analysis of autocatalytic reactions with michaelis-menten kinetics in an isothermal continuous...

Pergamon Chemical En~lineeriny Science. Vol. 52, No. 20, pp. 3455-3462, 1997 1997 Elsevier Science Ltd. All rights reserved

Printed in Great Britain PII S0009-2509(97)00109-7 0009-2509/97 $17.00 + 0.00

Analysis of autocatalytic reactions with Michaelis-Menten kinetics in an isothermal

continuous stirred tank reactor

Jyoti Kumar, Sunil Nath* Department of Biochemical Engineering and Biotechnology, Indian Institute of Technology,

Hauz Khas, New Delhi 110 016, India

(Received 20 January 1997; accepted 11 March 1997)

Abstract--A systematic analysis of the behavior of autocatalytic reactions obeying Michaelis-Menten kinetics and taking place in an isothermal continuous stirred tank reactor is presented. The mathematical model is solved for general nth-order autocatalysis and the dimensionless steady-state conversion as a function of dimensionless residence time, the local stability of the stationary states and the instability type and region in parameter space bifurcation diagrams are determined. The stability-instability diagrams are very different for n > 1 compared to n = 1 (quadratic autocatalysis). The simulations are extended over five decades of dimensionless residence time to characterize in detail the wealth of dynamic behavior exhibited by such systems. Analytical expressions are also derived for the maximum autocatalyst concentration obtained in such systems with both linear and Michaelis-Menten kinetics. Finally, some qualitative connections are drawn between the autocatalytic model employed here and the varied patterns of behavior observed in the biological process of glycolysis. ~ ' 1997 Elsevier Science Ltd. All rights reserved.

Keywords: Autocatalytic; Michaelis-Menten; CSTR; stability; dynamics; glycolysis.

INTRODUCTION

The analysis of a single, irreversible exothermic reaction in a continuous stirred tank reactor (CSTR) has been one of the intensively researched fields in fundamental chemical reaction engineering. Interest has focused on questions of unique and multiple stationary steady states, on local and global stability and possibilities of sustained oscillation, and on classifica- tions of behavior (Uppal et al., 1974, 1976; Vaganov et al., 1978; Kauschus et al., 1978). Studies of the corresponding isothermal prototypes, where feedback is not thermal but autocatalytic is of more recent origin (Lin, 1981; Heinrichs and Schneider, 1981; Gray and Scott, 1983; Gray and Scott, 1984; Scott, 1985; Sapre, 1989; Hu and Sapre, 1990; Grzesik and Skrzypek, 1993a, b). In autocatalytic reactions, a product of the reaction acts as a catalyst and aids in the conversion of reaction to product. This type of reaction will generally start with a slow rate and then greatly accelerate as the reactant is consumed and the autocatalyst is produced; as a result of this behavior, autocatalytic reactions may require special design considerations.

*Corresponding author. Fax: 91-11-6868521 or 6862037; E-Mail: [email protected].

The kinetic behavior of a wide range of autocatalytic systems can be well approximated by the prototype steps

A + n B ~ ( I + n ) B

B ~ C (1)

where n is the order of the autocatalytic system. With n = 1, the system is referred to as quadratic autocatalysis, while for n = 2, the system is cubic. While retaining the core reaction scheme of eq. (1), Prigogine and coworkers added a few steps and developed the Brusselator model (Glansdorff and Prigogine, 1971; Nicolis and Prigogine, 1977). This has been extensive- ly used to investigate temporal and spatio-temporal oscillations including dissipative structures in model systems. In previous treatments (Lin, 1981; Heinrichs and Schneider, 1981; Gray and Scott, 1983, 1984; Sapre, 1989; Hu and Sapre, 1990; Grzesik and Skrzypek, 1993a, b), only first-order kinetics is con- sidered for the decay reaction (B ~ C). Further, no solutions are available for general n. These equations have not been tested for Michaelis-Menten kinetics, the only exception being the work of Scott (1985) who devoted a short section of his paper to less than linear decay rates. However, even this treatment is limited to n = 1. This is surprising, considering that

3455

3456 J. Kumar

Michaelis-Menten kinetics is commonly followed by most enzyme reactions (Bailey and Ollis, 1986; Laidler and Bunting, 1973). It may also arise in heterogeneous catalysis if the removal of B occurs at a surface where the number of free sites available for adsorption becomes significantly reduced.

Examples of chemical reactions that exhibit autocatalytic behavior include the conversion of nickel oxide to nickel in hydrogen gas (Bandrowski et al., 1962), the reaction of methanol over zeolite catalysts to produce hydrocarbons (Chen and Re- agan, 1979; Ono et al., 1979), the oxidation of carbon monoxide on palladium supported catalysts (Jaegar et al., 1986), the catalytic cracking of paraffins on a zeolite catalyst to produce olefins (Abbot and Wojciechowski, 1989), the thermal cracking of polycyclic n-alkylarenes (Savage et al., 1989), and the acid-catalyzed hydrolysis of esters to carboxylic acids and alcohols (Boudart, 1991). Several complex systems in biochemistry, combus- tion chemistry, inorganic solution chemistry (e.g. the Belousov-Zhabotinskii reaction), and enzyme catalysis show autocatalytic behavior (Sapre, 1989). To take but one biological example, glycolysis is a fundamental autocatalytic process in which a six-carbon carbohydrate, such as glucose, is subjected to step-wise oxidation by a series of enzyme-catalyzed reactions to yield energy in the form of ATP for use by the cell. The glycolytic system presents an autocatalytic feedback loop which is indispensable for the appearance of self-organized behavior (Cortassa et al., 1991; Jou and Llebot, 1990). Further, the two key steps in the glycolytic pathway, i.e. those catalyzed by the enzymes phosphofructokinase and pyruvate kinase obey Michaelis-Menten kinetics.

This paper systematically analyzes the behavior of autocatalytic reactions obeying Michaelis-Menten kinetics for the decay step and taking place in an isothermal continuous stirred tank reactor. The mathematical model is solved for general nth order autocatalysis [eq. (1)] to determine the dimensionless steady-state conversion as a function of dimensionless residence time and the instability type and region in parameter space bifurcation diagrams. The stability-instability diagrams are very different for n > 1 compared to n = 1 (quadratic autocatalysis). The dynamics is shown to be more complex than that found by Scott (1985). The simulations are extended over five decades of dimensionless residence time to characterize in detail the wealth of dynamic behavior exhibited by such systems. Analytical expressions are also derived for the maximum autocatalyst concentration obtained (and the value of the residence time at which this maximum occurs) in such systems with both linear and Michaelis-Menten kinetics. Finally, some qualitative connections are drawn between the autocatalytic model employed here and the varied patterns of behavior observed in the biological process of glycolysis.

and S. Nath

MATHEMATICAL DEVELOPMENT

Mass balance equations We consider the reaction scheme

A + nB ~ (1 + n)B rate = k lAB" (19

B ~ C rate = k2B/(1 + rB).

The CSTR mass balance equations, which relate the rates of the chemical reactions to the input and output rates, are

dA d--[ = - k IAB" + ky(Ao - A) (2)

__dB= k lAB" ___k2B + k f (Bo - B) (3) dt 1 + rB

where k I is the inverse of the mean residence time tres and Ao and Bo are the concentrations of A (reactant) and B (autocatalyst) in the inlet stream. A characteristic time-scale tch for dynamic features emerges naturally from the rate of the principal autocatalytic step, i.e.

t¢~ = 1/(kxA~) (4)

We employ the following physically significant dimensionless variables

dimensionless concentration of A, ~ = A/Ao

dimensionless concentration of B, fl = B/Ao

dimensionless catalyst input, flo = Bo/Ao

dimensionless time, z = t/tch = k lA~t

dimensionless residence time, zr = tres/tch = k IA~/ks

dimensionless decay time, z2 = klAn~k2

dimensionless M - M constant, p = rAo (5)

With the above dimensionless variables, eqs (2) and (3) can be east into dimensionless form; [we assume there is no autocatalyst in the input, (i.e. flo = 0)]:

da (1 - a) - - = - aft" + - - (6) d~ ~r

d - ; = - (7)

Stationary states Stationary states of eqs (6) and (7) arise when da/dz

= dfl/dz = 0. A trivial solution is a = 1, fl = 0, which corresponds to zero conversion. For the nonzero solution, the dimensionless concentrations a and fl are related to each other as follows:

( 1 - a ) :~/~" = - - ( 8 )

Tr

~fl" = fl z2(1 + pfl) ~- " (9)

Analysis of autocatalytic reaction

Combining eqs (8) and (9) and eliminating the ~fl" we have term leads to

= l - f l z2 ( l+p f l ) + 1 (10) or

while eq. (8) itself can be recast into the computation- ally more convenient form

= ( 1 - ~ t ~ l:" (11)

For any value of z,, z2, p and the order of the autocatalytic system, n, eqs (10) and (I 1) can be solved iteratively to obtain the stationary steady-state solutions ~ and fl~.

Local stability of stationary states The character and local stability of the stationary

states determined using eqs (10) and (11) at any given value of the dimensionless residence time are determined by the eigenvalues /]'1 and '~2 of the Jacobian matrix evaluated at the stationary steady-state solution. The roots of this quadratic equation are given by

( d ~ ss ~ 0 . (12)

Using eqs (6) and (7), we can determine the individual terms of the Jacobian matrix, i.e.

= _ _ ;

( 1 3 )

___~ (dfl~ = 1 1

The matrix of eq. (12) now becomes

1 - ~ - _ _ ;~ - n ~ / r ~ - ~

l" r

1 1

z2(1 + pflss) 2 zr

Putting

1 j = _ / ~ . - _

17 r

K = - ncq~fl~"~- x

M = nO~ssflns 1 1 1

z2(1 + pfl~)2 "t'~

2 =0.

(14)

(15)

3457

J - M K- )~ = 0 (16)

)2 _ ( j + M)2 + (JM - LK) = 0 (17)

which is the final eigenvalue equation. The eigenvalues 2~ and 22 may be real or complex and positive or negative and are strong functions of the dimensionless residence time, z,. They may be classified as stable node (SN), stable focus (SF), saddle point (SP), unstable node (UN), and unstable focus (UF) etc. (Edel- stein-Keshet, 1988).

R E S U L T S A N D D I S C U S S I O N

In this section, we provide selected computational results based on our model. The organization of this section is as follows: we first present the variation of :q~ and flss and the character of stability with z~ for various values of the parameters "~2 and p followed by analytical expressions for the maximum concentration of fl~ and the residence time z . . . . . at which this maximum occurs. Next, we depict the regions of stability and instability on parameter space bifurcation diagrams. Finally, we discuss our model in the context of the glycolytic process.

Dependence of steady-state concentrations on residence time and character of stability

Figure l(a) and (h) show the dependence of the values of the dimensionless steady-state concentrations of A and B on the dimensionless residence time of the reactor. These figures are drawn for different values of the dimensionless decay time (z2 = 10, 100) in quadratic autocatalysis and constant dimensionless M - M constant (p = 15). In these figures, the bold line refers to ~tss and the dashed line to flss. ~ss shows a steep decline with increasing zr (at low z,) and reaches an asymptote for high values of z,. The value of this asymptote itself decreases with increasing values of z2. The variation of this asymptotic value with r2 can be mathematically determined by inspection ofeqs (9) and (11). For n -- 1, eq. (9) becomes EI+,I

: (= z2(1 +pfl) ~ . (18)

From eq. (11), for sufficiently large z, (for ct bounded between 0 and 1), fl tends to zero. Therefore, at large values of zr, eq. (18) reduces to

= 1/z2. (19)

Further, the minima in ~s shifts to higher z, values with increasing 152 (logt0zr - 1.8, 2.1, and 3.7 for ~2 = 10, 20, and 100 respectively), fls~ increases with

increasing z,, reaches a maxima, and then decreases with further increase in z,, as apparent from Fig. l(a) and (b). It finally reaches a value of zero at high z, in accordance with eq. (11). The maximum value of fl= increases and shifts to the right as TE increases.

1.0 1.0

0.75 U)

-,0.5 V)

0.25

0.0 0

SN / . sF / sN a

4~Nx 'SF UF UN

I | I I ! I t I I

% 2 4 6

log 10 ~r

SN / / 0.75

%~0.5 (/1 I/}

tS 0.25

0.0 0

/ / S N

SF UN

3458 J. Kumar and S. Nath

\ x\

2 4

I°glO gr

1 . 0 0 [ / " \

0 . 7 5 / s. \

~" 0.5" m m g

0.25

0.0 0 2

b

SF UN

\ 4

log 10 ~:r

1.0

0.75 ' U1 / SN ~

~0-5 I ~

O. 25

I

UF

b SP

/ UN

f

2 log 10 Z:r

0.( o 4

Fig. 1. Variation of the dimensionless stationary-state concentrations ~ (bold line) and fl~ (dashed line) and character of stability as a function of the dimensionless residence time, z, of the CSTR for n = 1 [quadratic autocatalysis; eq. (1)]

and p = 15. (a) z2 = 10; (b) z2 = 100.

Fig. 2. Variation of the dimensionless stationary-state concentrations ~s (bold line) and flss (dashed line) and character of stability as a function of the dimensionless residence time, z, of the CSTR for n = 1 [quadratic autocatalysis; eq. (1)]

and "r z = 5. (a) p = 1; (b) p = 100.

Although the curves for ~s~ and fl~, are shown to begin at a value of approximately 0.5, they begin, in reality, from 1 (for ~,,) and 0 (for fl~). The curves are extremely steep at low values of ~, and cannot be represented in this region with clarity. The point on the x-axis at which these curves begin can be exactly determined mathematically. From eq. (8), fl = 0 implies that

= 1 and vice versa. So, from eq. (18) we have

1 1 - - + -- = 1 (20) T 2 27 r

o r

z, = z 2 / ( z 2 - - 1). (21)

Equation (21) shows that at values of z2 ~ 1, no physically meaningful solution can be found for the autocatalysis model.

The dependence of the character of stability on the logarithm of the dimensionless residence time is also indicated in Fig. l(a) and (b). The abbreviations in the

figures are as follows: stable node (SN), stable focus (SF), unstable node (UN), unstable focus (UF), and saddle point (SP). These regions are determined by evaluating the roots of eq. (17). For ~ = 1 and fl = 0, the linear stability analysis also yields a region of saddle point (SP) between 0 and a very small positive value of z,. The vertical line demarcating this region lies very close to the y-axis; further it corresponds to zero conversion of A and does not provide any physically meaningful solution to our model (z, ~< 1). Hence this line is not indicated in the figures. For varying z2 but p fixed at a value of 15, the region in which the dimensionless concentrations of A and B change appreciably corresponds to a stable node (Fig. 1). The number of transitions in behavior is found to be greater at lower values of z2.

Figure 2(a) and (b) depict the variation of %s and flss and the character of stability with logtoT,. Here, z2 is kept constant and p is varied. The analysis of Fig. 1 also holds for these cases. We therefore only discuss some special features of the dynamic behavior.


The maximum value of fl~ increases dramatically with increase in p (from about 0.35 at p = 1 to 0.75 at p = 15 and 0.90 at p = 100). Further, the maximum shifts to the right with increasing p. Another interest- ing point is that the region in which ~,, and fl,, change appreciably is SN at higher p but shifts to a region of damped oscillation (SF) for lower values of p ['e.g. p = 1 in Fig. 2(a)]. The asymptotic value for high z, is constant in Fig. 2(a) and (b) as it depends only on z2 in accordance with eq. (19) but the way in which the asymptotic value is reached varies with p. At a low value of p, the curve approaches the asymptote from above [Fig. 2(a)] while at higher p values, the asymptote is approached from below as shown in Fig. 2(b).

Expressions for maximum fl~ We first derive the maxima for quadratic autocat-

alysis with linear kinetics in the reaction scheme of eq. (1) or in fact for a more general autocatalytic step

3459

Note that an erroneous result is given by Scott (1985) for 3, ...... The correct expression for fl, . . . . . achieved for a system with given z2 as z, is varied is provided by eq. (26) and the value of the dimensionless residence time at which this maximum in/3 occurs is given by eq. (25).

The derivation of the maxima for the case of Michaelis-Menten kinetics in quadratic autocatalysis is more complicated. Using the shorthand notation D, = 1/z, and D2 = 1/~:2 and eliminating • from eqs (8) and (18), we obtain the relationship

D, D 2 + D,. (27)

~ + D , l + p / ~

On rearrangement, we have

pD,fl z + (pD~ + D,(1 - p) + O2)fl

+ (D 2, - D, + D2Dr) = 0. (28)

ss =

Solving the above quadratic equation leads to

I( °2) l JI( °2) 12 - pO,+-~ + ( I - p ) + pD,+--~ - ( l + p ) + 4 p D 2 - D,

2p (29)

where only the positive sign corresponds to physical solutions. Differentiating fl~ with respect to Dr pro- vides the result

d f l s s - I P - - ~ 2 , ) I ( P D ' + ~ ) - ( I + P ) I ( P - - ~ 2 ~ ) +4D---2D 2, (30)

dD, 2p D2 4 D2' (2p) p D , + ~ - ( l + p ) + 4 p D 2 - D,

of the form A + B --, (n + 1)B. With flo = 0, the nonzero stationary state is given by the equations

a s s = - - Z2 + Zr

n~2"c r

nz2z, - - (~7 2 "~ 27r) / ~ =

For maximum fls,, 83~/8r, = O. Or

~ flss n'r 2 1 ~Tr (T2 -~- Tr) 2 "L "2 '

Equating this to zero, we have

T2 "Or,max

. n//-~Z~- 1

Substituting this value in eq. (23), we obtain the result

/~ . . . . . . ( , /~S~ - 1) 2 T2

Putting d3/dD, = 0 in eq. (30) and solving for "Cr.ma x leads to a very cumbersome expression and it is better to perform the calculations numerically (Figs

(22) 1 and 2). However, an approximate solution for Z,.max valid in the regime of high p is readily determined:

(23)

D . . . . . ~ / - ~ or r . . . . . ~ N / / ~ • (31)

Substituting the value of D ..... from eq. (31) into eq. (29) leads to the final result for fl ......

(24)

3 . . . . . = - E 2 x / ~ 2 p + (1 - p)] + ~p [8D2p - 8x/D2p

(25) - 4 P x / ~ + (1 + p)211/2. (32)

The results for the maxima obtained for large values of p (p > 10) by use of the analytical expressions [eqs (31) and (32)] matches extremely well with the results

(26) for the maxima obtained numerically (Figs 1 and 2) in all cases. Thus, for small values of p, we can assume


the kinetics to be predominantly linear and employ eqs (25) and (26) for determining the maxima while for large values of p (M-M kinetics) the results given by eqs (31) and (32) are to be preferred.

Parame te r space bi furcation d iagrams

Figures 3 and 4 portray the regions of stability and instability and the character of the patterns generated in the autocatalytic system with Michaelis-Menten kinetics for quadratic and cubic autocatalysis, respectively. The wealth of dynamic behavior is mapped on z r -p parameter space for various values of z2 in these figures. The dynamics is found to be far more complex than that determined by Scott (1985) and regions of SN, SF, SP, UF, and UN are observed. As the value of z2 is progressively increased from 5 to 10 to 100, the region of stability (SN) gets progressively enlarged (Fig. 3) for n = 1. The region of stable node also gets progressively enlarged upon progressive increase in p. The region of Fig. 3 marked by the dashed ellipse displays extremely complex dynamics and boundaries cannot be precisely identified. For low 27 2 ( ' t 2 ~ 10), extremely complex dynamics is also found for values of r, lying between 103 and 104.

Figure 4 displays the dynamics of the autocatalytic system for n = 2 and r2 = 10 and 100. The dynamics is significantly altered on changing the order of the system (Figs 3 and 4): although the figures appear geometrically similar, the stability characteristics of the regions (e.g. SN, SP) change considerably. The SN region in the cup portion of Fig. 3 (n = 1) has now been replaced by SP for n = 2. The SF region below the cup in Fig. 3 has been substituted by SN in Fig. 4. Similar changes were observed in numerical simulations of higher-order systems (n = 3, 4).

Appl icat ion o f the au tocata ly t ic model to 91ycolysis

Analysis of the glycolytic pathway (Mahler and Kordes, 1968; Jou and Llebot, 1990) shows that the kinetics of the overall process is governed by a few bottleneck steps, e.g. the reactions catalyzed by hexokinase, phosphofructokinase, and pyruvate kinase. The other reactions can be assumed to be fast and reversible. Using M - M kinetics for the rate limit- ing steps and assuming that the M - M saturation constants are larger than the concentrations of the A and B species (C6 and C3), the kinetic expressions simulating glycolysis work out to be (Nath and Goyal, manuscript in preparation)

dA - - = Vo - k x A B (33) dt

d B k2B - - = k l A B - - (34) dt B

I + - - K

where A represents the sum of the concentrations of glucose and fructose-6-phosphate, B the sum of the concentrations of 1,3-diphosphoglycerate and phosphoenolpyruvate, and C the concentration of

10 3 / '

i~/d I

10 2

I I

% 10

-i l.w" . 10 ,J.', . . . . . , • . . . . . : . . .~ 10 10 2 10 3 10 4

E'r Fig. 3. Regions of stability and instability of the stationary steady state ~,s and /~, mapped on the 1.,-p plane for an autocatalytic system of order n = 1 (quadratic autocatalysis) following Michaelis-Menten kinetics with 1.2 = 5 (bold line), 1.2 = 10 (dashed line), and 1.2 = 100 (dotted line). The region marked with a dashed ellipse displays complex dynamics. The vertical bold line at 1., = 2 stands for 1.2 = 5, 1.2 = 10 and

I" 2 = 100.

10 3

10 2

% 101

lo c

10-1

SN

sP

I

, , I . i I i I J

10 10 2

SN

-1 , i i I

10 3 10 4

Fig. 4. Regions of stability and instability of the stationary steady state ~, and fl,s mapped on the r,-p plane for an autocatalytic system of order n = 2 (cubic autocatalysis) following Michaelis-Menten kinetics with z2 = 10 (bold line)

and z2 = 100 (dashed line).

pyruvate. In the upper half of the glycolytic pathway, two molecules of ATP are consumed during conversion of A to B while in the lower half of the pathway, four molecules of ATP are produced during conversion of B to C. This production of excess ATP catalyzes the upper half of the pathway, i.e. conversion of A to B. In other words, a positive feedback loop through ATP operates in glycolysis which leads to self-organized behavior and justifies the autocatalytic


role of B. Equating the stoichiometric coefficients of D2 the ATP-producing and ATP-consuming reactions kl leads to a nonoscillatory system which suggests that k2 the positive feedback loop is indispensable for self- ky organized behavior. Other researchers have also K emphasized the role of ATP in glycolytic oscillations (Cortassa et al., 1991; Goldbeter, 1996). n

Inspection of eqs (33) and (34) for glycolysis shows r that they are similar to the equations of our auto- t catalytic model [eqs (2) and (3)] if the following rela- teh tions hold: Vo

k~Ao = Vo, r = l/K, /lout = O, Bin = Bout = O. Greek letters

(35) c~

Further, one molecule of A (C6) requires two molecu- fl les of ATP in order to produce two molecules of flo B (Ca) and one molecule of B produces two molecules

2 of ATP during conversion to C. Hence, in the first part of the reaction, one molecule of A and one P r molecule of B combine to produce two molecules of B with the understanding that B participates indirect- r, ly (through ATP), while in the second part of the r2 reaction, one molecule of B is converted to one mol- Subscripts ecule of C. The process of glycolysis can thus be max maximum modeled in the framework of the autocatalytic model ss steady state of this paper as

phosphofructokinase A + B , 2B

(36) pyruvate kinase

B , C

i.e. as a continuous and open system with input of A and output of C and resembles a CSTR. However, unlike a normal CSTR that contains A, B, and C in the output stream, the biological 'reactor' is more complex: it contains membranes which selectively per- mit only certain products (e.g. only C) to exit. The analysis of this problem presents a challenge for future research.

CONCLUSIONS

An analysis of autocatalytic reactions with Michaelis-Menten kinetics in an isothermal continuous stirred tank reactor is carried out. The wealth of dynamic behavior exhibited by such systems is characterized in detail. Analytical expressions for the maximum concentration of the autocatalyst and the residence time at which the maximum occurs are obtained and analyzed with respect to various parameters. Finally, the autocatalytic model is applied to the process of glycolysis.

A Ao

B C Dr

NOTATION

concentration of reactant in reactor, mol/m z concentration of reactant in input stream, mol/m 3 concentration of autocatalyst, mol/m 3 concentration of product, mol/m 3 reciprocal of dimensionless residence time

3461

reciprocal of dimensionless decay time rate constant, 1/s (mol/m3)" rate constant, 1/s inverse of residence time, 1/s Michaelis-Menten constant in glycolysis, mol/m 3 order of the autocatalytic system Michaelis-Menten constant, 1/(mol/m 3) time, s characteristic time, s input rate of reactant in glycolysis, mol/m 3 s

dimensionless concentration of A dimensionless concentration of B dimensionless autocatalyst concentration in input stream eigenvalue dimensionless Michaelis-Menten constant dimensionless time dimensionless residence time dimensionless decay time

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analysis of autocatalytic reactions with michaelis-menten kinetics in an isothermal continuous...

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