J.Mol.Evol.5,199-222 © by Springer-Verlag

(1975) 1975

Optimal Design of Feedback Control by Inhibition Dynamic Considerations* MICHAEL A. SAVAGEAU Department of Microbiology,

The University

of Michigan at Ann Arbor

Received July 12, 1974; March 14, 1975

Summary. The local stability of unbranched biosynthetic pathways is examined by mathematical

analysis and computer simulation using a novel non-

linear formalism that appears to accurately describe biochemical Four factors affecting the stability are examined: inhibition, parameters

equalization of the values among the corresponding for the reactions

of the pathway,

native patterns of feedback interaction. the pattern of feedback interactions steady-state

behavior.

systems.

strength of feedback

pathway length,

kinetic

and alter-

The strength of inhibition and

are important determinants

of

The simple pattern of end-product inhibition

unbranched pathways may have evolved because it optimizes state behavior and is temporally most responsive

in

the steady-

to change.

Stability in

these simple systems is achieved by shortening pathway length either physically or, in the case of necessarily a wide divergence

long pathways,

in the values of the corresponding

for the reactions of the pathway. light of available

experimental

These conclusions

kinetically by

kinetic parameters are discussed in the

evidence.

Key words: Natural Selection - Control Patterns - Biosynthetic Pathways

INTRODUCTION Of for

all biochemical the

organisms level.

control

biosynthesis For

are

the

several

of

most of

systems,

amino

acids

thoroughly these

the unbranched and

studied

pathways,

pathways

nucleotides at

e.g.

the

in

micro-

molecular

tryptophan

An earlier report of this work was given at the Annual Meeting of the American Society for Microbiology,

Miami Beach,

Florida,

1973

199

(Creighton & Yanofsky, 1970) and h i s t i d i n e (Brenner & Ames, 1971), all of the m o l e c u l a r c o m p o n e n t s have b e e n s t u d i e d and k i n e t i c a l l y c h a r a c t e r i z e d to some degree. D e s p i t e our r a t h e r e x t e n s i v e k n o w l e d g e at the m o l e c u l a r level, we are still rel a t i v e l y i g n o r a n t of the b e h a v i o r of the i n t a c t s y s t e m as it o p e r a t e s in the cell. We are only n o w b e g i n n i n g to u n d e r s t a n d the p r i n c i p l e s b e h i n d the d e s i g n of these m a r v e l o u s c o n t r o l systems. In the p r e v i o u s study of the s t e a d y - s t a t e b e h a v i o r of these types of s y s t e m s it was found that the e x p e r i m e n t a l l y o b s e r v e d p a t t e r n of e n d - p r o d u c t i n h i b i t i o n tends to be optimal w i t h r e s p e c t to a v a r i e t y of f u n c t i o n a l c r i t e r i a (Savageau, 1974). This is c o n s i s t e n t w i t h the p o s t u l a t e of o p t i m a l ity as the b a s i s for s e l e c t i o n (Rosen, 1967). One w o u l d expect these c o n t r o l s y s t e m s to have b e e n s e l e c t e d also for local stability, since p r e s u m a b l y one of the prime f u n c t i o n s of the amino acid b i o s y n t h e t i c p a t h w a y s is to p r o v i d e a r e l a t i v e ly c o n s t a n t s u p p l y of their e n d - p r o d u c t s for p r o t e i n synthesis. The s t a b i l i t y of linear m o d e l s of c h e m i c a l r e a c t i o n netw o r k s was e x a m i n e d by s e v e r a l a u t h o r s (e.g., see Hearon, 1953; Bak, 1963) b e f o r e the g e n e r a l m o l e c u l a r a r c h i t e c t u r e of control p a t t e r n s in b i o s y n t h e t i c p a t h w a y s had b e e n d i s c o v e r e d and a p p r e c i a t e d . C o n s e q u e n t l y , the b i o l o g i c a l q u e s t i o n s r a i s e d by m o r e r e c e n t d i s c o v e r i e s w e r e not s p e c i f i c a l l y add r e s s e d in these e a r l y analyses. O t h e r s have used m a t h e m a t i cal m o d e l s to i n v e s t i g a t e the c o n d i t i o n s for stable o s c i l l a tory b e h a v i o r in the c i r c u i t s for e n z y m e s y n t h e s i s as w e l l as in b i o s y n t h e t i c p a t h w a y s s u b j e c t to f e e d b a c k c o n t r o l by inh i b i t i o n (Goodwin, 1963; M o r a l e s & McKay, 1967; V i n i e g r a G o n z a l e z & M a r t i n e z , 1969; Walter, 1970). For these m o d e l s a p a r t i c u l a r a p p r o x i m a t i o n to the a c t u a l n o n l i n e a r systems is used, the s o - c a l l e d G o o d w i n e q u a t i o n s . All of the n o n l i n e a r i ty is lumped into the d e s c r i p t i o n of the first or a l l o s t e r i c enzyme. The r e m a i n d e r of the s y s t e m is a s s u m e d to be linear, c o m p o s e d of simple f i r s t - o r d e r r e a c t i o n s w i t h i d e n t i c a l kinetic p a r a m e t e r s . E v e n w i t h these s i m p l i f y i n g a s s u m p t i o n s , the r e s u l t i n g n o n l i n e a r m o d e l is d i f f i c u l t to a n a l y z e m a t h e m a t i cally and m u c h of the i n f o r m a t i o n has b e e n g a i n e d by c o m p u t e r simulation. Recently, H u n d i n g (1974) has r e e x a m i n e d the o r i g i n a l a n a l y s i s of V i n i e g r a - G o n z a l e z & M a r t i n e z (1969) and shown that theirs is not a n e c e s s a r y c o n d i t i o n for the e x i s t ence of an u n s t a b l e s t e a d y state or a stable l i m i t - c y c l e . The same c r i t i c i s m a p p l i e s to the i n d e p e n d e n t d e r i v a t i o n of this c o n d i t i o n by H i g g i n s et al. (1973). The m o r e r e c e n t a n a l y s i s by V i n i e g r a - G o n z a l e z (1973), however, avoids these d i f f i c u l ties for the m o s t part. H u n d i n g (1974) and H a s t i n g s & T y s o n

200

(1975) also have r e e x a m i n e d the c o m p u t e r s i m u l a t i o n r e s u l t s of W a l t e r and d i s c o u n t e d some of his claims. W a l t e r (1974) has gone on to s i m u l a t e a m o d i f i e d G o o d w i n m o d e l in w h i c h the u n r e g u l a t e d r e a c t i o n s are r e p r e s e n t e d by s p e c i f i c "hyperbolic" or " s i g m o i d a l " rate laws. The p o w e r - l a w f o r m a l i s m used in this p a p e r r e p r e s e n t s an a l t e r n a t i v e a p p r o a c h to the a n a l y s i s of b i o c h e m i c a l systems. S t a r t i n g w i t h the o b s e r v a t i o n that b i o l o g i c a l systems are c o m p o s e d of n e t w o r k s of e n z y m e - c a t a l y z e d r e a c t i o n s and u s i n g the w e l l - e s t a b l i s h e d p o s t u l a t e s of e n z y m e catalysis, one can in p r i n c i p l e w r i t e d i f f e r e n t i a l e q u a t i o n s w i t h r a t i o n a l function n o n l i n e a r i t i e s for an a r b i t r a r y system. It is p r a c t i c a l l y i m p o s s i b l e to deal w i t h these e q u a t i o n s in any g e n e r a l sense. N e v e r t h e l e s s , one can use a l o g i c a l a p p r o a c h a n a l o g o u s to l i n e a r i z a t i o n and d e r i v e a n o n l i n e a r a p p r o x i m a t i o n to the o r i g i n a l e q u a t i o n s that i n v o l v e s m u l t i d i m e n s i o n a l p o w e r - l a w s . This r e p r e s e n t a t i o n is g u a r a n t e e d to a c c u r a t e l y r e p r e s e n t the o r i g i n a l s y s t e m o v e r an a p p r o p r i a t e range of the d y n a m i c v a r i a b l e s in the same m a n n e r that a l i n e a r i z a t i o n will, e x c e p t t h a t the a p p r o p r i a t e r a n g e is g r e a t e r than that for any linear a p p r o x i m a t i o n (Savageau, 1969). The d y n a m i c a s p e c t s of o p t i m a l c o n t r o l by f e e d b a c k inhibition in b i o s y n t h e t i c p a t h w a y s , the i n t e g r a t i o n of these prop e r t i e s w i t h the s t e a d y - s t a t e p r o p e r t i e s d e s c r i b e d in the p r e v i o u s p a p e r (Savageau, 1974), and c o r r e l a t i o n s w i t h e x p e r i m e n t a l data w i l l be c o n s i d e r e d in this paper. For this purp o s e I shall A s s u m e that these s y s t e m s n o r m a l l y o p e r a t e in a s t a b l e s t e a d y state and focus on the c o n d i t i o n s for local stability. A n a l y t i c a l t e c h n i q u e s for e x a m i n i n g local s t a b i l i t y in terms of the p a r a m e t e r s of the p o w e r - l a w f o r m a l i s m are p r e s e n t e d in the A p p e n d i x . These t e c h n i q u e s , a l o n g w i t h comp u t e r s i m u l a t i o n s , are u s e d to e x a m i n e four factors a f f e c t i n g s t a b i l i t y : (i) s t r e n g t h of f e e d b a c k i n h i b i t i o n , (ii) d e g r e e of e q u a l i t y a m o n g the c o r r e s p o n d i n g k i n e t i c p a r a m e t e r s of the u n r e g u l a t e d r e a c t i o n s in the pathway, (iii) n u m b e r of reactions in the pathway, and (iv) a l t e r n a t i v e p a t t e r n s of feedb a c k i n h i b i t i o n . Finally, the r e s u l t s are c o m p a r e d and found to agree w i t h a v a i l a b l e e x p e r i m e n t a l evidence, w h i c h is cons i s t e n t w i t h the a s s u m p t i o n that local s t a b i l i t y is an import a n t f a c t o r in m e t a b o l i c r e g u l a t i o n and p r o v i d e s i n s i g h t into h o w such s t a b i l i t y is a c h i e v e d in nature.

STRENGTH

OF F E E D B A C K

INHIBITION

AND S T A B I L I T Y

The p a t h w a y r e p r e s e n t e d in Fig.1 w i l l be a n a l y z e d for local s t a b i l i t y first w h e n n=3. We shall d e r i v e an e x p r e s s i o n for the t h r e s h o l d of s t a b i l i t y w h i c h is a f u n c t i o n of the para-

201

Xo

~ X I

D .....

J X n

Fig.l. Model of an unbranched biosynthetic pathway. Xi's represent metabolite concentrations, the arrows from one X i to another represent enzyme catalyzed reactions, and the arrow from X n to the center of the arrow representing the first reaction denotes an allosteric modulation of the first reaction by the end-product X n. The initial substrate X O is subject to independent experimental manipulation

m e t e r s of the the p o w e r - l a w

(1)

system. The equations formalism are

Xl = ~lXo

gloX3gl3 -

BlXl

g21

describing

this

system

in

hll h22

X2 = ~2Xl

- B2X2

X3 = ~ 3 X 2 g32

- B3X3 h 3 3

T h e r a t e l a w for e a c h r e a c t i o n is r e p r e s e n t e d by a p r o d u c t o f power-laws, o n e for e a c h of the r e a c t a n t s a n d m o d i f i e r s ass o c i a t e d w i t h the r e a c t i o n . T h e e x p o n e n t gij r e p r e s e n t s the a p p a r e n t k i n e t i c o r d e r w i t h r e s p e c t to Xj, for t h e s y n t h e s i s of X i. ei is t h e a p p a r e n t r a t e c o n s t a n t for t h i s r e a c t i o n . W h e n a r a t e l a w a p p e a r s in a d e g r a d a t i v e t e r m of t h e s y s t e m ' s equations, the c o r r e s p o n d i n g parameters a r e hij a n d Bi, respectively. S i n c e t h e r a t e of d e g r a d a t i o n o f X i is the s a m e as the r a t e of s y n t h e s i s of X i + I in F i g . l , w e h a v e t h e f o l lowing equalities:

e2 = BI

;

g21

= h11

~3 = B2

;

g32

= h22

A l l the p a r a m e t e r s in Eq. (I) a r e p o s i t i v e e x c e p t f o r g13; it represents the a p p a r e n t k i n e t i c o r d e r of the f i r s t r e a c t i o n w i t h r e s p e c t to the e n d - p r o d u c t i n h i b i t o r a n d is t h e r e f o r e n e g a t i v e . X O is an i n d e p e n d e n t concentration variable.

202

The l i n e a r i z e d e q u a t i o n s d e s c r i b i n g the b e h a v i o r of this system for small v a r i a t i o n s about the normal o p e r a t i n g point are r e a d i l y o b t a i n e d from Eq. (I) by using the techniques dev e l o p e d in the Appendix. (2)

ul = FI [-hllul

+ g13u3 ]

~2 = F2 [ h11ul

- h22u2]

u 3 = F 3 [ h22u 2 - h33u3] The c h a r a c t e r i s t i c

equation

for this system is

3 (3)

l

2 +

(F1h11 + F2h22 + F3h33)l

+

(FiF2hllh22 + F i F 3 h l l h 3 3

+ FiF2F3hllh22(h33-g13)

+

+ F2F3h22h33)l

+

= O

or 3

2

+ %11

+ ~21 + %3 : O

F r o m Routh's c r i t e r i o n for s t a b i l i t y the critical c o n d i t i o n a m o n g these c o e f f i c i e n t s is ~i~2 > ~3' which implies

(4)

-g13


0

System for the n - v a r i a b l e

77-n = ~. I I 1 j=1

gij x. J

case

corresponding

n hij - Bi~ x. jl~1 j

i=I,

2,

to Eq.

....

n.

By a d i r e c t e x t e n s i o n s e c t i o n , we can w r i t e are (A11)

of the p r o c e d u r e s in the p r e v i o u s suba set of l i n e a r i z e d e q u a t i o n s . T h e s e

u.1 = F.I Eailul

+ ai2u2

+

"" "+ a i n U n ]

where gil F

for

of by

i

= ~

iXl 0

i = I, 2, The the

gi2

...,

gii -I ...X

X20

iO

gin ...X

no

n.

s t a b i l i t y of this s y s t e m is d e t e r m i n e d by the n a t u r e r o o t s of the c h a r a c t e r i s t i c e q u a t i o n , w h i c h is g i v e n

(A12)

(F1a11-l) F2a21

Flat2 (F2a22-l)

F1a13

.-.

F1aln

F2a23

.-~

F2a2n F3a3n

F3a31

F3a32

(F3a33-l)

...

Fnanl

Fnan2

F n a n3

...

(Fnann

=

0

I)

It is a n a l y t i c a l l y d i f f i c u l t to c h a r a c t e r i z e the r o o t s of this e q u a t i o n for an a r b i t r a r y case w h e n n is l a r g e r than 3. Nevertheless, for any p a r t i c u l a r case it is s t r a i g h t f o r w a r d to e x p a n d Eq. (A12) and a s c e r t a i n the n u m b e r of r o o t s w i t h p o s i t i v e r e a l p a r t s u s i n g the R o u t h c r i t e r i o n . In the p r e c e d i n g s e c t i o n s t h e s e t e c h n i q u e s are u s e d for the a n a l y s i s of the f a c t o r s a f f e c t i n g the d y n a m i c b e h a v i o r of biosynthetic pathways.

REFERENCES Bak,T.A. (1963). Contributions to the Theory of Chemical Kinetics. New York: W.A.Benjamin Brenner,M., Ames,B.N. (1971). In: Metabolic Pathways, 3rd Ed., VoI.V, Metabolic Regulation, H.J.Vogel, ed., p.349. New York: Academic Press Creighton,T.E., Yanofsky,C. (1970). In: Methods in Enzymology, VoI.XVIIA S.P.Colowick, N.O.Kaplan, eds., p.365. New York: Academic Press

221

Davis,B.D.

(1961). Cold Spring Harbor Symposia Quant. Bioi.26,

Ferdinand,W.

(19~6). Biochem.J.98,

i

278

Goodwin,B. (1963). Temporal Organization in Cells. London: Academic Press Hastings,S., Tyson,J. (1975). Biophys.J.15, 179a Hearon,J.Z. (1953). Bull.Math.Biophysics 15, 121 Higgins,J., Frenkel,R., Hulme,E., Lucas,A., Rangazas,G. (1973). In: Biological and Biochemical Oscillators, B.Chance, E.K.Pye, A.K.Ghosh, B.Hess (eds.), p.127, New York: Academic Press Hunding,A. (1974). Biophys. Struct.Mechanism i, 47 Klotz,I.M., Langerman, N.R., Darnall,D.W. (1970). Ann.Rev.Biochem.39, 25 Koshland,D.E., Neet,K.E. (1968). Ann.Rev.Biochem.37, 359 Monod,J., Changeux,J.-P., Jacob,F. (1963). J.Mol. Biol.6, 306 Monod,J., Wyman,J., Changeux,J.-P. (1965). J.Mol.Biol.12, 88 Morales,M., McKay,D. (1967). Biophys.J.7, 621 Rosen,R. (1967). Optimality Principles in Biology. London: Butterworths Savageau,M.A. (1969). J.Theoret.Biol.25, 370 Savageau,M.A. (1970). J.Theoret.Biol.26, 215 Savageau,M.A. (1971). Arch.Biochem.Biophys.145, 612 Savageau,M.A. (1974). J.Mol. E.vol.4, 139 Timothy, LaMar K., Bona,B.E. (1968). State Space Analysis: An Introduction. New York: McGraw-Hill Umbarger,H.E. (1956). Science 123, 848 Viniegra-Gonzalez,G. (1973). In: Biological and Biochemical Oscillators B.Chance, E.K.Pye, A.K.Ghosh, B.Hess, eds., p.41. New York: Academic Press Viniegra-Gonzalez,G., Martinez,H. (1969). Proc. Biophys.Soc.Abstracts 9, A210 Walter,C.F. (1970). J.Theoret.Biol.27, 259 Walter,C.F. (1974). J.Theoret.Biol.44, 219

Michael A.Savageau, Department of Microbiology The University of Michigan Medical School 6643 Medical Science Building II, Ann Arbor, Mich.48104, USA

222

Optimal design of feedback control by inhibition: dynamic considerations.

The local stability of unbranched biosynthetic pathways is examined by mathematical analysis and computer simulation using a novel nonlinear formalism...
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