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.
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Davis,B.D.
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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