A N ENGINEERING
STUDY OF BACTERIAL
KINETICS AND ENERGETICS
A . A . Esener
ár
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A N ENGINEERING
STUDY OF BACTERIAL
KINETICS AND ENERGETICS
Proefschrift
ter verkrijging van de graad van doctor in de
technische wetenschappen
aan de Technische Hogeschool Delft,
op gezag van de rector magnificus
prof. ir. B.P.Th. Veltman,
voor een commissie aangewezen
door het college van dekanen te verdedigen op
donderdag 1 oktober te 14.00 uur
door
A l i A y d i n E s e n e r
Chemical Engineer B.Sc. M.Sc.
geboren te Ankara
Dit proefschrift is goedgekeurd door de promotoren
PROF. DR. IR. N.W.F. KOSSEN
PROF. IR. J.A. ROELS
On t h e f r o n t c o v e r d e v i a t i o n s between t h e u n s t r u c t u r e d model p r e d i c t i o n s and t h e e x p e r i m e n t a l r e s u l t s a r e shown f o r oxygen u p t a k e and c a r b o n d i o x i d e p r o d u c t i o n r a t e s d u r i n g f e d - b a t c h g r o w t h ( p a r t o f F i g . 3 o f C h a p t e r 4)
I n t h e c o m p l e t i o n o f t h i s t h e s i s I g r a t e f u l l y acknowledge:
P r o f e s s o r s K o s s e n and R o e l s f o r t h e i r e x c e p t i o n a l g u i d a n c e and encouragement as my s u p e r v i s o r s
Dr. I r . J.C. v a n S u i j d a m f o r many d i s c u s s i o n s and t r a n s l a t i n g the summary t o Dutch
I r . G.C. v a n E y b e r g e n f o r t r o u b l e s h o o t i n g i n many computer programs M e s s r s . C. Ras and G. v a n d e r S t e e n f o r c h e m i c a l a n a l y s i s o f samples M e s s r s . J . P h . B r o n k h o r s t , A.L. de G r a a f and B.J.T. K e r k d i j k f o r h e l p i n t h e h a n d l i n g and m a i n t e n a n c e o f t h e b i o r e a c t o r s and a u x i l i a r y equipment
M e s s r s . F. Bolmann and C. Warnaar f o r d r a w i n g s and p h o t o g r a p h s
My s t u d e n t s , J . Roozenburg, G.M. B o l and T. Veerman f o r t h e i r c o n t r i b u t i o n s t o A p p l i c a t i o n 3, C h a p t e r 5 and C h a p t e r 7, r e s p e c t i v e l y G i s t B r o c a d e s N.V. o f D e l f t f o r f i n a n c i a l l y s u p p o r t i n g me d u r i n g t h i s work K r a u s - U i t h o f Fonds f o r t h e i r f i n a n c i a l c o n t r i b u t i o n towards the p r i n t i n g c o s t s o f t h i s t h e s i s
T A B L E OF CONTENTS
CHAPTER 1 INTRODUCTION
CHAPTER
I A i m and scope
I I O r g a n i z a t i o n o f t h i s t h e s i s
ON THE THEORY AND APPLICATIONS OF UNSTRUCTURED GROWTH MODELS I Development o f m i c r o b i a l e n e r g e t i c s I I M a c r o s c o p i c methods i n t h e s t u d y o f e n e r g e t i c s I I I An i n t r o d u c t i o n t o t h e m o d e l l i n g o f m i c r o b i a l growth IV A s i m p l e u n s t r u c t u r e d model f o r m i c r o b i a l growth V D i s c u s s i o n o f u n s t r u c t u r e d models and e n e r g e t i c s w i t h r e f e r e n c e t o e x p e r i m e n t a l r e s u l t s V I A d i s c u s s i o n on t h e c o n c e p t o f m a i n t e n a n c e V I I N o m e n c l a t u r e and r e f e r e n c e s 5 7 9 10 13 21 22
CHAPTER 3 MATERIALS AND METHODS
I D e s c r i p t i o n o f t h e e x p e r i m e n t a l s y s t e m and
a n a l y t i c a l methods 25 I I Developed and used t o o l s and methods 27
I I I A p p l i c a t i o n o f t h e s t a t i s t i c a l t e c h n i q u e s i n t h e
s t u d y o f m i c r o b i a l k i n e t i c s and e n e r g e t i c s 32
IV N o m e n c l a t u r e and r e f e r e n c e s 39
CHAPTER 4
CHAPTER 5
FED-BATCH CULTURE ; MODELLING AND APPLICATIONS IN THE STUDY OF MICROBIAL ENERGETICS
I Summary 41 I I I n t r o d u c t i o n 41 I I I Model 42 IV D e t e r m i n a t i o n o f b i o k i n e t i c and e n e r g e t i c p a r a m e t e r s 44 V M a t e r i a l s and Methods 45 VI R e s u l t s and d i s c u s s i o n 46 V I I C o n c l u s i o n s 55 V I I I A p p e n d i x 56 I X N o m e n c l a t u r e and r e f e r e n c e s 57
GROWTH OF MONO AND MIXED CULTURES IN SALINE ENVIRONMENT
I A b s t r a c t 59 I I I n t r o d u c t i o n 59 I I I M a t e r i a l s and Methods 60 IV R e s u l t s 61 V D i s c u s s i o n 63 V I C o n c l u s i o n s 66 V I I R e f e r e n c e s 67
CHAPTER 6 THE INFLUENCE OF TEMPERATURE ON THE KINETICS AND ENERGETICS I I n t r o d u c t i o n 69 I I Model 69 I I I M a t e r i a l s and Methods 70 IV R e s u l t s and D i s c u s s i o n 71 V N o m e n c l a t u r e and r e f e r e n c e s 73 V I Addendum i n f l u e n c e o f t e m p e r a t u r e on ks 75 i n f l u e n c e o f t e m p e r a t u r e on e n e r g e t i c p a r a m e t e r s 75 consequences f o r e n g i n e e r i n g o p e r a t i o n s and d e s i g n 77 V I I R e f e r e n c e s 78
CHAPTER 7 A STRUCTURED MODEL FOR BACTERIAL GROWTH
I I n t r o d u c t i o n 79 I I T h e o r e t i c a l development o f t h e g e n e r a l s t r u c t u r e d model 80 I I I D e s c r i p t i o n o f t h e two c o m p a r t m e n t a l system 81 IV D e r i v a t i o n o f t h e b a l a n c e e q u a t i o n s 83 V E v a l u a t i o n o f t h e v a l i d i t y o f t h e model 84 V I D i s c u s s i o n 85 V I I N o m e n c l a t u r e and r e f e r e n c e s 88 CHAPTER 8 APPLICATIONS I Comments on t h e d e s c r i p t i o n o f m a i n t e n a n c e m e t a b o l i s m d u r i n g a n a e r o b i c growth w i t h p r o d u c t f o r m a t i o n 89 I I B i o e n e r g e t i c c o r r e l a t i o n o f COD t o BOD 95 I I I D e s c r i p t i o n o f m i c r o b i a l growth b e h a v i o u r d u r i n g the wash-out phase; d e t e r m i n a t i o n o f t h e maximum
s p e c i f i c growth r a t e 101 IV On t h e s t a t i s t i c a l a n a l y s i s o f b a t c h d a t a 109 V Carbon d i o x i d e h o l d - u p as a s o u r c e o f e r r o r i n b a t c h c u l t u r e c a l c u l a t i o n s 117 SUMMARY 121 SAMENVATTING 123 OZET 125
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1 I
C H A P T E R 1
INTRODUCTION
I AIM AND SCOPE
I n t h i s t h e s i s some a s p e c t s o f b a c t e r i a l k i n e t i c s and e n e r g e t i c s a r e s t u d i e d w i t h r e f e r e n c e t o e n g i n e e r i n g a p p l i c a t i o n s . F o r e n g i n e e r i n g a p p l i c a t i o n s t h e v e r b a l model p r e s e n t e d i n F i g u r e 1 p r o v i d e s a good a p p r o x i m a t i o n t o r e a l i t y i n t h e d e s c r i p t i o n o f b a c t e r i a l g r o w t h and p r i m a r y p r o d u c t m e t a b o l i s m . product synthesis use of substrate A T P pool synthesis of biomass precursors biomass synthesis maintenance
Fig. 1: Distribution of substrate energy in microbial metabolism (from Roels and Kossen, 19 78, see Chapter 2, ref.16).
Here b a s i c a l l y t h r e e p r o c e s s e s a r e i d e n t i f i e d ; i ) b i o s y n t h e t i c p r o c e s s d u r i n g w h i c h p r e c u r s o r s a r e formed f r o m t h e s u b s t r a t e f o l l o w e d by t h e p o l y m e r i z a t i o n of them i n t o b i o m a s s , i i ) p r o d u c t f o r m a t i o n and i i i ) m a i n t e n a n c e p r o c e s s e s . The energy i n p u t i n t o t h e system i n t h e form o f c h e m i c a l energy i s d i s t r i b u t e d between t h e s e p r o c e s s e s . O f t e n t h e r e a r e i n t e r a c t i o n s between t h e s e p r o c e s s e s as i n d i c a t e d by t h e two way arrows i n F i g . 1.
mass i n t h e form of s u b s t r a t e a r e d i s t r i b u t e d between t h e s e p r o c e s s e s , i n an a t t e m p t t o m a n i p u l a t e t h i s d i s t r i b u t i o n as t o m i n i m i z e t h e o v e r a l l c o s t of t h e d e s i r e d p r o d u c t . T h i s can b e s t be a c h i e v e d by d e s c r i b i n g the whole p r o c e s s by a m a t h e m a t i c a l model and o p t i m i z i n g i t a c c o r d i n g t o an o b j e c t i v e f u n c t i o n .
W i t h t h e s e c o n s i d e r a t i o n s i n mind the f o l l o w i n g q u e s t i o n s were s e t and s t u d i e d i n an e f f o r t t o d e v e l o p a sound s t r a t e g y f o r r e s e a r c h , d e s i g n and c o n t r o l of m i c r o b i a l p r o c e s s e s .
1. How can a m i c r o b i a l s y s t e m be m o d e l l e d based on t h e e x i s t i n g knowledge?
2. How can the e x p e r i m e n t a l and c o m p u t a t i o n a l methods be improved and d a t a p r o c e s s e d i n o r d e r t o o b t a i n more o p t i m a l i n f o r m a t i o n ?
3. To what e x t e n t a r e the k i n e t i c and e n e r g e t i c p a r a m e t e r s i n f l u e n c e d by s e l e c t e d e n v i r o n m e n t a l changes ( s a l i n i t y and t e m p e r a t u r e ) ?
4. Does the mode of c u l t i v a t i o n ( b a t c h , f e d - b a t c h , c o n t i n u o u s ) i n f l u e n c e the e n e r g e t i c b e h a v i o u r of the system?
5. How can models c o n s i d e r i n g i n t e r n a l changes i n t h e m i c r o o r g a n i s m s be f o r m u l a t e d ? What a r e t h e i r p r o s p e c t s ?
I I ORGANIZATION OF THIS THESIS
C o n t e n t s of t h e c h a p t e r s a r e b r i e f l y o u t l i n e d i n t h e f o l l o w i n g .
Chapter 2, s t a r t s w i t h an i n t r o d u c t i o n t o t h e c u r r e n t s t a t e of m i c r o b i a l
e n e r g e t i c s . The t h e o r i e s t o be u s e d l a t e r on, a r e d e v e l o p e d h e r e . M i c r o b i a l e n e r g e t i c s and k i n e t i c s a r e d i s c u s s e d i n c o n n e c t i o n w i t h t h e f o r m u l a t i o n of u n s t r u c t u r e d models. The c h o i c e of k i n e t i c and e n e r g e t i c r e l a t i o n s i s
d i s c u s s e d . The d i s c u s s i o n s a r e i l l u s t r a t e d , s u p p o r t e d and/or t e s t e d w i t h the b a t c h and c o n t i n u o u s c u l t u r e d a t a o b t a i n e d d u r i n g t h i s s t u d y . F i n a l l y , a s h o r t comment on the c o n c e p t of m a i n t e n a n c e i s g i v e n .
Chapter 3, c o n s i s t s of t h r e e s u b - s e c t i o n s . The f i r s t d e s c r i b e s the e x p e r i m e n t a l
system and the a n a l y t i c a l methods u s e d . The second s u b - s e c t i o n o u t l i n e s some of t h e t e c h n i q u e s d e v e l o p e d f o r o b t a i n i n g more o p t i m a l i n f o r m a t i o n from the e x p e r i m e n t a l d a t a . The l a s t s u b s e c t i o n shows how t h e use of s t a t i s t i c a l p r o -c e d u r e s -can improve the e f f i -c i e n -c y of e x p e r i m e n t a t i o n and t h e r e l i a b i l i t y of the d a t a o b t a i n e d .
Chapter 4, d e s c r i b e s b a c t e r i a l growth i n f e d - b a t c h mode. I t i s shown t h a t t h e
u n s t r u c t u r e d model p r e s e n t e d e a r l i e r , b r e a k s down d u r i n g the t r a n s i t i o n from e x p o n e n t i a l t o s u b s t r a t e l i m i t e d growth phase. Use of f e d - b a t c h c u l t i v a t i o n i n the s t u d y of m i c r o b i a l k i n e t i c s and e n e r g e t i c s i s a l s o shown and d i s c u s s e d .
In Chapters 5 and 6, the i n f l u e n c e s of s e l e c t e d e n v i r o n m e n t a l changes on the s u b s t r a t e energy d i s t r i b u t i o n a r e s t u d i e d . F i r s t (Chapter S) t h e i n f l u e n c e of the p r e s e n c e of N a C l i s e v a l u a t e d a t d i f f e r e n t c o n c e n t r a t i o n s . The c o n c e n t r a -t i o n range i s from 0 -t o 40 kg/m3. The d a -t a o b -t a i n e d i s a l s o compared w i -t h those r e p o r t e d f o r a c t i v a t e d s l u d g e c u l t u r e s under the same c o n d i t i o n s . I n Chapter 6, t h e i n f l u e n c e of t e m p e r a t u r e i s s t u d i e d i n f e d - b a t c h c u l t u r e s . K i n e t i c d a t a a r e used f o r t h e e s t i m a t i o n of thermodynamic p a r a m e t e r s i n an A r r h e n i u s type of model extended t o d e s c r i b e a l s o t h e s u p e r o p t i m a l t e m p e r a t u r e range. Temperature e f f e c t s on e n e r g e t i c p a r a m e t e r s a r e a l s o p r e s e n t e d .
Chapter 7, i s an a t t e m p t t o f o r m u l a t e and t e s t a s i m p l e s t r u c t u r e d model i . e . ,
a model d e s c r i b i n g the i n t e r n a l s t r u c t u r e of the o r g a n i s m i n a d d i t i o n t o m a c r o s c o p i c v a r i a b l e s . The s i m p l e two compartmental model d e v e l o p e d i s shown t o d e s c r i b e biomass and s u b s t r a t e p r o f i l e s w e l l . E x t e n s i v e t e s t s w i t h i n t e r n a l c o m p o s i t i o n d a t a i n d i c a t e s the weakness o f the model. P r o s p e c t s o f t h e s e t y p e of models and the c o r r e c t approach t o t h e i r f o r m u l a t i o n and v e r i f i c a t i o n a r e s t r e s s e d .
Chapter 8, c o n s i s t s of f i v e s h o r t p u b l i c a t i o n s w h i c h a r e a l l more or l e s s
a p p l i c a t i o n s o f the c o n s i d e r a t i o n s p r e s e n t e d i n C h a p t e r 2 and 3. The f i r s t two i l l u s t r a t e the use of m a c r o s c o p i c methods i n d a t a a n a l y s i s and c o r r e l a t i o n . The o t h e r s a r e on t h e a p p l i c a t i o n of s t a t i s t i c s on b a t c h d a t a , e s t i m a -t i o n of -the maximum s p e c i f i c grow-th r a -t e by -the wash-ou-t -t e c h n i q u e and e s t i m a t i o n of the c a r b o n d i o x i d e r e t a i n e d i n b r o t h d u r i n g b a t c h c u l t i v a t i o n .
Klebsiella pneumoniae NCTC 418 f o r m e r l y known as Klebsiella aerogenes i s
chosen as the e x p e r i m e n t a l o r g a n i s m , s i n c e i t i s a t y p i c a l s o i l b a c t e r i u m o f t e n a l s o p r e s e n t i n w a s t e w a t e r s and i s c a p a b l e o f g r o w i n g a e r o b i c a l l y and a n a e r o b i c a l l y .
A l l u n i t s i n v o l v i n g biomass d r y w e i g h t a r e e x p r e s s e d on a s h - f r e e b a s i s . An e q u i v a l e n t o f any compound i s d e f i n e d as t h a t amount c o n t a i n i n g 12 grams o f e l e m e n t a l c a r b o n . F o r the biomass f o r m u l a e used h e r e , an e q u i v a l e n t o f biomass i s the same as one mole o f biomass. The y i e l d o f biomass on s u b s t r a t e i s sometimes e x p r e s s e d as C - e q u i v / C - e q u i v ( same as C-mole/C-mole ) , s i n c e t h i s i s a more fundamentel u n i t t h e n m o l a r or mass u n i t s . I t i n d i c a t e s d i r e c t l y t h e f r a c t i o n a l c o n v e r s i o n o f s u b s t r a t e c a r b o n t o biomass c a r b o n .
This thesis has been carried out within the Biotechnology Group of the delft University of Technology.
Postal address: Department of Chemical Engineering, Biotechnology Group, SBR, Jaffalaan 9, TH., Delft 2600, The Netherlands
C H A P T E R 2
ON THE THEORY AND APPLICATIONS OF UNSTRUCTURED GROWTH MODELS:
KINETIC AND ENERGETIC ASPECTS
I DEVELOPMENT OF MICROBIAL ENERGETICS
Development o f q u a n t i t a t i v e b a c t e r i a l e n e r g e t i c s c a n be assumed t o have commenced w i t h t h e work o f Monod'. Monod has d e f i n e d t h e m a c r o s c o p i c y i e l d o f b i o -mass on s u b s t r a t e as t h e r a t i o o f t h e bio-mass produced t o s u b s t r a t e consumed. He has p r o d u c e d h i s w e l l known k i n e t i c e x p r e s s i o n d e s c r i b i n g t h e dependence o f g r o w t h r a t e on t h e c o n c e n t r a t i o n o f t h e growth l i m i t i n g s u b s t r a t e ' . F o l l o w i n g the i n t r o d u c t i o n o f c o n t i n u o u s c u l t i v a t i o n t e c h n i q u e s , H e r b e r t (1958) has p r e -s e n t e d e v i d e n c e t h a t i n C - l i m i t e d c o n t i n u o u -s c u l t u r e -s , t h e m a c r o -s c o p i c g r o w t h y i e l d , Ys x was n o t c o n s t a n t b u t d e c r e a s e d as t h e d i l u t i o n r a t e d e c r e a s e d . * H e r b e r t a t t r i b u t e d t h i s e f f e c t t o what he c a l l e d t h e endogeneous metabolisrn. I n e f f e c t t h e f i r s t a t t e m p t was a c u r v e f i t t i n g e x e r c i s e . Then p h y s i o l o g i c a l c o n -s i d e r a t i o n -s were a t t a c h e d t o t h i -s o b -s e r v a t i o n . I t wa-s -s u g g e -s t e d t h a t , endoge-neous m e t a b o l i s m p r o c e e d s a t c o n s t a n t r a t e a t a l l p o s s i b l e g r o w t h r a t e s . The i n t r o d u c t i o n o f t h i s c o n c e p t m o d i f i e d Monod's e x p r e s s i o n t o a new form:
U - Ug { Cs / ( Ks + Cg ) } - Ue (1)
Here i s t h e r a t e o f endogeneous m e t a b o l i s m . When Cs >> Ks , p -> Pmax and hence Vmax = Pg ~ Ve • When Cs = 0 \i e g u a l s t o - pe , i . e . , n e g a t i v e growth i s a c h i e v e d . T h i s i s e q u i v a l e n t t o s e l f d e s t r u c t i o n . The o b s e r v e d y i e l d f o r a c o n t i n u o u s c u l t u r e system c a n now be shown t o be g i v e n by;3
y = ymaX {D / ( D + u )} (2) sx sx e P i r t (1965) c o n s i d e r e d t h e s u b s t r a t e r e q u i r e m e n t f o r g r o w t h a s s o c i a t e d and non-a s s o c i non-a t e d f u n c t i o n s s e p non-a r non-a t e l y non-and p o s t u l non-a t e d h i s w e l l known r e c i p r o c non-a l / l i n e non-a r r e l a t i o n ^ : 1 / Y = 1 / Ymax + m / y (3) S X S X s v ' He has f o r m a l l y i n t r o d u c e d t h e m a i n t e n a n c e c o e f f i c i e n t ms, .and a t t r i b u t e d i t t o
the s o - c a l l e d m a i n t e n a n c e f u n c t i o n s w h i c h i n c l u d e ; t u r n o v e r of c e l l m a t e r i a l s , o s m o t i c work t o m a i n t a i n c o n c e n t r a t i o n g r a d i e n t s , c e l l m o t i l i t y e t c . Eq.(3) p r e d i c t s a s t r a i g h t l i n e f o r 1/Ys x v s . 1/u . I n a number of c a s e s , however, s t r a i g h t l i n e s c o u l d not be o b t a i n e d and t h i s was shown t o be due t o the i n f l u ence of the growth r a t e on the f e r m e n t a t i o n p a t t e r n and ATP y i e l d of the p a r -t i c u l a r o r g a n i s m . Based on -t h i s o b s e r v a -t i o n , S-tou-thamer and B e -t -t e n h a u s s e n m o d i f i e d eq.(3) by c o n s i d e r i n g the g e n e r a l energy c u r r e n c y , ATP, and o b t a i n e d the f o l l o w i n g f o r m ^ :
qA T P = y 1 ATP + mA T P ( A )
L a t e r on, i n an a t t e m p t t o a c c o u n t f o r the d i s c r e p a n c y between t h e t h e o r e t i c a l and e x p e r i m e n t a l growth y i e l d s , the below r e l a t i o n has been p r o p o s e d 6 ;
q , ™ - U / (
Y
IJ1
) . . - + m\i
+ m (5)ATP M ATP ' t h e o r e t i c a l g e
T h i s e q u a t i o n has found l i m i t e d a p p l i c a t i o n s i n c e no means were o f f e r e d f o r the d e t e r m i n a t i o n of the growth a s s o c i a t e d (nig) and i n d e p e n d e n t (me) m a i n t e n a n c e c o e f f i c i e n t s . F u r t h e r m o r e , the Y ^ p v a l u e s can o n l y be c a l c u l a t e d f o r
a n a e r o b i c systems i f the m e t a b o l i c pathway and the a s s o c i a t e d s t o i c h i o m e t r y a r e e x a c t l y known. The r e s p i r a t o r y c h a i n of b a c t e r i a d i f f e r w i d e l y and depend on the growth c o n d i t i o n s . Thus, f o r a e r o b i c systems one must know t h e s o - c a l l e d P/0 r a t i o i n o r d e r t o c a l c u l a t e the Y^Tp v a l u e s , or v i c e v e r s a . I n some c a s e s ^ATP v alu e s o b t a i n e d from a n a e r o b i c s t u d i e s were used f o r the c a l c u l a t i o n of P/0 r a t i o s and Y ^ p v a l u e s . T h i s a p p r o a c h may not be v a l i d s i n c e a e r o b i c and a n a e r o b i c sysems a r e q u i t e d i f f e r e n t e n e r g e t i c a l l y . R e c e n t l y v a n V e r s e v e l d ^ has r e v i e w e d the methods a v a i l a b l e f o r d e t e r m i n i n g the P/0 r a t i o i n b a c t e r i a l s y s -tems. From h i s a c c o u n t and l i t e r a t u r e i t becomes c l e a r t h a t t h e r e i s y e t no r e l i a b l e method f o r t h e e s t i m a t i o n of t h e P/0 r a t i o s . Thus more o f t e n t h a n n o t , one has t o work i n terms of m a c r o s c o p i c y i e l d s and hence eq.(3) s t i l l f i n d s w i d e a p p l i c a t i o n , p a r t i c u l a r l y f o r a e r o b i c g r o w t h w i t h no b y - p r o d u c t f o r m a t i o n .
Under a v a r i e t y of c o n d i t i o n s the growth y i e l d s o b s e r v e d were much l o w e r t h a n e x p e c t e d . Senez^ s t u d i e d t h i s phenomenon and i n t r o d u c e d t h e term unbalanced
growth i m p l y i n g t h a t the two m a j o r p r o c e s s e s i n the m i c r o o r g a n i s m s ; a n a b o l i s m
and c a t a b o l i s m a r e sometimes n o t i n tune w i t h each o t h e r and c o n s i d e r a b l e amount of ATP produced c o u l d be w a s t e d . R e c e n t l y , N e i j s s e l and Tempest^i'O have d e m o n s t r a t e d t h e o c c u r e n c e of t h i s phenomenon i n a number of systems and c o n s i d e r e d energy s p i l l i n g r e a c t i o n s as an i n t e g r a l p a r t of the e v o l u t i o n a r y c o m p e t i t i o n c a p a b i l i t i e s of m i c r o o r g a n i s m s . These a u t h o r s have d e m o n s t r a t e d t h a t the p r e s e n c e of u n c o u p l e r s , e x c e s s energy and Csource and f o r c e d t r a n -s i e n t -s enhance the e x t e n t of energy -s p i l l a g e . 9 - 1 1 For a comprehen-sive a n a l y s i s of the c u r r e n t s i t u a t i o n i n m i c r o b i a l e n e r g e t i c s the r e a d e r i s r e f e r -r e d t o -r e c e n t -r e v i e w s ^ - 1 4
A l t h o u g h an o v e r w h e l m i n g body of i n f o r m a t i o n e x i s t s i n l i t e r a t u r e , the s t a t e of m i c r o b i a l e n e r g e t i c s i s s t i l l not advanced enough a t the f u n d a m e n t a l l e v e l to a l l o w e n g i n e e r i n g a p p l i c a t i o n s t o be based on them. A d d i t i o n a l l y , as has been p o i n t e d out by Stouthamer'3 enough a t t e n t i o n has not been p a i d by many w o r k e r s t o t h e i r e n e r g e t i c c a l c u l a t i o n s and t h i s c o u l d be one of t h e r e a s o n s f o r the a c c u m u l a t i o n of i n c o n s i s t e n t d a t a o v e r the y e a r s . M i c r o b i a l e n e r g e t i c s b e i n g a t an impasse a t the f u n d a m e n t a l l e v e l , r e c e n t l y much work has been done on the m a c r o - e n e r g e t i c b e h a v i o u r . S i n c e t h e s e s t u d i e s r e l y on b a l a n c i n g methods and p r i n c i p l e s of thermodynamics, they a r e f a v o u r e d f o r q u a n t i t a t i v e t e c h n o l o -g i c a l a p p l i c a t i o n s .
I I MACROSCOPIC METHODS IN THE STUDY OF MICROBIAL ENERGETICS
I n f o r m a t i o n o b t a i n e d by the a p p l i c a t i o n of e l e m e n t a l and energy b a l a n c e s and e n t r o p y i n e q u a l i t i e s can be c l a s s i f i e d as m a c r o s c o p i c i n f o r m a t i o n . A l t h o u g h such i n f o r m a t i o n p r o v i d e s u s e f u l t o o l s i n e n g i n e e r i n g a p p l i c a t i o n s , no m i c r o s c o p i c d e t a i l s a r e p r o v i d e d . These t e c h n i q u e s , n e v e r t h e l e s s , p r o v i d e the t e c h -n o l o g i s t w i t h a s t r o -n g s t a r t i -n g p o i -n t i -n i -n d u s t r i a l a p p l i c a t i o -n s . I f s t r u c t u r e d i n f o r m a t i o n i s supplemented and checked f o r c o n s i s t e n c y by t h e a p p l i c a t i o n of m a c r o s c o p i c methods v e r y u s e f u l i n f o r m a t i o n can be o b t a i n e d w i t h q u a n t i t a t i v e c o n f i d e n c e .
The s u b j e c t has been advanced by many w o r k e r s i n the r e c e n t y e a r s . Recent advancement of the s u b j e c t i s due t o R o e l s and K o s s e n ' ^ , R o e l s ' ^ , E r i c k s o n e t al l 8 - 2 0 and H e i j n e n and R o e l s 2 1 . I n t h i s s e c t i o n o n l y t h e r e l e v a n t r e l a t i o n s from t h e s e p u b l i c a t i o n s w i l l be g i v e n w i t h o u t p r o o f . These r e l a t i o n s w i l l be a p p l i e d t o e x p e r i m e n t a l and t h e o r e t i c a l a n a l y s e s , l a t e r on i n t h i s work.
Assuming t h a t C, H, N and 0 a r e the o n l y e l e m e n t s exchanged i n n o n - n e g l i g i b l e amounts i n the s y s t e m , the f o l l o w i n g s t o i c h i o m e t r i c growth e q u a t i o n can be w r i t t e n f o r growth on a s i n g l e C and energy s o u r c e . T h i s s o u r c e i s assumed t o be growth l i m i t i n g . 0„C H, 0 N, + <I>0„ + $.NH„ 2 b j C j 5 2 4 3 s u b s t r a t e >.C H, 0 Nj 1 b, c, d, biomass $QC H, 0 N, + $,C0„ + $7H.0 (6) 3 a^ b3 c3 d^ 6 2 7 2 p r o d u c t The m a c r o s c o p i c y i e l d f a c t o r i s now d e f i n e d a s : Ys x = I $2 I ^ a2 ( C - e q u i v / C - e q u i v ) (7)
Ys x s i m p l y i n d i c a t e s the degree of t r a n s f o r m a t i o n of the s u b s t r a t e c a r b o n i n t o b i o m a s s . Hence i t seems t o be a more f u n d a m e n t a l p a r a m e t e r t h a n y i e l d v a l u e s e x p r e s s e d on mass o r m o l a r b a s i s . The above e q u a t i o n has 7 f l o w s i n v o l v i n g 4 e l e m e n t s . Thus s p e c i f i c a t i o n of any 3 f l o w s f i x e s t h e s y s t e m a l g e b r a i c a l l y , i . e . , any 4 unknown f l o w s can be c a l c u l a t e d from t h e knowledge of any 3 f l o w s at s t e a d y s t a t e .
S t a r t i n g f r o m the p r i n c i p l e of t h e c o n s e r v a t i o n of a t o m i c s p e c i e s the f o l l o w i n g b a l a n c e s can be shown t o h o l d f o r t h e system d e s c r i b e d :
r = r - r - r (8) c s x p r = 1/4 ( y r - y r - v r ) (9) o 's s 'x x 'p p r.T = - d , r + d r + d r (10) N 2 s 1 x 3 p H e r e , Yx , ys Yp a r e d e f i n e d by t h e f o l l o w i n g :
= 4 + bl " 2 C, - 3 d. (11) = 4 + b2 " 2 c2 - 3 d2 (12) YP = 4 + b3 " 2 c3 " 3 d3 (13) Y i s a l i n e a r c o m b i n a t i o n o f e l e m e n t a l b a l a n c e s . I t c a n a l s o be d e r i v e d from a degree o f r e d u c t i o n b a l a n c e as d e f i n e d by E r i c k s o n and coworkers.'8>'* I t must a l s o be n o t e d t h a t Y d e f i n e d h e r e o n l y h o l d s f o r NH3 b e i n g t h e N - s o u r c e . R o e l s ' ? has i n t r o d u c e d t h e g e n e r a l i z e d degree o f r e d u c t i o n c o n c e p t w h i c h can be a p p l i e d t o growth w i t h any N s o u r c e .
Eq.(3) i n t r o d u c e d p r e v i o u s l y does n o t c o n s i d e r p r o d u c t f o r m a t i o n . Humphrey and J e f f e r i s * ^ and l a t e r on R o e l s and K o s s e n ' ^ have i n c l u d e d t h e c o n t r i b u t i o n o f p r o d u c t f o r m a t i o n p r o c e s s and m o d i f i e d eq.(3) t o :
, ..max , „max ,, ..
r = r / Y + r / Y + m C (14) s x s x p sp s x
S i m i l a r forms o f t h e above e q u a t i o n c a n be d e r i v e d f o r t h e c o n v e r s i o n r a t e s o f c a r b o n d i o x i d e and oxygen. By c o m b i n i n g e q s . ( 8 ) , (9) and (14) t h e f o l l o w i n g c a n be g i v e n :
r = ( 1 / Ymax - 1) r + ( 1 / Ymax - 1) r + m C (15)
c s x x sp p S X
r„
1/4 { ( Y / Ym 3 X - Y ) r + ( Y / Ym a X - Y ) r + Y m C } (16)S S X X x s sp p p s s x
From e q s . ( 1 5 ) and (16) a number o f u s e f u l r e l a t i o n s c a n be o b t a i n e d . A few a r e shown b e l o w . A more comprehensive l i s t has been g i v e n r e c e n t l y by H e i j n e n and R o e l s . 2 1 1 / Ymax = Y M ( 1 / Ym a x) - 7 / 4 (17) ox S S X X m = m (18) c s m = (y /4) m (19) o s s
E r i c k s o n e t a l . ' ^ have shown methods f o r d a t a a n a l y s i s and c h e c k i n g t h e con-s i con-s t e n c y by u con-s i n g r e l a t i o n con-s o f t h i con-s con-s o r t . A n o t h e r advantage o f t h e con-s e t o o l con-s l i e con-s i n t h e f o r m u l a t i o n o f t h e o r e t i c a l l i m i t s t o t h e e f f i c i e n c y o f c o n v e r s i o n p r o -c e s s e s . R o e l s ' ' has -c a l -c u l a t e d t h e maximum p o s s i b l e y i e l d v a l u e s a l l o w e d by the second Law o f Thermodynamics. C a l c u l a t i o n o f t h i s a l l o w s t h e d e f i n i t i o n o f the thermodynamic e f f i c i e n c y , l ^ h 'or t^l e growth p r o c e s s .
\ h =Ys x /U f • (20)
O t h e r n p a r a m e t e r s have a l s o been d e f i n e d based on oxygen and e l e c t r o n b a l a n c e s and i n e q u a l i t i e s . F o r growth w i t h NH3 as t h e ammonia s o u r c e , however, t h e r e are no s i g n i f i c a n t d i f f e r e n c e s between t h e v a r i o u s r e l a t i o n s . F o r t h e system d e s c r i b e d t h e f o l l o w i n g u s e f u l l i m i t s have been shown t o h o l d :
Y <
(22)
Y < Y / Y (23)
sx s x
A p p l i c a t i o n of m a c r o s c o p i c p r i n c i p l e s can a l s o p l a y an i m p o r t a n t r o l e i n p r o -c e s s -c o n t r o l , where the -c o n t r o l p a r a m e t e r -cannot be d e t e r m i n e d d i r e -c t l y e.g., e s t i m a t i o n of biomass c o n c e n t r a t i o n i n b r o t h w i t h suspended p a r t i c u l a t e subs t r a t e , l i k e subs t a r c h . E subs t i m a t i o n of h e a t o u t p u t , a v e r y i m p o r t a n t t a subs k i n p r o -c e s s d e s i g n and -c o n t r o l , -can a l s o be done, based on the measurement of a few o n - l i n e d e t e r m i n e d p a r a m e t e r s .
The m a c r o s c o p i c t o o l s were a p p l i e d t o the e x p e r i m e n t a l d a t a o b t a i n e d d u r i n g t h i s work f o r c h e c k i n g d a t a c o n s i s t e n c y and f o r the e s t i m a t i o n of e n e r g e t i c p a r a m e t e r s . A d d i t i o n a l a p p l i c a t i o n s a r e p r e s e n t e d i n C h a p t e r 8.
I l l AN INTRODUCTION TO THE MODELLING OF MICROBIAL GROWTH
I n g e n e r a l , b i o l o g i c a l systems a r e s u b - s e t s of c h e m i c a l systems. A w e a l t h of i n f o r m a t i o n e x i s t s on c h e m i c a l k i n e t i c s and d y n a m i c s . Thus, one can a t l e a s t i n t h e o r y , e x p e c t t o be a b l e t o d e s c r i b e b i o l o g i c a l systems i n terms of t h e dynamic b e h a v i o u r of i t s c o n s t i t u e n t s ; c h e m i c a l systems. I n p r a c t i c e , however, t h i s approach i n e v i t a b l y f a i l s due t o two r e a s o n s :
i . F i r s t l y , f o r an e x a c t d e s c r i p t i o n o f m i c r o b i a l m e t a b o l i s m one has t o c o n s i d e r a l l the c o n c e n t r a t i o n s o f the c h e m i c a l s u b s t a n c e s i n t h e imme-d i a t e environment of m i c r o o r g a n i s m s ( a - b i o t i c phase) as w e l l as i n the o r g a n i s m i t s e l f ( b i o t i c p h a s e ) . C o n s i d e r i n g t h a t E. ooli has more t h a n 2000 d i f f e r e n t p r o t e i n s , t h i s becomes an i m p o s s i b l e t a s k even a t the age of f a s t c o m p u t e r s ,
i i . S e c o n d l y , a l t h o u g h b i o c h e m i c a l k i n e t i c s has b a s i c a l l y t h e same t a s k s as c h e m i c a l k i n e t i c s i . e . , i d e n t i f i c a t i o n of r e a c t i o n s between m o l e c u l e s , d e t e r m i n a t i o n of the r a t e s of c h e m i c a l r e a c t i o n s and t h e development of t h e r e l e v a n t t h e o r i e s , i t has t o c o n s i d e r more complex k i n d of i n t e r a c -t i o n s , such as r e a c -t i o n s be-tween m o l e c u l e s and c e l l s , m o l e c u l e s and o r g a n e l l e s , c e l l s and c e l l s e t c . I n o t h e r w o r d s , b i o l o g i c a l k i n e t i c s i s not r e s t r i c t e d t o the s t u d y of r e a c t i o n s between e n t i t i e s b e l o n g i n g t o a s i n g l e l e v e l of o r g a n i z a t i o n b u t a l s o b e l o n g i n g t o d i f f e r e n t l e v e l s . 2 4
25 • R e c e n t l y , . Savageau has g i v e n a g e n e r a l g r o w t h e q u a t i o n t h a t i s based upon
t h e n a t u r e of the e l e m e n t a l mechanisms i n complex systems. The r e s u l t i n g s e t of d i f f e r e n t i a l e q u a t i o n s would be, however, v e r y l a r g e and complex f o r a complete system d e s c r i p t i o n . M o r e o v e r , as shown by P r i g o g i n e 2 6 t h e s e t y p e of e q u a t i o n s a r e not o n l y s p e c i f i c t o b i o l o g i c a l systems b u t a r e a p p l i c a b l e t o any system, u n i v e r s a l l y . S i n c e t h i s t y p e of complete d e s c r i p t i o n has p r o v e n t o be n o t p o s s i b l e , one aims f o r s i m p l i f i c a t i o n s t h r o u g h j u s t i f i a b l e a s s u m p t i o n s .
S i g n i f i c a n t s i m p l i f i c a t i o n s become p o s s i b l e v i a a s t u d y of t h e r e l a x a t i o n times of the v a r i o u s p r o c e s s e s t a k i n g p l a c e i n s i d e and the o u t s i d e of the b i o t i c phase. Thus one has t o c o n s i d e r and compare t h e time c o n s t a n t s o f t h e e n v i r o n m e n t a l changes and t h o s e of mechanisms i n s i d e the o r g a n i s m w h i c h f a c i l i t a t e the a d a p t a t i o n o f t h e o r g a n i s m t o t h e s e e n v i r o n m e n t a l c h a n g e s . I n two cases s i m p l i f i c a t i o n become p o s s i b l e :
i . F o r p r o c e s s e s w h i c h a r e c h a r a c t e r i z e d by v e r y l a r g e r e l a x a t i o n t i m e s , compared w i t h t h a t of the growth p r o c e s s , t h e mechanism and thus t h e c o n c e n t r a t i o n of t h e compounds i t r e g u l a t e s , do n o t change s i g n i f i c a n t l y . Thus t h e s e mechanisms and t h e i r e f f e c t s on the t o t a l , system b e h a v i o u r may
be c o n v e n i e n t l y n e g l e c t e d ,
i i . F o r p r o c e s s e s w h i c h a r e c h a r a c t e r i z e d by v e r y s h o r t r e l a x a t i o n t i m e s , b i o t i c mechanisms f o l l o w and r e s p o n d t i g h t l y t o t h e e n v i r o n m e n t a l changes and a g a i n t h e c o n c e n t r a t i o n s o f the b i o t i c components t h a t a r e a s s o c i a t e d w i t h t h e s e p a r t i c u l a r mechanisms can be c a l c u l a t e d from the a - b i o t i c c o n c e n t r a t i o n s . The s t a t e o f the mechanism c a n be r i g o r o u s l y d e s c r i b e d u s i n g o n l y e n v i r o n m e n t a l c o n c e n t r a t i o n i . e . , c o n c e n t r a t i o n i n t h e a - b i o t i c phase.
Based on t h e s e c o n s i d e r a t i o n s i . e . , most changes among t h e components o f a system o c c u r much f a s t e r t h a n t h e r a t e o f t h e growth f o r t h e system as a w h o l e , Savageau25 c o n c l u d e d t h a t t h i s m a t h e m a t i c a l l y i m p l i e s a s m a l l number o f r e l a t i o n s r e p r e s e n t i n g t h e s l o w e s t phenomena d e t e r m i n e t h e t e m p o r a l r e s p o n s e o f t h e e n t i r e system. A l l o t h e r r e l a t i o n s r e p r e s e n t i n g t h e f a s t e r phenomena c a n be assumed t o have reached a p s e u d o - s t e a d y s t a t e w i t h t i m e d e r i v a t i v e s e q u a l t o z e r o .
P r a c t i c a l growth models a r e u s u a l l y e x p r e s s e d i n terms of a r a t h e r a b s t r a c t u n i t s o f l i f e , t h a t i s , i n terms o f p o p u l a t i o n s . T h i s a p p r o a c h c o n s i d e r s t h e p o p u l a t i o n as an e n t i t y homogeneously d i s t r i b u t e d i n space and t i m e , and thus a v o i d s c o m p l i c a t i o n s t h a t m i g h t a r i s e due t o t h e s t o c h a s t i c phenomena a s s o c i a -t e d w i -t h -t h e e x i s -t e n c e o f i n d i v i d u a l o r g a n i s m s . I -t i s , however, i m p o r -t a n -t -t o n o t e t h a t t h i s a p p r o a c h i s o n l y v a l i d when the number o f o r g a n i s m s i n t h e s y s t e m i s v e r y l a r g e . T h i s was t h e c a s e f o r e x p e r i m e n t s t o be r e p o r t e d i n t h i s work.
Qcowth i s the p r o d u c t i o n o f new biomass by a p o p u l a t i o n when i t consumes a
s u i t a b l e l i v i n g o r non l i v i n g s u b s t r a t e f r o m i t s e n v i r o n m e n t and i n c o r p o r a t e s some o f t h i s s u b s t a n c e i n t o i t s own.24 Reproduction i s t h e i n c r e a s e i n t h e number o f d i s c r e t e i n d e p e n d e n t c e l l s o f a p o p u l a t i o n . Growth and r e p r o d u c t i o n a r e o b v i o u s l y c o u p l e d p r o c e s s e s , however, t h e d e g r e e o f c o u p l i n g may be d i f f e -r e n t f o -r each c a s e . I n t h i s s t u d y t h e s e two p -r o c e s s e s w i l l n o t be c o n s i d e -r e d s e p a r a t e l y b u t t h e t o t a l e f f e c t i s summed w i t h t h e d r y w e i g h t measurements.
One o f the most g e n e r a l a p p r o a c h e s f o r d e s c r i b i n g g r o w t h , was p r o v i d e d by P o w e l l 2 7 . I n h i s a p p r o a c h , t h e c u r r e n t s p e c i f i c growth r a t e o f a p o p u l a t i o n i s assumed t o depend n o t o n l y o n t h e c u r r e n t s t a t e o f t h e a - b i o t i c phase b u t a l s o on the e n t i r e h i s t o r y o f t h e a - b i o t i c phase s e e n by t h e b i o t i c phase. I n o t h e r words P o w e l l e x p r e s s e d t h e s p e c i f i c growth r a t e a t any i n s t a n t t o be a
' f u n c t i o n a l ' o f t h e s t a t e o f t h e a - b i o t i c phase. I n p r a c t i c e , however, t h i s a p p r o a c h i s d i f f i c u l t t o a p p l y and p a r t i c u l a r l y i n t h e c h o i c e o f f u n c t i o n a l s . A s i m p l e r a p p r o a c h would be t o assume t h a t t h e c u r r e n t growth r a t e s a r e
f u n c t i o n s o f t h e c u r r e n t s t a t e o f t h e a - b i o t i c and b i o t i c p h a s e s .
The most r i g o r o u s s i m p l i f i c a t i o n done i n t h e development o f p o p u l a t i o n models i s t h e a s s u m p t i o n t h a t t h e t o t a l amount o f the biomass i n t h e c u l t u r e i s
s u f f i c i e n t t o s p e c i f y t h e a c t i v i t i e s o f the m i c r o o r g a n i s m s . Model based on t h i s a s s u m p t i o n i . e . , i n w h i c h the v a r i a t i o n i n the biomass c o m p o s i t i o n i s t o t a l l y i g n o r e d , a r e c a l l e d UNSTRUCTURED WDELS.
IV A SIMPLE UNSRUCTURED MODEL FOR MICROBIAL GROWTH
The most p o p u l a r k i n e t i c e x p r e s s i o n used t o d a y i s the Monod r e l a t i o n . A l t h o u g h t h i s r e l a t i o n i s an homologue o f the M i c h a e l i s - M e n t e n e q u a t i o n , Monod a r r i v e d a t i t e m p i r i c a l l y . That i s , h i s r e l a t i o n p r o v i d e d good f i t f o r h i s e x p e r i m e n -t a l d a -t a . A l -t e r n a -t i v e l y , one c a n -t r y -t o p r o v i d e a m e c h a n i s -t i c f o u n d a -t i o n by r e a s o n i n g t h a t one enzymic r e a c t i o n t a k i n g p a r t i n a l o n g sequence m i g h t be t h e
b o t t l e n e c k and thus r a t e l i m i t i n g .
Now, c o n s i d e r a c o n s t a n t volume c u l t i v a t i o n system i n w h i c h t h e t o t a l m i c r o b i a l a c t i v i t y i s q u a n t i f i e d by t h e amount o f biomass (biomass w i l l i m p l y d r y w e i g h t t h r o u g h o u t t h i s work) and t h e r e i s a s i n g l e l i m i t i n g s u b s t r a t e (C and energy s o u r c e ) . To d e s c r i b e t h e s y s t e m , changes o f Cs and Cx and t h e i r i n t e r -dependence have t o be e v a l u a t e d . F o r t h e g e n e r a l c a s e t h e f o l l o w i n g b a l a n c e s can be f o r m u l a t e d : d C / d t (24) d C / d t s r + s (25) I f t h e r e l a t i o n o f l i n e a r s u b s t r a t e c o n s u m p t i o n i s chosen f o r r e l a t i n g r x t o rs 2,4 ( f o r t h e n o - p r o d u c t c a s e ) : r / Y x s x m C s x (26)
Eqs. (24) (25) and (26) w i l l be s u f f i c i e n t t o d e s c r i b e s i m p l e systems such as b a t c h , c o n t i n u o u s and f e d - b a t c h . $ i s t h e n e t f l o w term t o t h e system and i s f i x e d by t h e mode o f o p e r a t i o n , e.g., f o r c o n t i n u o u s c u l t i v a t i o n $ i s d e s c r i b e d by t h e f o l l o w i n g : D ( C (27) * = x D C (28) F o r b a t c h c u l t i v a t i o n , $ a r e z e r o , s i n c e i t i s a c l o s e d system as f a r as t h e non-gaseous phases a r e c o n c e r n e d . Thus t h e f o l l o w i n g p a i r o f eqs. d e s c r i b e t h e system: dC /dt = u C C / (K + C ) x max s x s s dC / d t = - y C C . / {(K + C ) Ym a X} - m C s max s x s s sx s x (29) (30)
No a n a l y t i c a l s o l u t i o n i s p o s s i b l e f o r t h i s s e t and hence n u m e r i c a l methods ' were used f o r s i m u l a t i o n p u r p o s e s d u r i n g t h i s s t u d y . Most o f t h e t i m e d u r i n g b a t c h growth o r g a n i s m s grow a t o r n e a r Um a x • S i n c e t h e e f f e c t o f m a i n t e n a n c e r e q u i r e m e n t s a r e e f f e c t i v e l y m i n i m i z e d a t n i g h u , a c o n v e n i e n t s i m p l i f i c a t i o n can be i n t r o d u c e d by n e g l e c t i n g t h e ms term i n t h e above model. I n t h i s c a s e an a n a l y t i c a l s o l u t i o n i s p o s s i b l e and c a n be shown t o be g i v e n by:
l n ( C /C ) + K Y /(Y C + C ) I n {(C /C ) / ( l + C / (Y C ) -x -xo s s-x s-x so -xo X xo xo sx so C / ( Y C ) ) } = y t (31) x sx so max The i m p l i c i t n a t u r e o f t h i s e x p r e s s i o n g i v e s problems i n p a r a m e t e r e s t i m a t i o n from e x p e r i m e n t a l d a t a by n o n l i n e a r r e g r e s s i o n . F u r t h e r s i m p l i f i c a t i o n s a r e p o s s i b l e by c o n s i d e r i n g v a r i o u s e x p e r i m e n t a l c o n d i t i o n s , e.g., i f C >> C Y and C >> K , t h e model reduces down t o :
C / C = exp ( U t ) (32) x xo max
Growth b e h a v i o u r i n f e d - b a t c h c u l t u r e s can a l s o be d e s c r i b e d by t h i s g e n e r a l u n s t r u c t u r e d model. T h i s i s p r e s e n t e d i n C h a p t e r 4.
I t i s i m p o r t a n t t o n o t e t h a t an u n r e a l i s t i c f e a t u r e ^0f the l i n e a r r e l a t i o n i s t h a t i t p r e d i c t s s u b s t r a t e u p t a k e even a f t e r s u b s t r a t e has been e x h a u s t e d . P a r t i c u l a r l y i n n u m e r i c a l s i m u l a t i o n s the e x p e r i m e n t e r must c o n s i d e r t h i s p o i n t c a r e f u l l y as t h i s m i g h t l e a d t o the c a l c u l a t i o n of n e g a t i v e s u b s t r a t e c o n c e n t r a t i o n v a l u e s . When Cs = 0, the c o n c e p t of endogeneous m e t a b o l i s m becomes handy. E x p r e s s i o n (1) p r e d i c t s z e r o growth r a t e when:
o r , C = K y / ( y ~ P ) (33) s s e g e C a K u / y (34) s s e g T h i s e x p r e s s i o n can a l s o a l l o w f o r n e g a t i v e g r o w t h ; a n a t u r a l phenomenon w h i c h can be o b s e r v e d e x p e r i m e n t a l l y , when Cs i s s m a l l e r t h a n the r i g h t hand s i d e of eq.(33) .
The d i f f e r e n c e between P i r t and H e r b e r t r e l a t i o n s stem from t h e i r d i f f e r e n t ways of i n t e r p r e t i n g the f u n c t i o n i n g of the m a i n t e n a n c e p r o c e s s e s .
The s i m p l e u n s t r u c t u r e d model has been a p p l i e d t o some p r a c t i c a l s y s t e m s , s u c c e s s f u l l y , p a r t i c u l a r l y f o r b a t c h growth where t h e growth i s n o t l i m i t e d by s u b s t r a t e , a n d f o r s u b s t r a t e l i m i t e d g r o w t h i n c o n t i n u o u s c u l t u r e s .
One wonders i f the Monod' r e l a t i o n i s t h e o n l y s u i t a b l e k i n e t i c e x p r e s s i o n f o r m o d e l l i n g . As has been shown by R o e l s 2 8 t h e d e t a i l e d n a t u r e of the k i n e t i c e q u a t i o n i s o n l y of s l i g h t i m p r o t a n c e f o r s u b s t r a t e l i m i t e d g r o w t h . T h i s i s because of the v e r y low Cg under s u b s t r a t e l i m i t i n g c o n d i t i o n s . A t such low Cs v a l u e s a p s e u d o - s t e a d y s t a t e h y p o t h e s i s w i t h r e s p e c t t o Cs h o l d s . Under t h e s e c o n d i t i o n s , i t can be shown t h a t the s t a t e e q u a t i o n f o r Cx i s o n l y d e f i n e d by t h e e n e r g e t i c and e x p e r i m e n t a l p a r a m e t e r s and n o t by the k i n e t i c r e l a t i o n . S i n c e Cx v s . t i m e , p r o f i l e i s n o t i n f l u e n c e d much by the r a t e e q u a t i o n , one s h o u l d n o t e x p e c t to o b t a i n a c c u r a t e i n f o r m a t i o n on the n a t u r e of the r a t e e x p r e s s i o n from the b i o m a s s - t i m e d a t a . Cs , however, may w e l l p r o v i d e u s e f u l i n f o r m a t i o n f o r the v e r i f i c a t i o n or r e j e c t i o n of t h e r a t e e x p r e s s i o n . U n f o r t u n a t e l y , Cs d a t a o b t a i n e d under s u b s t r a t e l i m i t a t i o n s u f f e r from l a r g e u n c e r t a i n i t i e s . Thus d i s c r i m i n a t i o n between t h e v a r i o u s k i n e t i c models become a d i f f i c u l t t a s k . I n n o n - s u b s t r a t e l i m i t e d systems e.g., b a t c h s y s t e m s , g e n e r a l l y the r a t e of growth i n c r e a s e s w i t h i n c r e a s i n g Cg up t o a p o i n t , t h e r e a f t e r rx r e m a i n s c o n s t a n t e.g., l i k e s a t u r a t i o n k i n e t i c s . I n t h i s case o n l y d a t a from the t r a n s i e n t phase from e x p o n e n t i a l t o s t a t i o n a r y phase can be u s e d f o r model d i s c r i m i n a t i o n . However, t h i s t r a n s i t i o n i s u s u a l l y q u i t e a b r u p t , s i n c e a t the p o i n t when the r e s i d u a l s u b s t r a t e , Cg, i s low ( Ks - Cs) the h i g h c e l l c o n c e n t r a t i o n r a p i d l y u t i l i z e s the r e m a i n i n g s u b s t r a t e .
I n t h e i r r e v i e w , R o e l s and Kossen'6 s t u i e d a number of u n s t r u c t u r e d models and have shown t h a t a l m o s t any o b s e r v a t i o n can be m o d e l l e d by any of them. Thus the c h o i c e of the k i n e t i c e x p r e s s i o n r e m a i n s t o be r a t h e r a r b i t r a r y . T h e r e f o r e , t h r o u g h o u t t h i s work Monod r e l a t i o n w i l l be used w i t h o u t any c o m p a r a t i v e j u s t i f i c a t i o n . Two o t h e r k i n e t i c e x p r e s s i o n s w i l l be compared w i t h t h a t of Monod :
T i e s s i e r Model u = U { 1 - exp(-C / K ) } max s (35) where K = K / I n 2 s Blackman M o d e l : u f o r max C > u A s — max C / A f o r C < u A s s — max (36)
Many more e x p r e s s i o n s have been r e p o r t e d and c l a i m e d t o be s u p e r i o r under c e r t a i n c a s e s . F o r a c o m p r e h e n s i v e l i s t r e c e n t r e v i e w s c a n be
consulted!6,24,25,31 in g e n e r a l a l l t h e proposed models a r e e m p i r i c a l o r s e m i
-e m p i r i c a l and hav-e mor-e o r l -e s s t h -e sam-e p r o p -e r t i -e s . I t has r -e c -e n t l y b-e-en shown i n l i t e r a t u r e t h a t most of t h e s e models c a n i n f a c t be g e n e r a l i z e d i n t o one f o r m , each model h a v i n g d i f f e r e n t p a r a m e t e r s . ^ 5 , 31
I n v i e w o f t h e s e c o n s i d e r a t i o n s most w o r k e r s f a v o u r Monod r e l a t i o n and do n o t g i v e any f u r t h e r a t t e n t i o n t o o t h e r r e l a t i o n s . I n some c a s e s , however, o t h e r e q u a t i o n s m i g h t be p r e f e r r e d from t h e p o i n t o f m a t h e m a t i c a l c o n v e n i e n c e . F o r i n s t a n c e , t h e u s e o f Blackman k i n e t i c s a l l o w s a n a l y t i c a l s o l u t i o n of f e d - b a t c h m o d e l s , w h i l e t h i s i s n o t p o s s i b l e w i t h Monod k i n e t i c s .
V DISCUSSION OF UNSTRUCTURED MODELS AND MICROBIAL ENERGETICS WITH REFERENCE TO EXPERIMENTAL RESULTS
Batch Cultures
K. •pneumoniae (aerogenes) was c u l t i v a t e d i n b a t c h mode, t o s t u d y t h e k i n e t i c
and e n e r g e t i c b e h a v i o u r . I n o c u l a used were a l w a y s a c t i v e l y g r o w i n g and c o n s e -q u e n t l y no l a g s were e n c o u n t e r e d . A t y p i c a l e x p e r i m e n t i s shown i n F i g . 1 where Cs and Cx p r o f i l e s a r e shown as f u n c t i o n s o f t i m e . A d d i t i o n a l l y t h e
s i m u l a t i o n p r o f i l e s by u s i n g t h e Monod, Blackman and T i e s s i e r models i n combin a t i o combin w i t h t h e l i combin e a r r e l a t i o combin f o r s u b s t r a t e c o combin s u m p t i o combin ( e q . ( 2 6 ) ) , a r e p r e s e combin -t e d ( s o l i d l i n e s ) .
The b r o k e n l i n e becomes a c o n t i n u a t i o n of l i n e 'a' i f t h e l i n e a r r e l a t i o n i s r e p l a c e d by H e r b e r t ' s endogeneous m e t a b o l i s m d e s c r i p t i o n . The p a r a m e t e r s used f o r s i m u a l t i o n s were o b t a i n e d from c o n t i n u o u s and b a t c h c u l t u r e d a t a . P a r a m e t e r s f o r Blackman and T i e s s i e r models were e s t i m a t e d r o u g h l y . Even t h e n , F i g . 1 shows t h a t a l l t h r e e models can d e s c r i b e the e x p e r i m e n t a l o b s e r v a t i o n s i n a more o r l e s s i d e n t i c a l way, p r o v i d e d a l l p a r a m e t e r s a r e e s t i m a t e d w i t h e q u a l c a r e . The endogeneous m e t a b o l i s m m o d e l , i n a d d i t i o n t o p r e d i c t i n g e x a c t l y the same b e h a v i o u r • as M o n o d + l i n e a r r e l a t i o n m o d e l , d e s c r i b e s the e a r l y decay phase w e l l . Thus as f a r as t h i s system i s c o n c e r n e d , H e r b e r t ' s model seems t o p r o v i d e a more comprehensive d e s c r i p t i o n of the r e a l b e h a v i o u r .
H a v i n g shown the r e l a t i v e s i m i l a r i t y of the p r e s e n t e d m o d e l s , Monod + l i n e a r r e l a t i o n model w i l l now be s u b j e c t e d t o a s e n s i t i v i t y a n a l y s i s w i t h r e s p e c t t o i t s p a r a m e t e r s . T h i s i s a f o u r parameter model ( Ks, Pm a x, Y ™x x, mg ). I n F i g s . 2,3,and 4 r e s u l t s of s i m u l a t i o n s c a r r i e d out by c h a n g i n g one parameter a t a t i m e , a r e shown.
Fiq. 2 : Sensitivity of the batch model to variations in u
v J J max
t(min)
J71CUC Fig. 4: Sensitivity of the batch model to variations in ^sx
The parameter v a l u e s were v a r i e d around t h e e s t i m a t e d t r u e v a l u e s . The c a s e of Ks i s d i s c u s s e d i n C h a p t e r 4.From t h e s e p l o t s one c a n c l e a r l y c o n c l u d e t h a t the k i n e t i c d e s c r i p t i o n o f b a t c h growth i n terms o f Cx v s . time p r o f i l e , i s not i n f l u e n c e d by c o n s i d e r a b l e changes i n e n e r g e t i c p a r a m e t e r s , ms and Ym a x . However, t h e k i n e t i c p a r a m e t e r , yma x ls shown t o be o f g r e a t i m p o r t a n c e . As p r e v i o u s l y d i s c u s s e d , t h i s p a r a m e t e r i s t h e most i m p o r t a n t f o r b a t c h g r o w t h as i t r i g i d l y f i x e s t h e growth b e h a v i o u r . A n o t h e r c o n c l u s i o n c a n be drawn f r o m F i g 4 i n r e l a t i o n t o parameter e s t i m a t i o n . That i s , i f Ym a.x i s t o be e s t i m a t e d from b a t c h d a t a , u s e o f Cs p r o f i l e would be more a c c u r a t e .
S i n c e mg has no s i g n i f i c a n t i n f l u e n c e on t h e outcome o f b a t c h s i m u l a t i o n s one can see t h a t t h e s i m p l i f i c a t i o n o f t h e g e n e r a l u n s t r u c t u r e d model ( e q s . ( 2 9 ) ,
(30) t o (31) ) has no s i g n i f i c a n t drawbacks. C o n s e q u e n t l y , i t c a n be c o n c l u d e d t h a t m v a l u e s cannot be d e t e r m i n e d f r o m b a t c h d a t a a c c u r a t e l y . More-o v e r , i f t h e e x p e r i m e n t a l b a t c h d a t a i s p l More-o t t e d More-on l More-o g - l i n e a r a x e s , More-one can see t h a t a f a i r l y good s t r a i g h t l i n e i s o b t a i n e d . That i s even a v e r y s i m p l e e x p r e s s i o n l i k e t h a t g i v e n by eq.(32) i s s u f f i c i e n t t o d e s c r i b e most o f the b a t c h g r o w t h . Thus t h e u s e o f c o m p l i c a t e d e x p r e s s i o n s l i k e t h a t g i v e n by eq.(31) may i n t r o d u c e u n n e c e s s a r y c o m p l i c a t i o n s p a r t i c u l a r l y i n the e s t i m a t i o n of model p a r a m e t e r s , i n w h i c h c a s e i m p l i c i t n o n l i n e a r r e g r e s s i o n i s n e c e s s a r y .
As p o i n t e d o u t p r e v i o u s l y , u n s t r u c t u r e d models do n o t c o n s i d e r changes i n c e l l u l a r c o m p o s i t i o n . Hence they a r e e x p e c t e d t o be s u c c e s s f u l a t s t e a d y s t a t e s o r d u r i n g t r a n s i e n t s t a t e s where t h e c e l l u l a r c o m p o s i t i o n r e m a i n s t h e same. F o r b a t c h growth i t has been shown t h a t f o r t h e c o m p o s i t i o n t o r e m a i n the same, each c o n s t i t u e n t compartment i n t h e c e l l must grow e x p o n e n t i a l l y a t t h e same r a t e as t h e t o t a l b i o m a s s . i . e . , s t e a d y s t a t e w i t h r e s p e c t t o w e i g h t f r a c t i o n s o f t h e components. When t h i s c o n d i t i o n i s s a t i s f i e d g r o w t h i s c a l l e d balanced.32 Thus when growth i s b a l a n c e d i t i s e x p o n e n t i a l t o o . The r e v e r s e i s however, n o t t r u e i . e . , e x p o n e n t i a l growth need n o t be b a l a n c e d .
I n an a t t e m p t t o check t h e v a l i d i t y o f t h i s m a t h e m a t i c a l s t a t e m e n t , macro-m o l e c u l a r c o macro-m p o s i t i o n was d e t e r macro-m i n e d d u r i n g e x p o n e n t i a l g r o w t h . As shown
i n F i g . 5, v i s u a l a n a l y s i s cannot r e j e c t t h e h y p o t h e s i s o f b a l a n c e d g r o w t h . However, t h i s m i g h t be a s i m p l i f i e d p i c t u r e , s i n c e , f o r i n s t a n c e t h e p r o t e i n
c o m p o s i t i o n m i g h t change w h i l e t h e t o t a l m e a s u r a b l e amount r e m a i n s t h e same.
Fig. 5: Macromolecular composition during batch growth.
I n F i g . 6, t h e gas exchange d a t a f o r a d i f f e r e n t b a t c h e x p e r i m e n t i s shown t o g e t h e r w i t h t h e s i m u l a t i o n p r e d i c t i o n s ( s o l i d l i n e s ) . H e r e , a d i s c r e p a n c y e x i s t s between t h e s i m u l a t e d and e x p e r i m e n t a l b e h a v i o u r towards t h e end of t h e e x p o n e n t i a l phase. The e x p e r i m e n t a l d a t a i n d i c a t e s i n c r e a s i n g Yo x and Yc x v a l u e s . No s a t i s f y i n g e x p l a n a t i o n f o r t h i s b e h a v i o u r c o u l d be o f f e r e d . T h i s i s f u r t h e r d i s c u s s e d i n C h a p t e r 4.
Continuous cultures
D u r i n g growth i n c o n t i n u o u s mode a t s t e a d y s t a t e t h e biomass c o m p o s i t i o n remains t h e same. Hence an u n s t r u c t u r e d model has a good chance o f s u c c e s s . However, such a model a l s o assumes t h e b i o t i c c o m p o s i t i o n t o r e m a i n t h e same at d i f f e r e n t d i l u t i o n r a t e s . As shown i n F i g . 7 t h e m a c r o m o l e c u l a r c o m p o s i t i o n , p a r t i c u l a r l y t h e RNA f r a c t i o n , changes as a f u n c t i o n o f t h e growth r a t e . M o r e o v e r , t h e e l e m e n t a l c o m p o s i t i o n o f biomass a l s o changes. S t a t i s t i c a l a n a l y s i s c a r r i e d o u t f o r 9 e l e m e n t a l c o m p o s i t i o n d e t e r m i n a t i o n s r e v e a l e d t h a t v a r i a t i o n i n C, H and N c o n t e n t s a r e s i g n i f i c a n t . Based on t h i s a n a l y s i s the e l e m e n t a l f o r m u l a o f biomass c a n be a p p r o x i m a t e d as a f u n c t i o n o f t h e growth r a t e , by : C H N 0 where b = (7.33 - 0.50 y ) / z 1 b c d c = (12.33 + 3.40 y ) / ( 1 4 z ) d = 26.97/(16z) z = (53.61 - 3.74 y ) / 1 2
For most c a l c u l a t i o n s an average f o r m u l a a t y = 0.5 h r_1 i s t a k e n ; ( yx = 4.16, m o l e c u l a r w e i g h t = 23.16, C H „N 9->oOn , „ , ) •
Fig. 7: Micvomolecular composition as a function of the growth rate, w 3 RNA = 0.11
wV n 3 Protein = 0.71 W]i~0 ' CaJb°Hld-Ta~te = 0.065
The s e n s i t i v i t y a n a l y s i s p r e s e n t e d f o r b a t c h growth model w i l l n o t be r e p e a t e d f o r c o n t i n u o u s c u l t i v a t i o n . The c o n c l u s i o n w i l l s i m p l y be s t a t e d a s : growth b e h a v i o u r d e s c r i b e d by t h e g e n e r a l u n s t r u c t u r e d m o d e l , e x c e p t near t h e wash-out r e g i o n , i s r i g i d l y f i x e d by t h e e n e r g e t i c p a r a m e t e r s , Y ™X X > m s
-Data o b t a i n e d from a c o n t i n u o u s c u l t u r e experiment were f i t t e d by t h e l i n e a r r e l a t i o n , as shown i n F i g s . 8 and 9. These p l o t s i n d i c a t e t h a t g r o w t h i n
c o n t i n u o u s c u l t u r e c a n i n d e e d be d e s c r i b e d by t h e p r e s e n t e d m o d e l . F i g . 8 shows a good s t r a i g h t l i n e f i t . However, s m a l l b u t d i s t i n c t d e v i a t i o n s c a n be seen f o r d a t a a t low growth r a t e s .
Fig. 8: Specific rate of svbstrate consumption as a fuction of the growth rate. Fig. 9: Specific OUR and CPR as fuations of the growth rate.
The e n e r g e t i c p a r a m e t e r s may be o b t a i n e d from c o n t i n u o u s c u l t u r e d a t a by p e r -f o r m i n g l i n e a r r e g r e s s i o n v i a t h e u s e o -f e q u a t i o n : q = y I Ym a X + m (38) s s x s or by p e r f o r m i n g n o n l i n e a r r e g r e s s i o n v i a t h e u s e o f t h e f o l l o w i n g e q u a t i o n : Y = y Ymax / ( y + m Ym a x ) (39) sx sx s sx I f t h e e x p e r i m e n t a l measurements a r e e r r o r f r e e , b o t h methods s h o u l d g i v e e x a c t l y t h e same p a r a m e t e r s . I f t h e r e a r e a s s o c i a t e d e r r o r s t h e s e approaches may r e s u l t i n d i f f e r e n t parameter e s t i m a t e s . An a n a l y s i s o f t h e two p r o c e d u r e s was c a r r i e d o u t and i t was found o u t t h a t i n a l l t h r e e c a s e s ( f o r s u b s t r a t e , oxygen, c a r b o n d i o x i d e d a t a v s . g r o w t h r a t e ) n o n l i n e a r r e g r e s s i o n gave a b e t t e r f i t f o r t h e e x p e r i m e n t a l d a t a . T h i s has been a s s e s s e d by c a l c u l a t i n g the scaled sum of residuals ( s e e T a b l e I ) . The d i f f e r e n c e m i g h t stem from t h e f a c t t h a t qs i s n o t a d i r e c t l y m e a s u r a b l e q u a n t i t y , b u t i s c a l c u l a t e d f r o m qs = y / Ys x . Thus i t may have a d i f f e r e n t e r r o r s t r u c t u r e . M o r e o v e r , i n a qs v s . y p l o t one has y i m p l i c i t l y i n b o t h axes and t h i s may be q u i t e u n d e s i -r a b l e f-rom a m a t h e m a t i c a l p o i n t o f v i e w . The dange-rs o f t h i s e x e -r c i s e i . e . , i n c l u d i n g the same v a r i a b l e i n b o t h axes i s d i s c u s s e d by Himmelblau.33
The e x p e r i m e n t a l d a t a have f i r s t been i n d i s c r i m a n e n t l y p r o c e s s e d by l i n e a r and n o n l i n e a r r e g r e s s i o n p r o c e d u r e s . The r e s u l t s a r e p r e s e n t e d i n T a b l e I .
I f t h e r e s i d u a l s a r e examined, one c a n d e t e c t a t r e n d ( F i g . 8 , 9 ) . T h i s i s p a r -t i c u l a r l y a p p a r e n -t i n -t h e qc v s . y p l o t . Here t h e r e s i d u a l s change t h e i r s i g n
T a b l e I : P a r a m e t e r e s t i m a t e s o b t a i n e d f r o m c o n t i n u o u s and b a t c h c u l t u r e d a t a . C o n t i n u o u s C u l t u r e d a t a : N o n l i n e a r R e g r e s s i o n L i n e a r Regresión A l l d a t a p o i n t s , n=27 / ( Z i e e r r n t i i r n Ymax 0.698 (0.869 - 0.707) 0.710 (0.701 - 0.720) sx Y ™a x 1.583 (1.542 - 1.624) 1.613 (1.570 - 1.659) Ymax 2.391 (2.289 - 2.493) 2.465 (2.367 - 2.570) cx m 3.342 (3.059 - 3 . 6 2 3 ) . 1 0 "2 4.072 (3.290 - 4 . 8 5 2 ) . 1 0- 2 s m 3.740 (3.365 - 4 . 1 1 5 ) . 1 0- 2 4.065 (3.373 - 4 . 7 5 7 ) . 1 0- 2 o m 3.668 (3.177 - 4 . 1 5 9 ) . 1 0_ 2 3.998 (3.321 - 4 . 6 7 5 ) . 1 0- 2 D a t a c o l l e c t e d above , y > 0.1, n=21 Ymax 0.710 (0.699 - 0.721) 0.7* 0.719 (0.706 - 0.732) 1.4* sx Ymax 1.620 (1.563 - 1.677) 2.5* 1.640 (1.570 - 1.701) 3.0* ox Y ™a x 2.513 (2.376 - 2.650) 5.3* 2.561 (2.428 - 2.709) 6.8* ms 4.241 (3.627 - 4 . 8 5 5 ) . 1 0- 2 4.997 (3.846 - 6 . 1 4 8 ) . 1 0- 2 m0 4.320 (3.627 - 5 . 0 1 2 ) . 1 0 ~2 4.609 (3.517 - 5 . 7 0 1 ) . 1 0 ~2 mc 4.530 (3.756 - 5 . 3 0 5 ) . 1 0- 2 4.819 (3.835 - 5 . 8 0 3 ) . 1 0 ~2 Thermodynamic e f f i c i e n c y v e r s u s g r o w t h r a t e d a t a , y> 0.1, n=63 Ymax 0.700 (0.693 - 0.707) mg X 4.146 (3.661 - 4.631). lö~1,B e r r r t t l J B a t c h C u l t u r e d a t a ( a v e r a g e o f t h r e e e x p e r i m e n t s ) : Y ™ £x 0.705 a l l Ymax i n C-eq/C-eq, C-eq/mole
Y™|x 1.544 a ll m i n c-eq/C-eq/hr, mole/C-eq/hr
Y ™a x 2.313
* s c a l e d sum o f r e s i d u a l s x 1 02
f i g u r e s i n p a r e n t h e s e s a r e t h e 95 % c o n f i d e n c e l i m i t s .
i n t h e t i m e sequence o n l y t h r e e t i m e s . Whereas i f t h e y had been r a n d o m l y d i s t -r i b u t e d t h e e x p e c t e d numbe-r o f s i g n change would have been ( n - I ) / 2 = 13. The d i s t i n c t d e v i a t i o n a t l o w y c a n be t h o u g h t t o be due t o t h e r e d u c e d v i a b i l i t y of t h e o r g a n i s m s . S i n c e above y=0.1, v i a b i l i t y i s more t h a n 95 %^4 a n ot h e r s e t
of p a r a m e t e r s were o b t a i n e d o n l y by p r o c e s s i n g d a t a c o l l e c t e d above y=0.1, (n=21, see T a b l e I ) . Note t h a t t h e r e a r e s i g n i f i c a n t d i f f e r e n c e s i n t h e m v a l u e o b t a i n e d by t h e two p r o c e d u r e s . F o r t h e second s e t , t h e goodness o f f i t i s a l s o shown f o r l i n e a r and n o n l i n e a r r e l a t i o n s . F o r d a t a above y=0.1
r e s i d u a l s o f t h e qs v s . y r e l a t i o n changes s i g n 11 t i m e s , qQ v s . y,9 t i m e s
e x c l u s i o n o f d a t a c o l l e c t e d a t v e r y low growth r a t e s , r e s t o r e d t h e l i n e a r i t y o f the d a t a i n r e l a t i o n t o t h e l i n e a r l a w . Thus i t c a n be c o n c l u d e d t h a t above p=0.1 t h e l i n e a r r e l a t i o n i s a r e a s o n a b l e d e s c r i p t i o n o f t h e c o n t i n u -ous c u l t u r e e n e r g e t i c s .
As d e s c r i b e d p r e v i o u s l y Y o r m v a l u e s c a l c u l a t e d from one e x p e r i m e n t a l r e s p o n -se c a n be c o n v e r t e d t o one ba-sed on a n o t h e r , by t h e u s e o f m a c r o s c o p i c methods Thus t h e most o p t i m a l e s t i m a t e o f t h e p a r a m e t e r s , Y and m c a n be o b t a i n e d by c o n s i d e r i n g d a t a o b t a i n e d f r o m a l l r e s p o n s e s , s i m u l t a n e o u s l y i . e . , when e v e r y measurement c o n t r i b u t e s t o t h e r e s u l t . T h i s c a n be done by p r o c e s s i n g I] v s . H d a t a , where r| i s c a l c u l a t e d f r o m Ys x, Yo x and Yc x d a t a (n=63) . The p a r a m e t e r s o b t a i n e d i n t h i s way a r e a l s o g i v e n i n T a b l e I . From t h i s t a b l e i t c a n be seen t h a t t h e 95 % c o n f i d e n c e l i m i t s o f m v a l u e s a r e q u i t e l a r g e when compared w i t h t h o s e o f Ymax v a l u e s . S i n c e t h e s e p a r a m e t e r s a r e d e t e r m i n e d s i m u l t a n e o u s l y , t h e i r e s t i m a t e s c a n be c o r r e l a t e d . A b e t t e r p i c t u r e c a n be o b t a i n e d about t h e a c c u r a c y o f t h e s e p a r a m e t e r s by c a l c u l a t i n g t h e i r a p p r o x i m a t e l o c u s o f t h e j o i n t c o n f i d e n c e l i m i t s ( F i g . 1 0 ) . From t h i s f i g u r e i t c a n be seen t h a t t h e e s t i m a t e s o f Y ™x x and mg a r e s l i g h t l y c o r r e l a t e d ; t h e p r i n c i p a l axes o f t h e e l l i p s e a r e a t an a n g l e t o t h e c o o r d i n a t e a x e s . I t has t o be emphasized t h a t the e s t i m a t e s o f Ymxx and ms may l i e o u t s i d e t h e i r i n d i v i d u a l c o n f i d e n c e l e v e l s . F r o m t h i s d i s c u s s i o n i t c a n be c o n c l u d e d t h a t t h e v a l u e o f Ymgx c a n be d e t e r m i n e d w i t h r e a s o n a b l e c e r t a i n i t y , w h i l e t h a t o f ms can o n l y be d e t e r -mined w i t h a l a r g e u n c e r t a i n i t y . Wide c o n f i d e n c e l e v e l s r e s t r i c t s one t o draw f i r m c o n c l u s i o n s from e x p e r i m e n t a l work c o n c e r n i n g m a i n t e n a n c e m e t a b o l i s m . Such w i d e c o n f i d e n c e l e v e l s may be one o f t h e r e a s o n s f o r t h e w i d e range o f m v a l u e s r e p o r t e d i n l i t e r a t u r e , f o r t h e same o r s i m i l a r s y s t e m ( s ) .
W i t h r e f e r e n c e t o T a b l e I , as f a r as t h e maximal y i e l d s a r e c o n c e r n e d , one c a n a l s o c o n c l u d e t h a t t h e r e i s no s i g n i f i c a n t d i f f e r e n c e between t h e e n e r g e t i c s of m i c r o b i a l growth i n b a t c h and c o n t i n u o u s modes.
Fig. 10: The loons of the joint confidence limits for energetic parameters as determined by the linear relation for substrate consumption; sq.(S&). (for data shewn in Fig.8)
