Hammes, Thermodynamics and Kinetics for the Biological Sciences, Wiley Interscience, New York, 2000. [11] I. Kefalogianni, G. Aggelis, Modeling growth and biochemical activities of Azospirillum spp, Appl. Microbiol. Biotehcnol. 58 (2002) 352–357. [12] K. Kovarova-Kovar, T. Egli, Growth kinetics of suspended microbial cells: from single-substrate-controlled growth to mixed-substrate kinetics, Microbiol. Mol. Biol. Rev. 62 (1998) 646–666. [13] A.L. Koch, M. Schaechter, The world and ways of E. coli, in: A.L. Demain, N.A. Solomon (Eds.), Biology of Industrial Microorganisms, 1–25, Butterworths, London, 1985. [14] J.C. Merchuk, J.A. Asenjo, The Monod equation and mass transfer, Biotehcnol. Bioeng. 45 (1995) 91–94. [15] J. Monod, The growth of bacterial cultures, Ann. Rev. Microbiol. 3 (1949) 371–394.
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