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Biochemical Reaction Engineering ARTHUR E. HUMPHREY Department of Chemical and Biochemical Engineering, University of Pennsylvania, Philadelphia, PA 19104
The literature dealing with the kinetic behavior of biological reactor systems surely must be as extensive, if not more so, than that for chemical systems. Hence, any reasonable review of biological reactor systems must of necessity be rather cursory in character. Consequently, I would like to pick and choose my topics in this review, focusing on those items I have personally found most important in dealing with fermenting systems. In particular, I w i l l deal with fermenting systems for the production of chemicals such as antibiotics, production of potential food and feed such as single c e l l protein, and biological conversion of wastes. This review will be divided into three parts: 1. basic biological kinetics 2. typical biological reactor configurations 3. specific examples Biological Kinetics The kinetics of biological systems may be expressed at four different system levels. These include 1. molecular or enzyme level 2. macromolecular or cellular component level 3. cellular level 4. population level Each level of expression has a unique characteristic that leads to a rather specific kinetic treatment. For example, biological reactions at the molecular level invariably involve enzyme catalyzed reactions. These reactions, when they occur in solution, behave in a manner similar to homogeneous catalyzed chemical reactions. However, enzymes can be attached to inert solid supports or contained within a solid c e l l structure. In this case, the kinetics are similar to those for heterogeneous catalyzed chemical reactions. Enzyme Kinetics. In their simplest form, enzyme catalyzed reactions, occurring in a well-mixed solution, are characterized 0-8412-0432-2/78/47-072-262$06.50/0 © 1978 American Chemical Society In Chemical Reaction Engineering Reviews—Houston; Luss, Dan, et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.
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HUMPHREY
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by the w e l l - k n o w n M i c h a e l i s - M e n t e n k i n e t i c e x p r e s s i o n (l) This r e l a t i o n s h i p d e p i c t s the s u b s t r a t e , S, combining r e v e r s i b l y w i t h the enzyme, E , t o f o r m a n e n z y m e - s u b s t r a t e c o m p l e x , E S , t h a t c a n i r r e v e r s i b l y decompose t o t h e p r o d u c t and t h e enzyme, i . e . #
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E+S _
k
+
1
^-
ES
^
E+P
(1)
This leads t o a k i n e t i c e x p r e s s i o n f o r the v e l o c i t y of the a c t i o n , v, of the f o l l o w i n g form: ν
X
k^ E S +2 ο -1 +2 0
max v
re
- fJfT -
7 k
/ Ο Ν
( 2 )
+S
+i
where ν k 2 * observable reaction rate at high s u b s t r a ? i c o n c e n t r a t i o n a n d , h e n c e , i s o n l y l i m i t e d by t h e i n i t i a l enzyme c o n c e n t r a t i o n , E . i s the d i s s o c i a t i o n c o n s t a n t . When a
=
x
+
E
i
s
t
i e
m
a
x
i
m
u
m
x
q
and
k
+2
«
k
-l> k
t
h
e
«M "
n
K
S
ES> E+P i s l i m i t i n g The s a t u r a t i o n c o n s t a n t , K g - k i s an i n d i c a t i o n o f the a f f i n i t y o f t h e enzyme a c t i v e s i t e f o r t h e s u b s t r a t e . B a s i c a l l y two k i n d s o f c a t a l y t i c p o i s o n i n g o r i n h i b i t i o n a r e c o n s i d e r e d . T h e s e i n c l u d e i n h i b i t i o n by c o m p e t i t i o n f o r t h e a c t i v e s i t e by a n o n - r e a c t i v e s u b s t r a t e and i n h i b i t i o n b y a s u b s t a n c e t h a t m o d i f i e s t h e enzyme a c t i v i t y b u t d o e s n o t compete f o r t h e a c tive site. This behavior i s i l l u s t r a t e d i n equation (3).
T h i s b e h a v i o r c a n be e x p r e s s e d by ν
r r
S
max
V
s
In Chemical Reaction Engineering Reviews—Houston; Luss, Dan, et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.
m
264
CHEMICAL REACTION ENGINEERING REVIEWS—HOUSTON
where (5)
and where K j i s d e f i n e d by
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k
- i
C e l l Growth K i n e t i c s . Enzyme k i n e t i c concepts have been u t i l i z ed by Monod (2) and others ( 4-6 ) t o express the k i n e t i c behavior of c e l l growth on a s i n g l e l i m i t i n g s u b s t r a t e . Monod (3 ) p o s t u l a t e d that the growth o f c e l l s by b i n a r y f i s s i o n on a s i n g l e l i m i t i n g s u b s t r a t e probably had a s i n g l e l i m i t i n g r e a c t i o n step and t h e r e f o r e behaved i n a manner analogous t o the M i c h a e l i s Menten enzyme k i n e t i c s , i . e . dX dt
S K +S
Y = y
max
X
>.v 7
K t }
s
where X=the c e l l concentrationS=the growth l i m i t i n g s u b s t r a t e conc e n t r a ttiico n , t=time and μ =the maximum growth r a t e . Since the the growth r a t e o f c e l l s , i n c r e a s i n g by b i n a r y f i s s i o n , i s def i n e d by m
1
d
x
(*\
equation (7) can be expressed as
»'*~4φ)
9
The behavior o f equation (9) i s d e p i c t e d i n F i g u r e 1. For a complete theory o f c e l l growth and s u b s t r a t e u t i l i z a t i o n , i t i s necessary t o know the r e l a t i o n s h i p between the growth of c e l l s and the u t i l i z a t i o n o f s u b s t r a t e . T h i s r e l a t i o n s h i p i s expressed as a y i e l d , Y, d e f i n e d as Y =§
(10)
The s i m p l e s t assumption i s t h a t Y i s constant. T h i s i s essen t i a l l y t r u e o n l y a t h i g h growth r a t e s as w i l l be shown l a t e r . From equations(9) and (10) one o b t a i n s the f o l l o w i n g r e l a t i o n s h i p for substrate u t i l i z a t i o n :
In Chemical Reaction Engineering Reviews—Houston; Luss, Dan, et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.
8. H U M P H B E Y
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Downloaded by UNIV OF CALIFORNIA SAN DIEGO on February 13, 2016 | http://pubs.acs.org Publication Date: January 19, 1978 | doi: 10.1021/bk-1978-0072.ch008
M o r e r e c e n t l y , i t h a s b e e n shown b y M a r r (7 ) , P i r t (8 ) a n d others (5,9,11)that the y i e l d i s not a constant. R a t h e r endogeneous r e s p i r a t i o n u t i l i z e s t h e energy y i e l d i n g s u b s t r a t e s f o r m a i n tenance f u n c t i o n s ; hence,the s u b s t r a t e u t i l i z a t i o n by c e l l s can be b e t t e r e x p r e s s e d b y dS
1 dX .
dr = Y^dT
«.
+ m X
( 1 2 )
or 1 dS
xdT
1
V
e
. +
,,.χ m
( 1 3 )
where m i s t h e maintenance r e q u i r e m e n t o f s u b s t r a t e p e r u n i t o f c e l l biomass p e r u n i t o f time and i s a true y i e l d constant representing the substrate u t i l i z e d only f o r growth. This r e l a tionship i s depicted i n Figure 2. R e c e n t l y i t h a s been suggested t h a t t h e r a t e o f c e l l i n c r e a s e s h o u l d be e x p r e s s e d as a n e t growth r a t e w h i c h i n v o l v e d b o t h g r o w t h , μ , a n d d e a t h , 6, ( 1 2 - 1 4 ) , i . e . :ΠΓ = μ Χ - δ Χ dt
(14)
where μ max and
«-«
max
U-irri?
(15)
I n e x p r e s s i o n s (14) and (15) c e l l d e a t h i s d e p i c t e d as h a v i n g a f i r s t - o r d e r k i n e t i c b e h a v i o r and as maximal, i . e . 6 , when t h e g r o w t h l i m i t i n g s u b s t r a t e i s z e r o , i . e . S=0, a n d m i n i m a l when t h e substrate i s i n excess, i . e . l - S / i K ' + S ^ O . In t h i s l a t t e r expres s i o n , K i s a constant used t o express t h e observed b e h a v i o r ( 1 4 ) . Over t h e y e a r s , numerous m o d e l s f o r d e p i c t i n g c e l l g r o w t h h a v e evolved. S e v e r a l w i l l be d i s c u s s e d h e r e . One s u c h m o d e l i s t h a t f o r growth under m u l t i p l e s u b s t r a t e l i m i t a t i o n . I t c a n be e x pressed as g
In Chemical Reaction Engineering Reviews—Houston; Luss, Dan, et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.
CHEMICAL REACTION ENGINEERING REVIEWS—HOUSTON
Downloaded by UNIV OF CALIFORNIA SAN DIEGO on February 13, 2016 | http://pubs.acs.org Publication Date: January 19, 1978 | doi: 10.1021/bk-1978-0072.ch008
266
S, g / L I T E R Figure 1.
Figure 2.
Behavior of Equation 9
Relationship between substrate utilization and growth
In Chemical Reaction Engineering Reviews—Houston; Luss, Dan, et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.
8.
HUMPHREY
Biochemical Reaction Engineering
267
μ = μ max __ m
Downloaded by UNIV OF CALIFORNIA SAN DIEGO on February 13, 2016 | http://pubs.acs.org Publication Date: January 19, 1978 | doi: 10.1021/bk-1978-0072.ch008
x
S
l
M u l t i p l e s u b s t r a t e l i m i t a t i o n f r e q u e n t l y occurs i n batch growth systems (15-17) . Another model f o r c e l l growth i s t h a t f o r s i t u a t i o n s of e x t r e m e l y l o w s u b s t r a t e c o n c e n t r a t i o n s , i . e . S
