Mathematical Models of the Growth of Individual ... - ACS Publications

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Mathematical Models of the Growth of Individual Cells

Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 19, 2015 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch005

Tools for Testing Biochemical Mechanisms

M. L. SHULER and M. M. DOMACH Cornell University, School of Chemical Engineering, Ithaca, NY 14853 The rationale for and development of mathematical models for single-cells are reviewed. The potential use of a computer model for Escherichia coli in ascertaining the plausibility of basic biological hypotheses is illustrated with respect to the control of the initiation of DNA synthesis and with respect to ammonium ion assimilation. Mechanisms postulated on the basis of in vitro enzymology can be tested for in vivo compatibility using the model. The transient behavior of single-cells to step-up and step-down in glucose or ammonium ion is shown to result in oscillatory responses and hysterisis. Methods to construct population models from single-cell models are discussed. Population models are important to engineering analysis and in relating data for experiments with large populations to the responses of a "typical cell". Why S i n g l e - C e l l Models? The b a s i c conceptual u n i t i n microbiology i s the s i n g l e c e l l . Although m i c r o b i o l o g i s t s and biochemists work with l a r g e populat i o n s of c e l l s , the goal i s g e n e r a l l y to understand the behavior of a " t y p i c a l " c e l l . There i s very l i t t l e d i r e c t data on the growth of i n d i v i d u a l c e l l s - and what data e x i s t s i s o f t e n of* questionable value because the c o n d i t i o n s r e q u i r e d f o r observat i o n s may lead to "unnatural responses". Thus, the behavior o f the " t y p i c a l " c e l l must be i n f e r r e d from the aggregated behavior of the t o t a l p o p u l a t i o n . B i o l o g i s t s u s u a l l y are i n t e r e s t e d i n a s i n g l e aspect of c e l l growth (e.g. p r o t e i n s y n t h e s i s ) . Cells c o n t a i n a complex, n o n l i n e a r , h i g h l y regulated s e r i e s of chemical r e a c t i o n s . Human l o g i c c o n s i s t s of a few l i n e a r s t e p s . I t i s e s s e n t i a l l y impossible f o r the unaided mind to i n t e r r e l a t e p r o t e i n s y n t h e s i s , DNA r e p l i c a t i o n , n u t r i e n t t r a n s p o r t , e t c . i n t o a coherent conceptual model of a c e l l . 0 0 9 7 - 6 1 5 6 / 8 3 / 0 2 0 7 - 0 0 9 3 $ l 1.50/0 © 1983 American Chemical Society

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


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Computer models of s i n g l e c e l l s act as an a i d In b u i l d i n g such conceptual models. Because such models make q u a n t i t a t i v e p r e d i c t i o n s - p r e d i c t i o n s which are experimentally v e r i f i a b l e they serve as good v e h i c l e s to r e v e a l e r r o r s i n e i t h e r b a s i c mechanisms or the manner i n which they are i n t e g r a t e d i n the conceptual model. Population models can be constructed from ensemb l e s of s i n g l e - c e l l models. Such models may be u s e f u l a i d s i n r e l a t i n g the behavior of whole populations to biochemical mechanisms w i t h i n a t y p i c a l c e l l . S i n g l e - c e l l models are p a r t i c u l a r l y w e l l s u i t e d to the conceptual and experimental needs of b i o l o gists. S i n g l e - c e l l models are a l s o of importance to biochemical engineers. The motivation i s d i f f e r e n t from b i o l o g i s t s . Engineers are i n t e r e s t e d i n manipulating a population of c e l l s ; such manipu l a t i o n i s f a c i l i t a t e d by mathematical models that can p r e d i c t the response of the population to perturbations i n the e x t e r n a l e n v i ronment. H i s t o r i c a l l y , the development of mathematical modeling i n biosystems has sprung from a need to model populations of cells. Population models have inherent l i m i t a t i o n s - l i m i t a t i o n s that can be circumvented by using ensembles of s i n g l e - c e l l models. Population Models. To understand why s i n g l e - c e l l models are u s e f u l i t i s important to review previous attempts at modeling populations. The h i s t o r y and philosophy of modeling as w e l l as the v i r t u e s and f a u l t s inherent i n previous models has been w e l l described i n the l i t e r a t u r e . Among these a r t i c l e s are those by Tsuchiya, F r e d r i c k s o n , & A r i s (1); P a i n t e r & Marr (2); Van Uden (3); G a r f i n k e l , eit a l . (4); F r e d r i c k s o n , Megee, & Tsuchiya (5); N y i r i (6); Boyle & Berthouex (7) ( f o r waste t r e a t ment) ; F r e d r i c k s o n (8) ( f o r s t r u c t u r e d models o n l y ) ; and Bailey (9). One method of c l a s s i f y i n g models i n v o l v e s the concept of " s t r u c t u r e " . Structured models have the inherent a b i l i t y to des c r i b e the p h y s i o l o g i c a l s t a t e of a microorganism or a c u l t u r e of such c e l l s . Unless the p h y s i o l o g i c a l s t a t e can be s p e c i f i e d , the dependence of a c u l t u r e on i t s previous h i s t o r y cannot be accounted f o r . Such dependence on h i s t o r y i s known to be important C5). T y p i c a l l y , s t r u c t u r e i s added to a model by c o n s i d e r i n g the c e l l or c u l t u r e to c o n s i s t of two or more components (e.g. n u c l e i c a c i d s and p r o t e i n , etc.) or to d i s t i n g u i s h between separate c e l l s on the b a s i s of s i z e or age. Any model not i n c o r p o r a t i n g the foregoing p r i n c i p l e i s u n s t r u c t u r e d — f o r example, the well-known Monod model. I t has been shown that unstructured models are a p p l i c a b l e only i n b a l anced growth s i t u a t i o n s (10), which, according to Campbell's (11) d e f i n i t i o n , r e q u i r e s that each component of the c u l t u r e be accumulated at the same r a t e . Unstructured models are never general, give very l i t t l e i n s i g h t i n t o c e l l u l a r mechanisms, and cannot be used to describe l a g and d e c l i n e phases i n batch growth, t r a n s i e n t



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response i n chemostats, or growth i n m u l t i s t a g e continuous c u l t u r e . I f the formation of a product i s dependent on c e l l h i s t o r y and i s not growth a s s o c i a t e d , then an u n s t r u c t u r e d model w i l l be u n s u i t a b l e . Unstructured models have been widely used, however, and have been s u c c e s s f u l when a p p l i e d to the commonly o c c u r r i n g balanced growth s i t u a t i o n s of e x p o n e n t i a l phase batch c u l t u r e and steady-state s i n g l e - s t a g e d continuous c u l t u r e . Two of the f i r s t s t r u c t u r e d models proposed were those by Williams (_L2, 13) and Ramkrishna, F r e d r i c k s o n , & Tsuchiya (14). Both models were d e t e r m i n i s t i c and d e a l t with a nonsegregated b i o mass. A nonsegregated model i s one which does not recognize exp l i c i t l y the e x i s t e n c e of i n d i v i d u a l c e l l s . For example, the contents of a fermenter can be thought of as c o n s i s t i n g of two p h a s e s — o n e b i o t i c and the other a b i o t i c . The b i o t i c phase can then be t r e a t e d as a homogeneous e n t i t y . Nonsegregated models, of course, cannot make any p r e d i c t i o n s about c e l l p r o l i f e r a t i o n or the e f f e c t s of c e l l geometry on growth. However, they are f a i r l y easy to handle mathematically, and the o v e r a l l biomass and i t s major components can be determined e x p e r i m e n t a l l y with reasonable accuracy. I t i s obvious to anyone who has m i c r o s c o p i c a l l y examined mic r o b i a l c u l t u r e s that c e l l s are i n d i v i d u a l s and can vary g r e a t l y i n observable p r o p e r t i e s such as c e l l s i z e . Only segregated mode l s can capture t h i s p r o p e r t y . Such models c o n t a i n s t r u c t u r e i n the sense that changes are allowed i n the d i s t r i b u t i o n of s t a t e s among the p o p u l a t i o n i n response to the e x t e r n a l environment. Examples of models u s i n g age as a measure of the index of a c e l l ' s stage i n c l u d e those by Von F o e r s t e r (15); Trucco (16); Yakovlev, et a l . (17); F r e d r i c k s o n & Tsuchiya (18); Kozesnik (19); and Lebowitz & Rubinow (20). C e l l s i z e can a l s o be u t i l i z e d as suggested by Koch & Schaechter (21) and Eakman, et a l . (22). In the above examples only one parameter was used as an index of s t a t e ; i t would be impossible to i n s e r t any i n f o r m a t i o n about biochemical mechanisms i n t o the model. Models which i n c l u d e both s t r u c t u r e and segregation lead to mathematical equations which are extremely d i f f i c u l t to s o l v e even with the a i d of modern high-speed computers. B a i l e y and co-workers (23, 24, 25) have made progress i n circumventing some of these problems. Nonetheless i t appears impossible to prepare p o p u l a t i o n models which would c o n t a i n a h i g h l e v e l of s t r u c t u r e (say more than f i v e components) and s e g r e g a t i o n , and which s t i l l are mathematically tractable. Population models which c o n t a i n both a h i g h - l e v e l of s t r u c ture and segregation are d e s i r a b l e . Such models have the potent i a l to make accurate p r e d i c t i o n s of t r a n s i e n t responses - such p r e d i c t i o n s are important not only f o r process c o n t r o l but may be u s e f u l i n d e s i g n i n g c y c l e d - r e a c t o r s which can have product y i e l d s greater than comparable s t e a d y - s t a t e c u l t u r e s (26, 27). There i s a l s o p r e l i m i n a r y evidence that under some circumstances only a small sub-population of c e l l s produce most of the product (24);



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thus the manipulation of the d i s t r i b u t i o n of sub-populations can be p r e d i c t e d only i f a structured-segregated model i s a v a i l a b l e . Tanner (28) has made persuasive arguments f o r the need f o r more s o p h i s t i c a t e d models t o optimize the design of commercial fermentation processes. He has s t a t e d that commercial fermentat i o n processes have not been s i g n i f i c a n t l y optimized "because there a r e no simple, g e n e r a l , and accurate mathematical models f o r p r e d i c t i n g the c a s u a l e f f e c t s of c o n t r o l v a r i a b l e changes". Furt h e r , he claims that "the establishment of j u s t one of these prof i l e s [pH, temperature, n u t r i e n t , e t c . ] t o optimize a batch f e r mentation i s a n e a r l y impossible task t o perform d i r e c t l y on a process". The use of mathematical models i n c o n j u n c t i o n with modern q u a n t i t a t i v e o p t i m i z a t i o n procedures could circumvent much of the experimental work. The need f o r a good model of b a c t e r i a l growth which could accommodate product formation has been w e l l recognized (e.g. 6^, 23, 28-34). A structured-segregated model i s s u f f i c i e n t l y general to be used f o r o p t i m i z a t i o n . A way to generate h i g h l y structured-segregated models which are mathematically t r a c t a b l e i s to b u i l d p o p u l a t i o n models u s i n g an ensemble of s i n g l e - c e l l models. Such an approach can avoid the generation of i n t e g r a l - d i f f e r e n t i a l equations which are so computationally d i f f i c u l t to s o l v e . Thus, s i n g l e - c e l l models f i l l c e r t a i n r e a l needs of both b i o l o g i s t s and engineers. The advantages of s i n g l e - c e l l models compared to normal p o p u l a t i o n models a r e : 1. e x p l i c i t accounting of c e l l geometry and i t s p o t e n t i a l e f f e c t s of n u t r i e n t t r a n s p o r t ; 2. the a b i l i t y to p r e d i c t temporal events during the d i v i sion cycle; 3. the a b i l i t y to consider the e f f e c t s of s p a t i a l arrangements w i t h i n a c e l l ; 4. and the ease i n which d e t a i l s about biochemical pathways and t h e i r i n t e g r a t i o n and metabolic c o n t r o l can be i n c l u d e d . These advantages make s i n g l e - c e l l models p a r t i c u l a r l y w e l l - s u i t e d to t e s t i n g the p l a u s i b i l i t y of hypotheses about metabolic mechanisms. S i n g l e - c e l l models i n v i t e complexity. T h e i r main disadvantage i s that s i n g l e - c e l l models represent only a " t y p i c a l " c e l l and are adequate r e p r e s e n t a t i o n s of the growth of c e l l populations only i f the moments o f d i s t r i b u t i o n of c e l l u l a r p r o p e r t i e s higher than the f i r s t - o r d e r can be ignored. The l a s t disadvantage i s circumvented by u s i n g an ensemble of s i n g l e - c e l l models t o b u i l d a p o p u l a t i o n model ( i . e . each s i n g l e - c e l l represents some small f r a c t i o n of the t o t a l p o p u l a t i o n ) . I d e a l Model. Having discussed why s i n g l e - c e l l models may be u s e f u l , what c h a r a c t e r i s t i c s should such models have? The f o l l o w ing f e a t u r e s are important: 1. the model must be c o n s i s t e n t with experimentally confirmed observations over a wide v a r i e t y of growth c o n d i t i o n s ,



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2. the model must have a s t r u c t u r e which can e a s i l y allow the i n c o r p o r a t i o n o f p o s t u l a t e d biochemical mechanisms, 3. the parameters o f the model must be determined d i r e c t l y from independent experiments o r estimated by an o b j e c t i v e s e r i e s of r u l e s (no a d j u s t a b l e parameters), 4. the e f f e c t of c e l l geometry and shape on n u t r i e n t uptake must be included (the e x t e r n a l environment must be e x p l i c i t l y accounted f o r ) , 5. the model must allow f o r the i n c l u s i o n o f randomness i n key metabolic systems, 6. the model must not i n c l u d e a r t i f i c i a l c o n s t r a i n t s such as growth must be e x p o n e n t i a l , the c e l l must maintain a given shape, the c e l l w i l l d i v i d e when the amount of given component doubles, etc., 7. the only s i g n a l s that a c e l l can u t i l i z e are the concent r a t i o n s of v a r i o u s biochemical s p e c i e s , and, 8. the model must be mathematically t r a c t a b l e . The model should be a model - not a c o l l e c t i o n of phenomenological equations based on curve f i t s . Examples of S i n g l e - C e l l Models One of the f i r s t examples of i n d i v i d u a l - c e 1 1 models i s that suggested by Von B e r t a l a n f f y (see 1). In h i s model growth was a r e s u l t of competition between the process o f n u t r i e n t a s s i m i l a t i o n and endogenous metabolism. N u t r i e n t uptake was p o s t u l a t e d t o be p r o p o r t i o n a l t o the c e l l ' s surface area and the c o n c e n t r a t i o n of n u t r i e n t i n the a b i o t i c environment and the r a t e of endogenous metabolism was p o s t u l a t e d to be p r o p o r t i o n a l t o c e l l mass. Heinmets (35) suggested i n 1966 a model f o r a s i n g l e c e l l (or the nucleus of an e u c a r y o t i c c e l l ) that incorporated 19 d i f f e r e n t i a l equations. The model contained an amino a c i d p o o l , a n u c l e o t i d e pool f o r RNA s y n t h e s i s , a general i n t r a c e l l u l a r metabolic p o o l , t o t a l p r o t e i n , RNA polymerase, genes f o r s y n t h e s i s of v a r i o u s RNA's, m-RNA (2 t y p e s ) , r-RNA, and t-RNA. The model could respond to step changes i n the e x t r a c e l l u l a r n u t r i e n t p o o l s . The main purpose o f the model was t o examine how a c e l l would change from normal t o abnormal growth. The mechanistic scheme r e f l e c t s the general understanding of c e l l u l a r biochemistry i n 1965 but d i f f e r s somewhat from the present conception of the c e l l . The constants chosen were a r b i t r a r y , and the model was not formulated t o any s p e c i f i c organism, and no attempt was made t o compare the p r e d i c t i o n s t o a c t u a l experimental data. E f f e c t s of c e l l geometry and s i z e on n u t r i e n t uptake were not considered. The c e l l d i v i s i o n hypothesis depends on an imposed c r i t e r i o n to do with the concent r a t i o n s of i n t r a c e l l u l a r components. C e l l d i v i s i o n i s not a " n a t u r a l " response of the model. Davison (36) has solved Heinmets' model with only the d i g i t a l computer and has compared the r e s u l t s with two experimental s i t u a t i o n s . The f i r s t was the



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recovery of an IS. c o l i f o r Mg s t a r v a t i o n , and the second was the e f f e c t of s p l i t r a d i a t i o n doses on Chinese Hamster c e l l s . In both cases reasonable q u a l i t a t i v e p r e d i c t i o n s were obtained from the model. Another example of s i n g l e - c e l l model i s that proposed by Simon (37) i n 1973. He considered only steady-state balanced growth. His model c e l l contained enzymes involved i n d i v i s i o n i n i t i a t i o n , i n the formation of DNA p r e c u r s o r s , and i n RNA synthesis. DNA s y n t h e s i s was i n i t i a t e d when a c r i t i c a l m a t e r i a l accumulated to a t h r e s h o l d value; DNA i n i t i a t i o n caused a l l the t h r e s hold m a t e r i a l to be consumed. Each p r o t e i n s y n t h e s i s r a t e was made p r o p o r t i o n a l to the degree of genome a c t i v a t i o n . The r a t i o of c e l l volume to p r o t e i n was considered constant, and volume growth was forced to be e x p o n e n t i a l . The surface to volume r a t i o was considered constant (a c y l i n d e r without ends). The e f f e c t s of the a b i o t i c environment were not e x p l i c i t l y accounted f o r ; the value of the growth r a t e was imposed on the c e l l . The c e l l was required to d i v i d e when the amount of each component i s doubled. I n i t i a l c e l l s i z e could be p r e d i c t e d . A short general q u a l i t a t i v e comparison of the model to experimental observations was given; the model was found to be s a t i s f a c t o r y f o r the parameters compared. Weinberg, Z e i g l e r , & L a i n g (38) and Z e i g l e r & Weinberg (39) have attempted to construct a model of an IS. c o l i c e l l . They have considered how the v a r i o u s components of the c e l l might be aggregated. T h e i r c e l l contains e x p l i c i t concentrations f o r c e l l w a l l , DNA, m-RNA, t-RNA, ribosomes, p r o t e i n , amino a c i d s , w a l l prec u r s o r s , n u c l e o t i d e s , ATP, and ADP are shown (38, 39) along with a pool c a l l e d glucose which i n c l u d e s glucose and other small metabolites made from glucose. They c l a i m that t h e i r model of a c e l l d i f f e r s from others i n that the a b s t r a c t i o n i n v o l v e d i n model formulation a r i s e s from the aggregation of v a r i a b l e s r a t h e r than the s e l e c t i o n of subsystems. D i f f e r e n c e and Boolean equations were used to d e s c r i b e the system. Hyperbolic ("saturation k i n e t i c s " ) r a t e forms were not used; f o r example, the r a t e of product i o n of amino a c i d s from glucose was made p r o p o r t i o n a l to the product of glucose, ATP, and enzyme 2 c o n c e n t r a t i o n s . Enzyme a c t i v i t y was modified to simulate an a l l o s t e r i c enzyme by use of Boolean equations. T h e i r c e l l was constrained to grow exponent i a l l y i n volume. A mechanism f o r i n i t i a t i o n of DNA r e p l i c a t i o n was included but none f o r c e l l d i v i s i o n . From the above papers (38, 39) i t i s not c l e a r whether n u t r i e n t uptake was e x p l i c i t l y accounted f o r or not. They (39) d e s c r i b e how they evaluated r a t e constants from the data i n the l i t e r a t u r e . A d i r e c t comparison to experimental data was made (38) f o r s h i f t experiments between v a r i o u s media; r-RNA and DNA concentrations were s a t i s f a c t o r i l y p r e d i c t e d by the model i f feedback c o n t r o l s were i n c l u d e d . Another approach to p r e d i c t i n g the response of a " t y p i c a l " c e l l to a s h i f t i n media has been suggested by Bremer and co-work-



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ers (40, 41, 42)· They have made extensive measurements of RNA metabolism and p r o t e i n s y n t h e s i s i n IS. c o l i and have c o r r e l a t e d t h e i r r e s u l t s with phenomenological expressions. These expressions coupled with s i m i l a r r e l a t i o n s h i p s suggested by other workers (e.g. Cooper and Helmstetter (43)) allow r a t h e r accurate c a l c u l a t i o n of c e l l s i z e and composition f o r u n r e s t r i c t e d growth i n media of v a r i o u s compositions. These expressions appear to be u n s a t i s f a c t o r y f o r growth r e s t r i c t e d by a l i m i t i n g n u t r i e n t as glucose (41). The above expressions do not e x p l i c i t l y depend on the e x t e r n a l environment; the growth r a t e which a media w i l l support must be known beforehand. These expressions are r e a l l y c o r r e l a t i o n s r a t h e r than a t r u e model. They are l i m i t e d to a narrow range of growth c o n d i t i o n s and are not p o t e n t i a l l y as general as the other models d i s c u s s e d . Another type of s i n g l e - c e l l model has been considered by Nishimura and B a i l e y (24). The model i s s t r u c t u r e d i n that both c e l l mass and DNA content are e x p l i c i t l y recognized. Rules concerning DNA r e p l i c a t i o n and the timing of i n i t i a t i o n DNA s y n t h e s i s are imposed on the c e l l ; the r u l e s are d e r i v e d from the observat i o n s of Cooper and Helmstetter (43) and Donachie (44). The s i n g l e - c e l l model was e s s e n t i a l l y the b a s i s f o r the c o n s t r u c t i o n of a p o p u l a t i o n model - a formulation among the most general c u r r e n t l y a v a i l a b l e . The main l i m i t a t i o n of the model i s i t s i n a b i l i t y to permit an e x p l i c i t c a l c u l a t i o n of each c e l l ' s growth r a t e i n terms of the e x t e r n a l environment and the c e l l ' s p h y s i o logical state. Ho and Shuler (45) proposed a mathematical model f o r the growth of an i n d i v i d u a l bacterium i n c o r p o r a t i n g feedback c o n t r o l of n u t r i e n t uptake. T h i s simple model could p r e d i c t the growth p a t t e r n f o r a c e l l of a given shape (filamentous, b a c i l l u s , or s p h e r i c a l ) . The model c e l l contained four components (ammonium ion, glucose, p r e c u r s o r s , and macromolecules). An a n a l y t i c a l s o l u t i o n was p o s s i b l e f o r filamentous c e l l s , but numerical s o l u t i o n s were r e q u i r e d f o r other c e l l shapes.t More r e c e n t l y Shuler, Leung, and Dick (46) have presented a more complete model f o r the growth of a s i n g l e - c e l l of E s c h e r i c h i a c o l i B/r A. The model contained 14 components. A l l of the c e l l ' s components were included i n one of the model components. The model c e l l r e l i e d on chemical concentrations as c o n t r o l s i g n a l s so that the growth p a t t e r n , timing of DNA s y n t h e s i s , c e l l shape, c e l l s i z e , c e l l composition, and c e l l d i v i s i o n could be considered n a t u r a l responses to e x p l i c i t changes (e.g. glucose concentration) i n the e x t e r n a l environment. An attempt was made to evaluate k i n e t i c parameters from independent measurements on e x p o n e n t i a l l y growing c e l l s . Four parameters, having to do with c e l l envelope tThe example c i t e d i n Equation B9-B14 of Ho and Shuler (45) i s f a u l t y . However, i t i s easy to demonstrate that the c o n c l u s i o n i s c o r r e c t . D e t a i l s of the c o r r e c t e d example are a v a i l a b l e upon request.



100

and c r o s s - w a l l formation were taken as a d j u s t a b l e . model i s being f u r t h e r developed.

T h i s prototype


The C o r n e l l S i n g l e - C e l l Model The prototype model has been revamped. The base model has 20 components; the a d d i t i o n a l components are necessary to allow an accurate d e s c r i p t i o n of the systems a l l o w i n g the i n c o r p o r a t i o n of ammonium i o n i n t o amino a c i d s , to allow more accurate estimates of c e l l u l a r energy expenditures, and to allow a more complete s i m u l a t i o n of systems c o n t r o l l i n g t r a n s c r i p t i o n and t r a n s l a t i o n . Some of the parameters i n the prototype model (46) were c a l c u l a t e d based on c e l l dimensions obtained f o r c e l l s f i x e d i n osmium t e t r o x i d e ; these have been r e c a l c u l a t e d u s i n g s i z e parameters obtained from g l u c o s e - l i m i t e d chemostat c u l t u r e s with g l u t e r aldehyde-fixed c e l l s . F i g u r e 1 and Tables I to IV d e s c r i b e the current model. J u s t i f i c a t i o n of parameter values have been given elsewhere (4650). Almost a l l parameters were estimated from independent measurements on e x p o n e n t i a l l y growing c e l l s or c e l l - f r e e systems. Values f o r η2 and η (parameters a s s o c i a t e d with the r a t i o of envelope used f o r extension and c r o s s - w a l l formation) were r e quired to be p o s i t i v e but were otherwise considered a d j u s t a b l e . Values f o r and K p ^ were adjusted w i t h i n the range of 0 to 3

A2

0.05

gm/cc of c e l l volume.

Values f o r η 2 , r\

39

, and

K

p

| > A 2

were estimated by comparing data to model p r e d i c t i o n s at y = 0.95 hr and μ = 0.5 h r " . A parameter f o r u n i d e n t i f i e d energy consumption, 6^ was i n c l u d e d to make the p r e d i c t e d growth y i e l d 1

1

9

1

at μ = 0.95 hr match experimental measurements. Values f o r iIF* * * * ZMi» l * experimental growth data f o r μ < 0.95 hr but were evaluated independently of the model's p r e d i c t i o n s . A l l parameters were set based on g l u c o s e - l i m i t e d growth o n l y . No f u r t h e r adjustments were made f o r p r e d i c t i o n of ammonium i o n - l i m i t e d growth. The model makes reasonable p r e d i c t i o n s of the dependence of c e l l s i z e , c e l l shape, c e l l composition, growth r a t e , and the timing of c e l l u l a r events on e x t e r n a l concentrations of glucose. Results f o r g l u c o s e - l i m i t e d growth are given elsewhere (47, 50). The c u r r e n t model makes use of recent observations of F r a l i c k (51), Messer, et a l . (52) , and Fayet & Louarn (53) to construct a mechanism f o r the c o n t r o l of the i n i t i a t i o n of DNA s y n t h e s i s . The scheme i s i l l u s t r a t e d i n Figure 2. The dnaA gene which i s l o c a t e d near the o r i g i n makes a gene product (RP) which represses the t r a n s c r i p t i o n of the 0-RNA gene. I n i t i a t i o n r e q u i r e s 0-RNA as a primer. An a n t i - r e p r e s s o r (ARP) i s made which i n a c t i v a t e s RP. Experimental evidence f o r ARP e x i s t s (51), and there are i n d i c a t i o n s that ARP production i s r e l a t e d to c e l l envelope formation (54). In the model we have made the r a t e of K

P

Z

a n
δ

(5)

ε Ρ2 + ω

+ .. .

δ Α Τ Ρ -> δ ( Α 0 Ρ

(6)

3 ει»Ρι + $ι*Α + ... -> Ρι»

+ ...

δ

2

2

3

Ό

•> P 2

Χ + ... + Ρ

3

r

2

(7)

Macromolecule Formation ΎιΡι Μι + ...

(8)

Ύ2?2

(9)

Ύ3Ρ3 J

J

£ Μ Μ

2

+ ...

3

+ ...

ρ ι

(11) γ 5A2 J

Μ

5

ρΐ(

A d d i t i o n a l D i r e c t Reductant Use Transport of ions Membrane recharge leakage

to o f f s e t

H

ρι

(ADP

ρ

ρ 3

ΑΤΡ -> δ

ATP

ΑΤΡ

Μ

->

(ADP

ρ

y

(ADp

->-

V

(

A

D

P

(ADP ΑΤΡ -+ δ Μ3

+ +

ν

+ +

p

i>

p

i>

p

i>

p

i>

+ + +

3

δ

Μ/

δ

Μ

W

(ADP + Ρ.) i

δ (ΑΟΡ

ρ 3

δ

+ ...

-V

Α

ρ 2

V

Ύ^Ρ*

Α2

Α

Μ (10)

ΑΤΡ -* δ

Α2

Τ

-> δ

Ρ

Mit

Α Τ Ρ 5

+

->

ω

r

V

(ADP

+

(ADP

+

ΐ Ο Ν *>* -

W

I0N

+


H 2

°

SHULER


5.

Table I ,

AND

DOMACH

Models

of Cell

103

Growth

continued.

A d d i t i o n a l B i o s y n t h e t i c Reductant Use SO" -> S

=

=

ω

8 0 ι +

Χ + SO" -> [ S ]

Mass use

ω

Β Ι ( )

A d d i t i o n a l ATP Coupling PG formation

ό ATP -> 6 „ ( A D P + P.)

Χ

+

M J * - M^*

D

PG

Non-Specified Size linked

PG

i

ATP Use oyATP -* δ (ΑΟΡ + P ) γ

±

* Coupled to e i t h e r phosphate bond energy or o x i d a t i v e r e a c t i o n s . Products of o x i d a t i o n i n c l u d e pmf and reducing e q u i v a l e n t s . ** 0X* **RED * l * reduced c e l l mass, r e s p e c t i v e l y . M

=

u

n

r

e

c

u

c

e

(

a n c


BIOCHEMICAL ENGINEERING TABLE I I :

Equations D e s c r i b i n g nent i n the C e l l .

the Rate of Change of Each Compo


The symbols are the same as l i s t e d i n Figure A d d i t i o n a l symbols used a r e :

1 and Table I .

μ, k, η = maximum r a t e s of synthesis of the macromolecules, p r e c u r s o r s , and enzymes, r e s p e c t i v e l y V = maximum transport r a t e Κ = s a t u r a t i o n constants = decomposition r a t e s i R = r a t e of transport F = number of f o r k s i n DNA molecule Ν = number of DNA o r i g i n s ο S c e l l u l a r surface area V = volume RI = mass of i d l i n g ribosomes moles acetate formed 7 L moles glucose d i s s i m i l a t e d GD = number of genes coding f o r s t a b l e RNA synthesis P = density f = p r o p o r t i o n a l i t y constant between Mi* and c e l l surface area SL = septum length CL = length of c y l i n d r i c s e c t i o n of the c e l l W = c e l l width SEPF = mass of septum IF = i n i t i a t i o n f a c t o r s f o r p r o t e i n synthesis RP = repressor p r o t e i n ARP = a n t i - r e p r e s s o r p r o t e i n (P/0) = maximum P/0 r a t i o max k

When S and D are used as s u b s c r i p t s , they denote r a t e of synthesis and r a t e of degradation r e s p e c t i v e l y . A and C as sub s c r i p t s r e f e r to anabolism and catabolism. dAi it

_ " V

r/

c

s

-

α

ι

[

(

d P l

\

ΐΓ>

/

d P l

N

ι

(

5

- -dT>

] D

,dGLN

N

a

(

- i,GLN ^r>

/ 1 Λ

(

1

)

s

NET

,dM . 5


(2)

SHULER A N D DOMACH

5.

Models

of Cell

105

Growth

Table I I , continued. .dATP dt

K

,dP!

;

rf i

s d M 2

ν

RTI


dt

[1 »

,dPG

s

l d t >

-

GLN ^ t

f

- "a!'V

s

+

< ω

ΐΟΝ

(3)

W~dt>

+

g

s

+

ω

ΒΙΟ

+

+

1 ?

±

+

l

H

±

)

/

d

t

]

+

dP Ρ

3

ν

δ

+

ω

(dATP/dt). + dX/dt(P/0) S max •180 4 + (12 - 4·Ζ)(Ρ/0) max

(dA /dt), 2

/ Ζ =μ

dA

(4)

3

(dMWdt)

^

2

(7) A

L»

GLN/V dt

" "A

SxGUI

+

A /V 2

G L N / V

«ΡχΑ,

+

A

(8a) l

/

V

2

Pi/V

Ai/V . W

+

A

l

/

V

W

+

P

GLN l

/

V J L

K

G L N

+ (GLN/V)^ (8b)

s ι · r. ( d t )

,dGLN.

Term i s equivalent t o glutamine d r i v e n Glutamate Synthase

Continued on next page.




106

dp *

Term i s equal to net Ρ χ formation,

N

("^~")

E

T

» where the f i r s t

term r e f e r s to Αχ i n c o r p o r a t i o n v i a glutamate ( d t

V

** T h i s term i s equivalent t o : e [ ( ^ f )

-

2

S t

dehydrogenase,

T h i s term i s equivalent t o :

1 D

ει

*(~ΠΓ*) S


SHULER A N D DOMACH

5.

Models

of Cell

107

Growth

Table II, continued. y

dMi


(12)

dM

2

RTI = P2 dt

iPG

P /V 2

^

P

P

^

* ^ 1 P G

+

PG/

7

\LIF rr K

iIF

+

[ (

dMi/dtv Μι >

•GD -

Mi Τ

l V

S, ι '

1

Max

k

TM

2

RTI (dMi/dt)

RTI1

rdM s

|

M 2

RTI

2

RTM dt

(13)

RTI dM

2

RTM dt

RTM RTM

c

2

- 2.0 χ 1 0 - i ^

i p G

l

V A - ^ * " I?

f!

ω _ ι

= 9.9 χ 10~* ^ P

10

ν •·° S Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 19, 2015 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch005

111

_

1

8X 1 0

Uh

r

_

"S

1

2 x10 3

=·° " S

^

1

l 0

3

- -° * ' f !

Continued on next page.


112


Table I I I , continued* enzymes = 1.0 χ 10

3

η

Π ι

2

η13 - 1.6 χ 10" h r "

== 3Q. 2ο χ 1iθn" 3

2

v

1

3

2

IL, = 2 . 0 χ ΙΟ" £ι|»


* P

k

β

10

(SE)'

= 0.01

n i f

ce

2

k

- 6 . 4 χ ΙΟ" - J g L -

TE, - ° ·

TPG «

1 2 5

h r

acetate

u = 0.9 ζ

oxidation

—. =1.5 0 'max

c e l l shape

ρ

X

'

k

k

±

p

h r

'

X

- 1 2 . 7 χ ΙΟ"»

3

p G

0 5

i

7

2

TPGA = ·

X

1 0

2

'

5

£

S

Κ „ = 1.0 χ ΙΟ" * ^ ζΜι» cc 1

ρ

= 0 . 2 5 8 *E* ce

cyto

ο env

f

= 259 χ 1 0

6

— gm

= 0.553 BS. cc 3

DNA i n i t i a t i o n k = 2.6 χ 1 θ " - ^ r ARP gm Mi* *** ·

s

3

·

6

5 x

1

0

-

1

7

S

IT = 0.22 - S * — iRP cc-hr

r

Stoichiometric Coefficients: «i

-

α l.GLN

=

β ι

=

Β

2

βι,

0.179 °· 1

>

1

1 2 2 2

2

8

= -0.456 -

1.28

ε Ε

2

=1.149

γ

χ

- 1.167

3 - 1·049

γ

2

= 1.057

ε„ = 0.128

γ

3

= 1.053 1

1

0

° · Υ5 = 1.11


5.

SHULER

AND DOMACH

Models

of Cell

113

Growth

Table I I I , continued*

glycogen

y = 2.0 χ 1 0 "

2

- 1

= .14 h r

5

0

5 K

M A 5

2

-°

x

1 0

" f

K

3

T M « 1-0 x 10"

g

3

5

3

enzymes Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 19, 2015 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch005

= 2

n i = 1.0 χ 1 0 " n = 1.6 x 1 0 " h r " 2

n = 3.2 χ 1 0 "

3

2

1

3

Κ

= 2.0 χ ΙΟ"» φ

Ει>

"

„„ .

%= 2

PpGpp

= 6.4 χ 1 0 "

P p G

k

k

= 125hr-

T p G

acetate

P = 0.9

oxidation

Pi ^| =1.5

3 H ?

-gl

i P i

K

1

>

0

1

= 0.05 h r '

TEit

K

r a o f

0

1

- 12.7 χ 10"» j g

TpGft2

- 7.2 χ 1(Γ« g l k

= 1.0 χ 1 0 " 3 2

Z

I

c e l l shape

P /

max . = 0.258 cyto

ce

f

= 5 9 χ 10 s

6

^ gm

2

2i env = 0.553 c3 c

P p n v

DNA i n i t i a t i o n

k

f t R p

- 2.6 χ 10"· ^

K

( R p

= 0.22 j ^ r

' . • ' • « • " ' " i Î

0

Stoichiometric oi a

= 0.179

l

6LN

'

β

β

Coefficients:

2

3H.

=

=

° ·

1

2

2

2

1 > 1 2 8

= -0.456 1.28

e = 1.149

Y i = 1.167

2

Ε

3

=

Ί

Μ

9

e« = 0.128

Ύ2

=

1-057

γ = 1.053 3

Ύ

=

1

Λ

0

" Ύ5 = 1 . U


114


TABLE IV:

Values of Parameters ments

f o r C a l c u l a t i o n of Energy

Compound or Process A , NH^+, Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on October 19, 2015 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch005

x

Mole ATP — gm

P,

Ribonucleotides

0.022

P,

Deoxyribonucleotides

P^,

Non-Protein Envelop Precursors

0.0016

M,

protein

0.039

M, 2

RNA

0.0067

M,

DNA

0.0071

M^,

Non-Protein Envelope

0.0081

M,

Glycogen

0.0124

3

x

3

5

'H^ gm

0.0056 0.0025

2

Mole

0.028

Amino Acids

X

ω, i

transport

A2, glucose transport +P ,

Mathematical Models of the Growth of Individual ... - ACS Publications

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