The monod equation: A revisit and a generalization to product inhibition situations

The Monod Equation: A Revisit and a Generalization to Product Inhibition Situations

OCTAVE LEVENSPI EL, Cliernical Engineering Depnrtment Oregon State Uni\.ersity, Coriwllis, Oregon 97331

Summary

This paper shows how to treat the substrate-limiting Monod equation in a straightforward manner for different types of fermentors (plug-flow, batch, and mixed-flow) using the general language of chemical reaction engineering. Straight-line plots are developed for directly finding the kinetic constants of the equation, and an example using Monod's original data illustrates the procedure. The Monod equation is then generalized to account for the effects of both substrate and inhibitory toxic wastes. Finally. for pure product inhibition performance. expressions are derived for various reactor types, and correlation graphs are developed for finding the kinetic constants of the reaction. An example from the recent literature shows that this equation form fits the data extremely well.

I. INTRODUCTION

Although the growth of microorganisms is an unusually complex phenomenon, it is often possible to represent this growth by rela- tively simple laws. In particular, Monod's pioneering proposed a very simple equation form whenever a single essential nutritional requirement is the growth-limiting factor. and where the presence of toxic metabolic products plays no role. In the general language of chemical reactors Monod's equation is expressed as

(sub y e ) +cells , (proFct) + (moreccells)

and

rc = kC,Cc/(CA + C,) (1)

where rc is the rate of production of cells: C , is the substrate concentration; C c is the cell concentration; C w is the Monod constant, the substrate concentration where the rate is one-half the maximum.

This paper proposes a simple generalization of the Monod equa-

Biotechnology and Bioengineering, Vol. XXII , Pp. 1671- 1687 (1980) @ 1980 John Wiley & Sons, Inc. 00063592/80/0022- 167 1$01.70

I672 LEV EN SPI EL

tion to account for the influence of inhibitory product R, as follows:

I ' r . = k ( l - C,/CR*)n.[CACC/(CA + C,)] (2a)

or

rC = k<, , , , [CACC/(CA + C,) ] with k,,,,, = k ( l - C,/C,")" (2b)

where C,* is the limiting concentration of inhibitory product above which cells cease to multiply. At the extreme where C R e CR*, inhibition by product does not slow the rate and so

k ( 1 - CR/CR*)n + k or kobs + k

in which case, eq. (2) reduces to the original inhibition-free substrate-limiting Monod expression of eq. ( I ) .

At the other extreme where CA %- Cw, the depletion of substrate does not slow the rate,

CA/(CA + CAW)+= 1

and eq. (2) reduces to substrate-independent and purely inhibitory kinetics, as follows:

rc = k ( 1 - C,/CR*pCc (3 ) Section I1 shows how to treat the substrate-limiting Monod

expression of eq. ( 1 ) in a straightforward manner for various reactor types: batch, plug-flow and mixed-flow [or continuous stirred tank reactor (CSTR), chemostat, turbidostat, etc.]. Section I11 does the same for the pure product inhibition expression of eq. (3), while Sec. IV considers situations where the generalized Monod expression of eq. (2) must be used. There, both depletion of nutrient and the accumulation of inhibitory product influence the progress of reaction.

Two examples, one using Monod's original data, the other re- quiring the use of the generalized expression, show how to find the kinetic constants simply and directly from reported data.

We make one simplifying assumption in this treatment, to the effect that every unit of substrate consumed will produce the same amount of cells, and of product R , at all conditions. Thus

THE MONOD EQUATION 1673

where [AIC], [AIR], and [CIR] are called the yields. Thus we assume throughout that the yields stay unchanged for any system. This is often a reasonable assumption, as was shown by Monod,’ Bazua and Wilke.7 and others.

11. SUBSTRATE-LIMITING MONOD KINETICS

Here we develop the performance equations for the Monod kinetics of eq. ( 1 ) occurring in various ideal reactor types. Since much of this is straightforward we -will be brief.

Batch or Plug-Flow Reactor

For a liquid-phase system the basic performance equation, given in any reactor design book (e.g., see Ref. 3), is as follows:

Inserting eq. ( I ) , converting to one variable with eq. (4), and then integrating gives the changing concentration of materials as

This expression gives an S-shaped concentration-time curve as shown in Figure 1 .

It is difficult to evaluate the kinetic constants from the integrated

- ‘run out of food

devia tes from exponential if both CA a n d Cc change a t

s t a r t appreciably during t h e run $ cco - tb Or T p

0

Fig. I . Concentration-time curve for substrate-limiting Monod kinetics in a batch or plug-flow reactor.

1674 LEVENSPIEL

expression of eq. (6), hence it is preferable to use the differential rate form of eq. ( I ) directly. For this take small time intervals At, and for each evaluate C,,, C(., and Fc. = ACJ A t . Then rearranging eq. ( I ) gives

C C / G = l / k + ( C . d W / C A ) (7 )

The plot of Figure 2 then shows how to evaluate the Monod constants from either batch or plug-flow experiments. They are given by the slope and intercept of the best straight line through the data.

Extension of these equations to plug flow with recycle and to optimum operations are given in Ref. 3.

Example of Substrate-Limiting Inhibition-Free Kinetics

Monod (Ref. I , p. 74) presents the following data on the growth of Escherichia coli cultures (cells C ) in a lactose medium (substrate A ) . Find a rate equation of the Monod type to represent the kinetics of this growth.

Solution

Make the tabulation given in Table I. Plotting the last two columns

( 8 4

The constants of this expression differ substantially from the values reported by Monod. His final rate expression, found by fitting a

as in Figure 3, we find

rC = 0.72 CACC/(CA + 26.3)

I 0 - - CM

Fig. 2. Graphical procedure for finding the Monod constants of eq. ( I ) from batch or plug-flow data. Data points are included for illustration only.


I , t I 1 I ,

0 01 0 2 03 0 4 0 5

Fig. 3. Plot of C J F , . vs. liC., for the batch reactor data of Table I .

curve to the four sets of rr . /Cp vs. c,, data of Table I is given as

rc = 1.2 CACc/(CA + 20) (8b)

The striking difference in eqs. (8a) and (8b) shows that the method of analyzing data can greatly influence the results obtained.

Mixed-Flow Reactor

The basic performance equation for steady-state mixed flow and any kinetics is given by (see Ref. 3)

(9 ) - - V Ci, emt - Ci, enter TI,, = - -

V T i , at vxi t wncl i t ion

where i is any reactant or product. For a feed containing substrate A but no cells C, insertion of eq.

( I ) into eq. (9), and converting to one variable with eq. (4) gives, in terms of A or of C

This expression was developed independently by Monod4 and by Novick and S ~ i l a r d . ~

The features of this remarkable equation are shown in Figure 4, which is a more convenient version of the graph first presented by Herbert? Note that all quantities such as washout, maximum pro-

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THE MONOD EQUATION I677

where

[CIA] CAO N N + I

N - - I c C , max possible N' I

\ \ \

production rote of ce l ls

I I - 4 - n

CAO \ \ \

------2. CC,opt imum

I washout maximum production here

rate of cells

Fig. 4. Concentration vs. mean residence time behavior for substrate-limiting Monod kinetics in a mixed-flow reactor where Cco = 0.

duction rate of cells, etc., are simple functions of k and of the ratio

To evaluate the kinetic constants from a series of steady-state CA 0 / CJ, .

mixed-flow runs, rearrange eq. (10) to give

I/CA = (k/C.~,w)..r,n - 1/CM ( 104

Plotting the concentration versus mean time data as in Figure 5, fitting the best line through the data, and finding its slope and intercept then gives the constants k and C , of the Monod equation.

Extension to operations where the feed contains cells, to optimum operations of a number of interconnected flow reactors, and to operations that use concentration and recycle of cells are all developed in Ref. 3.

111. PURE PRODUCT INHIBITION KINETICS

With sufficient food and a harmonious environment cells multiply freely. However, no matter how much food is present there always

I678 LEVENSPIEL

Fig. 5 . Plot used to evaluate the kinetic constants for substrate-limiting Monod kinetics in a mixed-flow reactor where Cco = 0. Data points are included for illustration only.

comes a point where either the cells crowd each other out, or their waste products inhibit growth. Hence Monod kinetics can be looked upon as a special case of a more general rate form that includes some sort of inhibition. This section develops the performance expressions for the special case of sufficient food, or C, %- C M , thus for the rate expression of eq. (3). Here it is most useful to develop all the equations in terms of C R and Cc, not C,.

Batch or Plug-Flow Reactor

In terms of CR or C c the performance expression is given as

cn dC, -

('no ( ' C O

f / , = T,, = 1 y - Inserting eq. (3) into eq. ( 1 I ) , writing all concentrations in terms of Cc or CR, and taking the special case where n = 1 gives, on integration

When plotted, this equation gives an S-shaped curve whose shape depends on Cco, as shown in Figure 6.

For n # 1, integration of eq. ( 1 I ) is messier and the final expression more complex; however, the shape of the concentration-time curve will still be S-shaped, somewhat similar to Figure 6.


Mixed-Flow Reactor

The basic performance expression that relates input to output from a mixed-flow reactor is still eq. (9). So inserting eq. ( 3 ) into it and converting all concentrations into C R with eq. (4) gives

(13) CR - CRO

(CR - CRO + [R/C]CCO)(l - cR/CR*Y kTm =

and for the special case where CcO = 0, or where the feed enters free of cells

kTfn = (1 - CR/CR*)-' for kTff8 > 1, Cco = 0 (14)

The properties of eq. (14) are displayed in Figure 7, which shows that washout occurs at k ~ , , ~ = 1 , and that the maximum production rate is dependent in a simple manner on CR* and n .

To find the kinetic constants from experiment first evaluate C R*

in a batch run using an excess of substrate A and letting r + 30.

Then rearrange eq. (14) to give

log Tj,, = - log k + n log [C.q*/(C,* - C R ) ] (15)

and plot as in Figure 8. The slope and intercept of the best line through the data will then give the kinetic constants of the reaction.

medium Cco

Fig. 6. Concentration-time curve for pure product inhibition kinetics in a batch or plug-flow reactor.

I680 LEVENSPIEL

C R Points X and Y have the

same production rate

washout J ' n l o p t i m u m conditions For n = I ' Topt = 2 Tw,,hout

Fig. 7. Concentration vs. mean residence time behavior for pure product inhibition kinetics in a mixed-flow reactor where Cr,, = 0.

IV. SITUATIONS WHERE BOTH SUBSTRATE AVAILABILITY AND PRODUCT INHIBITION AFFECT THE RATE

Here we must use the complete rate expression of eq. ( 2 ) , or

rC = [C/RlrR = kot,s*[CACr/(CA -k CM)] where

kdl, = k ( l - c,/c,x)n (16)

It turns out that it is impractical to use a batch or plug-flow reactor to unravel the interacting effects of the four rate constants of this equation, k , C,M, CR*, and n, and these are best found in a mixed- flow reactor. For this suppose we make a series of runs, each at different C R , using a cell-free feed. From the reactor composition we can find all the concentrations, rates, and T,,, values. Then the performance expression of eq. (9) combined with eq. (16) and rear- ranged gives

Trn = l / k o b s -k ( C A M / k o b s ) . ( 1 / C A ) with CCO = 0 (17)

Plotting the data as in Figure 9 then gives a family of curves, each at different C R . This graph gives C,w and k(1 - CR/CR*)". Then, knowing CR* (this is usually easy to evaluate beforehand) we can


0

I log 7; i.'

Fig. 8. Method for evaluating the rate constants for pure product inhibition kinetics from experiments in a mixed-flow reactor where C,., = 0. Data points are included for illustration only.

then find n and k from Figure 10; or else, if C,* is unknown, guess it until you get a straight line on Figure 10.

The following example illustrates the procedure.

Example on Product Poisoning

Bazua and Wilke' reported the following data on the fermentation of glucose ( A ) to form ethanol ( R ) , using Saccharomyces cerevisiae

slope =

intercept

C M

Fig. 9. Plot used to find the rate constants of the generalized Monod equation from a set of runs in a mixed-flow reactor where C,.,, = 0. Data points are included for illustration only.

I682 LEVENSPIEL

7 t find k f r o m t h i s intercept

Fig. 10. Evaluation of the toxic power n in the generalized Monod equation. Data points are included for illustration only.

(0, in a mixed-flow reactor, and in the presence of different levels of ethanol. Find a rate equation to represent this fermentation in- cluding the effect of inhibition by ethanol.

Solution

Using a portion of the reported data make the tabulation given in Table 11. Plot these data as in Figure I 1 from which we find slopes and intercepts and

C, , , = 0.222 giliter

Also from the reported data we have - [RIG] = 3.90

Since an experimental value for CR* is not reported in the paper, and is only said to be somewhere between 80 and 100, let us find it by the above recommended procedure. First guess CR* = 100. Then tabulate as in Table 111. A plot as in Figure 12 does not give a straight line. So try again with C,* = 90 and then 87.5. The latter value gives a reasonably straight line. So from this figure we have

n = 0.41 (slope)

k = 0.42 (from intercept)

CR’. = 87.5 g/liter


interceot I /

I I - -4 5 0 I0 20

I - (liter/grn) =A

Fig. I I . Plot of T , ~ vs. I/CA for the data of Table I1

The best-fit value reported by the authors is CR* = 93-94 @liter. The final rate equation is thus, in terms of cell production,

YC = 0.42(1 - CR/87.5)0.41[CA/(CA + 0.222)]

or in terms of alcohol production

rR = r c [ R / C ] = 1.64(1 - C,/87.5)0.41[CA/(CA -I- 0.222)]

This rate equation is of the generalized Monod type, and it accounts for both substrate availability and alcohol inhibition.

V. DISCUSSION

1 ) The similarity in shape of the graphs for substrate-limiting and for product inhibition kinetics (compare Figs. I and 6 for plug flow, Figs. 4 and 7 for mixed flow) may lead one to fit product inhibition systems with the convenient Monod equation. The fit probably will be good; however, extrapolation to new and different conditions will lead to error because the logic in the use of this equation will be quite wrong.

1684 LEVENSPIEL

TABLE I1 Bazua and Wilke's Mixed-Flow Reactor Data Reanalyzed

Reported From Figure I I Calculated from

7 tic* reported data (hr) (literlg) CIl [RlCI slope intercept

13.88 20.8

3.65 0.56 2.5 5.21 3.79 3.70 2.50 2.38 0.22

16.67 22.20 7.58 3.97 3.21 2.66 0.15

13.66 1 I .49]

> 29.19 3.76 0.62 2.8

26.04 12.05 14.37 13.44 10.42 9.92 7.25

8.13 4.88 1.60 7.2 2.08 2.08 0.17

One must first find which of these two factors is limiting and then use the corresponding correct equation form.

2) Let us compare an expression proposed by Aiba et a1.8 for substrate availability plus product inhibition kinetics with the generalized Monod equation proposed here.

Exponential-type equation

rr = kle-k2CR.[CACr/(CA + C , v ) ] (18)

Generalized Monod equation

TC = k(1 - CR/CR*p[CAC, / (C, + C,v)] (19)

For given cell and substrate concentration, eq. (18) does predict a slowing of the rate as the concentration of toxic product R rises. But it also says that cell action will always continue no matter how


intercept = log k = -0.377 :_ k = 0.42

s l m e = n = 0.41

for cl,

--06

log - -0 7

--08

- - 0 9

[k (I - 37 R

-12 -10 - 0 8 -06 -04 - 0 2 0 l o g ( l - + ) C

CR

Fig. 12. Plot of log [k(l - Cfi/Cfi*)n] V S . log ( I - C R / ~ R * ) .

high the concentration of R becomes. Thus in alcohol production it says that even when the alcohol concentration is 20, 40, or 60%, fermentation continues. This is not what is observed, hence eq. (18) cannot be used for such situations.

As opposed to this, eq. (19) says that there is a definite upper concentration limit for inhibitory product CR* above which fermentation ceases.

The empirical constant n in eq. (19) accounts for the fact that one can approach this limit in various ways: linearly ( n = I ) , rapid

TABLE 111 Evaluation of the Toxic Power, n

1 - 2 log 1 -_ 1 log[k(l -c,/c,*)n1= CR CR* ( :I*) k ( l - Cfi/Cfi*)" log k +nlog(l - C R / C R * )

4.37 0.956 -0.0195 2.5 22.19 0.778 -0.1090 2.8 61.29 0.387 -0.4123 4.0 81.30 0.187 -0.7282 7.2

-0.3979 - 0.4472 -0.6021 -0.8573

I686 LEVENSPIEL

initial drop in rate followed by a slow approach to CR* ( n > I ) , or vice versa ( n > 1) . This is shown in Figure 13. If one does not approach too close to CR*, eqs. (18) and (19) with n B I could both give approximately the same fit to data.

Holzberg et in effect, used the special case of eq. (19) where n = I to fit their experimental findings.

3) In this paper we have used the normal language of reactor design. The relationship to the corresponding language of biotechnology is as follows:

Dilution rate for mixed flow

D = l /Trp , = V / V

Specific growth rate

t-l = rc/Cc = k ( l - cR/cR*)nrcA/(cA + C,,)I

Maximum specific growth rate in the presence of R

rr Pinax = - - - k ( 1 - CR/C,*Y’ = k,,b\, CA * CJI

CC

Maximum specific growth rate in the absence of R

~ l . 0 = t . ( . /C( . = k , C.4 * C,M, C , = 0

In simple situations either language can be used; however, in more complex applications involving interconnected reactors, recycle, or cell concentration, it probably is easier to follow what is happening

k

kobs

reaction stops

Fig. 13. Power n shows how the observed rate constant of the Monod equation [eq. (2)] decreases as toxic product R rises.


if one did not combine distinct quantities into an overall quality such as p.

Extension to some of these operations can be found in Ref. 3.

Nomenclature

substrate, cells, rate-depressing product concentration of A , C , and R in any convenient units (such as moliliter: @liter) concentration ofA, C , and R initially in a batch reactor or in the feed entering a flow reactor Monod or saturation constant: see eq. ( I ) (moliliter) concentration of product R at which fermentation ends dilution rate (hr-l) flow or production rate (gihr or molIhr) reaction rate constant: see eq. ( I ) (hr-l) toxic power: see eq. ( I ) reaction rate (quantity formed/liter.hr) time of reaction in a batch reactor (hr) volumetric flow rate of fluid into and out of a steady flow reactor (literihr) reactor volume (liter) yield, defined in eq. (4)

space time or residence time of fluid in a plug-flow reactor (hr) space time or mean residence time of fluid in a mixed- flow reactor (hr)

The author would like to recognize Goran Jovanovic and Adonis Stephanakis for deriving the expressions appearing in Figures 4 and 7.

References

I . J . Monod. Rechrrches sur la Croissnnce des Cultures Bacterienne (Hermann

2 . J. Monod, Ann. Re\.. Microbiol.. 3 , 371 (1949). 3. 0. Levenspiel, Chemical Rerrctor. OfnnihooX (OSU Book Stores, Corvallis,

4. J. Monod. Ann. I n & / . Ptr.\tertr-. 7 9 , 39U (1950). 5 . A. Novick, and L. Szilard, Proc. Ntrrl. Actrd. Sci., 36, 708 (1950). 6. D. Herbert, S. I .C . Monograph. 12, 21 (1959). 7. C. D. Bazua and C. R. Wilke, Biotechnol. Bioeng. Symp. , 7 , 105 (1977). 8. S. Aiba. M. Shoda, and M. Nagatani. Biorechnol. Bioeng., 10, 845 (1968). 9. I . Holzberg, R. Finn, and K. Steinkraus. Biotechnol. Bioeng., 9 , 413 (1967).

et Cie, Paris, 1958).

OR. 1979).

Accepted for Publication October 25, 1979

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The monod equation: A revisit and a generalization to product inhibition situations