Foundations of Biochemical Engineering - American Chemical Society


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2

Kinetics

of

Enzyme

Systems

DAVID F. OLLIS

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University of California, Chemical Engineering Department, Davis, CA 95616

The many circumstances leading to the Henri equation for enzyme conversion of soluble substrates are first noted, followed by some kinetic forms for particulate and polymer hydrolysis. Effects common to immobilized enzyme systems are summarized. Illustrative applications discussed include metabolic kinetics, lipid hydrolysis, enzymatic cell lysis, starch liquefaction, microenvironment influences, colloidal forces, and enzyme deactivation, a l l topics of interest to the larger themes of kinetics and thermodynamics of microbial systems. Enzyme k i n e t i c s i s by now a c l a s s i c a l f i e l d , even i n the arena o f immobilized enzymes, and d e t a i l e d t r a n s i e n t as w e l l as steady s t a t e k i n e t i c models of c e l l u l a r and s u b c e l l u l a r functions have been advanced f r e q u e n t l y which hinge h e a v i l y on the k i n e t i c forms appropriate to s i n g l e enzyme systems. (See l a t e r conference papers of Urabarger, Shuler and Domach, B a i l e y , Agathos and Demain, Marc and Engasser, Costa and Moreira, and Frederickson, f o r examples.) We open t h i s d i s c u s s i o n with a review of causes f o r success and u n c e r t a i n t y regarding the s i n g l e enzyme r a t e equation, f o l l o w i n g which we w i l l touch upon r e s u l t s f o r p a r t i c u l a t e substrates, and c l o s e with some r e s u l t s f o r immobilized enzyme systems. 1.

Soluble

substrates 1

The Henri form f o r r e a c t i o n v e l o c i t y , v, of an enzyme c a t a l y z e d r e a c t i o n was proposed i n 1902 as eqn (1.1):

v

- W ï f s ï

(

1

·

0097-6156/83/0207-0027$07.50/0 © 1983 American Chemical Society

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

υ

28

BIOCHEMICAL

ENGINEERING

where ν = maximum v e l o c i t y , Κ = constant, and S = concentra­ t i o n of f r e e (uncomplexed) s u b s t r a t e . T h i s form was l a t e r d e r i v e d by M i c h a e l i s and Menten i n 1913 using an e q u i l i b r i u m assumption i n eqn (1.2a) and an assumed s i n g l e slow step, eqn (1.2b), which y i e l d s form (1.1) above: 2

S + Ε

«k

ES

(1.2a)

Ε + products

(1.2b)

2

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ES

->

where Michaelis-Menten:

ν

Κ Ξ k / k i = d i s s o c i a t i o n constant of ES 2

= k E ο 3

max

(1.3a)

where Ε = t o t a l (1.3b) ο enzyme c o n c e n t r a t i o n (E + ES)

3 The subsequent Briggs-Haldane treatment recognized that i n general the r a t e constants ( k i , k ) and k could be of a r b i t r a r y r e l a t i v e magnitudes, r a t h e r than only ( k i , k ) » k assumed i m p l i c i t l y by eqns (1.2a,b), l e a d i n g to the form (1.1) again but with Κ and ν ^ given by eqns. (1.4a,b): 2

3

2

3

χ

Briggs-Haldane:

Κ =

(1.4a)

= k E (1.4b) max ο Thus, the constant Κ i s no longer a simple e q u i l i b r i u m constant^ Extensions of form (1.1) a r e apparently endless i n number. For example, f o r a common case of a (two substrate)-(one enzyme) system g i v e n below, ν

Reaction

3

3

D i s s o c i a t i o n Equibrium

Constant

Ε + S i / J ESi

Ki

(1.5a)

Ε + S

K

(1.5b)

2

J ES

ESi + S

2

2

J ESiS

ESi + S ->ESiS 2

ES S

£

2

2

2

Ki

2

Κ

Ρ + E

eqn (1.1) i s again recovered where ν eqns (1.6a,b);

(1.5c) (1.5d) (1.5e) and Κ are i n d i c a t e d i n

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

2.

OLLis

Kinetics

of Enzyme

=

kE S — - — Ki2 + S

Systems

29

2

ν max

substrate

Κ =

(1.6a)

2

Κ

2

1

^ + *

l K l 2

(1.6b)

S2 + K i 2

Thus, i f S2 i s approximately constant as expected f o r S2 = s o l ­ vent or S2 = b u f f e r e d Η or OH , the Henri form of constant parameters ν and Κ w i l l appear to f i t the data f o r S i . S i m i l a r l y , i¥ ?he order of s u b s t r a t e b i n d i n g r e a c t i o n s (5a) and (5b) i s o b l i g a t o r y , form (1.1) i s again obtained (e.g., by e l i m i n a t i o n of eqn (1.5d)). When the second binding e n t i t y , S2 i s not a s u b s t r a t e , but instead an enzyme a c t i v a t o r or competitive i n h i b i t o r , form (1.1) i s again regained with both ν and Κ dependent upon a c t i v a t o r / i n h i b i t o r l e v e l as i n d i c a t e d 6y eqns (1.6a,b). The extension of the network i n eqns (1.5 a-e) to m u l t i p l e intermediates i s important i n d e v i s i n g p l a u s i b l e multi-enzyme r a t e forms f o r m i c r o b i a l metabolism. The Henri form obtained i n eqns (1.5a-e) i n v o l v e s three d i f f e r e n t enzyme complexes, a l l e q u i l i b r a t e d with each other "upstream" to the k i n e t i c a l l y slow step (eqn (1.3)). The n e g l i g i b l e f r e e energy changes evident f o r many^of the steps i n the Embden-Meyerhof g l y c o l y t i c sequence (Figure 1) i n d i c a t e that the three r e a c t i o n sequences (1.7a,b,c),

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a

X

glucose-6-P

£ fructose-6-P ->

(1.7a)

f r u c t o s e - d i - P J · · · J 2(phosphoenolpyruvate)

(1.7b)

pyruvate j l a c t a t e

(1.7c)

and

w i l l behave k i n e t i c a l l y l i k e the simpler r e a c t i o n sequence (1.7d): A (glucose) •> B(fructose-6-phosphate) •+ C(phosphoenolpyruvate) -* D ( l a c t a t e )

(1.7d)

Moreover, the a c t u a l feedback i n h i b i t i o n on the conversion of Β by the ( l a t e r ) products c i t r a t e and ATP (not shown) leads to the yet f u r t h e r s i m p l i f i c a t i o n given by the network below:

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

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BIOCHEMICAL ENGINEERING

50

40

< CD

CD

30 Figure 1. Free energy of intermediates of Embden-Myerhoff glycolytic sequence in human erythrocytes. Reproduced, with permission, from Ref. 6. Copyright 1970, Worth Publishers.

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

2.

OLLis

Kinetics

of Enzyme

A ( e x t r a c e l l u l a r glucose) 0

membrane permease ( E i )

B(fructoee-6-phoephate)(Si) -

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products ( i n c l . ATP,

31

Systems

phosphofructokinase

citrate)

Thus, the k i n e t i c s of conversions i n metabolic c e l l u l a r sequences, and even i n whole c e l l k i n e t i c s , at or near steady s t a t e may be expected to resemble the k i n e t i c r a t e form appro­ p r i a t e to one or a very small number of s e q u e n t i a l enzyme c a t a ­ l y z e d steps. The i m p l i c a t i o n s of t h i s point i n k i n e t i c models of s t r u c t u r e d c e l l systems are r e f l e c t e d i n l a t e r c o n t r i b u t i o n s i n t h i s conference. The Henri form thus has a n e a r l y u n i v e r s a l appeal and u t i l i t y i n r e p r e s e n t i n g the v a r i a t i o n of r e a c t i o n v e l o c i t y ν with Si. The i n t e r p r e t i v e ambiguity accompanying the success of t h i s u n i v e r s a l " f i t t i n g " f u n c t i o n leads us to r e c a l l G i l e a d i ' s c h a r a c t e r i z a t i o n of the Langmuir form, i d e n t i c a l to ( e q n ( l . ) ) i n n o n - b i o l o g i c a l heterogeneous c a t a l y s i s : The Langmuir a d s o r p t i o n isotherm may now be regarded as a c l a s s i c a l law i n p h y s i c a l chemistry. I t has a l l the i n g r e d i e n t s of a c l a s s i c a l equation: i t i s based on a c l e a r and simple model, can be d e r i v e d e a s i l y from f i r s t p r i n c i p l e s , i s very u s e f u l now, about 50 (now 60) years a f t e r i t was f i r s t d e r i v e d and w i l l probably be u s e f u l f o r many years to come, and i s r a r e l y ever a p p l i c a b l e to r e a l systems, except as a f i r s t approximation. A l l r e a c t i o n s , c a t a l y z e d or otherwise, are r e v e r s i b l e at the molecular l e v e l . T h i s f a c t i s most important i n k i n e t i c s when o v e r a l l conversion i s l i m i t e d thermodynamically. For example, the i s o m e r i z a t i o n of glucose to f r u c t o s e by glucose i s o m e r a s e 5

25

G + E

ki k J EXJE + F k-i k-

(1.8)

2

gives (glucose) ^ ( f r u c t o s e ) at e q u i l i b r i u m . Applying the pseudo-steady s t a t e assumption to the s i n g l e intermediate gave eqn (1.9a) ν [G - G*] Π Q) Κ + [G-G*] * m

a

x

a

U

where the parameters ν

,

a

;

and Κ are found i n eqns (1.9b,c) max

Γ

Κ*+1

K Ί r

F

(1.9b)

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

BIOCHEMICAL

32

κ = [ ΐ ς ^ Π ν ν

κ

]

*

G

**

+

9 K

*ρ G

ENGINEERING

( L E 9 C )

and K* = g l u c o s e / f r u c t o s e e q u i l i b r i u m constant Kp, K

G

= Briggs-Haldane constants f o r f r u c t o s e to glucose and glucose to f r u c t o s e , i . e . ,

Kj, = ( k

e l

+k /k 2

and K

=

Q

- 2

(k^+k^/^

G = glucose c o n c e n t r a t i o n a t e q u i l i b r i u m .

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Here we note t h a t , even when s t a r t i n g with G » G * , the equation (1.9a) w i l l s t i l l r e f l e c t a r e v e r s i b l e kinetîc form through eqn (1.9c) where Κ = f ( G * ) , although the i n i t i a l r a t e w i l l appear to s a t i s f y eqn (1.1), with Κ _ = (Κ - G*). apparent 2. P a r t i c u l a t e substrates The use of enzymes to l y s e c e l l s , hydrolyze f a t emulsions, s o l u b i l i z e proteinaceous c o l l o i d s , l i q u i f y or s a c c h a r i f y s t a r c h g e l s and granules, and degrade v a r i o u s components o f c e l l u l o s i c s u b s t r a t e s i n d i c a t e s that many s u b s t r a t e s a r e present i n a par­ t i c u l a t e form. K i n e t i c forms f o r such enzyme c a t a l y z e d r e a c t i o n r a t e s a r e here noted, and w i l l be r e v i s i t e d i n the subsequent d i s c u s s i o n of immobilized enzyme k i n e t i c s . A f i r s t example i s the h y d r o l y s i s of a uniform s u b s t r a t e , i n the form of a g i v e n t r i g l y c e r i d e emulsion, by a l i p a s e . Thus, the data of Sarda and D e s n e u e l l e * ( F i g u r e 2a,b) i n d i c a t e c l e a r l y that p u r i f i e d p a n c r e a t i c l i p a s e p r e p a r a t i o n s provide a c t i v i t y only when a substrate:water i n t e r f a c e i s present. The corresponding r e a c t i o n v e l o c i t y v s . s u b s t r a t e s u r f a c e area f u n c t i o n (Figure 2a,b) can be d e s c r i b e d by the form (2.1), which i s the analog to (1.1) f o r the case where the i n i t i a l enzyme l e v e l Ε i s much l a r g e r than the i n i t i a l s u b s t r a t e l e v e l ( s u r f a c e area). 7

ν = ν · bfzr]; ν max K+E max L

J>

8

= kS ο

(2.1)

where Ε = f r e e (uncomplexed) enzyme c o n c e n t r a t i o n and ν , Κ = constants as f o r eqn. (1.1). In c o n t r a s t to s o l u b l e s u B s i r a t e s where a s i n g l e r e a c t i o n product i s t y p i c a l l y i d e n t i f i a b l e , m u l t i p l e r e a c t i o n products r o u t i n e l y occur i n p a r t i c u l a t e degra­ d a t i o n and/or enzymatic depolymerizations. A p a r a l l e l / s e q u e n t i a l conversion network appears to be provided f o r t r i g l y c e r i d e s by the data o f Constantin et a l . (Figure 3) which a r e c o n s i s t e n t with the f o l l o w i n g network (2.2): 9

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983. 2

Saturation

LOS 1.5S (0.328 M) 2.0S

3.5S

S

4

0

-

-

-



Ι 0.1

/

Saturation

LOS 1.5S (0.153 M)

1JA

i

I

0.3

ι—ι—τ- ι

Soluble ι Insoluble

0.5S

5

2.0S

I

1

ι—ι 1 0.5 0.7

Interfacial Area (Methyl Butyrate) ,10 cm

Figure 2. Variation of the rate of lipase hydrolysis of triacetin and methyl butyrate with the concentration of the substrate. The concentration is expressed as a fraction of the saturation con­ centration (S) and the hydrolysis rate as a percentage of the rate of triolein hydrolysis under optimal conditions. Key: O, impure lipase plus esterolytic activity; A, lipase purified by electrophoresis. Reproduced, with permission, from Ref. 8. Copyright 1965, Academic Press.

0.5S

5

(Triocatin), 10 cm

Interfacial Area

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2

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34

BIOCHEMICAL ENGINEERING

Hydrolysis, % Figure 3. Liberation of diglycerides, monoglycerides, and glycerol during hydrolysis of a triglyceride by pancreatic lipase in the presence of sodium taurocholate and calcium ions. Key: O, triglyceride; 0, diglyceride; • , monoglyceride; X, glycerol. Reproduced, with permission, from Ref. 8. Copyright 1965, Academic Press

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

2.

Kinetics

OLLIS

of Enzyme

35

Systems

,triglyceride^ diglyceride

monoglyceride

(2.2)

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glycerol' The s i m p l i c i t y of k i n e t i c r e s u l t s (Figure 3) i s aided by the presence of s u r f a c t a n t (deoxycholate) and d i v a l e n t c a t i o n Ca which a s s i s t s i n remova^ of f a t t y a c i d product from the r e a c t i o n i n t e r f a c e . Lacking Ca presence, f a t t y a c i d w i l l accumulate at the i n t e r f a c e and compete with enzyme Ε f o r the l i m i t e d s u b s t r a t e surface area; the r e a c t i o n w i l l consequently e x h i b i t product inhibition. F u r t h e r , the extent of t r i g l y c e r i d e decomposition i s s t r o n g l y dependent upon s o l u t i o n composition: l a c k i n g c a l c i u m , only t r i - t o d i g l y c e r i d e conversion was noted, and even with Ca present and high enzyme l e v e l s , monoglyceride i s not converted to g l y c e r o l u n t i l the t r i g l y c e r i d e i s consumed. A simple depolymerization of c e l l w a l l m a t e r i a l apparently occurs i n l y s i s of Micrococcus l y s o d e i k t i c u s by lysozyme. The homogenous r e a c t i o n k i n e t i c s of enzyme a t t a c k of using oligomers of t h i s monopolymeric c e l l w a l l example are conveniently studied by using oligomers of the monomer [disaccharide:(GlcNac-MurNac) (N-acetyl-D-glucosamine-N-acetylmuramic a c i d ) ] . The endoenzyme can cleave an oligomer of i u n i t s , at any of the s u s c e p t i b l e bonds measured from a r e f e r e n c e (e.g., non-reducible) end. A model r e a c t i o n network a l l o w i n g f o r such random i n t e r n a l cleavage (but f o r b i d d i n g cleavage of the f i r s t s i t e from e i t h e r end) was proposed by Chipman: t

9

10

κ

Ε + S

« J

±

(ES^j

j = 0,1,1

1 1 i + i

(2.3a)

k (ES ).

-

±

C

Aj + S

( i - j )

j = 1, . . . , i

(2.3b)

k [(ES.). -

V

A, + H 0 (2.3c)] k. Α..(+Η 0) V Ε + S (2.3d) *Τ A + S^ Ε + S (2.3e) Here, the (i+2)-mer S^, i s complexed at the j * * s i t e , numbered 0 to i+1, but only b i n d i n g at s i t e s 1< j < i can r e s u l t i n r e a c t i o n , as a s i t e at an end i s u n r e a c t i v e . The model f i t k i n e t i c data from e x p e r i m e n t s s t a r t i n g with (uncleavable) dimer only. Other measures of r e a c t i o n r a t e f o r c e l l l y s i s are t y p i ­ c a l l y more convenient: o p t i c a l d e n s i t y of p a r t i a l l y l y s e d c e l l s ?

1

3

±

2

Jj

± + j

1

11

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

36

BIOCHEMICAL ENGINEERING

i n a whole c e l l r e a c t i o n mixture may s u f f i c e to represent the overall c o n v e r s i o n . The h y d r o l y s i s of starches by amylases i s most conveniently considered i n terms of sugar production ( f o r s a c c h a r i f i c a t i o n ) or v i s c o s i t y r e d u c t i o n ( f o r l i q u e f a c t i o n ) . Starch pastes are c h a r a c t e r i s t i c a l l y p s e u d o p l a s t i c (shear t h i n n i n g ) , and thus s a t i s f y , over an intermediate range of shear r a t e s , a power law r e l a t i o n s h i p between shear s t r e s s , τ, and shear r a t e , γ, of the form (2.4): 2 0 , 2 2

τ = K(^)

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where

N

Κ = apparent

(2.4) u n i t shear v i s c o s i t y

. , x l Ν = power law index ( =1 XT

r

pseudoplastic „ . ) Newtonian r

N

fc

12

As s t a r c h pastes were hydrolyzed by α-amylase , the para­ meters Κ and Ν were observed to change with r e a c t i o n progress i n a coupled manner according to the equation (2.5): log Κ - log

- Ν log

(2.5)

Here the p o i n t s (τ^,γ^) have the s i g n i f i c a n c e of being a l i m i t i n g h i g h shear r a t e (and shear s t r e s s ) at which the s o l u t i o n i s pre­ d i c t e d to achieve the Newtonian h i g h shear l i m i t . Thus, i f reducing ends of product were measured to provide an average molecular weight of remaining s t a r c h molecules, i t would be p o s s i b l e to r e l a t e Κ to the extent of r e a c t i o n and, through eqn (2.5), the consequent rheology of the p a r t i a l l y l i q u e f i e d product. (See l a t e r paper of R o l l i n g s , Okos, and Tsao f o r a d i s c u s s i o n of s t a r c h h y d r o l y s i s products.) The a c t i o n of the enzyme rennet on milk, known to d e s t a b l i z e a K-casein and t r i g g e r agglomeration of a m u l t i - c a s e i n m i c e l l a r suspension (a,β,γ,ε,κ) to produce coagulated m i l k f o r cheesemaking, has been shown to produce a r e a c t i o n suspension with power law behavior a l s o described by eqns (2.4-2.5)(Tucznicki and Scott B l a i r ) . * Where production of p a r t i c u l a r mono- or d i s a c c h a r i d e s i s d e s i r e d , employment of an a p p r o p r i a t e exoenzyme, such as 3-amylase i s noted. The r a t e of depolymerization i s again e s s e n t i a l l y an Henri form, unless modified by a product i n h i b i ­ t i o n term. Non-productive b i n d i n g , l e a d i n g to apparent i n h i b i t i o n , occurs f r e q u e n t l y f o r endoenzymes. For example, lysozyme h y d r o l y s i s of poly (GlcNac-MurNAc) y i e l d s dimer (GlcNac-MurNAc)2 which cannot be cleaved (being an end bond), j = 0 , i n eqn 2.3b above), but i t s non-productive b i n d i n g decreases the number of enzyme s i t e s a v a i l a b l e f o r a c q u i r i n g a c l e a v a b l e oligomer or polymer. S i m i l a r l y , the h y d r o l y s i s of ( i n s o l u b l e ) c e l l u l o s e [G] 13

1 1

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

2.

Kinetics

OLLis

of Enzyme

Systems

37

to c e l l o b i o s e , Gz i n eqns 2.6a-d and thence t o glucose, G, i n v o l v e s a c e l l o b i o s e noncompetitive b i n d i n g to the c e l l u l a s e ; t h i s i n h i b i t i o n i s r e l i e v e d by c e l l o b i o s e conversion to glucose by β-glucosidase : 9

15

k

[G] + Ej^ ( c e l l u l a s e ) k ->

1

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E

l

+

G

Ε

3

E [G] 2

*

E

+

+ G

χ

i

G

2

(2.6a)

(2.6b)

2

c e l l o b i o s e

1 2 (

E (B-glucosidase) + G

E[G]

binding)

2G + E

£

(2.6c) (2.6d)

£

15

The form of the i n h i b i t i o n i s n o n - c o m p e t i t i v e for cello­ b i o s e , thus Κ i s unaffected ( o r i g i n a l s u b s t r a t e s t i l l binds with same K) but the maximal v e l o c i t y i s diminished according t o eqn (2.7): i 1 + (

N

max

=

k

E

3 o

K^

k

3 k

l+î/Κ^

[

1

(

2

'

7

)

C e l l u l o s i c residues c h a r a c t e r i s t i c a l l y c o n s i s t of a mixture of c e l l u l o s e , h e m i c e l l u l o s e , and l i g n i n . Even the i n d i v i d u a l components are heterogeneous: the i n s o l u b l e c e l l u l o s e i t s e l f con­ s i s t s of a s t r u c t u r e d m a t e r i a l of v a r y i n g degrees of c r y s t a l linity. An i l l u s t r a t i v e s t r u c t u r e d substrate d e s c r i p t i o n i n c l u d i n g a k i n e t i c model and supporting data was provided by Ryu, Lee, T a s s i n a r i and Macy. These i n v e s t i g a t o r s modelled the substrate heterogeneity by assuming that the c e l l u l o s e c o n s i s t e d p r i m a r i l y of two phases: "an impermeable, denser, and h i g h l y ordered p a r a c r y s t a l l i n e or amorphous phase." Correspondingly, each phase possesses a d i f f e r e n t r e a c t i v i t y . With S (amor­ phous), S ( c r y s t a l l i n e ) and S ( r e s i d u a l i n e r t , i n c l u d i n g l i g n i n ) making up the t o t a l ceïlulose mass, S , the k i n e t i c model i n c l u d i n g binding of s o l u b l e product Ρ So the enzyme i s the system of r e a c t i o n s given by eqns (2.8a-3): 16

(enzyme adsorption)

k

Ε

ad

E*

(2.8a)

des k

(amorphous conversion)

E* + S

la J k Za

M

„ E*S

k

3 ·>

E*+P

(2.8b)

0

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

BIOCHEMICAL ENGINEERING

Downloaded by UNIV LAVAL on October 25, 2015 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch002

38

(crystalline conversion)

E* +S

(inert binding)

E* + S

( s o l u b l e pro­ duct b i n d i n g to s o l u b l e enzyme)

Ε + Ρ

(2.8c)

E*+P

(2.8d)

(2.8e)

EP

These authors determined c r y s t a l l i n i t y (and thus S ) from x-ray d i f f r a c t i o n data; molecular a c c e s s i b i l i t y and surrace area of v a r i o u s l y m i l l e d samples were obtained from i o d i n e adsorption and BET measurements. Degree of c e l l u l o s e p o l y m e r i z a t i o n was determined from viscometry of cadoxen-dissolved s o l u t i o n s , with the s p e c i f i c v i s c o s i t y extrapolated to zero c o n c e n t r a t i o n to obtained the i n t r i n s i c v i s c o s i t y , [η], from which i n turn the v i s c o s i t y average molecular weight M was estimated from the MarkHouwink equation: [η] = 38.5

χ 10

-5 0.76 M J

W

(2.9)

The experimental r e s u l t s supported both the p a r t i t i o n of sub­ s t r a t e i n t o three sub-classes, and the product i n h i b i t i o n as well. Immobilized

Enzymes

The attachment of c a t a l y t i c a l l y a c t i v e s i t e s to m a t e r i a l s which may be e a s i l y recovered from a r e a c t i o n mixture has been the s i n e qua non of most u s e f u l examples of c a t a l y s i s o u t s i d e of enzymology, and t h i s l a t t e r area has begun to f o l l o w s u i t (see f o r example, the s i x Enzyme Engineering C o n f e r e n c e s , as w e l l as s e v e r a l summary t e x t s ). In a p l e a s a n t l y exhaustive review of the k i n e t i c s of immobilized enzyme systems, G o l d s t e i n s e v e r a l years ago assigned "the e f f e c t s of i m m o b i l i z a t i o n on the k i n e t i c behavior of an enzyme" to four s i t u a t i o n s : 17

1 9

"1. Conformational and s t e r i c e f f e c t s : the enzyme may be conformâtionally d i f f e r e n t when f i x e d on a support; a l t e r n a t i v e l y i t may be attached to the s o l i d c a r r i e r i n a way that would render c e r t a i n p a r t s of the enzyme molecule l e s s a c c e s s i b l e to substrate or e f f e c t o r . These e f f e c t s are ( i n 1976) w e l l understood."

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

2.

OLLis

Kinetics

of Enzyme

39

Systems

Downloaded by UNIV LAVAL on October 25, 2015 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch002

"2. Partitioning effects: the e q u i l i b r i u m s u b s t r a t e , or e f f e c t o r concentrations w i t h i n the support may be d i f f e r e n t from those i n the bulk s o l u t i o n . Such e f f e c t s , r e l a t e d to the chemi­ c a l nature of the support m a t e r i a l , may a r i s e from e l e c t r o s t a t i c or hydrophobic i n t e r a c t i o n s between the matrix and low-molecular weight species present i n the medium, l e a d i n g t o a modified microenvironment, i . e . , to d i f f e r e n t concentrations of s u b s t r a t e , product or e f f e c t o r , hydrogen and hydroxyl ions, e t c . , i n the domain of the immobilized enzyme p a r t i c l e . "3. Microenvironmental e f f e c t s on the i n t r i n s i c c a t a l y t i c parameters o f the enzyme: such e f f e c t s due t o the p e r t u r b a t i o n of the c a t a l y t i c pathway of the enzymic r e a c t i o n would r e f l e c t events a r i s i n g from the f a c t that enzyme-substrate i n t e r a c t i o n s occur i n a d i f f e r e n t microenvironment when an enzyme i s immobi­ l i z e d on a s o l i d support. "4. D i f f u s i o n a l or mass-transfer e f f e c t s : such e f f e c t s would a r i s e from d i f f u s i o n a l r e s i s t a n c e s to the t r a n s l o c a t i o n of s u b s t r a t e , product, or e f f e c t o r to or from the s i t e of the enzymic r e a c t i o n and would be p a r t i c u l a r l y pronounced i n the case of f a s t enzymic r e a c t i o n s and c o n f i g u r a t i o n s , where the p a r t i c l e s i z e or membrane thickness are r e l a t i v e l y l a r g e . An immobilized enzyme f u n c t i o n i n g under c o n d i t i o n s o f d i f f u s i o n a l r e s t r i c t i o n s would hence be exposed, even i n the steady s t a t e , to l o c a l concentrations of s u b s t r a t e product or e f f e c t o r d i f f e r e n t from those i n the bulk s o l u t i o n . " These i n f l u e n c e s have been evaluated s u c c e s s f u l l y , i n d i ­ c a t i n g a pleasant grasp of these fundamental i n f l u e n c e s on enzyme k i n e t i c s . We consider s e v e r a l i l l u s t r a t i v e r e s u l t s . The i n f l u e n c e of the number of covalent couplings t o an enzyme on i t s a c t i v i t y i s shown f o r lysozyme, l i p a s e , and chymotrypsin f o r s o l u b l e enzyme (Figure 4a) and immobilized enzyme (Figure 4 b ) . Here, the s i m i l a r i t y i n a c t i v i t y p a t t e r n changes f o r a l l three enzymes, as the r a t i o of s o l u b l e or i n s o l u b i l i z e d c o u p l i n g groups (Figures 4a and 4b, r e s p e c t i v e l y ) to enzyme i s i n c r e a s e d , suggests c l e a r l y that a l t e r a t i o n of r e l a t i v e enzyme a c t i v i t y upon immobilization (on only the e x t e r n a l surfaces of these polyacrylamide supports) i s governed by conformation or s t e r i c changes e f f e c t e d by the extent of enzyme covalent c o u p l i n g . The microenvironment e f f e c t due to Donnan e q u i l i b r i a i s shown by the now c l a s s i c r e s u l t s (Figure 5) f o r p H - a c t i v i t y p r o f i l e s of chymotrypsin on p o l y c a t i o n i c , and p o l y a n i o n i c sup­ p o r t s vs. that of the s o l u b l e enzyme. Here, the charged s u b s t r a t e p a r t i t i o n i n g i n bulk s o l u t i o n (S ) and i n the porous support (S ) i s given by ο 2 0

(3.1a) where ze = s u b s t r a t e charge, ψ = e l e c t r o s t a t i c p o t e n t i a l of

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

40

BIOCHEMICAL ENGINEERING

8;^

>— Downloaded by UNIV LAVAL on October 25, 2015 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch002

E S

MOLAR RATIO (REACTIVE GROUPS/ ENZYME LOADING) χ Ι Ο

>>

1.0

I

^

ι­ ο

±

\ \

ο.

χ

Lysozyme^ \

0.01

^

ο Κ 1.0

MAcylated) L( Diazotized)

°\ 3

2

α- -Chymotrysin

V

0.1

-

10

MOLAR RATIO

ο ο \

Lipase >

100

1000

(REGENT/ENZYME)

Figure 4. Top, relative specific activity of immobilized enzyme versus the molar ratio (ratio of immobilized surface coupling groups to enzyme immobilized). Key: · , diazotized lysozyme; X, diazotized lipase; Δ, acylated a-chymotrypsin. Bottom, relative specific activity of modified soluble enzymes versus the molar ratio (ratio of soluble coupling reagent to enzyme). Key: O, diazobenzenesulfonic acid lysozyme; X, diazobenzenesulfonic acid-lipase; | , diazobenzenesulfonic acid-chymotrypsin; Δ, acetic anhydride-chymotrypsin. Reproduced, with permission, from Ref. 20.

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

2.

OLLis

Kinetics

of Enzyme

Systems

Downloaded by UNIV LAVAL on October 25, 2015 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch002

ι — ι — ι — ι — ι — ι — ι — Γ

Figure 5. Activity-pH curves. Key: O, chymotrypsin; · , a polyanionic ethylenemaleic acid (EMA) copolymer derivative of chymotrypsin, and A, a polycationic derivative, polyornithyl chymotrypsin. Substrate is acetyl-L-tryrosine ethyl ester. Ionic strength is 0.01 mol/L. Reproduced, with permission, from Ref. 19. Copyright 1976, Academic Press, Inc.

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

41

42

BIOCHEMICAL ENGINEERING

support, and k = Boltzman constant. With Henri's eqn (1.1) assumed t o apply f o r the a c t u a l i n t e r n a l c o n c e n t r a t i o n S.^ adjacent t o the enzyme, the observed r a t e expressed i n terms of the measurable S value i s s t i l l i n agreement with the Henri form, but the apparent K i s given by eqn (3.1b). f

K

f

= Kexp(zei|;/kT)

(3.1b)

D

as seen from using eqn (3.1a) i n eqn (1.1): ν

K+S

V

Downloaded by UNIV LAVAL on October 25, 2015 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch002

max i _ ±

ν S max ο Kexp(zei|;/k T) + S B

,~ , ν U-ic;

q

Donnan e q u i l i b r i u m , being an example of charge-charge i n t e r ­ a c t i o n , i s a l s o i n f l u e n c e d by i o n i c s t r e n g t h ; the apparent M i c h a e l i s constant K' of the Henri form (1.1) was shown to vary with i o n i c s t r e n g t h according to eqn (3.2) 21

K

f

,

= γΚ[1 - K Zm /KI]

l$

(3.2)

c

i n n e r

b u l

where the a c t i v i t y c o e f f i c i e n t r a t i o i s γ = y /y k^ i s the e f f e c t i v e f i x e d charge c o n c e n t r a t i o n i n s i d e the support, and I the bulk s o l u t i o n i o n i c s t r e n g t h . The i n f l u e n c e of c o l l o i d a l f o r c e s on r e a c t i o n s i n v o l v i n g immobilized enzymes a c t i n g on i n s o l u b l e s u b s t r a t e s has r e c e i v e d l e s s a t t e n t i o n , y e t i t appears to o f f e r some c l e a r examples of fundamental phenomena important i n enzyme k i n e t i c s . Datta examined l y s i s of Micrococcus l y s o d e i k t i c u s by s o l u b l e and (polyacrylamide) immobilized lysozyme. He noted that the decrease i n s o l u b l e enzyme a c t i v i t y with decreasing i o n i c strength (Table 1) p a r a l l e l e d the measured decrease i n c e l l l y s i s measured i n flow through a packed bed r e a c t o r of immobilized enzyme. Z

m

2 2

Table l

I o n i c Strength [M]

Soluble Enzyme Activity

2

2

Immobilized Enzyme Rela­ tive activity exp't (theory)

C e l l Surface Potential ψ

(mV)

+83.5mV

ΙΟ"

4

1700

ΙΟ"

3

4000

2.3

(^4.3)

ΙΟ"

2

12000

13.3

(^8.3)

^0

81.0mV -36mV

-37.3mV 22.6 (^8.3) 12000 icf * ( f i r s t numbers a r e experimental r e s u l t s , second ( i n parentheses) are values p r e d i c t e d by p a r t i c l e c o l l e c t o r theory, i n c l u d i n g i n ­ fluence of i n t e r c e p t i o n ( p a r t i c l e s i z e ) and Brownian motion on mass t r a n s f e r of p a r t i c l e s to immobilized enzyme surface.) 1

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

2. OLLis

Kinetics

of Enzyme

Systems

43

The d e c l i n e i n r a t e with decreasing i o n i c strength i s e v i ­ dently a s s o c i a t e d with a s i g n r e v e r s a l i n net c e l l charge, from a t t r a c t i v e (ψ - +44.5mV, ψ -, - -(36 to 37.3mV)) to r e p u l s i v e (ψ = ¥feï to 83.5mV) as determined from e l e c t r o p h o r e t i c m o b i l i t y measurements. A M i c h a e l i s constant (K) modif i c a t i o n may a l s o be involved (eqn 3.2 above). The agreement between observed r a t e and the c a l c u l a t e d mass t r a n s f e r l i m i t e d r a t e ( l a s t column i n parenthesis) means that the f l u i d mechanical and c o l l o i d a l f o r c e s which a c t to a t t r a c t / r e p e l enzyme and c e l l p a r t i c l e s are, a t the highest i o n i c strengths, dominating the slow step k i n e t i c s , i . e . , that every c e l l which encountered the immobilized enzyme surface was subsequently h y d r o l y z e d . Simil a r l y , the d e c l i n e i n r a t e f o r s o l u b l e enzyme i s to be a s s o c i a ted with a more d i f f i c u l t binding o r lessened substrate a f f i n i t y occasioned i n part by the s h i f t from a t t r a c t i v e to r e p u l s i v e c o l l o i d a l f o r c e between enzyme and c e l l . The mass t r a n s f e r of a c e l l to an immobilized enzyme p a r t i c l e brings a r e l a t i v e l y modest number of hydrolyzable bonds per c e l l p a r t i c l e to the c a t a l y s t . The above lysozyme data i n d i c a t e that the r e a c t i o n r a t e i s e s s e n t i a l l y mass t r a n s f e r c o n t r o l l e d , i . e . , the enzyme can cleave the small number of c e l l w a l l bonds per u n i t volume of c e l l rather r a p i d l y . S o l i d and emulsion p a r t i c l e s of s i m i l a r s i z e , such as c e l l u l a s e p a r t i c l e s or l i p i d emulsion d r o p l e t s , b r i n g a much l a r g e r number of c l e a v a b l e bonds per p a r t i c l e to the immobilized enzyme. Immobilized p a n c r e a t i c lipase shows an observed r e a c t i o n r a t e which i s 10 to 100 times l e s s than that p r e d i c t e d from c o l l o i d a l p a r t i c l e t r a n s port (Figure 6 ) . The p a r t i c l e c o l l e c t i o n e f f i c i e n c y u t i l i z e d i n c a l c u l a t i o n of the p a r t i c l e mass t r a n s f e r c o e f f i c i e n t i s composed of only two important terms, i n t e r c e p t i o n and Brownian motion ( g r a v i t a t i o n a l s e t t l i n g and i n e r t i a l impactions a r e n e g l i g i ble). Thus, the p a r t i c l e mass e f f i c i e n c y of c o l l e c t i o n , η, i s the simple sum of two terms: * ** s

u

r

f

a

c

e

a

Downloaded by UNIV LAVAL on October 25, 2015 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch002

22

2 3

2 3

23

η

overall

^Brownian • °·

9(

2

+

^interception )

ΐ Τ Τ Ί Γ e enz ο

2

/

3

+

x

-

5

(

2

d^> enz

(3

3

' >

where d , d ^ are substrate and immobilized enzyme p a r t i c l e d i a ­ meter, r e s p e c l i v e l y , and Ν i s the f l u i d v e l o c i t y e n t e r i n g the enzyme r e a c t o r . The observed r a t e of r e a c t i o n (Figure 6) i s a l s o orders of magnitude higher than the r a t e c a l c u l a t e d by assuming that only a few molecules of product are formed per emulsion p a r t i c l e c o l l i s i o n with the enzyme surface. Thus, p a r t i c l e - c a t a l y s t encounters r e s u l t , on the average, i n the cleavage of thousands to m i l l i o n s of l i p i d bonds per p a r t i c l e - c a t a l y s t c o l l i s i o n event. The above p a r t i c u l a t e examples i n v o l v i n g lysozyme and pan­ c r e a t i c l i p a s e concerned substrates with uniform reactant e

n

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

BIOCHEMICAL ENGINEERING

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44

£ S3

100 M

^ ^ 2 0 0 - 2 5 0 Mesh 3

ι , - —325 Mesh

s CD

i

10

• #

—50-100 Mesh^ /

1 ^

325 Mesh 3

r

£

«

*

50-100 Mesh Figure 6. Comparison of calculated mass transfer rate and experimentally observed reaction rates at various substrate concen­ trations. Broken lines are calculated values. Experimentally observed values for stain­ less steel are: X, 50-100 mesh; O, 200250 mesh; and Δ, 325 mesh. Reproduced, with permission, from Ref. 23. Copyright 1975, John Wiley and Sons, Inc.

200-250 Mesh

0.1 43

1

0

* ' ' ' ' •' ' ' 0.4 0.8 1.2 1.6 2.0 2.4 2.8 1

É

1

1

Substrate Concentration (Volume %)

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

Downloaded by UNIV LAVAL on October 25, 2015 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch002

2.

OLLis

Kinetics

of Enzyme

45

Systems

s t r u c t u r e : Micrococcus l y s o d e i k t i c u s ( p r o c a r y o t i c ) and t r i b u t y r i n , r e s p e c t i v e l y . More complex p a r t i c u l a t e s t r u c t u r e s provide a v a r i e t y of hydrolysable bonds per p a r t i c l e , l e a d i n g to a greater d i f f i c u l t y i n reaching a d e s c r i p t i o n of enzyme h y d r o l y t i c kinetics. Examples here i n c l u d e e u c a r y o t i c c e l l w a l l s (e.g., yeast, with c r o s s l i n k e d mannan and glucan components), and the c e l l u l o s i c substrates discussed i n the e a r l i e r s e c t i o n . Enzyme immobilization leads always to the need to consider d i f f u s i o n a l l i m i t a t i o n s , as the l a t t e r can give r i s e to r a t e l i m i t a t i o n s and d i s g u i s e s . The general k i n e t i c r e s u l t s are w e l l e s t a b l i s h e d , and we touch on only a few i l l u s t r a t i v e examples i n t h i s short paper; d i f f u s i o n a l l i m i t a t i o n s , enzyme e l e c t r o d e design, and polystep conversions. Internal d i f f u s i o n a l r e s i s t a n c e i s widely appreciated now, and t e s t s f o r freedom from such i n t e r n a l d i s g u i s e s a r e commonly used i n k i n e t i c s t u d i e s . The l e v e l of complexity of d i f f u s i o n a l i n t r u s i o n s i s i l l u s t r a t e d with c o n s i d e r a t i o n of an immobilized dual system: glucose oxidase (E^) and c a t a l a s e (E«) a r e used to o x i d i z e glucose (G) to o x i d i z e d glucose ( G ^ ) , and to decompose ox hydrogen peroxide ( H 0 ) : v

G

+

0

2

G

Î E

H

2

2

ox

+

H



( 3

2°2

2

t

2°2

H

-

1 2°2

+

(

4 a )

3

,

4

b

)

(In a developing commercial process, the o x i d i z e d glucose i s reduced subsequently to f r u c t o s e using a t h i r d c a t a l y s t . ) The enzymes E^ and E« are both d e a c t i v a t e d by hydrogen peroxide, l e a d i n g to a d i f f u s i o n - r e a c t i o n s i t u a t i o n which changes slowly i n time. K i n e t i c equations f o r t h i s system have been widely studied; the equations (3.5 a,b,c,d,e,f) used r e c e n t l y by Chang a r e c h a r a c t e r i s t i c of the l i t e r a t u r e phenomena examined: 2 6

27

[G = glucose, A = oxygen, Β = i ^ C ^ , and Ρ = o x i d i z e d glucose] Glucose consumption r a t e (by oxidase ν

-

V

l•

Γ

E

ox

/ ( 1 + K

G

/ C

G

+

(ox)) :

V a> c

( 3

-

5 a )

Peroxide conversion r a t e (by c a t a l a s e ( c a ) ) : = V

z

- k

Ε C ca ca D

(3.5b)

Oxidase d e a c t i v a t i o n : dE O

X

dt

- - k

0

3

C E_ Β ox B

(3.5c)

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

BIOCHEMICAL ENGINEERING

46 Catalase

activity:

dE dt

= -k.C^E 4 Β ca

Product formation dP dt

=

(3.5d) (at pseudo-steady s t a t e ) :

^dG dt

(3.5e)

In commercial l a r g e s c a l e c a t a l y t i c conversions, a c t i v i t y maintenance i s important. C a l c u l a t i o n s by C h a n g i n d i c a t i n g the glucose oxidase a c t i v i t y p r o f i l e s vs. d i s t a n c e i n the c a t a l y s t p e l l e t are shown i n Figure 7 f o r v a r i o u s c a t a l a s e enzyme loadings

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27

l

9

20

0 = / R

k

Ε /D ca ca A i n F i g s . 7a (t=0), 7b (t=100), 7d(t=150), 73(t=20C0, and 7f(t=300). Lack of any c a t a l a s e (φ = 0, F i g s . 7a-3, curve a) r e s u l t s i n the most r a p i d d e a c t i v a t i o n of Ε , while the highest E^ l o a d i n g , case glucose oxidase system i s probably the best studied example of c a t a l y s t a c t i v i t y maintenance. E v a l u a t i o n of system behavior, based upon H e n r i - r e l a t e d r e a c t i o n s (3.5a, 3.5b) and d e a c t i v a t i o n s (eqns 3.5c, 3.5d) o c c u r r i n g simultaneously with i n t e r n a l (eqns 3.5a,b) d i f f u s i o n and e x t e r n a l mass t r a n s f e r pre­ sents as complete a k i n e t i c a n a l y s i s as i s c o n v e n t i o n a l l y a v a i l ­ able i n other w e l l studied commercial examples of c a t a l y s i s . A second a p p l i c a t i o n of immobilized enzymes i n v o l v e s enzyme e l e c t r o d e s . An i l l u s t r a t i v e example of k i n e t i c fundamentals i s provided by the urease electrode,which uses a urease membrane to cleave urea i n t o bicarbonate and ammonium i o n , coupled to an ammonium i o n s p e c i f i c pontentiometric e l e c t r o d e . As with the glucose oxidase/catalase example above, the homogeneous phase k i n e t i c s can be a p p l i e d to describe the response of the immobi­ lized urease: 2

2

2 8

i n "planar" membrane: d i f f u s i o n + r e a c t i o n of urea s u b s t r a t e :

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

OLLis

Kinetics

of Enzyme Systems

47

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2.

Figure 7. Activity profiles for glucose oxidase (E ) coimmobilized with catalase (Eg) in an initially uniform concentration. The glucose oxidase loading is constant; the catalase loading increases as reflected by increased Thiele modulus, = R V Ε,ο/DSo, from Curve a (E = 0) to Curve d (E (max)). Dimensionless time appears at the lower right-hand corner of each graph (27). t

s

M0

t0

American Chemical Society Library 1155 16th St., N-W.

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983. Washington, D.C. 20036

BIOCHEMICAL ENGINEERING

48

d

P

2

k kE S

g

3

Q

s ^ 2 " (1+KS) (1+K- [NH +]

=

( 3

°

4

'

6 a )

production of ammonium i o n product ,2 +, 2k-KE S r M H

v;

+ (

n«)(i«'[HH^i)

with boundary c o n d i t i o n s (at steady

d[NH+] Downloaded by UNIV LAVAL on October 25, 2015 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch002

—j

d

V

d

0

( 3

V

6 b )

state)

at ζ = L (ammonium i o n electrode)

z

(3.6c)

~ = L ( S - S. ) a t ζ = 0 ( e x t e r n a l s αζ Z D - ν surface)

.

·

s

= 0 =— d z

=

(3.6d)

dNHT = k

( [ N H

l M H

(

3

6

β

)

N H4 ^ " îlb> · 4 -d^ Using publisheddata of L a i d l e r et a l . , the c a l c u l a t e d ammonium ion concentration p r o f i l e i s presented i n Figure 8 f o r v a r i o u s urease loadings i n the t h i n "planar" membrane g e l surrounding the t i p of the ammonium i o n e l e c t r o d e . These r e s u l t s c o n t a i n u s e f u l design information. I f ^e wish to use the e l e c t r o d e to measure substrate (urea), the NH^ s i g n a l a t the ammonium e l e c t r o d e sur­ face (z/L = 1.0, F i g . 8) should be independent of Ε , thus we should produce an e l e c t r o d e with Ε > 20-30 mg urease per ml of g e l . I f we wish t o detect an i n h i b i t o r , then we should load the enzyme membrane e l e c t r o d e very l i g h t l y (e.g., Ε = 2 mg/ml) so that a 50% a c t i v i t y inhibition,corresponding to E^ = Ε /2, produces a considerable drop i n NH^ s i g n a l . The experimental voltage response o f an urease enzyme e l e c t r o d e with v a r i o u s urease loadings was determined by G u i l b a u l t and M o n t a i v o ; F i g . 9 i n d i c a t e s that the i n i t i a l l o a d i n g of 20-30 mg c a l c u l a t e d above i s i n reasonable agreemen£ with the experimental r e s u l t s (note that c a l c u l a t i o n gives NH^ (Z=0) while measurement i s reported as ammonium i o n e l e c t r o d e voltage.) Many other sub­ stances have been analyzed experimentally with s i m i l a r enzyme e l e c t r o d e probes ( G u i b a u l t ) , i n d i c a t i n g major p o t e n t i a l f o r analogs of eqns (3.6a-e). A

2 9

30

3 2

Summary The k i n e t i c s of s o l u b l e and immobilized enzymes, involved i n r e a c t i o n s of s o l u b l e and i n s o l u b l e substrates appears to be s u f f i c i e n t l y w e l l studied over the l a s t 20 years that r e a c t o r design procedures based on fundamental k i n e t i c s r a t e equations may be executed with considerable confidence. The a p p l i c a t i o n o f such emzyme k i n e t i c s forms to s t r u c t u r e d models of m i c r o b i a l metabolism has a l s o progressed, as t h i s book documents.

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

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2.

OLLis

Kinetics

0

0.1

of Enzyme

0.2

0.3

Systems

0.4

49

0.5

0.6

0.7

0.8

0.9

1.0

DIMENSIONLESS THICKNESS Figure 8. Influence of enzyme loading (mg enzyme/cc gel) on product profiles for 50μ membrane; bulk urea concentration, 0.0833 M; bulk ammonium ion, negligible (10 M). Reproduced, with permission, from Ref. 28. Copyright 1972, Plenum Publishing Corp. 1

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

BIOCHEMICAL ENGINEERING

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50

τ

1

1

1

1

Γ

UREASE (mg//tgel) Figure 9. Comparison of experimental and calculated electrode steady-state responses versus enzyme loading. Key: Δ, experimental data (L = 350μ, bulk urea = 0.0833 M, bulk ammonium ion negligible); O, calculation from model given in text (L = 50μ, bulk urea = 0.0833 M, bulk ammonium ion negligible (10 M). Reproduced, with permission, from Ref. 28. Copyright 1972, Plenum Publishing Corp. 1

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

2.

OLLIS

Kinetics

of Enzyme

Systems

51

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18. 19. 20.

21. 22. 23. 24.

Henri, V. Comptes Rendus Acad. Sci., Paris, 1902, 135, 916. Michaelis, L; Menten, M. L., Biochem. Ζ., 1913, 333, 49. Briggs, G. E.; Haldane, J. B. S. Biochem. J., 1925, 19, 338. Dixon, M.; Webb, E. C. "Enzymes", Academic Press, 1964; Chapter IV). Gileadi, E. (ed), "Electrosorption", Plenum Press, New York, 1967. Lehninger, A. L. "Biochemistry", Worth Publishers, New York, 1970, p. 327. Sarda, L.; Desnuelle, P. Biochim. Biophys. Acta, 1958, 30, 513. Wills, E. D. Adv. in Lipid Research, 1965, 3, 219. Constantin, M. J.; Pasero, L.; Desnuelle, P. Biochim. Biophys. Acta, 1960, 43, 103. Chipman, D. M. Biochemistry, 1971, 10, 1714. Chipman, D. M.; Pollock, J . J.; Sharon, N. J. Biol. Chem., 1968, 243, 481. Cruz, Α., Russel; W. B.; Ollis, D. F. AIChE J., 1976, 22, 832. Whitaker, J . "Enzymology of Foods", John Wiley and Sons, New York. Tuczinski, W.; Scot Blair, G. W. Nature, 167, 216, 367. Howell, J . Α.; Stuck, J . D. Biotech. Bioeng., 1975, 17, 873. Ryu, D. D. Y.; Lee, S. F . ; Tassinari, T.; Macy, C. "Effect of Compression Million on Cellulose Structure and on Enzyma­ tic Hydrolysis Kinetics", private communication, 1981. "Enzyme Engineering I", L. B. Wingard (ed.), Plenum Press, 1972; II, Pye Ε. K. and L. B. Wingard (eds), Plenum Press, New York, 1974; III, Pye Ε. K. and Weetall, H. H. (eds), Plenum Press, New York, 1977; IV, Brown, G. and Mannecke, G. (eds), Plenum Press, New York; V, Plenum Press, New York, 1980; VI, 1982 (in press). Zaborsky, O. "Immobilized Enzymes", CRC Press, 1973; Chibata, (ed), "Immobilized Enzymes: Research and Development", Kodansha Press, Tokyo, 1978. Goldstein, L. "Adv. In Enzymology, XLIV, (K. Mosbach (ed.)), Academic Press, New York, 1976, pp. 397-443. Datta, R.; Ollis, D. F., "Immobilized Biochemicals and Affinity Chromatography, (R.B. Dunlap (ed.)), Plenum Ρress, New York, 1974, p. 293; "Adv. in Enzymol.", XLIV, pp. 444450 (K. Mosbach (ed.)). Wharton, C. M.; Crook, Ε. M.; Brocklehurst, K. Eur. J. Biochem., 1968, 6, 572. Datta, R., PhD Thesis, Princeton Univ., 1974. Lieberman, R.; Ollis, D. F . , Biotech. Bioengig, XVII, 1401 (1975). Bailey, J . B.; Ollis, D. F. "Biochemical Engineering Funda­ mentals", McGraw H i l l , 1977, p. 469.

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Fratzke, A. R.; Lee, Y. Y.; Tsao, G. T. G.V.C./AIChE - Joint Meeting, Munich, 1974, 4, F2-1. 26. Prenosil, J . F., Biotech. Bioeng., 1979, 21, 89; Buchholz, K.; Godelman, . Biotech. Bioeng., 1978, 20, 1201; Reuss, M.; Buchholz, K. Biotech. Bioeng., 1975, 17, 211. 27. Chang, T. MSE Thesis, Univ. of California, Davis, March, 1982. 28. Ollis, D. F.; Carter, R. "Kinetic Analysis of a Urease Elec­ trode: in "Enzyme Engineering II", F. K. Pye and L. B. Wingard (eds), Plenum Publishing Corp., New York, 1972, p. 271. 29. Hoare, J . G.; Laidler, K. J. J . Am. Chem. Soc., 1950, 72, 2487; Wall, M. C . ; Laidler, K. J. Arch. Biochem. Biophys., 1953, 43, 299; Wall, M. C . ; Laidler, K. J., Arch. Biochem. Biophys., 1953, 43, 307. 30. Guilbault, G. C. and Montalvo, J., J. Am. Chem. Soc., 1970, 92, 2533. 31. Wills, E. D. Adv. In Lipid Research, 1965, 3, 197. 32. Guilbault, G. C. "Biochemical Engineering II, A. Constantinides, W. R. Vieth and K. Venkatasubraminian (eds), Ann. Ν. Y. Acad. Sci., 1981, 309, 285. RECEIVED

July 7, 1982

In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.