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A C y b e r n e t i c P e r s p e c t i v e of
Microbial
Growth
DORAISWAMI RAMKRISHNA
Downloaded by SUNY STONY BROOK on October 25, 2014 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch007
Purdue University, School of Chemical Engineering, West Lafayette, IN 47907
What goes by cellular metabolism is an immense class of chemical and physical rate processes within and without the cell marked most strikingly by their diversity and specificity. It forms the basis of the response of microorganisms to their environment, the modelling of which must of necessity suffer some oversimplification if it is to be of any value. In this context, kinetic models of microbial growth within the framework of interaction between lumped biochemical species and the environment have had considerable appeal in the past. However, the subtle facilities which derive from the elaborate internal machinery of the cells pose a challenge that no meager expansion of the kinetic framework will ever meet. We examine here a cybernetic perspective of microbial growth which contends that the net asset of the cell's internal machinery is the facility to make 'rational' (optimal) decisions in responding to its environment, one that seems markedly manifest in the situation of diauxic growth. The growth of microbial cells in the presence of multiple substrates is addressed in this work within a cybernetic framework which lays emphasis on the optmial allocation of existing 'resources' among parallel enzyme-synthesis systems. The work to be presented will discuss the main issues connected with this outlook, what the chief assets of the framework might be and the interpretation of several experimental results within the cybernetic scheme.
0097-6156/83/0207-0161 $06.00/0 © 1983 American Chemical Society
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
Downloaded by SUNY STONY BROOK on October 25, 2014 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch007
162
BIOCHEMICAL ENGINEERING
Mathematical models of v a r y i n g degrees of s o p h i s t i c a t i o n are the main instruments with which q u a n t i t a t i v e understanding has been secured of engineering processes both i n a n a l y s i s and design. In d e a l i n g with m i c r o b i a l systems, an important development i n the past has been of the adoption of the chemical k i n e t i c framework viewing the c e l l mass as lumped biophase (the soc a l l e d non-segregated models) and i n t e r a c t i n g as a whole with i t s environment. Such models have been the b a s i s f o r design and a n a l y s i s of i n d u s t r i a l fermentations. Several i n t e r e s t i n g features of the methodology of chemical r e a c t i o n engineering have found t h e i r way i n t o biochemical engineering. Whether or not p r e c i s e l y q u a n t i t a t i v e understanding f o l l o w s from modelling, an e n t i r e l y u s e f u l aspect of i t has been the e f f i c i e n t assessment of the i m p l i c a t i o n s of s p e c i f i c hypotheses concerning the behavior of a given system thus formulating new r e v e a l i n g experiments. I t i s perhaps i n t h i s sense more than any other that k i n e t i c mode l l i n g has been a v a l u a b l e t o o l i n the i n v e s t i g a t i o n of m i c r o b i a l systems, and i t i s i n t h i s sense too that the ideas to be presented here must be i n t e r p r e t e d . In the adoption of a s t r i c t l y chemical k i n e t i c framework f o r modelling m i c r o b i a l systems, an important d i s t i n c t i o n a r i s e s . Microorganisms a r e not a "dead bag" of biochemical c o n s t i t u e n t s responding to t h e i r environment i n s t r i c t conformity with a k i n e t i c p r e s c r i p t i o n that i s c h a r a c t e r i s t i c of r e a c t i o n systems. (In t h i s connection, i t i s i n t e r e s t i n g to a l l u d e to the remark of F r e d r i c k s o n and Tsuchiya [1] that "... organisms ... are not t i n s o l d i e r s ! " ) . I t i s germane that we quote Demain [ 2 ] i n t h i s regard, "Microorganisms have evolved over the years, developing b e t t e r and b e t t e r mechanisms to prevent overproduction of t h e i r metabolites. Yet we m i c r o b i o l o g i s t s and bioengineers are dedicated to i n c r e a s i n g the i n e f f i c i e n c y of fermentation organisms as we continue to work toward the goal of almost complete conversion of n u t r i e n t i n t o product with as l i t t l e as p o s s i b l e going i n t o the m i c r o b i a l protoplasm (except, of course, i f we are i n the s i n g l e c e l l p r o t e i n b u s i n e s s ) . " F u r t h e r , Demain observes t h a t , " A l l microorganisms must possess r e g u l a t o r y ( c o n t r o l ) mechanisms i n order to s u r v i v e . Very e f f i c i e n t organisms are tightly controlled. In fermentation organisms, c o n t r o l s are l e s s r i g i d but nevertheless present." The i m p l i c a t i o n s of the f o r e going a s s e r t i o n s a r e deep and f a r - r e a c h i n g . The biochemical engineer's o b j e c t i v e i s i n c o n f l i c t with that of the organism and an attempt to ' c o n t r o l the organism without f a m i l i a r i t y with i t s i n t e r n a l c o n t r o l f e a t u r e s could lead one away from p r o j e c t e d optimal goals. ( I t must of course be mentioned here that there e x i s t s e v e r a l instances of dramatic improvements i n the formation of fermentation products through e f f o r t s based on a q u a l i t a t i v e understanding of the r e g u l a t o r y processes w i t h i n the c e l l s . Some of these i n v o l v e genetic v a r i a t i o n s while others do not.) Demain p o i n t s out that c o o r d i n a t i o n of m i c r o b i a l metabolism i s a n e c e s s i t y born out of the tremendous d i v e r s i t y which per1
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
Downloaded by SUNY STONY BROOK on October 25, 2014 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch007
7.
RAMKRISHNA
Cybernetic
View of
Growth
163
vades c e l l u l a r c o n s t i t u t i o n and a c t i v i t y . The genetic c a p a c i t y of a b a c t e r i a l c e l l can accommodate over 1000 enzymes and that only under some s t r i c t s u p e r v i s i o n can the resources of the c e l l be invested f r u g a l l y . Such a c o o r d i n a t i o n i s accomplished c h i e f l y by ( i ) i n d u c t i o n , ( i i ) c a t a b o l i t e r e g u l a t i o n and ( i i i ) feedback r e g u l a t i o n . Induction provides the c e l l with the mechanism to form enzymes r a p i d l y when needed. C a t a b o l i t e r e g u l a t i o n becomes u s e f u l , f o r example, i n the i n h i b i t i o n of formation of c e r t a i n enzymes ( c a t a b o l i t e r e p r e s s i o n ) . Feedback r e g u l a t i o n i n c l u d e s feedback i n h i b i t i o n and feedback r e p r e s s i o n . Feedback i n h i b i t i o n i s d i s t i n g u i s h e d from r e p r e s s i o n i n that the former involves an end product which i n h i b i t s the a c t i o n of an enzyme somewhere "upstream" the pathway, while the l a t t e r implies the prevention of the formation of one o r more upstream enzymes by a d e r i v a t i v e of the end product. The a c t u a l accomplishment of these r e g u l a t o r y processes r e q u i r e s the i n t e r v e n t i o n of an elaborate genetic mechanism. The b i o s y n t h e s i s of a p a r t i c u l a r enzyme i s i t s e l f an elaborate and complex process i n v o l v i n g s e v e r a l c e l l u l a r com ponents. The genetic information f o r any p a r t i c u l a r enzyme i s c a r r i e d i n a s t r e t c h of DNA which i s the ' s t r u c t u r a l gene' f o r that p r o t e i n . The 'pattern' i s t r a n s c r i b e d i n a s t r i p of the messenger RNA that d i c t a t e s the proper sequence of amino acids i n the synthesis of the enzyme. Van Dedem and Moo-Young have made an i n t e r e s t i n g beginning i n t o i n c o r p o r a t i o n of the operon theory of Jacob and Monod i n t o a k i n e t i c model f o r enzyme syntheses [ 3 ] . Indeed even a r e l a t i v e l y simple model leads to many u n i d e n t i f i able k i n e t i c constants. Our o b j e c t i v e i n t h i s work i s to present an approach a t variance from modelling of m i c r o b i a l response based purely on k i n e t i c considerations. We base t h i s approach on the viewpoint that while the d e t a i l e d modelling of r e g u l a t o r y processes (accounting f o r t h e i r underlying genetic mechanisms) i s i n t r a c t ably complicated, i t may be p o s s i b l e to i n t e r p r e t them as being i n s p i r e d by an optimal motive. I t would seem that such f a c i l i t y f o r an optimal response of an organism to i t s environment would be an a c q u i s i t i o n c o n s i s t e n t with the theory of e v o l u t i o n . While t h i s viewpoint i s evident i n the statement of Demain quoted e a r l i e r , i t has been a popular aspect of contemporary b i o l o g y (see f o r example [ 4 ] ) . This approach based on p o s t u l a t i n g the existence of an o p t i m a l i t y c r i t e r i o n i s what we have termed as the 'cybernetic' p e r s p e c t i v e . (Cybernetics, which a r i s e s from the Greek word χ υ β ε ρ ν η τ η Β meaning steersman, has been defined by Wiener as c o n t r o l and communication i n the animal and the machine [5]). The b a s i c merit of the cybernetic approach i s that i t adopts a mathematically simple d e s c r i p t i o n of a complex organism but compensates f o r the o v e r s i m p l i f i c a t i o n by a s s i g n i n g an o p t i mal c o n t r o l motive to i t s response. The i m p l i c a t i o n i s that the elaborate i n t e r n a l machinery of the c e l l provides the organism with the f a c i l i t y to implement the c a l c u l a t e d c o n t r o l p o l i c y .
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
Downloaded by SUNY STONY BROOK on October 25, 2014 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch007
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While a t r u l y d e t a i l e d model would a c t u a l l y be concerned with d e t a i l s of t h i s implementation (even i f i t i s not viewed as such) they are of no concern to the cybernetic model. There i s of course no d i r e c t way of confirming an optimal s t r a t e g y so that i t s a c c e p t a b i l i t y must be based on experimental evidence p e r t a i n ing to i t s i m p l i c a t i o n s . T h i s has been the b a s i s of some c r i t i c ism of the c y b e r n e t i c approach [6], I t i s emphasized here that formulations of optimal p o l i c i e s must be construed merely as i n n o v a t i v e d e s c r i p t i o n ( p r e d i c t i o n ) of observed phenomena r a t h e r than as " u l t i m a t e " explanations. In t h i s connection, i t i s important to recognize that the approach could lead to many new and i n t e r e s t i n g experiments. An example of the cybernetic approach to m i c r o b i a l growth may be found i n the work of Swanson, A r i s F r e d r i c k s o n and Tsuchiya [7,8] , who were s p e c i a l l y concerned with the l a g phase i n s i n g l e s u b s t r a t e systems. I t may be seen that the cybernetic approach i s based on an i n v a r i a n t s t r a t e g y r a t h e r than an i n v a r i a n t k i n e t i c response i m p l i c i t i n the framework of k i n e t i c models. Thus thaapproach envisaged here i s more l i k e l y to d e s c r i b e t r a n s i e n t semi-continuous e x p e r i ments, i n which the c e l l s environment may be v a r i e d at w i l l , t h a n a purely k i n e t i c approach. There are however important cons t r a i n t s only w i t h i n which the framework suggested here may be considered. Since the organisms are deemed to respond based on an i n v a r i a n t optimal s t r a t e g y ( i . e . , the c e l l s never l o s e s i g h t of t h e i r optimal goal while experiencing the v a r y i n g environment) we are not addressing s i t u a t i o n s i n which any a b e r r a t i o n i n the r e g u l a t o r y mechanisms (leading to non-optimal goals) are encountered. Thus only environmental manipulations i n which no genetic changes are brought about i n the organisms can be l e g i t i m a t e l y considered, s i n c e the mutants cannot be expected to r e t a i n the optimal framework of the o r i g i n a l genotype. Having s a i d t h i s , we wish to emphasize the d i s t i n c t i o n between s t r i v i n g f o r an optimal goal and a c t u a l l y accomplishing i t . I f we grant that the f a c i l i t y f o r optimal behavior has been acquired f o r c e r t a i n types of environmental v a r i a t i o n s , then one may expect that as long as the organisms are subjected to the same c l a s s of environmental changes, they would not only s t r i v e f o r an optimal goal but a l s o accomplish i t . On the other hand, i f we subject the microorganisms to environmental changes of a type e n t i r e l y d i f f e r e n t from those with which they can cope i n an optimal way, then the r e s u l t could w e l l be a non-optimal behavior even i f there have occurred no genetic changes. Such a b e r r a t i o n s from optimal beh a v i o r would seem to come w i t h i n the purview of the c y b e r n e t i c approach. In f a c t t h i s i s an important i s s u e which we w i l l address again at a l a t e r stage. A c l a s s i c example i n which the i n t e r n a l r e g u l a t o r y processes of the c e l l s play a very important r o l e i s the phenomenon of d i a u x i c growth discovered by Monod [ 9 ] i n m u l t i p l e s u b s t r a t e systems. In d i a u x i c growth there i s p r e f e r e n t i a l u t i l i z a t i o n of c e r t a i n s u b s t r a t e s over o t h e r s , although each s u b s t r a t e by i t s e l f 1
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
Downloaded by SUNY STONY BROOK on October 25, 2014 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch007
7.
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would have been acceptable to the organism. The s i t u a t i o n appears to f i t the c y b e r n e t i c approach r a t h e r w e l l s i n c e p r e f e r ence f o r a p a r t i c u l a r s u b s t r a t e could w e l l be the r e s u l t of an optimal s t r a t e g y . We quote again from Demain [2]. "The c e l l faces a problem when more than one u t i l i z a b l e growth s u b s t r a t e i s present. Enzymes could be formed to c a t a b o l i z e a l l s u b s t r a t e s , but t h i s would be w a s t e f u l . Instead, enzymes are made which u t i l i z e the best s u b s t r a t e ( u s u a l l y glucose) and only a f t e r exhaustion of the primary s u b s t r a t e are enzymes formed which c a t a b o l i z e the poorer carbon source." That one s u b s t r a t e i s " b e t t e r " than another i s unmistakably implied although a q u a l i f i c a t i o n as to what makes one b e t t e r than the other i s cons p i c u o u s l y absent. THE CYBERNETIC FRAMEWORK.
SOME ISSUES
Viewing the c e l l as a c o n t r o l system r a i s e s s e v e r a l important i s s u e s . We assume that the c o n t r o l system i s much l i k e that conceived f o r c o n t r o l l i n g i n d u s t r i a l systems. F i r s t , an essent i a l part of c o n t r o l of a process i s measurement of one or more process v a r i a b l e s . Thus the transformation of n u t r i e n t m a t e r i a l to protoplasmic mass and a l s o products which are released i n t o the environment provides a spectrum of q u a n t i t i e s f o r "measurement". I f c o n t r o l i s based on sensing concentrations of environmental components p r i o r to (such as n u t r i e n t s ) or a t e a r l y stages of t r a n s f o r m a t i o n , i t comes w i t h i n the category of feedforward c o n t r o l . In c o n s t r a s t i f c o n t r o l i s based on c o r r e c t i o n of performance by measuring v a r i a b l e s a t the downstream end of the transformation process then we have feedback c o n t r o l . I n deed we could have both feedforward and feedback s t r a t e g i e s . Feedback c o n t r o l i s p o s s i b l e , f o r example, i n muscular a c t i o n ( i n higher organisms) to accomplish a p a r t i c u l a r g o a l . In the s i t u a t i o n s of i n t e r e s t to us here one i s i n c l i n e d towards a feedforward s t r a t e g y because an e f f i c i e n t system would gear i t s i n t e r n a l machinery to prepare ahead i n a complex process of breakdown of n u t r i e n t s . I t i s recognized however that the p o s s i b i l i t y of feedback strategies cannot be e n t i r e l y overlooked. The next i s s u e i s that of the c o n t r o l o b j e c t i v e . Prior d i s c u s s i o n s i n t h i s paper and r e f e r e n c e to other works such as Demain [2] g r a v i t a t e towards the concept of maximizing biomass p r o d u c t i v i t y . Again t h i s suggestion i s more of an i n t e r e s t i n g p o s s i b i l i t y than with a view to exclude others that may appear more a t t r a c t i v e i n subsequent stages of t h i s work. Suppose we grant that the c e l l ' s goal i s i n maximizing c e l l mass. Two b a s i c questions a r i s e . I s the d e s i r e d maximization of the c e l l mass p r o d u c t i v i t y at every i n s t a n t , or over a f i n i t e time i n t e r v a l i n which case the average p r o d u c t i v i t y i s maximized? We may r e f e r to the f i r s t case as a "short term" (or instantaneous) perspective. In the second case, we have a "long term" p e r s p e c t i v e . In examining the r e l a t i v e p l a u s i b i l i t y of the two schemes we must
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
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bear i n mind that growth represents a complex sequence of react i o n s o c c u r r i n g n e c e s s a r i l y over a f i n i t e time i n t e r v a l . Thus an organism e v o l v i n g i n c e r t a i n patterns of environmental v a r i a t i o n s could conceivably account f o r t h i s i n formulating i t s optimal p o l i c y much i n the s p i r i t of "saving ( i t s resources) f o r a r a i n y day". This argument i s i n favor of a 'long term object i v e . Of course the organism's p r o j e c t i o n of i t s f u t u r e could prove to be i n e r r o r (since the experimenter can vary i t s e n v i r onment at w i l l ) the r e s u l t of which would then be a non-optimal response. D h u r j a t i [10] has i n v e s t i g a t e d a 'long term' model f o r d i a u x i c growth which w i l l be discussed subsequently. A 'short term' p e r s p e c t i v e i s r e f l e c t i v e of a d e c i s i o n to make the best use of the environment at the i n s t a n t (without concern f o r the f u t u r e ) . Here one has the option of assuming that there may or may not be a time delay i n the implementation of that p o l i c y . I f a small time delay e x i s t s , (a long time delay does not r e f l e c t an e f f i c i e n t organism) i t may not g r e a t l y a f f e c t the organism's response i n slowly v a r y i n g environment. However, f o r r a p i d environmental changes, s u b s t a n t i a l l y nonoptimal behavior may be expected from such models. Another i n t e r e s t i n g s i t u a t i o n to i n v e s t i g a t e here i s the e f f e c t of how l i m i t a t i o n s on c o n t r o l v a r i a b l e s might a f f e c t a p o l i c y based on a short term p e r s p e c t i v e . To make t h i s c l e a r e r , suppose the c o n t r o l o b j e c t i v e i s to be implemented by a l l o c a t i n g e x i s t i n g resources of the c e l l . Suppose f u r t h e r that resources are cons t a n t l y r e p l e n i s h e d by growth processes. A short term perspect i v e makes no a n t i c i p a t i o n about subsequent replenishments so that i t i s p o s s i b l e to conceive of s i t u a t i o n s where, i n the i n t e r e s t of an immediate optimal o b j e c t i v e , an impending c r i s i s (such as inadequate resources to u t i l i z e a l a t e r f a v o r a b l e environment) i s not accounted f o r . Another important i s s u e i s that of the proper c o n t r o l v a r i a b l e s . We have j u s t a l l u d e d to the p o s s i b i l i t y of viewing m i c r o b i a l response as a resource a l l o c a t i o n problem. The r e sources might, f o r example, c o n s t i t u t e only a c r i t i c a l resource such as ATP that i s the primary source of energy f o r metabolic purposes. I t i s d i f f i c u l t to be any more s p e c i f i c about t h i s i s s u e at t h i s i n c e p t i v e stage of our work. In what f o l l o w s we w i l l present two models, one based on a 'long term' p e r s p e c t i v e and another based on a 'short term' pers p e c t i v e both addressing the phenomenon of d i a u x i c growth i n m u l t i p l e substrate systems.
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1
Growth i n M u l t i p l e Substrate Systems Monod's pioneering work on d i a u x i c growth [ 9 ] i s an i n t e r e s t i n g s i t u a t i o n i n which i n t e r n a l r e g u l a t o r y processes appear to play an important r o l e . Here, the c e l l s are confronted with a mixture of two substrates ( g e n e r a l l y sugars) such as, say g l u cose and xylose. While growth could occur f o l l o w i n g a normal
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
Downloaded by SUNY STONY BROOK on October 25, 2014 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch007
7.
RAMKRISHNA
Cybernetic
View of
167
Growth
lag phase with e i t h e r of the substrates i n the absence of the other, i n the mixed substrate environment the c e l l s show an ex c l u s i v e preference f o r glucose u n t i l a l l of i t i s v i r t u a l l y ex hausted before xylose i s u t i l i z e d . This preference f o r glucose could be i n t e r p r e t e d as the r e s u l t of an optimal d e c i s i o n . D h u r j a t i [10] has considered a cybernetic model based on a long term p e r s p e c t i v e of m i c r o b i a l response to a m u l t i s u b s t r a t e en vironment. Thus he proposes that the c e l l s i n the inoculum, on exposure to a mixture of two substrates (such as glucose) and (such as x y l o s e ) , a t some i n i t i a l i n s t a n t (t=0) must decide on a program of u t i l i z a t i o n of the two substrates such that a l l the obtainable biomass i s r e a l i z e d i n the s h o r t e s t time. The optimal o b j e c t i v e i s t h e r e f o r e the maximization of the average biomass p r o d u c t i v i t y over the e n t i r e period of growth. The t a c i t i m p l i c a t i o n here i s that a batch growth i s presumed a t the outset. T h i s " p r o j e c t i o n of the f u t u r e " i s e s s e n t i a l to a long term p o l i c y . Thus the i n t e r n a l machinery of the c e l l i s presumed to commit i t s e l f to a p o l i c y of a l l o c a t i o n of i t s r e sources based on the assumption that no subsequent replenishment of e i t h e r substrate would be a v a i l a b l e . Of course a c t u a l ex perience could be otherwise s i n c e the experimenter could vary the environment a t w i l l . The question a r i s e s then as to how the model would view the growth of the c u l t u r e i n an environment thus manipulated. Could the c e l l s not review the s i t u a t i o n a t any i n s t a n t when there i s an e x t e r n a l manipulation and make a r e a l l o c a t i o n of i t s resources ( i . e . , r e v i s e i t s c o n t r o l p o l i c y ) ? If so could t h i s occur c o n t i n u a l l y r e g a r d l e s s of the speed with which the substrate environment i s varied? D h u r j a t i s model i s predicated on a long term p o l i c y which i s provoked only when a step change i n s u b s t r a t e concentration occurs. While there a r e some conceptual d i f f i c u l t i e s with t h i s model connected with the questions j u s t r a i s e d , i t does lead to the d i a u x i c growth curve. To h i g h l i g h t on the model, the c e l l growth process i s viewed as comprising the growth r e a c t i o n Β + S + E + 2B i = 1, 2,... where Β i s the biomass, S i i s the i substrate, E i i s consider ed as a 'key enzyme' i n the uptake of the i s u b s t r a t e . The enzyme E i i s synthesized a t a r a t e dependent on the a l l o c a t i o n of i t s resources and D h u r j a t i ' s [11] equation i s reproduced here de. - r - = ot.Ru. - B.e. (1) dt i l i l where e . i s the enzyme concentration, a i and $^ are r a t e con s t a n t s , R i s a f i x e d r a t e of t o t a l resource a v a i l a b i l i t y the f r a c t i o n of which a l l o c a t e d f o r enzyme E i i s given by U i . The resource a v a i l a b i l i t y r a t e R i s assumed to be f i x e d here a l though i n a more elaborate model i t i s p o s s i b l e to i n c l u d e r e plenishment of resources during growth. Note that E q . ( l ) im p l i e s a maximum l e v e l of enzyme concentration because of the first order breakdown r a t e . The substrate consumption r a t e i s 1
±
±
t
h
t
n
1
#
#
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
BIOCHEMICAL ENGINEERING
168
assumed to i n v o l v e the enzyme c o n c e n t r a t i o n i n the f o l l o w i n g man ner ds. V e.s dt^
=
"
b
K~+~s7 1
i - l .
2.·..
(2)
1
while growth on the substrates i s w r i t t e n as
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J = - b E Y . ^
(3)
dt i ι dt The biomass, whose concentration i s represented by b, i s viewed as comprising key enzymes and the r e s t of the biomass. For the sake of s i m p l i c i t y no m a t e r i a l resources have been e x p l i c i t l y i n cluded i n the biomass f o r the present. D h u r j a t i [10] based the optimal a l l o c a t i o n of resources on minimizing the time r e q u i r e d f o r r e a l i z i n g a l l of the biomass from the d i f f e r e n t s u b s t r a t e s . Thus the c o n t r o l o b j e c t i v e i s given by Min t Zu. = 1 (4) i î where t f i s the time required f o r the substrate l e v e l s to drop to some preassigned values (since a r b i t r a r i l y small values would lead to a r b i t r a r i l y l a r g e times mathematically). This completes D h u r j a t i s model. The mathematical techniaue of computing the optimal a l l o c a t i o n r a t e s to accomplish the object i v e (4) i s contained i n the well-known minimum p r i n c i p l e of Pontryagin [11] which w i l l not be discussed here. The l i n e a r f u n c t i o n a l i t y of the c o n t r o l v a r i a b l e s u^ leads to what i s r e f e r red to as a "bang-bang" p o l i c y which i n the present context imp l i e s e x c l u s i v e u t i l i z a t i o n of one of the s u b s t r a t e s . ( D h u r j a t i [10] has i n v e s t i g a t e d the p o s s i b i l i t y of s i n g u l a r c o n t r o l which could accommodate simultaneous consumption of d i f f e r e n t subs t r a t e s and f i n d s the c o n s t r a i n t s to be i m p l a u s i b l e ) . The d i a u x i c growth s i t u a t i o n i s thus described by t h i s model. D h u r j a t i was able to adapt model constants to d e s c r i b e growth data obtained by him on K l e b s i e l l a pneumoniae grown on a mixture of glucose and xylose. The preference f o r glucose with a higher growth r a t e on i t i s c o n s i s t e n t with the i n t e r p r e t a t i o n s of the model. The r e s u l t i s d i s p l a y e d i n F i g u r e 1. The question a r i s e s as to whether the organism i n f a c t consumes both glucose and xylose a f t e r the former has dropped to s u f f i c i e n t l y low l e v e l s . While t h i s i s not r e a d i l y v e r i f i a b l e because of the d i f f i c u l t i e s inherent i n measuring small d i f f e r e n c e s i n small substrate conc e n t r a t i o n s some a s s e r t i o n s can be made i n regard to D h u r j a t i s model. The l i n e a r i t y i n the c o n t r o l v a r i a b l e w i l l permit only e x c l u s i v e u t i l i z a t i o n of one substrate or the other. I f f o r example 'more p r e f e r r e d * s u b s t r a t e , say drops to s u f f i c i e n t l y low l e v e l s , the preference would switch to the 'less p r e f e r r e d ' substrate ( S ) a f t e r which e x c l u s i v e u t i l i z a t i o n of S w i l l f o l l o w u n t i l again i t i s p r o f i t a b l e to switch to Si and so on. This d i d not occur i n the c a l c u l a t i o n s presented because of the high discrepancy between growth r a t e s on the two substrates and f
l
1
1
1
2
2
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
RAMKRISHNA
Cybernetic
View of
Growth
169
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7.
Figure 1. The diauxic growth curve predicted by Dhurjati's model for Klebsiella pneumoniae grown on a glucose-xylose mixture. Dashed lines, model; solid line, experimental.
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
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170
BIOCHEMICAL ENGINEERING
the preassigned l e v e l s of substrate concentrations at t f . Undoubtedly, f o r s u b s t r a t e p a i r s on which the d i f f e r e n c e i n the growth r a t e i s not very great, the model would p r e d i c t a l t e r n a t e switching between substrates u n t i l t f i s reached. Unless nonl i n e a r r e l a t i o n s h i p s are entertained i n the c o n t r o l v a r i a b l e s one i s constrained to t h i s s o r t of p r e d i c t i o n . I t w i l l be of i n t e r e s t to i n v e s t i g a t e experimentally whether such a l t e r n a t e switching i s i n f a c t encountered. I t i s i n t h i s connection that the measurement of d i s s o l v e d oxygen becomes a very i n t e r e s t i n g diagnostic tool. For example, a change i n p o l i c y (such as switching substrates) i s r e a d i l y detected by a r e l a t i v e l y abrupt change i n d i s s o l v e d oxygen while no corresponding change i s l i k e l y to be apparent i n e i t h e r the biomass or the substrate concentrations. In Figure 2, are presented some r e s u l t s from D h u r j a t i s experiments. The organism's growth on xylose i s i n terrupted by v a r y i n g amounts of glucose at d i f f e r e n t i n s t a n t s of time. The d i s s o l v e d oxygen shows an abrupt drop i n oxygen conc e n t r a t i o n at the i n s t a n t of glucose a d d i t i o n s i n c e the growth r a t e i s higher on glucose and c a l l s f o r increased oxygen consumption. The oxygen concentration shoots up again when the added glucose i s exhausted. The concentration to which t h i s upshoot occurs depends on time elapsed f o l l o w i n g glucose a d d i t i o n before the the organism i s ready to r e v e r t to xylose again. If t h i s time i s short, growth resumes on xylose at the same l e v e l at which i t was i n t e r r u p t e d . I f the time span i s longer the r e sumption occurs at a somewhat reduced l e v e l p o s s i b l y i n d i c a t i n g that some breakdown of the key enzyme f o r xylose may have occured i n the i n t e r i m . There appears to be l i t t l e doubt that the measurement of d i s s o l v e d oxygen w i l l play a key r o l e i n the diagnosis of switching p o l i c i e s of the c e l l ' s i n t e r n a l machinery. Kompala [12] has considered a model based on a short term p e r s p e c t i v e f o r the growth s i t u a t i o n with which D h u r j a t i d e a l t . This model adopts the viewpoint that the c e l l o p t i m a l l y a l l o c a t e s i t s resources at every i n s t a n t . Thus the s t a t e of the c e l l and i t s environment r e s u l t s i n an immediate resource a l l o c a t i o n with the o b j e c t i v e of maximizing the a c c e l e r a t i o n ( or minimizing the d e c e l e r a t i o n ) i n growth r a t e . The i m p l i c a t i o n i s that the c e l l i s ever a l e r t to environmental changes and always does the best. Some p o s s i b l e o b j e c t i o n s may a r i s e . F i r s t , i t i s not c l e a r , whether i t i s reasonable to expect that regardless of the r a t e at which environmental changes occur, the c e l l s would o p t i m a l l y gear i t s enzyme synthesis r a t e unhampered by any metabolic i n e r t i a ; f o r example, could c o n t r o l a c t i o n taken at a p a r t i c u l a r i n s t a n t r e s u l t i n an instantaneous change i n enzyme synthesis rate? An attempt to cure t h i s would i n essence be a renewed quest f o r a 'long term' model. Second, with the imposition of a proper penalty f o r d e c i s i o n , i t does not n e c e s s a r i l y f o l l o w that what i s best f o r each i n s t a n t i s best on the whole, an i s s u e that has been r a i s e d before. Kompala's model constants do adapt to produce the d i a u x i c curve. His model equations are v i r t u a l l y 1
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
RAMKRISHNA
Cybernetic
View of
Growth
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7.
TIME (HOURS) Figure 2. Dissolved oxygen curve for Klebsiella pneumoniae grown on xylose with periodic addition of glucose. G, 250 mg/mL glucose; X, 250 mg/mL xylose (t = 0, 50 mLX).
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
171
172 the sis
BIOCHEMICAL ENGINEERING same a s t h o s e and breakdown
o f D h u r j a t i [10] e x c e p t f o r t h e e n z y m e s y n t h e r a t e s i n Eq.(1) w h i c h must be r e p l a c e d by de. α!s.
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dT
-
φ τ γ ι
"
6
e
(
i i
5
)
At any i n s t a n t t , the s t a t e o f t h e biomass i s determined by t h e c o n c e n t r a t i o n s b , a n d {e^} w h i l e t h a t o f t h e e n v i r o n m e n t b y { s ^ } . The growth r a t e a t t h i s i n s t a n t i s d e t e r m i n e d by t h e above v a r i ables. However, t h e o r g a n i s m g e a r s t h e enzyme s y n t h e s i s r a t e a t t h a t i n s t a n t t o maximize d^b/dt by m a n i p u l a t i n g the c o n t r o l v a r i a b l e s {u^}. Because u ^ occurs l i n e a r l y i n Eq.(5) t h e o p t i m a l decision is readily arrived at. By l e t t i n g V.a's. 2
j
we
find
=
{ i :
the optimal policy
m a x
(K'Vs.)
2 }
to be the "bang-bang"
( 6 )
result
"ι =
<
7)
This remarkably simple r e s u l t as against the use of the Pontryagin p r i n c i p l e r e q u i r e d f o r D h u r j a t i ' s model i s indeed a great asset f o r t h i s model. ( P a r a l l e l to the p o s s i b i l i t y of s i n g u l a r s o l u t i o n s i n D h u r j a t i ' s f o r m u l a t i o n , one needs to i n v e s t i g a t e s i m u l t a n e o u s u t i l i z a t i o n h e r e w h e n Vjxl^s^ /^ + s^ versus intersect for different i ) . F i g u r e 3 shows t h e r e s u l t s o f c a l c u l a t i o n s f o r b a t c h g r o w t h w i t h two s u b s t r a t e s . The s e q u e n t i a l u t i l i z a t i o n of t h e sub s t r a t e s c h a r a c t e r i s t i c of the d i a u x i c growth s i t u a t i o n i s r e produced by the model. T h i s by i t s e l f i s no more t h a n e s t a b l i s h i n g t h a t t h e model p r e m i s e s a r e n o t i m p l a u s i b l e and i s thus more a b e g i n n i n g t h a n t h e e n d . I n F i g u r e 4, t h e m o d e l ' s perspective of an experiment i n w h i c h t h e growth on t h e ' l e s s preferred' s u b s t r a t e i s i n t e r r u p t e d b y a d d i t i o n o f t h e more p r e f e r r e d s u b strate. The immediate s w i t c h t o glucose observed i n D h u r j a t i ' s e x p e r i m e n t s i s accommodated b y t h i s m o d e l . I f i t i s agreed that a model based on a long term p e r s p e c t i v e such as that of D h u r j a t i w i l l ignore environmental changes other than those on which t h e o p t i m a l p o l i c y was b a s e d , t h e n t h e p e r t u r b e d b a t c h e x p e r i m e n t j u s t r e f e r r e d to cannot be described by D h u r j a t i ' s model. In t h i s sense, Kompala's model i s an improvement on D h u r j a t i ' s model. 2
2
N e i t h e r o f t h e above models e x p l i c i t l y i n c l u d e t h e o r i g i n a l c l a i m that the resources a r e a part of the biomass and t h e i r a v a i l a b i l i t y may t h u s b e c o n s t r a i n e d . This feature i s readily b u i l t into either model. F o r e x a m p l e , we may d e n o t e t h e r e sources by R, t h e i r c o n c e n t r a t i o n by r , and p o s t u l a t e that t h e i r formation occurs by growth, and consumption by t h e i r a l l o c a t i o n f o r enzyme s y n t h e s i s . A d m i t t e d l y t h i s i s somewhat o f a n a r r o w v i e w o f how r e s o u r c e s may b e u s e d i n m e t a b o l i c p r o c e s s e s b u t t h i s s i m p l i f i c a t i o n i s v i r t u a l l y a t t h e same l e v e l a s t h a t i n h e r e n t i n the lumped k i n e t i c models o f t h e p a s t . The d i f f e r e n t i a l equation f o r r may b e w r i t t e n a s
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
7.
RAMKRISHNA
Cybernetic
View of
173
Growth
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UNDISTURBED BATCH EXPERIMENT
0.000
Figure 3.
1.000
2.000
TIME
3.000
4.000
The diauxic growth curve predicted by Kompala's short term model. AG (glucose) = 90.0; AX (xylose) = 60.0.
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
174
BIOCHEMICAL ENGINEERING
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PERTURBED BATCH EXPERIMENT
0.000
Figure 4.
1.000
2.000
TIME
3.000
4.000
Growth on xylose interrupted by glucose addition as predicted by Kompala. AG = 90.0; AX = 60.0.
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
7.
RAMKRISHNA
β
Cybernetic
View of
175
Growth
where ^Λ ^9 i ) i s the r a t e of consumption of resources f o r the synthesis of i enzyme E^; i t i s r e l a t e d to the r a t e of synthesis of E-^ to w i t h i n a s t o i c h i o m e t r i c m u l t i p l e . In Eq.(8) γ i s a s t o i c h i o m e t r i c constant. In p a r t i c u l a r i t i s of i n t e r e s t to note that Σ u. < 1 (9) i u
r
t
n
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1
where the i n e q u a l i t y s i g n i s to account f o r the p o s s i b i l i t y that not a l l of the a v a i l a b l e resources need n e c e s s a r i l y be invested at any i n s t a n t . A model based on a long term p e r s p e c t i v e could i n f a c t c a l l f o r a p o l i c y of u n d e r u t i l i z a t i o n of resources at a given i n s t a n t i f a subsequent stage c a l l s f o r increased r e sources. Propose, f o r example, that a f
i
(
s
i ' i
with Eq.(5) modified de. dt
U
r
)
by a.
=
6
ί(Κ| +
S i
V )
(κ. + u.r)
( 1 0 )
u r
S i
(iq +
S
I i
±
S i
) (κ
±
+ u.r)
- 6!e. i i M
c
(11)
where the δ.^ i n Eq.(lO) represents a r e a d i l y i n t e r p r e t e d s t o i chiometric constant. In E q . ( l l ) i s a M i c h a e l i s constant. The n o n l i n e a r i t y with respect to u^ i n E q . ( l l ) i s to be noted p a r t i c u l a r l y , which may e l i m i n a t e the e x c l u s i v e bang-bang optima l i t y r e s u l t . Thus simultaneous u t i l i z a t i o n of substrates could a l s o be p r e d i c t e d by such models. I f ample resources are pre sent, then the above Michaelis-Menten k i n e t i c expression i n Eq. (11) prevents o v e r - a l l o c a t i o n f o r any p a r t i c u l a r enzyme. Whether the modelling be done based on a short or long term p e r s p e c t i v e , the issue r a i s e d e a r l i e r i n regard to provoking non-optimal responses by s u i t a b l e environmental v a r i a t i o n s i s an important one. The short term models would not n e c e s s a r i l y pre d i c t , f o r example, the maximum average p r o d u c t i v i t y of biomass over the e n t i r e period of growth. S i m i l a r l y the long term models would not p r e d i c t the maximum average p r o d u c t i v i t y i f changes o c c u r r i n g i n the environment are other than those which were accounted f o r i n the o p t i m i z a t i o n . Thus an i n t e r e s t i n g method of e v a l u a t i n g the responses p r e d i c t e d by the models i s to compare them with those i n which " p e r f e c t optimal behavior" i s considered. Obviously p e r f e c t optimal behavior may be p r e d i c t e d by using the Pontryagin p r i n c i p l e on the s t a t e equations i n which the imposed environmental v a r i a t i o n s are b u i l t i n . One need hardly point out that the s u p e r p o s i t i o n of the a c t u a l m i c r o b i a l response r e l a t i v e to that of the model and that based on p e r f e c t o p t i m a l i t y would then be of the most v i t a l i n t e r e s t .
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
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Concluding Remarks Our attempt here has been to f i n d a mathematical framework for c e r t a i n well-expressed ideas concerning m i c r o b i a l behavior i n regard to the r o l e of i n t e r n a l r e g u l a t o r y processes. This framework e s s e n t i a l l y draws on the formulation of k i n e t i c models i n that biomass i s lumped i n t o a small number of component masses, and growth being regarded as increase i n t h e i r q u a n t i t i e s by i n t e r a c t i o n with n u t r i e n t s . The component masses i n c l u d e key enzymes whose s y n t h e s i s i s c o n t r o l l e d by optimal a l l o c a t i o n of c e r t a i n c r i t i c a l resources. Various important i s s u e s have been r a i s e d connected with t h i s c y b e r n e t i c p e r s p e c t i v e . The main m o t i v a t i o n i s to evaluate the consequences of hypotheses concern ing i n t e r n a l r e g u l a t o r y processes, and to f i n d experimental s i t u a t i o n s i n which they manifest i n the most unambiguous ways. It i s not e s s e n t i a l that t h i s p u r s u i t be i n consonance with pre c i s e q u a n t i t a t i v e f i t s of model p r e d i c t i o n s with experimental data. Rather the f u n c t i o n of such model b u i l d i n g i s simply to develop b e t t e r understanding of r e a l systems. Unfortunately, i t i s not o f t e n that t h i s aspect of modelling i s perceived as such.
Ackrawledgnents The work presented here owes much to the p a r t i c i p a t i o n of graduate students Prasad D h u r j a t i and Dhinakar Kompala, and to my colleagues Michael C. F l i c k i n g e r and George T. Tsao, and i s the subject of a f u t u r e j o i n t p u b l i c a t i o n under p r e p a r a t i o n . P a r t i a l support by the N a t i o n a l Science Foundation under Grant No. Eng 7820964 i s g r a t e f u l l y acknowledged.
Nomenclature Β b Ε e f Κ Κ» R r S s t u V
Biomass Concentration of biomass Key enzyme Concentration of key enzyme Rate of comsumption of resources for s y n t h e s i s of key enzyme. M i c h a e l i s constant f o r growth M i c h a e l i s constant of enzyme syn thesis T o t a l resource a b a i l a b i l i t y r a t e Concentration of l i m i t i n g resource Substrate Concentration of s u b s t r a t e Time Fractional allocation Rate constant i n growth
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
7.
RAMKRISHNA
Cybernetic
View of
Growth
Yield Greek
symbols α,a β Ύ
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1
177
coefficient
Rate constants i n enzyme s y n t h e s i s Rate constant f o r enzyme breakdown S t o i c h i o m e t r i c constant f o r r e plenishment of resources by growth S t o i c h i o m e t r i c constant f o r con sumption of resources f o r enzyme synthesis Kroenecker d e l t a equals u n i t y when i=j and vanishes otherwise M i c h a e l i s constant f o r consumption of resources f o r enzyme s y n t h e s i s
Subscripts Refers to i substrate Refers to f i n a l s t a t e of growth when s u b s t r a t e s drop to p r e a s s i g n ed l e v e l s
Literature Cited 1. Fredrickson, A.G., and H.M. Tsuchiya, in "Chemical Reactor Theory," E. L. Lapidus and N.R. Amundson, PrensticeHall, N . J . , 1975. 2. Demain, A . L . , Adv. Biochem. Eng. Ed. T.K. Ghose, A. Fiechter; N. Blakebrough Springer Verlag, Heidelberg, 1971. 3. Van Dedem, G., and M. Moo-Young, Biotech, and Bioeng., (17), 927, 1975. 4. Rosen, R., "Foundations of Mathematical Biology," Ed. R. Rosen, Academic Press, Vol II, 1972. 5. Wiener, Ν., "Cybernetics or Control and Communication in the Animal and the Machine," M.I.T. Press, Mass. 1975. 6. Oster, G. and E.O. Wilson, Monographs in Population Biology 12, ch. 8. "A critique of optimization theory in evolutionary biology," Princeton U. Press, 1970. 7. Swanson, C.H., R. Aris, A.G. Fredrickson and H.M. Tsuchiya, J. Theor. Biol., (12), 228, 1966. 8. Tsuchiya, H.M., Fredrickson, A.G., and Aris, R., Adv. Chem. Eng., (6), 125, 1966. 9. Monod, J., "Recherches sur la croissance des cultures bacteriennes," Herman & Cie. Paris, 1942.
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
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BIOCHEMICAL ENGINEERING
10. Dhurjati, P., Ph.D. Dissertation, Purdue University, West Lafayette, Indiana, 1982. 11. Pontyagin, L . S . , V.G. Boltyanski, R.V. Gamkrelidze and E.F. Mischenko, "The Mathematical Theory of Optimal Processes," Interscience, Ν.Y., 1962. 12. Kompala, D.S. M.S. Dissertation, Purdue University, West Lafayette, Indiana, 1982. June 29, 1982
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RECEIVED
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
