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Kinetic amplification of enzyme discrimination
Kinetic amplification of enzyme discrimination. J a c q u e s NINIO .,@ .
The Salk Institute [or Biological Studies P.O. Box 1809, San Diego, Ca!i[ornia 92112. (12-12-197/t). Summary - - The dependance of the accuracy of enzymatic systems on the mechanism of the catalyzed reaction is investigated, using a prohabilistic approach. Certain mechanisms of reaction, involving a delay in one of the steps act as kinetic amplifiers of molecular discriminations. The relationship between our scheme for a delayed reaction [1] and Hopfield's scheme [2] is discussed.
INTRODUCTION. Many enzyme systems discriminate with remark a b l e e f f i c i e n c y b e t w e e n t h e i r s u b s t r a t e a n d clos e l y r e l a t e d m o l e c u l e s . It is g e n e r a l l y a c c e p t e d t h a t t h i s d i s c r i m i n a t i o n d e p e n d s in l a r g e m e a s u r e on an a c c u r a t e m a t c h i n g of the s h a p e of t h e e n z y m e to that of t h e s u b s t r a t e e i t h e r d u r i n g b i n d i n g of the l i g a n d o r at a s u b s e q u e n t step of t h e r e a c t i o n ~3, 4]. T h e e x a m i n a t i o n of r e c e n t r e s u l t s (to be disc u s s e d later) on the a c c u r a c y of D N A p o l y m e r i zation, on r i b o s o m a l s e l e c t i o n a n d on the d i s c r i m i n a t i v e a b i l i t i e s of the a m i n o a e y l - t R N A ligases led us to m o d i f y t h a t p o i n t of v i e w , at least for c e r t a i n classes of e n z y m a t i c s y s t e m s . Our p i c t u r e involves two elements : a) An e n z y m e h a s c e r t a i n d i s c r i m i n a t i v e abilities b y v i r t u e of its m a t c h i n g to t h e s u b s t r a t e . T h i s m a y be r e f l e c t e d t h r o u g h a d i f f e r e n c e in the d i s s o c i a t i o n c o n s t a n t s k s a n d k~ of t h e e n z y m e s u b s t r a t e a n d the e n z y m e - a n a l o g u e a s s o c i a t i o n s , or t h r o u g h d i f f e r e n c e s in o t h e r k i n e t i c p a r a m e t e r s of the r e a c t i o n . b) T h e e n z y m a t i c r e a c t i o n is c o n s i d e r e d as a p r o c e s s i n g of t h e s u b s t r a t e a n d t h e a n a l o g u e l e a d i n g to p r o d u c t s p s a n d pA. T h e p r o c e s s as a w h o l e c o n t a i n s o n e (or m o r e ) d i s c r i m i n a t i v e step, a n d t a k e s m o r e o r less a d a v a n t a g e of the discriminative potentialities r e f l e c t e d in t h e ratio k 2/k s O u r e x a m i n a t i o n of s e v e r a l cases s h o w in a g r e e m e n t w i t h i n t u i t i o n , t h a t in m o s t cases, the d i s c r i m i n a t i o n of t h e w h o l e p r o c e s s is s m a l l e r Present address where all correspondence shouht be addressed: Institut de Biologic Mol6culaire, Tour 43, 2 Place Jussieu, Paris (58).
t h a n t h a t of t h e d i s c r i m i n a t i v e step. T h u s , if t h e c o n c e n t r a t i o n s of s u b s t r a t e a n d a n a l o g u e a r e equal, w e h a v e in g e n e r a l [ p s ] / E p A ] ~-- kaA / k as. H o w e v e r , w e l o o k e d for a n d f o u n d a class of r e a c t i o n s w h i c h a l l o w the p r o c e s s as a w h o l e to be m o r e d i s c r i m i n a t i v e t h a n its d i s c r i m i n a t i v e step. W e are a b l e to c o n s t r u c t s c h e m e s f o r w h i c h [ p s ] / [ p A ] m a y a p p r o a c h (]~c~/ks )~ o r e v e n h i g h e r p o w e r s of t h e r a t i o of t h e t w o d i s s o c i a t i o n c o n s t a n t s (*). T h u s , in p r i n c i p l e , N a t u r e has at least t w o s t r a t e g i e s for a c h i e v i n g a c c u r a c y . It m a y d e s i g n a v e r y s o p h i s t i c a t e d b i n d i n g site, o r it m a y s e l e c t a s h r e w d m e c h a n i s m of r e a c t i o n w h i c h t a k e s a d v a n t a g e of s m a l l d i f f e r e n c e s in k i n e t i c p a r a meters. The second strategy would appear more i n t e r e s t i n g in s i t u a t i o n s w h e r e t h e e n z y m e s y s t e m n e e d s to s o l v e several a c c u r a c y p r o b l e m s . T h i s is p r e c i s e l y the case e n c o u n t e r e d in D N A p o l y m e r i z a t i o n a n d in r i b o s o m a l s e l e c t i o n of tRNA.
DNA polymerization. A s a m e e n z y m e is i n v o l v e d in the r e p l i c a t i o n of e a c h of t h e f o u r bases A, T, G, C. d C T P w h i c h is a s u b s t r a t e in one case ( w h e n a G is c o p i e d ) b e c o m e s a d a n g e r o u s a n a l o g u e w i t h r e s p e c t to t h e t h r e e o t h e r cases. M u t a n t s of p h a g e T4 D N A p o l y m e r a s e h a v e b e e n o b t a i n e d , d i s p l a y i n g e i t h e r m n t a t o r [5] or a n t i m u t a t o r [6] p r o p e r t i e s . T h e s e m u t a n t s h a v e been extensively characterized both genetically [7-91 a n d b i o c h e m i c a l l y [10-13]. T h e i n t e r e s t i n g (*) Our findings were briefly presented at <
J. N i n i o .
588
feature is that different p o i n t m u t a t i o n s in the DNA p o l y m e r a s e gene have the effect of l o w e r i n g or r a i s i n g s i m u l t a n e o u s l y most of the error-levels that can be m e a s u r e d : s p o n t a n e o u s A.T. ~ G.C or G . C - - - - ~ - A . T t r a n s i t i o n s , t r a n s v e r s i o n s , errors due to the i n c o r p o r a t i o n of base-analogues such as 2-amino p u r i n e or bromo-U, etc. [9]. Yet, the a m i n o a c i d substitutions resulting from the m u t a t i o n s should have different geometrical consequences. This suggests that T4 DNA p o l y m e r a s e makes use of a d i s c r i m i n a t i v e p r o c e d u r e w h i c h can be a p p l i e d w i t h more or less vigour. More precisely, it was proposed that the l d n e t i c b a l a n c e between two steps of the reaction ( p o l y m e r i z i n g activity versus exonuclease activity) was more i m p o r t a n t for DNA p o l y m e r a s e selectivity than the discrim i n a t i v e potentialities of the i n d i v i d u a l steps [13]. U n f o r t u n a t e l y , the authors reached that conclusion on the basis of a kinetic t r e a t m e n t w h i c h is actually i r r e l e v a n t (see the Discussion).
R i b o s o m a l selection. Here again, a n u m b e r of r i b o s o m a l m u t a n t s are k n o w n w h i c h have the effect of c h a n g i n g i n a parallel m a n e r <
UNDELAYED REACTIONS. We start our study of the d i s c r i m i n a t i v e <
ld ke,
E + S ~ "
ES . . . .
~,-E 4. ps
(1)
We assume t h r o u g h o u t the article that such m a c r o s c o p i c r e p r e s e n t a t i o n s are exact, i.e., they reflect a c c u r a t e l y the processes at the m i c r o s c o p i c level. T h e n the k i n e t i c s can be expressed in terms of probabilities. IS? k s dt is the p r o b a b i l i t y dp that a free enzyme molecule w i l l form an enzymesubstrate complex in the i n f i n i t e s i m a l time dr, w i t h s i m i l a r expression for k s a n d k_s. W h e n the complex ES is formed, it disappears exponentially w i t h time a c c o r d i n g to exp (-kS t &st). k S is the r e c i p r o c a l of the mean s t i c k i n g time
BIOCHIMIE, 1975, 57, n ° 5.
~s of the complex, that w o u l d be measured i n the absence of the p r o d u c t - f o r m i n g reaction. Once an enzyme-substrate or an enzyme-analogue association is formed, h o w e v e r t r a n s i t o r y it may be, what is the p r o b a b i l i t y of c o m p l e t i o n of the reactio.n ? F o r a n y short i n t e r v a l of time dt, a fraction k~ decays, w h i l e a fraction k s dt leads to p r o d u c t formation. Thus, the average probability of c o m p l e t i o n of the reaction is quite s i m p l y given by P = k s / ( k s -]- kS1) and by the e q u i v a l e n t expression for the analogue A. If we c o m p a r e the rates at w h i c h substrate and analogue are t r a n s f o r m e d , we have the ratio : Vs ESI k s ps - -
~ -
VA
(2)
[A] ki~ pA
We will define the d i s c r i m i n a t i o n term D as the velocity ratio of Eq (2) divided by the conc e n t r a t i o n ratio :
Vs / VA
1)
(3)
E s I . EAI
In the case of Michaelis kinetics, D is a product of two terms : the ratio of the collision efficien~cies of substrate and analogue, a n d the ratio of the p r o b a b i l i t i e s of completion of the reaction after a collision : k s pS D -(4) k~ pa
This expression can be extended to a n y other m e c h a n i s m of r e a c t i o n for w h i c h there is only one c o n f o r m a t i o n a l state of the enzyme a l l o w i n g an e n t r y of the ligands. Consider the special ease w h e r e k s = k la a n d k s = k 2 = A k2 " W h e n the reaction time I/ko is very short c o m p a r e d to both s t i c k i n g times I/kSl and I / k ~ , D = I. The enzyme does not d i s c r i m i n a t e b e t w e e n substrate and analogue. As the r e a c t i o n time increases w i t h respect to the s t i c k i n g times, d i s c r i m i n a t i o n also increases and reaches the l i m i t D = k~/1Csl = {-sfiA for infinite r e a c t i o n times. Thus, w h e n the only kinetic difference w h i c h is exploited by the enzyme is that of the sticking times, d i s c r i m i n a t i o n is at best i n the ratio of the stictdng times. (Of course, a larger D value can be o b t a i n e r if kS2 > k2)'a The result can be generalized to the case w h e r e there are a d d i t i o n a l i n t e r m e d i a t e stages in the f o r m a t i o n of the p r o d u c t :
k~ kn + I . . . . . ~ : : _ " ~ ' E S ( ~ , ) - - - ~ E ÷ ps
(5)
Enzgme
W h e n tile c o m p l e x ES(~) is f o r m e d , it m a y either dissociate, or yield product with a probab i l i t y P w h i c h is g i v e n b y the e l e g a n t f o r m u l a of K n o r r e a n d M a l y g u i n e [20~. I
P
k 1 (
--I+
[
I +
~-2
....
k
i I+
-" ka
+ k-(n-j~) (t + - - - - ) kn
n]i
589
Discrimination.
(6)
of c o n v e n t i o n a l c h e m i c a l events. T h e n , t o is n o t a fixed time, but a random variable with a mean v a l u e t h a t c a n be c o m p u t e d . T h e f o l l o w i n g is t h e s i m p l e s t e x a m p l e of d e l a y e d r e a c t i o n s w e h a v e b e e n a b l e to c o n s t r u c t . W e s t a r t w i t h a t w o - s u b s t r a t e r e a c t i o n in w h i c h t h e b i n d i n g of o n e s u h s t r a t e h a s no i n f l u e n c e on t h e b i n d i n g of the o t h e r s u b s t r a t e (see Eq. (7)).
kn+l
F r o m t h e r e , it is easy to s h o w that w h e n all the k i n e t i c p a r a m e t e r s of s u b s t r a t e a n d a n a l o g u e s a v e o n e p a i r (k s a n d k ; ' or kS a n d k A) are equal, the d i s c r i m i n a t i o n c a n n o t be b e t t e r t h a n t h e r a t i o of t h e k i n e t i c p a r a m e t e r s : k ~ / k s o r k Ai / k . iS. T h e r e s u l t a p p l i e s to a n y p a i r of k i n e t i c p a r a m e t e r s , a n d to a r e a c t i o n i n v o l v i n g a n y n u m b e r of steps.
DELAYED REACTIONS. In this section we consider the case where b i n d i n g is the d i s c r i m i n a t i v e step. W e c o u l d h a v e said, for a M i c h a e l i s s c h e m e , t h a t t h e e n z y m e is c eomparing>> t h e s t i c k i n g t i m e s ~s and Ta. S u p p o s e n o w t h a t t h e r e a c t i o n s u b s e q u e n t to b i n d i n g of the s u b s t r a t e or t h e a n a l o g u e is d e l a y e d b y a t i m e t o. T h e t i m e s d u r i n g w h i c h the r e a c t i o n s m a y o c c u r are n o w ~'s = ~ s __ to and ( a = ( a __ to . It is c l e a r t h a t t h e r a t i o t ' s / t ' a of t h e s e a p p a r e n t s t i c k i n g t i m e s can be c o n s i d e r a b l y l a r g e r t h a n the r a t i o t s / t a. T h i s r a i s e s t h e q u e s t i o n : a r e snell d e l a y e d r e a c t i o n s c o n c e i v a b l e c h e m i c a l l y ? At first, o n e m a y t h i n k that t h e r e a l i z a t i o n of d e l a y s s h o u l d i n v o l v e effects w h i c h a r e o u t s i d e t h e r a n g e of v a l i d i t y of tile t r a d i t i o n a l c h e m i c a l n o t a t i o n . It t u r n s out h o w e v e r t h a t d e l a y s c a n be o b t a i n e d b y s e q u e n c e s
and we make the E ~ E S 2 b r a n c h of the r e a c t i o n i r r e v e r s i b l e b y c o u p l i n g it to a v e r y rapid destructive process, for instance : ES.)_ + A T P
(very rapid)
>
E -t- $2 + AMP + PP (8) or, if S 2 is A T P , this last r e a c t i o n can be r e p l a c e d b y the in situ d e g r a d a t i o n of A T P . T h e r e a s o n f o r the a d d i t i o n a l (and r e a c t i o n w i l l be m a d e i n t u i t i v e later.
essential)
Let us i n t r o d u c e t h e v a r i o u s p r o b a b i l i t i e s of the d i r e c t t r a n s i t i o n s f r o m one state to a n o t h e r : p : k 2 !S,,!/(k 2 [$2] + k l ) f o r E S i q = k a / ( k a + k 2 + k l) for ES1S 2 r -- k e/(k a + k_2 + k l) for ES1S 2
-~- ESaS 2 _~P, and ~ ES 1
A g a i n w e ask the q u e s t i o n : o n c e ES t is f o r m e d , w h a t is the p r o b a b i l i t y t h a t it w i l l l e a d to P ? W e may have several back and forth motions froul ES1S.~ to ES 1 a n d ES~ to ES1S 2 b e f o r e f o r m i n g the p r o d u c t . T h u s , the o v e r a l l p r o b a b i l i t y is g i v e n as the s u m of the s e r i e s : P
=
1)q + p r p q
+
(pr)2pq
+
......
(9)
Pq 1 --
pr k.~ k s [$21
(lO)
- - k , 2 k a [$2] + k_1 (k 3 + k_2 -}- k 2 IS2]) + k-'21
ES1
ka
ES z BIOCHIMIE, 1975, 57, n ° 5.
5
E + 1'
(7)
J. Ninio.
590
T h e i n t e r e s t i n g f e a t u r e is t h e p r e s e n c e of a q u a d r a t i c k_21 t e r m . Fo.r v e r y s h o r t s t i c k i n g t i m e s ( l a r g e v a l u e s of k 1), a n d if t h e o n l y k i n e t i c differ e n c e w h i c h is e x p l o i t e d is t h a t of s t i c k i n g t i m e s , we have :
I) ~
(ts/ta)2
p r o b a b i l i t y of p r e s e n c e is zero. Sz m a y c o m e , leave, c o m e again, etc. Its p r o b a b i l i t y of p r e s e n c e r i s e s g r a d u a l l y u n t i l it r e a c h e s a m a x i m u m l e v e l
(11)
Thus, analogues with very weak binding c o n s t a n t s a r e p r o c e s s e d a c c o r d i n g to t h e s q u a r e s of t h e s t i c k i n g t i m e s . In o r d e r to d e t e r m i n e m o r e p r e c i s e l y the d o m a i n of v a l i d i t y of t h e q u a d r a t i c effect, o n e has to t a k e i n t o a c c o u n t t h e t w o b r a n c h e s of t h e r e a c t i o n . L e t us c a l l a t h e c o n c e n t r a t i o n of E r e l a tivetoES~: a = [ E ] / ( [ E ] -4- [ E S 2 J ) - - u c a n b e c o m p u t e d t h r o u g h t h e s t e a d y - - state e q u a t i o n s . T h e c o m p l e t e e x p r e s s i o n for P i s : P
Pq = a -I-pr
q -4- ( I - a ) - - I-pr
(12)
T h e q u a d r a t i c e f f e c t is f o u n d in t h e p q / ( I - p r ) t e r m , a n d n o t in t h e q / ( I - p r ) t e r m . T h e i r r e v e r sible c o u p l i n g of E q u a t i o n (8) h a s t h e effect of n m k i n g a close to o n e a n d t h u s is a n e c e s s a r y c o n d i t i o n for t h e v a l i d i t y of E q u a t i o n (10). T h e q u a d r a t i c effect r e q u i r e s a l a r g e v a l u e of k. 1 ( s u p e r i o r to k 3, k 2 a n d k 2 [$2]) b u t n o t too l a r g e a v a l u e e i t h e r , f o r t h e n t h e (I-~) q / ( I - p r ) t e r m w o u l d c e a s e to be n e g l i g i b l e c o m p a r e d to t h e u p q / ( I - p r ) t e r m . T h u s , t h e q u a l i t y of t h e i r r e v e r s i b l e c o u p l i n g ( r e f l e c t e d in t h e v a l u e of a) d e t e r m i n e s h o w f a r in t h e d o m a i n of s h o r t s t i c k i n g times, the a m p l i f i c a t i o n effect m a y h o l d . T h e q u a d r a t i c effect is a b s e n t w h e n t h e t w o stcp r e a c t i o n f o l l o w s a m e c h a n i s m of s h ' i c t l y o r d e r e d a d d i t i o n s u c h as : kl E -4- $ 1 ~
ESj
I~k2 ES1 -4- S,~~.-~>ES1S2 k_2
k3 ~-E -4- P
(13)
T h i s r e a c t i o n c a n be r e w r i t t e n as : kl
E+ 1
ESI
k'2 [S2] >ES1S,,.
K
k3
~ E + P
timo
Fro. 1. - - Intuitive explanation of the kinetic amplification effect. We consider a two-substrates reaction (Equations (7) and (8)). The first substrate binds to the enzyme with a sticking time 0 s larger than the sticking time 0A of the analogue A. ~Ve take as origin of the timeaxis the instant w h e n the complex ES~ or EA is forreed. The probability of presence of $2 on the enzyme (P, in ordinate) increases 'with time. The asymptotic v~tlue of P corresponding to equilibration between arrivals and departures of S~ is k2 [S~]/(k~ [S~l + k_2). The overall probability of the reaction w h e n $1 or A is on the enzyme can be obtained by adding the probabilities of the reaction during consecutive intervals of time. For every small interval of time St, the probability of the reaction is the product of the probability of presence of $2 (the value of P taken on the curve) by a constant ~dt. ~ht is the probability of forming a product during a small interval of time 8t vchen both S~ and S~ are on the enzyme. Thus, the overall probabilities corresponding to sticking time 0 s and 0A are proportional to the areas under the curve. For small O's they vary as the squares of sticking times.
c o r r e s p o n d i n g to e q u i l i b r a t i o n (fig. 1). S u p p o s e w e d i v i d e t h e t i m e - a x i s in s h o r t s u c c e s s i v e i n t e r vals o,f d u r a t i o n St. T h e s t i c k i n g t i m e s of snbs t r a t e a n d a n a l o g u e can be w r i t t e n as NS fit a n d NA 8t r e s p e c t i v e l y , w h e r e N s a n d N A c a n b e t a k e n as i n t e g e r s . D u r i n g an i n t e r v a l of t i m e St, a n d p r o v i d e d b o t h S~ a n d S 1 a r e p r e s e n t on t h e e n z y m e , t h e r e is a p r o b a b i l i t y 8p of m a k i n g t h e p r o d u c t . I n i t i a l l y , t h e p r o b a b i l i t y of p r e s e n c e of S 2 is l o w and what happens makes only a small contribut i o n to t h e total p r o b a b i l i t y of f o r m i n g t h e product.
_
(14) and the discrimination E q u a t i o n (6).
can
be
obtained
from
T h e r a t i o of t h e t w o p r o b a b i l i t i e s f o r s u b s t r a t e a n d a n a l o g u e (if t h e i r s t i c k i n g t i m e s a r e s h o r t e n o u g h ) c a n be a p p r o x i m a t e d b y (see F i g . 1) :
ps/pA Intuitive explanation of the delayed reaclion effect. Let us c o n s i d e r t h a t t h e c o m p l e x E S 1 h a s b e e n f o r m e d at t i m e 0, a n d let us f o l l o ~ t h e e v o l u t i o n of t h e e n , z y m e - s u b s t r a t e c o m p l e x w i t h time. I n i t i a l l y , S 2 is absent f r o m t h e c o m p l e x : its
BIOCHIMIE, 1975, 57, n ° 5.
= (1.4.2.4.3.4. . . . . -4- N s) 8 P / (1.4.2.4.3.4. . . . . -4- NA)SP (NS,/NA) 2 (15) (This was indeed the intuitive consideration w h i c h led us to the c o n s t r u c t i o n of t h e r e a c t i o n scheme).
Enzyme Discrimination. The p r e c e d i n g p r o b a b i l i s t i c d e s c r i p t i o n does not a p p l y to strictly ordered reactions. F o r then, u p o n b i n d i n g of So, S1 is not allowed to leave the complex p r i o r to the d e p a r t u r e of S,), a n d the s t i c k i n g time of S 1 c a n n o t he defined as an entity i n d e p e n d a n t of the b i n d i n g of S~. Thus, the p r o b a b i l i s t i c model i m p l i e d r a n d o m d e p a r t u r e s for S1 a n d S.,. If this c o n d i t i o n is realized, the p r i n c i p l e of m i c r o s c o p i c reversibility requires that the order of b i n d i n g of S1 and S2 be also r a n d o m , a n d we are led to scheme (7). However, we are interested in the u p p e r b r a n c h of the reaction, exclusively. The lower b r a n c h can d i s c r i m i n a t e at best in the ratio of sticking times. F r o m there, the necessity of E q u a t i o n (8) follows n a t u r a l l y .
COMPARISON W I T H HOPFIELD'S SCHEME.
591
In both schemes, there are two gates of exit for the substrate, a l l o w i n g a double-cheek mechanism. Since the substrate could bypass the doublecheck if e n t e r i n g directly through the second gate, an i r r e v e r s i b l e energetic c o u p l i n g is introduced in both schemes. It has the effect of m a k i n g the second gate p r a c t i c a l l y inaccessible for substrate entry. The m a i n difference b e t w e e n the two schemes c o n c e r n s the p o s i t i o n n i n g of the i r r e v e r s i b i l i t y w i t h respect to the last step of the reaction. Physically, while our scheme c o r r e s p o n d s to an a u t h e n t i c time-delay, Hopfield's scheme corresp o n d s to a p u m p i n g effect analogous to the Overhauser effect in N.M.R., a n d was discovered t h r o u g h this analogy (Hopfield, p e r s o n a l communication). DELAYED ESCAPE OF THE PRODUCT. In the p r e c e d i n g example b i n d i n g was the d i s c r i m i n a t i v e step. We p r e s e n t n o w a scheme
Hopfield's scheme can be w r i t t e n a s : ES
(t~i) E AMP + P P i
ES"
>
Let us intrc~duce the e l e m e n t a r y p r o b a b i l i t i e s of t r a n s i t i o n : p = ke./(k,~ -[- k l) for ES -~'ES* q -- k3/(k 3 + k_,2 + k. 4) for ES* ---~" P r : k_z/(k 3 q-- k_,~ + k_~) for ES* --~- ES W h e n the substrate b i n d s to E, it m a y go either to ES w i t h the f r e q u e n c y a : k l / ( k 1 q- k 4) or to ES* w i t h the f r e q u e n c y I-a. Thus, the overall p r o b a b i l i t y of f o r m i n g the p r o d u c t after a collision b e t w e e n the enzyme a n d the substrate is : P :
Pq ~-I-pr
q + (I-a)-I-pr
(17)
Kinetic amplification is o b t a i n e d as i n our example through the p q / ( I - p r ) term. T h e r e is a complete m a t h e m a t i c a l analogy b e t w e e n the two schemes.
BIOCHIMIE, 1975, 57, n ° 5.
E+P
w h e r e i n the amplification effect is exerted u p o n a step of a different nature. The model is directly i n s p i r e d by k n o w n biological facts. Some DNA polymerases Ell 12, 21, Z2] and some aminoaeyl-tRNA ligases [23, 24] are k n o w n to possess a h y d r o l y t i c activity t o w a r d s the p r o d u c t of their reaction, w h i c h is i n d e p e n d a n t of their synthetic activity (the latter b e i n g d r i v e n by the splitting of a p y r o p h o s p h a t e bond). Consider an isoleucyl-tRNA ligase w h i c h has just aminoacylated a molecule of tRNAne w i t h either isoleucine or valine. While the aeylated tRNA is still on the enzyme it is subjected to a h y d r o l y t i c activity with rates kS and k A for Ile-tRNAne a n d val-tRNAI~e respectively. Let us call t the time that the loaded tRNA s p e n d s on the enzyme. The p r o b a b i l i t i e s of h y d r o l y s i s for the two molecules are I-exp(-kSt) and I-exp(-kAt) respectively. It is
J. Ninio.
592
easy to show from there that the ratio of h y d r o lyzed Val-tRNAn~ to hydro!yzed Ile-tRNAIlo~ c a n n o t exceed kA/kS. Howver, this ka/kS ratio is not the relevant one. What is really i m p o r t a n t is the ratio of the surviving molecules of the two kinds. Then, the a n s w e r is quite different. If the ratio of correct to i n c o r r e c t molecules b o u n d to the enzyme before the h y d r o l y t i c step is pS/pA, it becomes after that step :
ps/pa
= pS/p.& e-(kS-kA)t
(18)
This suggests the possibility that an enzyme m a y achieve an e x p o n e n t i a l exploitation of differences in k i n e t i c parameters. Here, we r e a s o n n e d as if the time t available for the h y d r o l y s i s of the p r o d u c t was a fixed time. This c o n d i t i o n is not realized i n the most simple situatic, n, w h e n ~he p r o d u c t leaves the enzyme a c c o r d i n g to firstorder kinetics with a r a t e - c o n s t a n t k. We w o u l d obtain :
ps/pA
= pS/pA. (k ÷ k a ) / ( k + kS)
(19)
a n d the c o n t r i b u t i o n of the p r o o f - r e a d i n g function to d i s c r i m i n a t i o n w o u l d not be better t h a n the ratio kA/kS. However, if the escape of the p r o d u c t is delayed, although t r e m a i n s a r a n d o m variable, its d i s t r i b u t i o n is more stepwise a n d one o b l a i n s an amplification effect. As a practical w a y of a c h i e v i n g a (random) delay, c o n s i d e r the foliow i n g model. Upon a m i u o a c y l a t i o n , the tRNA is firmly hooked to the enzyme. Its d e p a r t u r e r e q u i r e s a e o n f o r m a t i o n a l change of the IRNA, or of the enzyme, or the b i n d i n g of a n o t h e r molecule to the enzyme, for i n s t a n c e the b i n d i n g of a second a m i n o acid [~6, 27~. The overall situation can be described b y the scheme :
the p r e c e d i n g section. W h e n ES~ is formed, the p r o b a b i l i t y of escaping h y d r o l y s i s is o b t a i n e d r e p l a c i n g i n E q u a t i o n (10) k 1 by kS. Now, the amplification effect bears on the h y d r o l y s i s rates, and not on sticking times. THE RANDOM WALK OF A DNA POLYMERASE. The situation w i t h the b i f u n c t i o n a l DNA polymerases presents a d d i t i o n a l i n t e r e s t i n g features. After the template-directed i n c o r p o r a t i o n of a b.ase (n) the enzyme moves one step f o r w a r d for the i n c o r p o r a t i o n of the next base (n-+-1). Meanwhile, the base i n c o r p o r a t e d last is (
ES~
"
Hi
? ES.,-t- Si
(20)
H., w h e r e S1 represents the acylated tRNA, S o stauds for the second a m i n o a e i d and the H's refer to states of the enzyme i m m e d i a t l y after the occnr e n c e of a hydrolysis. (We are not interested here i n k n o w i n g what are the substrates that r e m a i n t e m p o r a r i l y on the enzyme after hydrolysis), The situation presents a formal s i m i l a r i t y to that of
BIOCH1MIE, 1975, 57, n o 5.
forward, go b a c k w a r d several steps, etc. We have to c o n s i d e r all possible trajectories of the DNA p o l y m e r a s e along the template. We have started s t u d y i n g the p r o b l e m both e x p e r i m e n t a l l y and theoretically (J. N. e F. Bernardi, w o r k in progress). Our t r e a t m e n t leads to the following expression :
Enzyme
p s = pS (I_qS)/Ei_K(i.qS) (22) where K is the small root of E q u a t i o n (23) : Kz (I-qS) (I-qa) - - K (I-qSqA) + pSqS + paqA = 0 (23) Setting : a s ~- [I-K(I-qA)l/[I-K(I-qS)~
(24)
one gets ps/pA
= tt SA pS(I_qS)/[pA(I_qX)~
(25)
a s appears as a corrective factor w h i c h may be called the <
(26)
Therefore one can make the peelback term as i m p o r t a n t as one "wishes. However, i n most practical situations, the peelback t e r m is r a t h e r u n i m p o r t a n t , and w i l l be neglected in the forthc o m i n g considerations. The e l e m e n t a r y prc~babilities p's and q's are related to the e l e m e n t a r y kinetic p a r a m e t e r s (the k's). Suppose that k s and k A are the M n e t i c constants for the h y d r o l y s i s of a correct basep a i r a n d the c o r r e s p o n d i n g i n c o r r e c t base-pair as measured in the absence of synthesis. The c o n t r i b u t i o n of the h y d r o l y t i c f u n c t i o n to discrim i n a t i o n (I-qS)/(I-qA) may be larger t h a n kA/k s only if there is a delay in the escape from the exonucleolytic site. W h e t h e r or not a delayed escape effect holds ~,ould d e p e n d on the mechan i s m of i n c o r p o r a t i o n at the head of the enzyme ; that is on the n u m b e r of reversible or i r r e v e r s i b l e steps involved and the b a l a n c e b e t w e e n the kinetic p a r a m e t e r s of the p r o p a g a t i o n reaction and those of the h y d r o l y t i c reaction. D i s c r i m i n a t i o n b e i n g related to the time w h i c h separates two i n c o r p o r a t i o n s , it should depend on the c o n c e n t r a t i o n s of the dNTP's. More precisely, the errors of in.corporation at a n y p o s i t i o n along the c h a i n would i n c r e a s e w i t h i n c r e a s i n g c o n e e n trations of that dNTP w h i c h is to be i n c o r p o r a t e d at the n e x t position. Such an effect can in p r i n ciple be revealed in classically designed experiments [29, 30~. Practically, the exonuelease activity results in the c o n v e r s i o n of nueleoside t r i p h o s p h a t e s into nneleoside m o n o p h o s p h a t e s . If Hopfield's scheme applies to DNA p o l y m e r i z a t i o n , one should also be able to observe a release of nucleoside monophosphate, but governed by different laws than the release due to the exonuclease activity. One may also w o n d e r "why Hop field's scheme could not be applied twice, the p o l y m e r a s e cleaving BIOCHIMIE, 1975, 57, n ° 5.
593
Discrimination.
two phosphates in a row. A c o n s e q u e n c e of a repeated Hop field scheme for DNA p o l y m e r i zation would be the formation of both n u c l e n s i d e d i p h o s p h a t e s and nucleoside m o n o p h o s p h a t e s in the course of the p o l y m e r i z a t i o n reaction. We have p e r f o r m e d a n u m b e r of e x p e r i m e n t s w i t h E. c o l t DNA p o l y m e r a s e I, E. c o l t RNA p o l y m e r a s e and two e u k a r y o t i c DNA polymerases (J. N., F. Bernardi, G. Brun, A. Assairi, M. L a u b e r a n d F. Chapeville, m a n u s c i p t in p r e p a r a t i o n ) . In most cases, we are able to detect a release of nucleoside diphosphate and nucleoside m o n o p h o s p h a t e w h i c h parallels the i n c o r p o r a t i o n reaction. F o r instance, u s i n g E. c o i l DNA p o l y m e r a s e I, Poly (dC) as a template i n the p r e s e n c e of m a n g a n e s e , dGTP a n d dATP as c o m p e t i n g substrates, we followed in parallel e x p e r i m e n t s dGMP i n c o r p o r a t i o n and dGDP and dGMP release a n d also dAMP m i s i n c o r p o r a t i o n and dADP a n d dAMP release. We observed a perfect p a r a l l e l i s m b e t w e e n dGTP c o n s u m p t i o n (resulting m a i n l y in dGlV[P i n c o r p o r a t i o n ) and dATP c o n s u m p t i o n ( m a i n l y converted into dADP). By <
>, we m e a n that the c o n d i t i o n Log (UdATP ( t ) ] / [dATP ( O ) l ) / L o g ([dGTP ( t ) ! / [ d G T P (O)]) = c o n s t a n t held true t h r o u g h o u t the reaction. Furthermore, the c o n s t a n c y of the ratio [ d A D P ] / ([dAMP i n c o r p o r a t e d ] + [dAMP released]) was also verified to a very good a p p r o x i m a t i o n . This a n d other evidence suggest that nueleoside incorp o r a t i o n and c o n v e r s i o n of nucleoside triphosphate into nueleoside d i p h o s p h a t e or monophosphate are alternative outcomes of the i n t e r a c t i o n of a nucleoside t r i p h o s p h a t e v:ith the polymerasetemplate complex in the course of p o l y m e r i z a t i o n . They do not prove yet that Hopfield's i n c o r p o r a tion scheme is correct. The possibility r e m a i n s that nucleoside mono and d i p h o s p h a t e formation are due to a p a r a s i t i c abortive b r a n c h of the p o l y m e r i z a t i o n reaction. DISCUSSION. The p r o b a b i l i s t i c a p p r o a c h that we have used a n d the steady-state t r e a t m e n t of enzyme kinetics are strictly equivalent in their consequences. T h e y are two `ways of expressing the same basic p h e n o m e n a . I n that respect, w h e n G o o d m a n e t al. [28] compute by two different methods errorfrequencies, and then <
J. Ninio.
594
steady-state t r e a l m e n t of the Michaelis equation one w o u l d obtain for i n s t a n c e : VS __~ IS] [E.] k s k s (k~ + k~)
+ kSO
+ (27)
W h e n m a k i n g the ratio vS/vA, most of the terms cancel out a n d we are left with just the terms that we o b t a i n e d directly by p r o b a b i l i t y reasonnings. P r o b a b i l i s t i c equations were developped b y K n o r r e and Malyguine [20] for c o m p a r i n g the rates of two reactions o c c u r r i n g on a same enzyme (for i n s t a n c e the p y r o p h o s p h a t e exchange r e a c t i o n versus the formation of the a m i n o a c y l a d e n y l a t e on an aminoacyl-tRNA ligasc). However, K n o r r e ' s group did not deal w i t h d i s c r i m i n a t i o n problems. A n u m b e r of reports, often misleading, a p p e a r e d r e c e n t l y l i n k i n g a c c u r a c y to kinetics. Thus, Yarus [31] claimed that <
BIOCHIMIE, 1975, 57, n ° 5.
their E q u a t i o n (2). The source of their e r r o r can be traced back to a confusion i n the theoretical p a p e r [28]. W h e n w r i t i n g (p. 425) R(Bj) = t u r n B i / ( i n c Bj + t u r n Bj), the authors identified a p r o b a b i l i t y of h y d r o l y s i s in the e l e m e n t a r y step w i t h an overall p r o b a b i l i t y of h y d r o l y s i s i n the complete process. It is difficult to judge at the onset w h e t h e r or not some e n z y m a t i c systems take advantage of the amplification possibilities that "we have discussed. F o r one t h i n g our analysis stresses the fact that a c c u r a c y m a y d e p e n d c r i t i c a l l y u p o n details to w h i c h no attention w o u l d be paid n o r m a l l y . F o r instance, a n y analysis of the h y d r o l y t i c activity in the aminoacyl-tRNA ligases should take into c o n s i d e r a t i o n the sequence of the events that lead to the d e ? a r t u r e of the tRNA from the enzyme. The reaction of d e g r a d a t i o n of ATP described in E q u a t i o n (8) if observed w o u l d be c o n s i d e r e d as ~'aste, or as due to in vitro artifacts~ One may ask w h y the cell is not w o r k i n g at the highest achievable level of accuracy. A possible a n s w e r is that efficiency a n d a c c u r a c y a p p e a r s o m e h o w a n t i n o m i c . Consider the case of Equation (10). T h e r e is a good d i s c r i m i n a t i o n w h e n k~ is large c o m p a r e d to k 3 and k_~ a n d k2[$2]. T h e n a c c u r a c y may be controlled t h r o u g h the c o n c e n t r a t i o n of S 2. By lo:wering it one makes the reaction more accurate, but at the same time it appears less e f f i c i e n t : it takes more trials i n order to have a successful i n c o r p o r a t i o n . All the reaction schemes that we have consid e r e d involve a last i r r e v e r s i b l e step. T a k i n g into a c c o u n t the terminal r e v e r s i b i l i t y makes the analysis far m o r e complicated, but is a necessity for a valid evaluation of the energetic cost of accuracy. A p a r t i a l solution to the p r o b l e m m a y be p r o v i d e d by m a k i n g the last step of the reaction reversible and i n c l u d i n g an a d d i t i o n a l irreversible step w h i c h represents the net consumption of the p r o d u c t by the next step of the metabolic net into w h i c h the c o n s i d e r e d r e a c t i o n is imbedded.
Acknolwledgments. I am indebted to L. E. Orgel for the discussions from which this paper stemmed, and for going patiently through many of its earlier versions. The work was supported by the Edna MeConnell Clark Foundation and by the C.N.R.S. (France). R~sn~r~. On a 6tudi6 par une approehe probabiliste la relation entre la pr6cision dont un syst6me enzymatique est capable et le m6canisme r6actionnel. Certains rod-
Enzyme Discrimination. eanislnes dans lesquels nne des 6tapes est r e t a r d e r agissent comme des amplificateurs ein6tiques des d i s c r i m i n a t i o n s mol6culaires. La relation entre notre schema de r6action retard6e (1) et le sch6ma de Hopfield (2) est diseut6e. REFERENCES. 1. Ninio, J. (1975) in <(L'Evolution des Macromoldcures Biologiques )> (C. Sadron, Ed.) Editions du C.N.R.S., Paris, in the press. 2. Hopfield, J. J. (1974) Proc. Nut. Acad. Sci., USA, 71, 4135-4139. 3. Koshland, D. E. (1958) Proc. Nat. Acad. Sci., USA, 44, 98-104. 4. W o l t e n d e n , R. (1972) Accounts Chem. Res., 5, 1018. 5. Spryer, J. F., Karam, J. D. ,¢ Lenny, A. B. (1966) Cold Sprin 9 Harbor Syrup. Ouant. Biol., 31, 693-697. 6. Drake, J. W. & Allen, E. F. (1968) Cold Sprinff Harbor Sgmp. Quant. Biol., 33, 339-344. 7. Freese, E. B. & Freese, E. (1967) Proc. Nat. Acad. Sci., USA, 67, 650-657. 8. de Vries, F. A. J., S'wart-Idenburg, Ch. J. H. & de V~Zaard, A. (1972) Molec. Gen. Genet., 117, 60-71. 9. Drake, J. W. (1973) Genetics Supplement, 73, 45-64. 10. Hershfield, M. S. (1973) J. Biol. Chem., 248, 14171423. 11. Nossal, N. G. & Hershfield, M. S. (1973) in DATA Synthesis in Vitro (Wells, R. D. & Inman, R. B., eds.), pp. 47-62, University P a r k Press, Baltimore. 12. Muzyezka, N., Poland, R. L. ~ Bessman, M. J. (1972) J. Biol. Chem., 247, 7116-7122. 13. Bessman, M. J., Muzyczka, N., Goodman, M. F. z: Schnaar, R. L. (1974) J. Mol. Biol., 88, 409-421.
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