Biotecbnol. Prog. 1995, 11, 712-716
712
Derivation of Kinetic Equations for Growth on Single Substrates Based on General Properties of a Simple Metabolic Network J. J. Heijnen"and B. Romein Department of Biochemical Engineering, Delft University of Technology, Julianalaan 67, 2628 BC Delft, The Netherlands
Research on microbial growth has resulted in a large number of different equations to describe growth kinetics, the most famous of which is that of Monod. In the present work, the case of single substrate limitation in the absence of substrate inhibition is studied. Metabolism can then be regarded as a simple network consisting of a single reaction for anabolism and another for catabolism. These reactions are interconnected by cofactors of rather constant concentration. Growth is described as a combination of substrate uptake kinetics and growth kinetics. The notion that enzymatic activities of anabolism and catabolism depend on growth rate leads to a number of substrate uptake models, which fall into a few families. Each of them can be described by a general and flexible three-parameter equation. Well-known equations like that of Monod and Blackman turn out to be special cases. It could be concluded that the presented equations show more or less the same behavior in a scaled uptake rate versus substrate concentration plot, although the (powered) Monod equation might in general be preferred because its overall shape seems to resemble experimental observations most.
Introduction Kinetic equations which describe the specific growth rate p of microorganisms as a function of the concentration of substrates are crucial in understanding and describing many phenomena in biotechnological processes. In case the conditions can be chosen such that only one compound is growth limiting, the Monod equation is often used. However, systematic deviations from the Monod equation may occur. Although the experimental data are scarce, it can be noted that at low substrate concentrations the actual growth rate lies above the Monod prediction, whereas at higher values, the approach to the p m value proceeds more slowly than expected (Lendenmann, 1994;Rutgers, 1990; Senn, 1989). However, a.0. the inconsistency of experimental data (Senn 1994)has led to the development of a large amount of alternative equations, as proposed by Blackman, Moser, Teissier, Dabes et al. (19731, and Powell (see also Roels (1983)), which are listed in Table 1. All these growth equations can be written, as indicated, in the dimensionless concentration c (c = CJKJ and growth rate p' = p/pm) in the form
+'
The number of parameters is also indicated. g(c) is different for the various proposals, but all share the properties that if no substrate inhibition occurs, g(c) increases monotonously from 0 to 1if c increases from 0 to infinity. The first four equations have no biochemical background, although the Monod equation is originally based on the concept of a single rate-limiting enzyme which controls growth via Michaelis-Menten kinetics. The equations of Dabes and Powell are very similar, which is not surprising, because both propose two relevant processes in series (a slow and fast enzyme step and a transport and enzyme step, respectively) to describe growth. Notably both have three parameters, in
Table 1. Well-Known Growth Rate Equations
name Monod Blackman Moser Teissier Dabes
+
p' = c/(l c ) p' = c c < 1 p ' = 1 C'l p' = cV(l C R ) p' = 1 - exp(-c)
+
'-
,-l+c+a 4a
[ l - (1 Powell
no. of parameters
relation
+ +
,-l+c+a I- 2a
[l-(1-
>'"I >'"I
(1 8ac c al2
+ +
(1 4ac c a)'
3
contrast to the other equations that have only two. A more extensive discussion on these models is given by Senn (1989) and Senn et al. (1994). In this paper a general concept will be outlined which leads to a collection of different expressions for substrate uptake kinetics, qs, of the general dimensionless type:
which results in a complete growth model by combining with the Herbert-% equation as shown by Rhamkrishna et al. (1966).
Characteristic Structural Features of Metabolic Networks in Growing Micro-organisms In metabolic networks a multitude of biochemical reactions occur, all of which are catalyzed by enzymes. This aspect has been used by Dabes et al. (19731, who
8756-7938/95/3011-0712$09.00/0 0 1995 American Chemical Society and American Institute of Chemical Engineers
Biotechnol. Prog., 1995, Vol. 11, No. 6 S
713 Table 2. Exponents Used for kl and ks Relations (Eqs 6a and 6b), Which Were Substituted in Eqs 7a and 7b, and the Expression Obtained for qa (Table 3)
,X
VI
Figure 1. Growth as a coupled network of anabolism and catabolism. Substrate S is converted into biomass ( X ) at the expense of ATP ( X I )a t a rate V I . The resulting ADP (XZ)is transformed into ATP a t a rate uz using an electron donor (edonor), which leads to the formation of COz.
assumed a linear sequence of enzyme-catalyzedreactions to derive a growth kinetic equation. However, recently it has been extensively argued by Reich and Sel'kov (1981) that the kinetic behavior of metabolic networks can be understood to a large extent by only recognizing the coupling of biochemical pathways by moiety-conserved cofactors like ATP/ADP or NADH H+/NAD+,etc. Furthermore, it has become clear that in metabolic networks the activities of enzymes are not constant but depend very much on the growth rate p . Enzyme levels can increase and decrease as a function of p (Egli et al., 1980; Matin, 1981; Roitsch and Stolp, 1986). In contrast, it is known that the sum concentrations of the cofactor pairs are not very dependent on growth rate (Reich, 1981). These features are sufficient to derive a whole class of simple expressions for substrate uptake kinetics as will be shown below.
+
Substrate Uptake Kinetics Derived from Metabolic Network Structural Features In a simplified view, microbial growth may be divided into a catabolic and an anabolic process, which are coupled by the cofactor couple ATPIADP. In anabolism substrate S is converted into biomass X at the expense of ATP. Catabolism generates metabolic energy by combustion of an electron donor to C02 under consumption of ADP and production of ATP. This is schematically shown in Figure 1. In general both reactions are irreversible and the turnover of cofactors is in a pseudo-steady state (due to their often low concentration). Further it is assumed that only one substrate, S, is limiting. This means that the rate of anabolism, VI, depends on the concentrations of S and ATP, denoted by C , and X I , respectively, whereas u2 depends only on the concentration of ADP, denoted by X2. It is assumed that u1 depends on C , according to a simple hyperbolic saturation function. This leads to the following rate equations for u1 and UZ: fi
v,=kX'k,
L,
exponent nl (for kl)
exponent n2 (for k2)
0 0 0 0 1 -1
0
'I2
-1 1 -1 0
1
-1 '12
0 0 0 -1 -1 1
'f2
- '12
1
or 7b
a a a a a a a a a a b b b b b
-1
-2 -2 1
substituted in eq 7a
-2
-2 '13
obtained expression (see Table 3) 111.1 111.2 111.3 111.4 111.5 111.4 111.6 111.7 111.8 111.9 111.1 111.7 111.10 111.11 111.12
a a a
exponent n in eq 8 1
-1
'12
- 'I2
2 '13
'13 '13
- '13
-3
This coupling model of anabolism and catabolism directly leads to a well-known hyperbolic relation for q, in C,. However, this is only so if k1, Kz, and k , are assumed to be constant. From numerous measurements it is known that the activities of catabolic (which relate to k2) and anabolic enzymes (which relate to kl) are functions of growth rate (Egli et al., 1980; Matin, 1981; Roitsch and Stolp, 1986). For a quantitative analysis it is necessary to formulate these functions, where it is convenient to make kl and kz dependent on q, instead of p . The following, generally applicable functions are considered: (6a)
(6b) For various values of the exponent n, the functions (eqs 6a and 6b) can be substituted in eq 5, which can subsequently be solved to obtain q, versus C,. Such substitutions do however in general lead to equations which cannot be solved analytically. Therefore two situations can be considered. First it is assumed that k2 > k1, meaning that substrate uptake is mostly influenced by anabolism. For this situation, eq 5 becomes 4, = klCJ(ks + CJ
(7b)
In both situations, the hyperbolic relation is still obtained. Substitution of a number of k l and kz dependencies in eqs 7a and 7b-using different values of nl and nz in eqs 6a and 6b-leads to algebraic equations which can be solved analytically. The equations that are obtained all contain two parameters. They can be made dimension-
714
Biotechnol. Frog., 1995, Vol. 11, No. 6
Table 3. Substrate Kinetic Equations Obtained from Analysis of Cofactor-CoupledNetwork with Variable Enzyme ContenP
equation no. 111.1
111.2 111.3 111.4 111.5 111.6
remarks Monod Blackman
unscaled expression for q‘ c/(l c)
+
C
threshold
c value where q’ = 0.5 1 ‘12
+ 1/c2)1/2- l/c
(1
4/3
(c2 + 2€)1/2 - c 1 - l/c
114 2
scaled expression for q’ c/(l + c) 0.5~ [ l + 9/(16~~)1’/~ -
family M
&
+
(~‘116 1/g)112 - 0.2% 1 - 1/(2€)
J M
4/15
111.7 111.8
[c/(l + c)1’/2 (1 - 1/c)”2
threshold
111.9
M M
112
B M
111.10 111.11
M
0.547
111.12 111.13 111.14 a They are divided into families, as indicated in the last column: M, Monod type; B, Blackman type; J, square root type. For discussion, see text.
less by introducing convenient definitions of a dimensionless substrate uptake rate q‘ and a dimensionless concentration c. The resulting equations are listed in the third column of Table 3. For easy comparison the equations can be rescaled by introducing a dimensionless concentration c‘ by requiring that always for c‘ = 1,q‘ = 0.5. For all scaled dimensionless equations (fifkh column of Table 3) it also holds that q’ 0 for c’ 0 and that q‘ = 1 for c‘ =.
-
-
-
Classification of Kinetic Equations for Substrate Uptake From Table 3, it is clear that a number of equations is obtained that can be divided into several categories. Powered Monod Equation. The first family contains equations which are of the Monod-type (eq 111.1: c/(c 1))or exponents thereof, as indicated in the last column of Table 3. Similarly, equations in (1- l/c) are obtained (eq 111.5)or exponents thereof. On close inspection it can be shown that all of these equations can be written in a single, scaled, dimensionless form:
+
The scaled equations in Table 3 are obtained from eq 8 as follows: n = 1gives eq 111.1; n = -1 gives eq 111.5; n = 2 gives eq 111.10; n = 2/3 gives eq 111.11;n = l/3 gives eq 111.12; n = 4 3 gives eq 111.13; n = -3 gives eq 111.14. It can be concluded that eq 8 represents a general threeparameter equation. This equation takes through the exponent n the influence of a variable enzyme concentration as a function of growth rate into account, according to Table 2. Equation 8 shows that, because q’ > 0, a restriction on c’ exists: c’ > 1 - 21”
(9)
For n > 0, this condition poses no restriction on c’, whereas for n < 0, this restriction gives a threshold value. Equation 8 is shown in Figure 2 for a number of values of n. A wide range of q‘(c‘) behavior can be covered,
including threshold behavior (Button, 1985) and the Monod equation (n = 1). In general-comparing to the Monod behavior-curves are characterized by a fast rise in the concentration region 0 < c’ 1 and a rather slow approach of the asymptote for c’ > 1. Note that in general positive values for the exponent lead to Monodlike behavior which show in all but the simplest case (n = 1)a sigmoid curve, whereas negative exponents give rise t o threshold kinetics. Blackman-QpeEquations. Table 3 also shows that Blackman kinetics (eq 111.2) and its exponents (eq 111.9) are obtained. A generalized scaled dimensionless threeparameter equation then reads =
6)”
(10)
This result is obtained from eq 7a if exponents n2 = 1 and nl < 0 in eqs 6a and 6b. The exponent n in eq 10 equals then n = 141 - nl),which shows that 0 < n < 1. Because q’(c’) is maximal equal to 1, eq 10 shows that this equation only holds for the dimensionless c’ < 2lIn. For higher c’ values, q’ = 1. Figure 3 shows the behavior of eq 10 compared to that of the Monod equation. Obviously the behavior of eq 10 shows much resemblance to that of eq 8, as can be seen in comparing Figures 2 and 3. Square Root Type Equations. Finally Table 3 shows so-called square root equations (eq 111.3, eq 111.4, and eq 111.6). In Figure 4, these are compared to the Monod equation. Again the behavior displayed is not much different from the powered Monod equation, as can be seen in Figures 2 and 4.
Discussion By comparing the different types of equations, it is easily seen that eq 8 with either positive or negative exponent shows a q‘ behavior which covers also the behavior of eq 10 (Figure 3) or the square root type
Biofechnol. Prog., 1995, Vol. 11, No. 6
I
1.00
715 I.oo
n-4
Monod
0.80
0.80
0.60
0.60
U
U
0.40
0.40
0.20
0.20
0.00 0
1
2
3
4
5
0.00 0
1
3
2
4
5
C
C
Figure 4. Rescaled specific substrate uptake rate as function of the rescaled dimensionless substrate concentration according Monod kinetics and square root equations. Equations 111.3, 111.4, 111.6, and Monod kinetics are shown.
1 .oo
0.80
0.40
initially rises faster t h a n t h e Monod curve, b u t for c’ < 1, q‘ rises slower, which is often observed (Lendenmann, 1994). For n < 0, a threshold value of c’ is found. For n < 1 a sigmoid behavior is observed which according to Reich and Sel’kov (1981) is observed i n t h e growth kinetics of E. coli.
0.20
Conclusions
0.60 U
0.00 0
1
3
2
4
5
C
Figure 2. Rescaled specific substrate uptake rate as a function of the rescaled dimensionless substrate concentration according to Monod-type and threshold-type kinetics (eq 8). Several values for the coefficient n of eq 8 are shown to illustrate its effect on curve shape. (a) Monod-type kinetics: n = 0.25, 0.33, 0.50, 1 (Monod),2, 3,4. (b) threshold-type kinetics: n = -0.25, -0.33, -0.50, -1, -2, -3, and 1 (Monod). Note that for some values of n the corresponding equations are listed in Table 3.
Notation
0.80
0.60 U
0.40
0.20 0.00 y 0
In t h e above, it could be shown that on t h e basis of a very simple conceptual model of growth, i.e. t h e division of metabolic activities into catabolism and anabolism which a r e coupled by conserved moieties, a series of equations to describe growth kinetics can be derived. The rate of substrate uptake can be described by a general three-parameter equation. This powered Monod equation also displays t h e interesting properties of sigmoidal shape, threshold behavior, and fast rise/slow approach to its asymptote compared to t h e Monod equation. Combination of this substrate uptake equation with t h e Herbert-Pirt relation with growth kinetics gives a complete growth model.
1
3
2
4
5
C
Figure 3. Rescaled specific substrate uptake rate as a function of the rescaled dimensionless substrate concentration according to eq 10 and Monod kinetics. Several values for the coefficient n of eq 10 are shown to illustrate its effect on curve shape: n = 0.25, 0.33, 0.50 (eq 111.9) and 1 (eq 111.2).
equation. Therefore, eq 8 seems to offer a flexible threeparameter equation which should cover most observed growth curves. It is noted t h a t for 0 < n < 1 t h e q’
dimensionless concentration rescaled dimensionless concentration substrate concentration (mol,.L-l) affinity constant for substrate (mol,-L-l) rescaled affinity constant for substrate (mol,.L-l) kinetic rate constant for anabolism (mols~molx-l~h-l) maximal value of the kinetic rate constant for anabolism (mol,*mol,- * hkinetic rate constant for catabolism (mol,*molx-l~h-l) maximal value of the kinetic rate constant for catabolism (mol,*molx-l-h-l) exponent, representing growth-rate-dependent enzymatic activity exponent, representing growth-rate-dependent anabolic activity exponent, representing growth-rate-dependent catabolic activity rescaled dimensionless specific substrate uptake rate specific substrate uptake rate (mols*molx-l*h-l) dimensionless specific substrate uptake rate
Biotechnol. frog., 1995, Vol. 11, No. 6 maximal specific substrate uptake rate (mol,. mol,- h- l) specific reaction rate of anabolism (mol,.molx-l*h-l) specific reaction rate of catabolism, expressed per mol of substrate (mol,*mol,-'*h-') dimensionless, normalized cofactor concentration
-
(ATP) dimensionless, normalized cofactor concentration
(ADP) kinetic constant occurring in the equation of Dabes specific growth rate (h-l) maximum specific growth rate (h-l) dimensionless specific growth rate dimensionless growth rate function dimensionless substrate uptake function
Literature Cited Button, D. K. Kinetics of nutrient-limited transport and microbial growth. Microbiol. Rev. 1985, 270-297. Dabes, J. N.; Finn, R. K.; Wilhe, C. R. Equation of substratelimited growth: The case for Blackman kinetic. Biotechnol. Bioeng. 1973,15, 1159-1177. Egli, Th.; Van Dijken, J. P.; Veenhuis, M.; Harder, W.; Fiechter, A. Methanol metabolism in yeast: regulation and synthesis of catabolic enzymes. Arch. Microbiol. 1980, 124, 115-121. Lendenmann, U. Growth kinetics of Escherichia Coli with mixtures of sugars. Ph.D. Thesis 10658, Swiss Federal Institute of Technology Zurich, 1994.
Matin, A. Regulation of enzyme synthesis as studied in continuous culture. In Continuous cultures of cells; Calcott, P. H., Ed.; CRC Press: Boca Raton, FL; 1981; pp 64-97. Ramkrishna; Fredrickson; Tsuchiya. Dynamics of microbial propagation models considering endogenous metabolism. J . Genet. Appl. Microbiol. 1966, 12, 311-327. Reich, J. G.; Sel'kov, E. E. Energy metabolism of the cell-a theoretical treatise; Academic Press: London, 1981. Roels, J. A. Energetics and kinetics in biotechnology; Elsevier Biomedical: Amsterdam, 1983. Roitsch, T.; Stolp, H. Synthesis of Dissimilatory enzymes of serine type methylotrophs under different growth conditions. Arch. Microbiol. 1986, 144, 245-247. Rutgers, M. Control and thermodynamics of microbial growth. Ph.D. Thesis, University of Amsterdam, 1990. Senn, H. P. Kinetik und Regulation des Zuckerabbaus von Escherichia coli ML30 bei tiefen Zuckerconcentrationen. Ph.D. Thesis, Swiss Federal Institute of Technology Zurich, 1989. Senn, H. P.; Lendenmann, U.; Snozzi, M.; Hamer, G.; Egli, T. The growth of Escherichia coli in glucose-limited chemostat cultures: a re-examination of the kinetics. Biochim. Biophys. Acta 1994,1201, 424-436. Accepted August 25, 1995.@ BP950058V Abstract published in Advance ACS Abstracts, October 15, 1995. @
