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Appl Microbiol Biotechnol (1993) 38:610-614 App//ed Microbiology Biotechnology © Springer-Verlag 1993 An error estimation of Michaelis-Menten (Monod)-type kinetics Laszlo Szigeti 1, Robert D. Tanner z 1 Department of Agricultural Chemical Technology, Technical University Budapest, 1521 Budapest Gellert ter 4, Hungary 2 Department of Chemical Engineering, Vanderbilt University, Nashville, TN 37235, USA Received: 13 April 1992/Accepted: 16 September 1992 Abstract. Due to research on biochemistry and genetic engineering, mathematical models of microbial growth have become more complicated but Michaelis-Menten or Monod type expressions have still been used for conver- sion rates, uptake rates, etc. It is worth examining the error that can be caused by these quasi-steady-state hy- potheses. This paper presents a simple but very effective rationale function that describes the error of the quasi- steady-state hypothesis in enzyme kinetics. A simplified fermentation kinetic model was used for comparison of microbial growth but no analytical error function has been found for batch cultivation. In the case of contin- uous fermentation the error can be given in an analytical form. Many simulations, based on real SCP experi- ments, show a significant effect of the quasi-steady-state hypothesis. Since the rate constants of intracellular events are not really known, we have to be very careful when taking into account Michaelis-Menten type expres- sions in the building of complicated models. Introduction By studyingrecently published papers it can be seen that attempts are made to describe very complicated, com- plex events, processes, and phenomena by very compli- cated, complex mathematical models. Due to the results of genetic engineering, modelling of the growth of re- combinant microorganisms (Coppela and Dhurjati 1990; Nielsen et al. 1991) or plasmid instability (Park et al. 1991) are necessarily contemporary. Description of the growth of mycelial fungi (Aynsley et al. 1990) or de- velopment of general models (Straight and Ramkrishna 1991) are also very important. In recently published papers (Aynsley et al. 1990; Coppela and Dhurjati 1990; Dussap et al. 1991; Moresi et al. 1991; Nielsen et al. 1991; Park et al. 1991; Straight and Ramkrishna 1991) modelling of these above-men- tioned problems use Michaelis-Menten- or Monod-type Correspondenceto: L. Szigeti expressions for conversion rates, uptake rates, and spe- cific rates. Before the application of these well-known hyperbolas, we think it is worth examining the errors that can be caused by this simplification. To make the situation clear, we have to return to the derivation of these hyperbolas. Results Error analysis if a quasi-steady-state hypothesis is used Enzyme situation. It is well known that the starting point for obtaining a Michaelis-Menten type [or Briggs- Haldane (Crooke et al. 1979)] hyperbola is the reaction scheme: E + S ~ [ES] k3 ~ ,E+P where S = substrate, P = product, E = enzyme, [ES] = en- zyme-substrate complex, and ki = rate constant i. Assuming that [ES] = constant, with the quasi-steady- state hypothesis: k3E*S vmS VBH - - -- (1) k~+k3 Km+S --+S kl results, where E*=the total enzyme amount-~(g mol), P = product concentration (g tool 1-1), S = substrate concentration (g mol 1- 1), VBH= production rate (veloci- ty) (g mol 1-1 s- 1), Km= Michaelis constant (g mol 1-1), and v~ = maximum velocity (g tool 1- 1 s - 1). By not using the steady-state hypothesis it can be proved (Crooke et al. 1979) that the v= dP/dt product formation rate satisfies the differential equation: d(dP) (k2+k3+kiS)v-klk3E*S ~-~ ~ =-k3 (kz+kiS)v-klk3E*S (2) Equation 2 can be derived by using the reaction equa- tions of the reaction scheme. Applying the materi~l balance then the differential equation for [ES] can

An error estimation of Michaelis-Menten (Monod)-type kinetics

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Appl Microbiol Biotechnol (1993) 38:610-614 App//ed

Microbiology Biotechnology © Springer-Verlag 1993

An error estimation of Michaelis-Menten (Monod)-type kinetics

Laszlo Szigeti 1, Robert D. Tanner z

1 Department of Agricultural Chemical Technology, Technical University Budapest, 1521 Budapest Gellert ter 4, Hungary 2 Department of Chemical Engineering, Vanderbilt University, Nashville, TN 37235, USA

Received: 13 April 1992/Accepted: 16 September 1992

Abstract. Due to research on biochemistry and genetic engineering, mathematical models of microbial growth have become more complicated but Michaelis-Menten or Monod type expressions have still been used for conver- sion rates, uptake rates, etc. It is worth examining the error that can be caused by these quasi-steady-state hy- potheses. This paper presents a simple but very effective rationale function that describes the error of the quasi- steady-state hypothesis in enzyme kinetics. A simplified fermentation kinetic model was used for comparison of microbial growth but no analytical error function has been found for batch cultivation. In the case of contin- uous fermentation the error can be given in an analytical form. Many simulations, based on real SCP experi- ments, show a significant effect of the quasi-steady-state hypothesis. Since the rate constants of intracellular events are not really known, we have to be very careful when taking into account Michaelis-Menten type expres- sions in the building of complicated models.

Introduction

By studyingrecently published papers it can be seen that attempts are made to describe very complicated, com- plex events, processes, and phenomena by very compli- cated, complex mathematical models. Due to the results of genetic engineering, modelling of the growth of re- combinant microorganisms (Coppela and Dhurjati 1990; Nielsen et al. 1991) or plasmid instability (Park et al. 1991) are necessarily contemporary. Description of the growth of mycelial fungi (Aynsley et al. 1990) or de- velopment of general models (Straight and Ramkrishna 1991) are also very important.

In recently published papers (Aynsley et al. 1990; Coppela and Dhurjati 1990; Dussap et al. 1991; Moresi et al. 1991; Nielsen et al. 1991; Park et al. 1991; Straight and Ramkrishna 1991) modelling of these above-men- tioned problems use Michaelis-Menten- or Monod-type

Correspondence to: L. Szigeti

expressions for conversion rates, uptake rates, and spe- cific rates. Before the application of these well-known hyperbolas, we think it is worth examining the errors that can be caused by this simplification. To make the situation clear, we have to return to the derivation of these hyperbolas.

Results

Error analysis if a quasi-steady-state hypothesis is used

Enzyme situation. It is well known that the starting point for obtaining a Michaelis-Menten type [or Briggs- Haldane (Crooke et al. 1979)] hyperbola is the reaction scheme:

E + S ~ [ES] k3 ~ , E + P

where S = substrate, P = product, E = enzyme, [ES] = en- zyme-substrate complex, and ki = rate constant i.

Assuming that [ES] = constant, with the quasi-steady- state hypothesis:

k3E*S vmS VBH - - - - (1)

k~+k3 Km+S - - + S

kl

results, where E * = t h e total enzyme amount-~(g mol), P = product concentration (g tool 1-1), S = substrate concentration (g mol 1 - 1), VBH = production rate (veloci- ty) (g mol 1-1 s - 1), Km= Michaelis constant (g mol 1-1), and v~ = maximum velocity (g tool 1 - 1 s - 1).

By not using the steady-state hypothesis it can be proved (Crooke et al. 1979) that the v= dP/dt product formation rate satisfies the differential equation:

d ( d P ) (k2+k3+kiS)v-klk3E*S ~-~ ~ = - k 3 (kz+kiS)v-klk3E*S (2)

Equation 2 can be derived by using the reaction equa- tions of the reaction scheme. Applying the materi~l balance then the differential equation for [ES] can

611

dP dt Briggs-Haldane

~ ~ , tg * :-k3 ",~ Sp s* \ ~

Fig. ]. A typical simulated trajectory of the single i~te[~ediate enzyme-substrate model, with substrate S being converted to prod- uct P. Here ~ / d t is the rate of formation of product where t=time. The corresponding Briggs-Haldane model is compared with the simulated trajectory in the region O~S=S~, where S~ is the substrate value corresponding to the peak of the ~ / d t trajec- tory. The tangent to the trajectory of S=S* is tangent ~ tg (8).= - ks

5 , 0 "

4.5- 4.0. 3.5. 3.0. 2.5 2.0 1.5 1.0 0.5-

0 0 0;1 012 013 OiL 0.5 016 0'.7 018 0'.9 1[0

Substrete Fig. 2. Error estimation in enzyme kinetics. The simulation pa- rameters are the constants kl = 1, kz= 10, ks = 10, total enzyme amount (E*) = 10, and initial substrate amount (S*) = 1. v, product formation rate; VSH, production rate; a . . . . maximum error; e, er- ror

be eliminated and by exploiting the expression - dS

dP / dt dS/dt ' Eq. 2 can be obtained.

It easy to show (Crooke et al. 1979; Szigeti et al. 1989) that v ~ passes th rough the m ax im um point o f the solution o f the Eq. 2 as illustrated in Fig. 1.

Denot ing the intersection point by Sp the quest ion is: how much is the error in interval [0, Sp] caused by the steady-state hypothesis d[ES]/dt = 0, i.e.

e = max I v - v ~ I = ? (3) Se[O, Sp]

Since v > v ~ (Crooks et al. 1979; Szigeti et al. 1989) if Se[0, Sp], then the absolute value can be omit ted in Eq. 3.

s can be approached by the m ax im um value o f

~ (S) = v (S) - VBH (S) (4)

Taking the derivative o f Eq. 4

d~(S) dv dvsH (k2+k3+kaS)v-kak3E*S - - ~___

dS dS dS -k3 (k2+k~S)v-k lk3E*S VmS

(Km + S) 2

substituting v (S) = ~ (S) + VBH (S) into the r ight -hand side d~(S)

and f rom the max im um requirement - = 0, and ap- dS

plying Eq. 1, after a little t iresome calculation the fol- lowing expression can be obtained:

Vm S Vm Km

~max(S) - K m + S k~(Km+S)~+KmE*(k2+k~S) (5)

Selecting two simulations, Figs. 2 and 3, out o f more than fifty carried out, it can be seen that rat ionale func- t ion ~m~ (S) is a very good approximat ion of error func- t ion ~(S). Theoretical ly the max imum o f ~m~(S) is an est imation o f e, but it is more impor tan t that

G~(S) v~K~ er(S) - - - - (6)

VnH(S) kl(Km+S)3+gmE*(k2+kiS)

0.18

0,16

0,14

0.12

0.10

0.08

O.06

0.04

O.02

0 0

~ / v /

# '5 ~ " - ~ , ~ , i , ~ , - - ~ ,

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Substrate

Fig. 3. Error estimation in enzyme kinetics. The simulation pa- rameters are kl = 4, k2 = 1, ks = 1, E* = 0.25, and S* = 1

describe the relative error (er) caused by the quasi-stea- dy-state assumption.

To illustrate the magni tude o f the relative error let us consider the case when the E* total enzyme amoun t is large enough and the substrate is small, i.e. mathemat i - cally:

k3 lira lira er(S) - (7)

E*~o S~0 k2

It is easy to see that in this case if the rate constants are equal then the relative error is 100% or if k2 is small enough then the relative error can be very large.

Fermentation situation. To carry out an analysis for fer- menta t ion processes, we have examined the generally ac- cepted reaction scheme (Tanner 1970):

R + S ~ [ R S ] k S • R + R (8)

where R represents the cell machinery o f the cell, S the substrate and [RS] the cell machinery-substrate com- plex.

We assume that the microbial cell concentra t ion de- noted by x, is related to R and [RS] by the equat ion

612

x = K ( R + [RSI) (9)

where K is a conversion factor. (Enzyme analogy: E* = E + [ES])

Introducing the specific growth rate (B):

l d x ~ - (10)

x d t

the following equation can be proved (Crooke and Tan- ner 1980):

dl~ (JJ+k6)[k4k6S-(ks+k6+k4S)~-B 2] - - = ( 1 1 ) dS (S* +R*-S)(-k4k6S+k41gS+ksp)

where S* = S (0), R* = R(0) initial conditions. The main point of the proof of Eq. 11 is very similar

to the proof of Eq. 2. Let us use the reaction equations of the reaction scheme (Eq. 8) and apply the material balance to eliminate the differential equation for [RS]. By the application of Eqs. 9 and 10, two differential equations can be obtained for/~ and S. Exploiting the expression:

dl.t dlt/dt - - - then Eq. 11 can be deduced.

dS dS/dt

The quasi-steady-state hypothesis is in this case:

[RS] - - = constant (12) R + [RS]

It has been shown (Crooke and Tanner 1980) that Eq. 12 is equivalent to:

dU - - = 0 ( 1 3 ) dS

From Eqs. 12 and 13 the steady-state equation for/~ is:

-(ks+k6+k~S)+]/(ks+k6+k4S)2+4k4k6S ~(S) = (14)

2

Multiplying Eq. 14 by the conjugate of the numerator and introducing notations:

ks + k6 Ks - - - tim=k6 (15)

k4 the following equation can be obtained:

p~S 2 /2(S) - (16)

K s + S ¢ 4pm S 1 + 1 + k4(Ks+S) 2

k6 can be considered the maximal growth rate (Pm) be- cause it can be proved (Crooke and Tanner 1980; Tan- ner 1970) that the solution of Eq. 11, p(S), satisfies the inequality:

p (S) < k6 (17)

In Eq. 16 if S = 0 or S~oo , then the second factor of the right-hand side is equal to 1. If we want to use a hyperbola as a steady-state model instead of the root function (Eqs. 14 or 15), then the first factor of the right-hand side can be used and the function:

0,6-

0,5-

0.4-

"~ 0.3-

=L0.2-

0.1

0 0 0'.2 0'.4 0'.6 0'.8 110 112 114 116 1.8 2.0

Substmte (g/I)

Fig. 4. Simulation of different batch fermentation models ~,/~, fi) based on real SCP experiments. The simulation parameters are Ks=0.1 (g/l), pro=0.54 (I/h), Y=0.38, x(0)=0.116 (g/l), and S (0) = 2 (g/l)

0.6-

0.5

0.4

~ 0.3 ̧

0.2

0.1

O.

f

012 014 0'.6 018 110 112 114 116 1.'8 2.0 Substrate (g/I)

Fig. 5. Simulation of different batch fermentation models (p,/~, fi) based on real SCP experiments. The simulation parameters are Ks=0.01 (g/l), //m=0.54 (l/h), Y=0.38, x(0)=0.116 (g/l), and S (0) = 2 (g/l)

S /2(S) - - - (18)

K s + S

can be considered as the Monod model. Equations 16, 18 and the solution of Eq. 11 can be seen in Figs. 4 and 5. Except for Ks the parameters of these are real and come from methanol-based fermentations (Nyeste un- published data).

Otherwise, it is easy to estimate the maximal relative error between/2 (S) and/2 (S). Since the denominator o f the second factor of Eq. 16 has a maximum at S = K s therefore:

2 2 fi(S) >/2(S) >/2 (S) - - (19)

1+]/~ 1 + ] / 1 +

B~m [ ks +Bm

From Eq. 19

/2(S)-f i(S) < = 0.2 (20) fi(S) 2

The problem of obtaining a function like Eq. 6 for the relative error of/~ (S) and fi(S) or/~(S) and/~(S) has

not been solved. However, in the case of continuous fer- mentation, we can show the difference in the steady states between the Monod model (quasi-steady-state hy- pothesis) and a model based on Eq. 8.

Equation 8 results in the following equations (Szigeti et al. 1989):

dx - - ~--- k 6 [(X* - X ) +K(S* - S)] - D x (21) dt

dS 1 - [ ( k s + 2 k 4 S ) ( x v - x - K S )

dt K - k4 S (xv - K S ) + D (So - S) (22)

where XF=X* +KS* , x* =x(0) , S* =S(0) are the initial conditions, D is the dilution rate, and So is the substrate concentration at the inlet. The continuous version of the Monod model is well known; we call (21) and (22) basic models.

Let us denote the steady states of the basic model and Monod model by Sb, 2b, SM, 2M respectively. Introduc- ing

e = S b - S M (23)

then f rom Eqs. 21 and 22

#m Y (So - gM) #,,, Y e 2b = + (24)

( k5 "~- # m ) • -t- # M (ks -}- # m ) e q- # m

l /k4 Ye / k4 X b = X M -I- (25)

Kse/#m + 1/k4 Kse/#m + 1/k4

Equation 25 shows that even if a very small error occurs in the substrate steady states (which is conceivable be- cause of the steady-state condition # ( S ) = D ; see Figs. 4 and 5) the steady-state values of cell concentration might be very different depending first of all on Ks. These situations can be seen in Figs. 6-8. The paramet- ers come f rom the same SCP fermentations as above (Nyeste unpublished data).

2.0-

613

1.8 ̧

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2-

0 0

\~s.

5 iO i5 ~0 215 3'0 35 Time (h)

Fig. 7. Simulation of Monod and Eqs. 21 and 22 continuous fer- mentation models based on real SCP experiments. The simulation parameters are Ks=0.1 (g/l), /~m=0.54 (I/h), Y=0.38, x(0)=0.116 (g/l), S(0)=2 (g/l), So=2 (g/l), and D=0.3 (l/h)

1.8. 1.6 ̧ 1.4 -~,.. 1.2 ~

0.8 :M 0.6 ~

0.41 ,

0 .2 - i s

0 5 10 I5 20 25 30 35 Time (h)

Fig. 8. Simulation of Monod and Eqs. 21 and 22 continuous fer- mentation models based on real SCP experiments. The simulation parameters are Ks=0.01 (g/I), pro=0.54 (i/h), Y=0.38, x (0) = 0.116 (g/l), S (0) = 2 (g/l), So = 2 (g/l), and D = 0.3 (1/h)

2.0

1.6 ~\ 1.4 1.2 SM

1.0 S~'\; 0.8 ~ 0.6 - 0.4 0.2

0 ~ 1'0 i5 ~0 i5 3~) 35

Time (h)

Fig. 6. Simulation of Monod and Eqs. 21 and 22 continuous fer- mentation models based on real SCP experiments. The simulation parameters are Ks=0.5 (g/l), #m=0.54 (I/h), Y=0.38, x(0)=0.116 (g/l), S(0)=2 (g/l), So=2 (g/l), and dilution rate (D) =0.3 (l/h)

D i s c u s s i o n

Despite the huge development of computer science - it is easy to solve complicated differential equations by com- puters - we have to be careful in the building of a struc- tural model because the rate constants of the intracellu- lar events are not really known. Therefore, the quasi- steady-state hypothesis (Michaelis-Menten, Briggs-Hal- dane, Monod) may lead to incorrect results. In those cases, if the error caused by the quasi-steady-state as- sumptions is large then the simplification cannot be al- lowed and the full set of differential equations should be used. (This is just one of the reasons why a model is not able to describe reality. A more complex model is likely to be more adequate than a simpler one.)

614

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