II. Transient phase of two-substrate enzyme systems

J. theor. Biol. (1981) 89, 19-35

II. Transient

Phase of Two-substrate

J. GALVEZ?,

R. VARoNt

Enzyme Systems

AND F. GARCIA

CANOVA&

University of Murcia, Murcia, Spain (Received 24 September 1979, and in revised form 29 February 1980) The time dependencein the transient phaseof the two-substrateenzyme systemswhich evolve accordingto the random ternary-complex, ordered ternary-complex, ping-pong bi-bi and Theorell-Chance mechanismshas beenstudied.With this purpose,the equationsderived in paperI have been applied.This hasallowedto proposea method in order to obtain the rate constantvaluesof theseenzyme reactions.In addition, we showthat, from experimental knowledgeof the induction periodsof the ligandspecies,it is possibleto give a method which allows’to discriminate between these mechanisms. 1. Introduction

We have undertaken, with the aid of the method previously describld in paper I of this series, the problem due to the time-dependent behaviour in the transient phase of two-substrate enzyme systems (random ternarycomplex, ordered ternary-complex, ping-pong bi-bi and Theorell-Chance mechanisms). However, we must point out that Hijazi & Laidler (1973a) have already reported on the same problem, although the results here exposed have not been accomplished by these authors. On the other hand, various authors (Dondoroff, Barker & Hassid, 1947; Boyer, 1959; Silverstein & Boyer, 1964; Laidler & Bunting, 1973; Hijazi & Laidler, 1973b) have approached the problem to discriminate the mechanisms of the diverse two-substrate systems. The fundamentals of some methods (Dondoroff et al., 1947; Silverstein & Boyer, 1964; Boyer, 1967) are isotopical interchange; whilst others e.g. Laidler & Bunting (1973) are the study of the steady state of these systems but with inhibition of some of the enzyme species which participate in them and, finally, Hijazi & Laidler (19733) have also applied the premixing techniques in the study of the transient phase of these systems. Here we suggest a new method based on the experimental knowledge of the induction periods of some of the -FLaboratory of Physical Chemistry, Faculty of Science. f Department of Biochemistry. 19

0022-5193/81/050019+17$02.00/0

@ 1981 Academic Press Inc. (London) Ltd.

20

J.

GALVEZ

ET

AL.

ligand species which has the advantage that it is not necessary to subject the species which take part in the enzyme reaction to any previous treatment (isotopical labelled, inhibition or premixing). 2. Mechanisms (A)

DESCRIPTION

OF

THE

MECHANISMS

The mechanisms we will study are:

I k,

E+X+Y

Random ternary-complex mechanism

Ordered ternary-complex mechanism E

Ping-pong

bi-bi mechanism

Theorell-Chance

mechanism

In these mechanisms we make the following changes:

(1)

II.

In the addition,

TRANSIENT

PHASE

21

SYSTEMS

we have for the random ternary-complex

mechanism (2)

ordered ternary-complex

mechanism EY EAB

ping-pong

bi-bi mechanism

Theorell-Chance

EA’= X, EA’B=XXq > ’

(4)

EY =X3.

(3

mechanism

When we refer to these mechanisms, we will call them 1, 2, 3 and 4, respectively. In all mechanisms the diagram of classes remains reduced to one circle. In Fig. 1 we indicate the directed graphs corresponding to these mechanisms. In the four mechanisms the set C of the classes consists of one single class,

X4

X4

(a)

(b)

xQJx4 @lx3 X3

(c) FIG.

1. (a), (b), (c) and (d) directed

Cd)

graphs

of mechanisms

1, 2, 3 and 4.

22

J.

GALVEZ

ET

AL.

Ci, to which belong all the enzyme species involved, so that in ail them

In mechanisms

a =O,

(6)

b =O,

(7)

c = 1.

@I

1,2 and 3 Cl

= {Xl,

x2,

x3,

(9) (10)

x41

m = (41, and in mechanism 4 Cl

= {Xl,

x2,

(11)

X3)

Co= (3).

(B)

INITIAL

(12)

CONDITIONS

We will suppose that at the start of the reaction exists only free enzyme, being x7 its initial concentration (other initial conditions for the enzyme species would not create any difficulty). In order to apply the results of paper I, it is necessary to suppose that substrates Y1 and Y2 are in excess with respect to free enzyme, so that during the whole course of the reaction it is verified, approximately: y1=

YY

y2=y:

The initial restriction.

conditions

(C)

(13)

>*

of products Y3 and Y4 are yt and yi, without any

SYSTEM

OF

DIFFERENTIAL

EQUATIONS

The evolution of the enzyme species with time mechanisms 1, 2 and 3 by the matricial equation

is determined

in

II.

TRANSIENT

being the secular determinant

PHASE

of the matrix K

I&l--S

K22-s

]

K13

K23

K14

K24

K34

K13

= kiyi

K22

= -(K21+

K41=

(1%

K43

’

K44-s1

So, in mechanism

1 we have:

+Kn)

K24)

Kzl = k-1 K24 = k&y: K33 = -(K31+ K31 = k’l K34 = k& K44

1

K42

K32 K33-s

The Kij values depend on the mechanism. Ku = -Wn Kn= kd

K41

K31

KZl

K12

D(s) =

23

SYSTEMS

K34)

= -(K41

+K42

(16)

+K43)

k3

Kd2= kl_;? K43 = k-2 K32=K23=K,4=0 In mechanism

2, the Kij values are: KII

= -Kl2

KIZ

= klv?

K22

= -(K21+

Kzl= K24

K24)

k-1 = kzv;

K33=-

>

K31

K31=

k4

K44

= -(K42

K42

=

+ K43)

k-2

K43

= k3

K32

= K13

= K14

= K41=

K23

= K34

=

0,

07)

24

J.

In mechanism

GALVEZ

ET

AL.

3, Kij is, KII

= -K12

K12

=

kd

Kzz

=

-W21

+K23)

K21 = k-1 K23

=

k2

K33

=

-K34

K34

=

k&

K44

=

-K41

K41=

K13

1.

(18)

k4

= K14

= K24

= K31=

K32

= K42

= K43

= 0,

In mechanisms 1,2 and 3, D(S) = 0 has three non-zero roots: Al, A2 and A3 and one zero root h4, so that from the properties of the determinants it is verified: ~D’1’(0)=A1+A2+A3, (19) ~D(2)(0)=A~A2+A1A3+A2A3,

(20)

1 DC3)(0) = AlA2A3.

(21)

If we make = -CD’l’(0),

(22)

Pj = 1 Dc2’(0),

(23

Qj = -1 D’3’(O),

(24)

Nj

from equations (22)-(24) we obtain: Mechanism

1 N=nlo+nllY~+nl2Y~,

(25)

pl

(26)

=Plo+plly~+p12(y~)2+p13y20+p14(y~)2+PlSy~y~,

QI =

410+q11y~+q12(y~)2+q13y~+q14(y20)2+q15Y~Y20

+416(Y~)2Y~+q17Y~(Y~)2

(27)

being nlo = k-2 + kl2 + kwl + kLl + k3 nll=kl+k2 n12= k; +ki

(28)

II.

TRANSIENT

plo = k-~kl~+ PII= kt(k-z+

PHASE

SYSTEMS

25

(k-1 + k’l )(k-2 + k’z + k3) k’l + kLz + k3) + kz(kLz + k-1 + k3)

p12=k1k2 ~13 ~14 ~15

= k; (k-2 + k::! + k-1 + k3) + k;(k-2 = k’lk; = ktk; +k;k,+kzk;

410 = q11= qn = q13 = q14 =

k-ikL1 (k-z+ kl2 + k3) (kL + ks)(k-lk2+ k’,kl)+ klkz(kLz + k3) (k-z+ kd(k:lk; + k-,k;) k;k; (k-z + k3)

k;k;?(kL = klkzk; q17 = k;k,k;

+ kL, + k3)

>

(29)

J

k-2kLlkl + kl_,kwlk; (30)

+k-l)+k~kh(k-z+k’~)+k~k;k~

415= q16

Mechanism

\

J

2 (31)

(32) (33)

with = km2 + kwl + k3 + k4 n21= kl n22 = k2 1

n2o

(34)

P20 = (k-2 + kd(k-1 + kz,)+ kplk4 ~21= ~22 p23

kl(k-2 + k3 + k4) kz(k3 -t k4) = klk2

=

qzo = k-xkdk-2 + k3) qzz = klkdk-2 + k3) m = kzk3k4 qm = klkz(k3 + k4)

(35)

(36)

26

Mechanism

J.

GALVEZ

ET

AL.

3 &

=n30+ nw?+n3d

(37)

p3

=P30+p31Y~+p32Y20+p33Y~y20

(38)

Q3

=q31~:+q32~:+q33y:y;

(39)

with n30=k-1+kZ+k4 n31= kl n32 = k3 ~30

= kdk-I+

p31=

kl(kz

p32

= ks(k-1

~33

= k&3

(40)

+ kd

(41)

+ k2 + k4) i

q31= k&L q32= k&&-t m = W&+ In turn, the matricial

I

k2)

+ k2) . kd

equation corresponding

(42)

to mechanism

4 is:

(43)

being K11= -K12 K12 = kly: K22 = -(K21

+K23)

K21= k-1

(44)

= by! = -K31 k3

K23 K33 K31=

and the secular determinant: K11-S D(s) =

K12

0

K21

K31

0

K22-s K23

K33-s

(45)

II.

TRANSIENT

PHASE

27

SYSTEMS

whose roots A1 and AZ are non-zero and A3, zero, verifying that: ~D”‘(O)=Al+A2,

(46)

1 d2’(0)

(47)

= Alhz.

For convenience, in the following deductions we will make: P4 = - 1 D’l’(O),

(4%

Q-A= C d2’(0),

(49)

so that from equations (45)-(49) it results: p4=p4o+p41Y~+p42Y2o

(50)

Q4=q4o+q41y~+q42y~+q43y~y~

(51)

being (52)

(53)

(D)

CONCENTRATION

OF

THE

ENZYME

SPECIES

In mechanisms 1,2 and 3 all enzyme species have as precursors X1, X2, X3 and X4, so that for all of them it is verified: &={1,2,3,4}

(i=1,2,3,4)

(54)

vi = {1,2,31

(i = 1,2,3,4)

(55)

and, since Cr is final, Aio # 0. Applying the methods exposed in our paper I of this series, we obtain: xi =Aio+Ail

eh”+Ai2

eAzt+Ai3 eA3’ (i = 1,2,3,4),

(56)

being (57)

28

J. A,

=

GALVEZ

ET

(--lmi(Ah)d

(i = 1,2,3,4;

zh

Ah jIih In mechanism that

AL.

h = 1,2,3).

(58)

(A, -AhI

4, all enzyme species have as precursors Xi, XZ and Xx, so $f ={l,

2,3}

(i = 1,2,3),

(59)

21

(i = 1,2,3).

(60)

={I,

rli

Besides, being Ci the only final class, Aio # 0, and so, xi =Aio+Ail

eA2’ (i = 1,2,3),

eh1’+Ai2

(61)

where A,

=(Ll)‘+‘(~li)OxY

ci =

1

10

2

3)

3,

9

(62)

h = 1,2; p = 1,2; p f h).

(63)

AlAz A,

=

(--l)i&i(Ah)X:

ih

(i = 1,2,3;

Ah&--Ah)

(E)

In mechanisms

CONCENTRATION

OF

THE

PRODUCTS

1,2 and 3 for Y3 and Y4 it is held

u4,

4, = {L&3,41

=

(64)

and thus, ~,={1,2,3}

(s=3or4).

(65)

Besides, as class C1 is a final one, it results asof

(s=3or4)

(66)

which signifies that Y3 and Y4 reach the steady state. As shown in paper I of this series we obtain: y,-y~=a,0t+~,+~eA1r+~eAz’+~e”3’ In mechanism

(s=3or4).

(67)

4, for Y3 as well as for Y4, it follows:

u4* = 44 = {1,2,3)

(68)

Ps = {La.

(69)

and therefore,

II.

TRANSIENT

PHASE

29

SYSTEMS

On the other hand, class Cr is final and, in consequence, both Y3 and Y4 reach also the steady state: y,-y~=a,Ot+Pr+~eA1’+~eAzr

(s=3or4).

(70)

Applying the corresponding equations of paper I of this series, we have elaborated Table 1, where the values of aSo, & and (Y,~ (S = 3,4; h = 1,2,3 TABLE

1

Values of (Y,o, pS and (Y&, in mechanisms [echanism

~30

Fl QI

1

ff40

P3

P4

Ql

P3

a3h

F4--P1(~30 ff30

F2

04h

Fl a3h Ah

2

1, 2, 3 and 4

F3-p2(y30

,j+h

PZa40

@,-Ah)

FS

F2

0130

Q2

Qz

Q2 Ah

F2

3

Fs

-

P3a40

ff30

Q3

Q3

--

pjiihi+b)

P3%0

F3

Fh--P4~~40

p4a40

FZ

Q4

Q4

(A,

-Ad

Ah ,=i,,

(A,-&,)

FIO

0130

Q4

y=~ih(Ap-l\h)

F¶

Q3 Ah ,j+h

4

Ah

(pfh)

(lJfh)

-Ad+Ad

in mechanisms 1,2 and 3; h = 1,2 in mechanism 4) for the four mechanisms have been consigned, being Pi and Qi (i = 1, . . . ,4) defined by equations (26), (32), (38), (.50), (27), (33), (39) and (51). The Fi (i = 1,2,. . . , 10) are: Fl = k3(k--lk;k2+

kLlklk;

+ k;kzk;y;+

klk;kzy:)y:y;x;,

(71)

F2

= kJ&k4yh&~,

(72)

F3

= hk&&~d,

(73)

F4

= kdk;kz+k&bb%~,

(74)

Fs = kMk&+

k-h:d,

(75)

F6 = klkzy:y;x:, F7 = k3{[k;kz(k-I+

(76) k;y;)

+ klk;(k’l

+(k~k2+klk;)hhly~y~x~},

+ kzy:)

(77)

30

J. Fa

GALVEZ

ET

AL.

(78)

= k&Mk4+bAy$&~,

I%= klk2(k3y~+hh)(k~+hh)y~x~, FIO = klk& (F)

(7%

+bJ&%:.

INDUCTION

(80)

PERIODS

OF

THE

SUBSTRATES

AND

THE

PRODUCTS

Let rl, TV, r3, r4 be the induction periods of Y1, Y2, Y3 and Y4 in each of these mechanisms. If we take into account the results exposed in paper I, we obtain the induction periods indicated in Table 2. In this table R, S and W are rather complicated expressions (given in the Appendix), where the rate constants as well as y: and yi are involved. 2

TABLE

Values of the induction periods in mechanisms Mechanism

71

1, 2, 3 and 4

72

73

tV+g k,+k,

2

( -xc+--+-

k-,+k, k&3

k2+k4

3

1

--+-3+-

( k&4

1

P2

YZ >

Qz

1

--+-k3+k4 k&4

P3

k3 YZ >

--+- 1 k4

Q3

74

,+g

1

1 p2 -k,+a,

Pz

Qz p3

QJ

4

At the same time, if dkl = rk - 71 (k, 1 = 1,2,3,4; k # l), and we consider the values of the induction periods given in Table 2, we can build Table 3 of whose utility we will speak later. Hereafter, we will make lim rk =rt

(k = 1,2,3,4),

(81)

(k = 1,2,3,4).

(82)

YT-

lim Tk = ?-z* YS-

(G)

CURVES

OF

THE

VARIATION PRODUCTS

OF WITH

THE THE

CONCENTRATION

OF

THE

TIME

In Fig. 2 the possible curves of the concentration of Y3 and Y4 vs. t are represented. In these plots we have supposed that the roots are real. We

II. TRANSIENT (a)

PHASE

31

SYSTEMS

lb)

(cl

FIG. 2. Possible curves of ys - yp vs. t (s = 3,4). The form of curve ys - yi vs. f can be, in all mechanisms, any of those indicated in (a), (b) and (c) depending on rs being negative, zero or positive. The form of the curve y4- yi vs. t may be any of those indicated in (a), (b) and (c) in mechanism 1, and that indicated in (c) for the others. In mechanism 1, curves y3 - y: and y4 - yi vs. t coincide.

must take in mind that the plots are schematics and, consequently, not made on scale. (H) DETERMINATION

they are

OF THE CONSTANTS

In the determination of the constants, the method suggested by Maguire, Hijazi & Laidler (1974) is used, and so, we must suppose that the roots of D(S) = 0 are real and sufficiently separated (in the practice, one can try to obtain both requisites changing the initial concentrations of the substrates, since the roots depend on those concentrations). In some cases, we also have suggested other methods for the determination of these constants. Mechanism 1 We make 73 74

I =7.

(83)

From equations (81), (82), (83) and from Table 2, 7* =

PI2

q12+416YY

T** =

PI4 414+ql,Y(:’

A plot of l/r*

034) (85)

vs. yx gives a straight line with slope k; and an intercept

32

J.

GALVEZ

ET

AL.

k,+ kL2. A plot of l/7** vs. y? is a straight line with slope k2 and an intercept k3 + kp2. If we represent In IyS- yy --asot -&I (S = 3 or 4) vs. t, we obtain three straight portions with slopes hi, AZ and h3. A plot of -(Al + AZ+ A3) vs. y? gives a straight line with slope kl+ kZ and an intercept kel + kLI + kez+ kla + k3+ (k; + k;)yi. With an analogous plot vs. yi, we obtain a straight line with slope k’l + kk and an intercept k-l+ki,+k-:!+kFz+kj+(k+kz)y:.

In turn, if we represent (S’i/8y?),; vs. y?, we obtain a straight line with an intercept p11+pi5y;. A representation of this intercept vs. yi allows to obtain pll. In the same form, from the representation of (8P1/8y$),: vs. yz” we can determine p13. With these facts ail the constants may be determined. Mechanism

2

Here ~-2 is 74

* -P**+P23Yi q21 +q23yi’

036)

A plot of l/7: vs. yi gives the curve represented in Fig. 3 which allows to determine k3 + k4. If we represent -d i3 vs. l/y; we obtain a straight line with slope (k-2+ k3)/k2k3 and an intercept l/k3. A plot of -(Al +A2+A3) vs. yy, remaining yz constant, is a straight line with slope kl and an intercept k-2 + kwl + k3 + k4+ kzy;. A representation of this intercept vs. yg gives a straight line with slope k2 and an intercept kp2 + kMI + k3 + k4. From these facts all constants may be determined.

kq ( k-2 +k3) k-2+k3+k4

FIG.

3. Plot of l/72

vs. yg in mechanism

2.

II.

Mechanism

TRANSIENT

PHASE

33

SYSTEMS

3

If we represent --& vs. l/y: we obtain a straight line with slope l/k3 and an intercept l/k4. On the other hand, -d13 coincides with l/kz. A plot of -(A1 + A2 + h3) vs. y 7, with remaining y: constant, gives a straight line with slope kl and an intercept kpI + kz + kq + k3yi. From these facts all constants may be determined. Mechanism

4

A plot of l/7$ vs. yi coincides with a straight line with slope kZ and an intercept kJ. A representation of l/7:* vs. y? gives a straight line with slope kI. A plot of -(hl+A2) vs. y?, with y: remaining constant, gives a straight line with an intercept kA1 + k3 + kzy:. From these facts all constants may be determined. 3. Discrimination

Between These Mechanisms

If we take into account the fact that expressions R - S, R - W and S - W are not linear functions of l/y:, Table 3 is of great utility for discriminating TABLE

Values [echanism

d12

R-S

1

2

k_,+k, -k,k,s

1

1 - ( -+zz k3

of dkl in

3

mechanisms

dn

44

R-W

R-W

k-,+k,‘l

1

( k,+k,

--+--ii

1,2,3

and 4 dzs

d24

s-w k-,+k,

1

s-w 1

>

--

-- k,+k,

between the mechanisms studied above and with this purpose we give some examples. So, for example, if we plot d12 and d13 (for known T], 72 and TV)vs. l/y: we obtain in both cases the same straight line through the origin. There is therefore no doubt that the mechanism according to these results is mechanism 4. Contrary to this, if the same representation provides in the first case a straight line of negative slope not passing through the origin and in the second one a straight line parallel to the abscissa, the mechanism in accordance with these is that of mechanism 3. The possibilities of the horizontal use of this table are, as we see, very great.

d 34 0 1

J. GALVEZ

34

TABLE

ET

AL.

4

Additional criteria of discrimination if dr2 is known Mechanism 1 2 3 4

42 vs. 11~; Not straight Straight on the origin Straight not passing through the origin Straight on the origin

slope of l/72* vs. l/y: Variable Constant

If we dispose of only one value of dkl, we must use the table by columns. So, knowing d13, and representing this value vs. l/y;, we obtain a straight line parallel to the abscissa, the mechanism may be that of 3. If the straight line passes through the origin, the possible mechanism is 4; if the straight line does not pass through the origin and has a negative slope, the possible mechanism is 2 and, finally, if none of these occurs, the mechanism may be 1. Knowing d13 or d23 gives immediate information about the possible mechanisms. Knowing d14or dz4 does not provide, in an indicative way, any information. Knowing dlz or ds4provides information, but it is necessary to recur to other additional facts. In short, from Table 2 and equations (32), (33), (35), (36), (50)-(53) and (81) we obtain the criteria of discrimination shown in Tables 4 and 5. TABLES

Additional criteria of discrimination if d34 is known Mechanism 1 2 3 4

44 0 Constant Linear function of l/y; Constant

slope of l/r:

vs. yz

Variable Constant

REFERENCES BOYER, P. D. (1959). Arch. Biochem. Biophys. 82, 387. DONDOROFF,M.,BARKER,H. A.&HAssID,W.Z.(~~~~).J. biol. Chem. 168,725. HIJAZI, N. H. & LAIDLER, K. J. (1973a). Can. J. B&hem. 51,832. HUAZI, N. H. & LAIDLER, K. J. (19736). Biochim. biophys. Actu 315,209. LAIDLER, K. J. & BUNTING, P. S. (1973). The Chemical Kinetics of Enzyme Action, 2nd edn., p. 125. Oxford: Clarendon Press. MAGUIRE,R.J.,HIJAZI,N.H. & LAIDLER,K.J. (1974).Biochim. biophys. Acta 341,l. SILVERSTEIN,E.& BoYER,P.D.(~~~~).J. bioi. Chem.239,3908.

II.

TRANSIENT

PHASE

SYSTEMS

35

APPENDIX Taking into account the methods illustrated obtain:

in paper I, it is possible to

(Al) WI 63) (~ll)l=-k’lk-l-(k~l+k-l)(k’z+k-~+k~)-(k-lkz+kzk,+k1~k~)y~ - (k’lk; (a&

= ki(k12

(aI&

= -k;

(a14h

= (k&b

+ k;k3 + k-zk;)y;

- k;kzy;y;,

+ kL1 + k-2+ k3+ kzy:)y:,

(A4) (A5)

(kl_z + k-1 + k-2+ k3+ k;y;)y;, +k;kdyi’y;,

646) (A7)

(~11)o=-kf-lk-l(k~2+k-2+ks)-k-lkz(k,+k’,)y~ - k’lk;(k-2+ (a&=

kJk’l(k’:!

k3)y;-

khkzk3y:y;

+k-2+ks)+kz(k’2

VW +k3)y:]y;+k;klzkzy:y;

(A9)

(uls)o=-k;[k-l(kl2+kk-2+k3)+k~(k-2+k3)y~]y~ - kLk;y:y;, (a1410

= [klk;fkL1 +kzy~)+k~kz(k-lk;y~)ly~y~.

WO)

(All)