ORIGINAL ARTICLE

Re-interpretation of the logistic equation for batchmicrobial growth in relation to Monod kineticsF. Kargi

Department of Environmental Engineering, Dokuz Eylul University, Buca, Izmir, Turkey

Introduction

Microbial growth can simply be described by the follow-

ing bioreaction

RSþ X �! nX þ RP ð1Þ

where S, X and P are the substrate, biomass and the

product respectively. The reaction is autocatalytic as the

biocatalyst (biomass) is generated by the reaction.

The rates of microbial growth, substrate utilization and

product formation have the following forms,

dX=dt ¼ f ðS;X; PÞ ð2Þ

�dS=dt ¼ gðS;X; PÞ ¼ ð1=Yx=sÞðdX=dtÞ ð3Þ

dP=dt ¼ yðS;X; PÞ ¼ Y p=sð�dS=dtÞ ð4Þ

Different equations were used to describe batch micro-

bial growth (Shuler and Kargi 2002). Among the most

widely used growth rate expressions are the logistic equa-

tion and the Monod kinetics. The logistic equation devel-

oped by Verhulst considers only biomass concentration

disregarding the substrate utilization. Variation of bio-

mass concentration with time is sigmoidal for the logistic

equation (Verhulst 1845). Monod kinetics considers both

the biomass and the rate limiting substrate concentra-

tions in growth rate expression and is the most widely

used kinetic equation for batch microbial growth

(Monod 1949). However, the logistic equation has also

been used by many investigators to describe batch micro-

bial growth (Simkins and Alexander 1984; Schimidt

et al.1985; Kingsland 2002). The kinetic constants in

those two equations were considered to be different and

not interrelated. While the Monod kinetics is mechanistic

Keywords

batch microbial growth, kinetic constants,

logistic equation, Monod kinetics.

Correspondence

Fikret Kargi, Department of Environmental

Engineering, Dokuz Eylul University, Buca,

Izmir, Turkey. E-mail: fikret.kargi@deu.edu.tr

2008 ⁄ 1013: received 16 June 2008, revised

and accepted 31 October 2008

doi:10.1111/j.1472-765X.2008.02537.x

Abstract

Aims: To determine the underlying substrate utilization mechanism in the

logistic equation for batch microbial growth by revealing the relationship

between the logistic and Monod kinetics. Also, to determine the logistic rate

constant in terms of Monod kinetic constants.

Methods and Results: The logistic equation used to describe batch microbial

growth was related to the Monod kinetics and found to be first-order in terms

of the substrate and biomass concentrations. The logistic equation constant

was also related to the Monod kinetic constants. Similarly, the substrate utiliza-

tion kinetic equations were derived by using the logistic growth equation and

related to the Monod kinetics.

Conclusion: It is revaled that the logistic growth equation is a special form of

the Monod growth kinetics when substrate limitation is first-order with respect

to the substrate concentration. The logistic rate constant (k) is directly propor-

tional to the maximum specific growth rate constant (lm) and initial substrate

concentration (S0) and also inversely related to the saturation constant (Ks).

Significance and Impact of the Study: The semi-empirical logistic equation can

be used instead of Monod kinetics at low substrate concentrations to describe

batch microbial growth using the relationship between the logistic rate constant

and the Monod kinetic constants.

Letters in Applied Microbiology ISSN 0266-8254

398 Journal compilation ª 2009 The Society for Applied Microbiology, Letters in Applied Microbiology 48 (2009) 398–401

ª 2009 The Author

with meaningful constants, the logistic equation is a

mathematical approximation with a rather meaningless

rate constant.

Researchers investigated the suitability of both models

at different stages of microbial growth and found one

model or the other more suitable (Simkins and Alexander

1984; Schimidt et al. 1985). However, the relationship

between the two models has not been revealed theoreti-

cally to guide the researchers for the use of the right

model. Therefore, the major objective of this article is to

reveal the relationship between the logistic and Monod

kinetic equations for batch microbial growth and to

express the logistic rate constant in terms of Monod

kinetic constants.

Theoretical background

The logistic equation used to describe the batch microbial

growth has the following form (Shuler and Kargi 2002;

Weisstein 2008).

dX

dt¼ kX 1� X

Xm

� �ð5Þ

where Xm is the maximum biomass concentration at the

end of batch growth and X is the biomass concentration

at any time during batch growth which are given by the

following equations.

Xm ¼ X0 þ Yx=sðS0 � SeÞ ð6Þ

X ¼ X0 þ Y x=sðS0 � SÞ ð6aÞ

where X0 is the initial biomass concentration (gX l)1),

Yx ⁄ s is the growth yield coefficient (gX g)1S), S0 and Se

are the substrate concentrations at the beginning (t = 0)

and at the end of batch growth (gS l)1) and S is the

substrate concentration at any time during batch growth

(gS l)1). X0 and Se values are usually negligible as

compared with Xm and S0.

With negligible X0 and Se, the eqns (6) and (6a) can be

approximated to

Xm ¼� Yx=sS0 ð6bÞ

X ¼ Yx=sðS0 � SÞ ð6cÞ

The integral form of the logistic growth equation has

the following form with the initial condition of X = X0 at

t = 0.

X ¼ X0ekt

1� ðX0=XmÞ 1� ektð Þ ð5aÞ

Substrate concentration is not considered as a parameter

in the logistic growth equation. However, it is well known

that the microbial growth rate is related to the rate-limit-

ing substrate concentration according to the Monod

kinetics as follows (Monod 1949; Shuler and Kargi 2002).

dX

dt¼ lmS

ðKs þ SÞX ¼ lX ð7Þ

where l and lm are the actual and the maximum specific

growth rates (h)1), S is the rate limiting substrate (nutri-

ent) concentration (gS l)1), Ks is the saturation or half-

rate constant (g l)1) and X is the biomass concentration

(gX l)1). Equation (6a) can be used to relate the biomass

and substrate concentrations during batch growth.

Substitution of

S ¼ S0 �X � X0

Yx=s

into eqn (7) yields the following growth rate equation.

dX

dt¼

lmðY x=sS0 þ X0 � XÞðKsY x=s þ Y x=sS0 þ X0 � XÞX ð7aÞ

Integration of eqn (7a) yields the following equation

describing the variation of the biomass concentration

with time during batch growth according to the Monod

kinetics (Shuler and Kargi 2002).

ALnðX=X0Þ�BLnððYx=sS0þX0�XÞ=Yx=sS0Þ ¼ lmt ð7bÞ

where A = (KsYx ⁄ s + S0Yx ⁄ s + X0) ⁄ (Yx ⁄ sS0 + X0) and

B = KsYx ⁄ s ⁄ (X0 + Yx ⁄ sS0).

Both the differential and the integral forms of the

logistic and Monod kinetic equations are quite differ-

ent. Monod kinetics is a mechanistic model where the

growth rate is related to both the substrate and

biomass concentrations as presented in eqn (7). The

logistic equation is independent of the substrate con-

centration and is only related to the biomass concentra-

tion. The growth rate is proportional to the actual

biomass concentration (X) and the difference between

the maximum and the actual biomass concentrations

(Xm ) X) which is known as the ‘carrying capacity’.

The logistic equation does not reveal any mechanism

for substrate utilization for batch growth unlike the

Monod kinetics.

The relationship between the logistic and Monodkinetics for batch microbial growth

The logistic equation [eqn (5)] can be modified as follows

by using eqns (6) and (6a).

F. Kargi Batch microbial growth kinetics

ª 2009 The Author

Journal compilation ª 2009 The Society for Applied Microbiology, Letters in Applied Microbiology 48 (2009) 398–401 399

dX

dt¼ kX 1� X

Xm

� �¼ kðX0 þ Yx=sðS0 � SÞÞ

1�ðX0 þ Yx=sðS0 � SÞÞðX0 þ Yx=sS0Þ

� � ð5bÞ

As X0 > X, X0 can be neglected and eqn (5b) reduces to

eqn (5c),

dX

dt¼ kY x=sðS0 � SÞ S

S0¼ kX

S

S0ð5cÞ

Equation (5c) indicates the inherent first-order kinetics

assumption in the logistic equation as the eqn (5c) is

first-order in terms of both the substrate and the biomass

concentrations.

By using the following relationship,

dX

dt¼ �Y x=s

dS

dtð8Þ

eqn (5c) can be written in terms of the substrate concen-

tration as follows:

� dS

dt¼ kðS0 � SÞ S

S0¼ kS 1� S

S0

� �ð9Þ

Equation (9) is the logistic equation for substrate utili-

zation in batch growth. Variation of the substrate concen-

tration with time during batch growth can be estimated

by the following equation,

S ¼ S0 �X � X0

Y x=s

ð8aÞ

where X = X0 and S = S0 at t = 0 yielding X = Xm and

S = Se at the end of batch growth. Biomass concentra-

tion (X) in eqn (8a) is to be estimated by using

eqn (5a).

The Monod kinetics for batch microbial growth as rep-

resented by eqn (7) reduces to the first-order kinetics at

low substrate concentrations (S > Ks) as follows:

dX

dt¼ lX ¼ lmS

KsX ð7cÞ

by comparing eqns (5c) and (7c) one can easily find out

that the rate constant for the logistic equation is related

to the Monod kinetic constants as follows:

k ¼ lm

KsS0 ¼ k1S0 ð10Þ

where k1 (lm ⁄ Ks) is the first-order rate constant for the

Monod kinetics. The rate constant for the logistic equa-

tion is directly related to the first-order rate constant and

the initial substrate concentration (S0). Substitution of

eqn (8a) into eqn (7c) yields eqn (5) which is the differ-

ential form of the logistic equation.

Equation (7c) can be further arranged as follows by

neglecting X0

dX

dt¼ lX ¼ lmS

KsX ¼ lmS

KsY x=sðS0 � SÞ ð7cÞ

or in terms of substrate utilization rate

� dS

dt¼ dX

dt

1

Y x=s¼ lmS

K sðS0 � SÞ ð7dÞ

As lm

Ks¼ k

S0, then eqn (7d) takes the following form

� dS

dt¼ kSð1� S=S0Þ which has the same form as eqn

(9).

Therefore, the Monod kinetics reduces to the logistic

equation with the assumption of first-order kinetics with

respect to the rate limiting substrate. In other words, the

logistic equation is a special form of Monod kinetics for

batch microbial growth with the assumption of the first-

order kinetics which holds for low substrate concentra-

tions (S > Ks).

The batch growth and substrate utilization curves as

estimated by the Monod [eqn (7b)], logistic [eqn (5a)]

and the first-order Monod [eqn (7c)] equations were

compared in Fig. 1 by using the following constants.

X0 ¼ 0�5 g l�1; S0 ¼ 10 g l�1; lm ¼ 0�25 h�1;

K s ¼ 2 g l�1; Yx=s ¼ 0�35 gX g�1S

The first-order Monod and the logistic equations

yielded the same curves for the variation of biomass and

12

10

8

6

4

X, S

(g

l–1)

2

00 2

Time (h)4

S

X

6 8

Figure 1 Batch microbial growth and substrate utilization curves as

predicted by the logistic ( ), Monod ( ) and the first-order Monod

(—) kinetics. X0 = 0Æ5 g l)1, S0 = 10 g l)1, lm = 0Æ25 h)1, Ks = 2 g

l)1, Yx ⁄ s = 0Æ35 gX g)1S.

Batch microbial growth kinetics F. Kargi

400 Journal compilation ª 2009 The Society for Applied Microbiology, Letters in Applied Microbiology 48 (2009) 398–401

ª 2009 The Author

the substrate with time. However, the growth and sub-

strate consumption curves estimated by the Monod equa-

tion were quite different and sluggish as compared with

the logistic curve due to saturation type substrate con-

sumption in Monod kinetics.

Conclusion

The semi-empirical logistic batch growth equation was

related to the mechanistic Monod kinetics and the mech-

anism of substrate utilization was revealed for the logistic

growth curve. The logistic equation was found to be a

special case of the Monod kinetics with the first-order

substrate utilization kinetics. The Monod kinetics reduces

to the logistic equation when the growth kinetics is first-

order in terms of the substrate and the biomass concen-

trations. The logistic rate constant (k) was found to be

proportional to the maximum specific growth rate con-

stant (lm) and the initial substrate concentration (S0) and

also inversely proportional to the saturation constant

(Ks). The logistic growth equation can be used instead of

complicated Monod kinetics to describe batch microbial

growth at low substrate concentrations where the rate of

substrate utilization is first-order with respect to substrate

concentration.

References

Kingsland, S.E. (2002) Modeling Nature. Chicago, IL, USA:

The University of Chicago Press.

Monod, J. (1949) The growth of bacterial cultures. Annu Rev

Microbiol 8, 371–374.

Schimidt, S.K., Simkins, S. and Alexander, M. (1985) Models

for the kinetics of biodegradation of organic compounds

not supporting growth. Appl Environ Microbiol 50, 323–

331.

Shuler, M.L. and Kargi, F. (2002) Bioprocess Engineering: Basic

Concepts, 2nd edn. NJ, USA: Prentice Hall.

Simkins, S. and Alexander, M. (1984) Models for mineraliza-

tion kinetics with the variables of substrate concentration

and population density. Appl Environ Microbiol 47, 1299–

1306.

Verhulst, P.F. (1845) Recherches mathematiques sur la loi

d’accroissement de la population. L’Academie Royale des

Sci. Et Belles-Lettres de Bruxelles 18, 1–41.

Weisstein, E.W. (2008) Logistic equation. MathWorld—A Wol-

fram Web Resource. http://mathworld.wolfram.com/Logistic

equation.html.

F. Kargi Batch microbial growth kinetics

ª 2009 The Author

Journal compilation ª 2009 The Society for Applied Microbiology, Letters in Applied Microbiology 48 (2009) 398–401 401