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Relaxation spectra of two-step mechanisms as measured by the nadh-binding to glutamate dehydrogenase
BULLETIN
OF
MATHEMATICAL
BIOLOGY
VOLUME 39, 1977
R E L A X A T I O N SPECTRA OF TWO-STEP MECHANISMS AS MEASURED BY THE N A D t t - B I N D I N G TO GLUTAMATE D E H Y D R O G E N A S E
[]MANFRED KEMPFLE
Physikalisch-chemische Abtcilang des Physiologisch-ehemisehen Instituts der Universitgt, Nussallee 11, 5300 Bonn, Germany
Temperature-jump relaxation experiments performed with glutamate dehydrogenase and the reduced coenzyme nicotinamide adenosine dinueleotide show two clearly separated relaxation times. Three of the simplest and most plausible mechanisms attributable to two relaxation times are treated here theoretically and the concentration dependence of the relaxation times is determined in each case with different experimental conditions. This makes it possible to distinguish between the mechanisms and also to determine the rate constants of the different reaction steps.
Relaxation kinetic experiments are not only a powerful tool in determining the single rate constants of chemical reaction steps but also in deciding between different possible reaction mechanisms. This paper deals with the second case: it will be shown from the concentration dependence of the measured relaxation times we are able to conclude which reaction mechanism is present. For those who are not familiar with relaxation kinetics a very brief introduction is given. For more detailed information they should refer to recent publications (Eigen and DeMaeyer, 1963; Czerlinsky, 1966; Hammes, 1968; Yapel and Lumry, 1971). The concept of relaxation in a physical process relates to the time delay between a sudden change in external conditions and the readjustment of the system in equilibrium to the change. Almost any chemical equilibrium can be 297
298
MANF~ED KEMPFLE
perturbed rapidly by stepwise or periodical changes of external parameters according to the van t'Hoff equation
(31nK~ /
(3 In K'~ ,
lnK
=
/O l n K \ ...,
where K represents the equilibrium constant of the reaction, T the temperature, p the pressure, and E the electric field strength. The following internal re-equilibration (the "relaxation process") is now characterized by a time constant z, the so-called relaxation time, and can be monitored by suitable techniques. I f this chemical re-equilibration process includes several reactions, there is a spectrum of time constants (the relaxation spectrum) which can be expressed in terms of the rate constants by welldefined mathematical transformations (Eigen, 1957). The analysis of the relaxation phenomena becomes fairly simple if the perturbation is kept sma]l to admit the linearization of the rate equations in terms of deviations of the concentrations from their reference values (deviations not larger than 10%). The one-step equilibrium reaction between an enzyme E and a ligand L is a useful introduction to the general principle in the derivation of the expression for the relaxation times, because of its simplicity:
~,2 E+L ~EL,
K =-
/~2,
k~: k12
-
[E][L] [EL]
(1)
where [E], [L], [EL] are the equilibrium concentrations and k:2, k2: are the rate constants for the forward (from left to right in (1)) and for the backward reaction step respectively. As a consequence of a rapidly applied perturbation the system is suddenly produced in a nonequilibrium state. From it the system relaxes to a new equilibrium. The expression for the occurring single relaxation time of this mechanism is obtained by analyzing the time-dependence of system (1) as it shifts to achieve its new equilibrium conditions. Let us assume that the perturbation has been carried out by a temperature jump and that the equilibrium constant K for the reaction (1) is dependent on temperature (due to the van t'Hoff equation AH 0 ¢ 0). It is immaterial whether AH ° is positive or negative, since the mathematical expression for the relaxation time is independent of the direction in which the equilibrium shifts after the perturbation. Using the law of mass action (the general principle of reaction kinetics) we can describe the rate equations for the equilibrium dE dt
dL dt
d[EL] dt
-
k:2E.L-k~:C,
C = EL.
(2)
RELAXATION If we
SPECTRA
OF TWO-STEP
carry out only small perturbations,
MECHANISMS
the instantaneous
concentrations
E, L, and EL may be expressed as the sum of the time-independent concentration and a small time-dependent concentration term: E = E-F6F,,L = L-t-6L, EL = EL+sEL,
299
C -= EL.
equilibrium (3)
E, L, and C are the equilibrium concentrations at the higher T-jump temperature. From the conservation expressions Eo=
E+C
= E+C,
LO = L + C
= L+C,
(4)
where E ° and L ° represent the initial total concentrations of E and L in our system, we obtain the relation between the small time-dependent concentrations 6~, 6L, 6~L by substituting (3) into (4):
E+6~+C+6EL = L+JL+C+JEL
E+C,
-= L q - C .
Therefore, 6E-{-6EL -~ 0;
6L-{-6EL --~ O.
(5)
From this we can express the 6 z , 6L, 5~L by a single term 6 : 6 = 6~ = 6L = -- 6EL. With this from (3) we get E = ETS,
L = L-FS, C = C-6.
(6)
Substituting this again into the rate equations (2) gives d(E + 6) dt
k12(E + 6)(L + 6) - k 2 1 ( C - 6).
-
(7)
Expanding and rearranging results in d6 - ~l-t = kl~E.L - k21C + [k~(E + L) + ~.fl. 6 + k~2- 63. The first two terms (not containing 6) cancel out since they are equal to each other (see (1)). Now the great mathematical simplicity of the method is due to the small value of 6~' which can then be neglected. This procedure results in a single first-order differential equation d6 -
d--t = [ k ~ 2 ( E + L ) + k z ~ ] . 6
1 = k.6 = -.6.~
(8)
The coefficient k of 6 is the reciprocal relaxation time given by 1
k = - = kl~(E+L) +kzl.
(0)
300
M A N F I ~ E D KEIVfPFLE
This is a concentration-dependent term for 1/z allowing the separate determination of the rate constants k12 and k21 of system (1). A plot of the reciprocals of the measured relaxation times 1/z vs the sum E + L of the correspondent concentrations of the reaction partners yields the recombination rate constant k12 from the slope of the straight line and the dissociation rate constant k21 from the intercept with the ordinate (Figure 1). But since it is not always possible to calculate the equilibrium concentrations E + L, the relaxation times m a y also be expressed in terms of the total "weighed in" concentrations. However these terms are much more complicated (Czerlinsky 1966).
1 1;
E.L
k12 ~ EL k21
1 -_ k21 " k12 ( [E]f + [C]f) ~-K d = k2.__[ k12
(
[~]f ° [tYlf)
F i g u r e 1. L i n e a r relationship b e t w e e n t h e first reciprocal relaxation t i m e and the s u m of t h e free concentrations of the reactants at equilibrium. This plot allows t h e d e t e r m i n a t i o n of t h e rate constants for t h e first .(fast) isolated step a n d t h u s t h e calculation of the dissociation (or association) constant
After this simple example and the calculation of its relaxation time we change to a much more complicated system. T-jump relaxation experiments of the reduced nicotineamide-adenine-dinucleotide (NADH) binding to glutamate dehydrogenase (GIuDH) from beef liver show two clearly separated relaxation times when we measure the fluorescence change of the coenzyme due to binding (Kempfle et al., 1974). In general, each relaxation time corresponds to at least one reaction step (whereas the opposite m a y not be valid--see Eigen, 1967). Therefore we have to consider at least a two-step mechanism in our case. In recent publications such two-step reactions have been treated (Czerlinsky, 1966 ; Hammes, 1968; Yapel and Lumry, 1971) and the concentration dependence of the relaxation times in those cases was shown. Our case is different from those of recent publications,
RELAXATION
SPECTRA OF TWO-STEP MECHANISMS
301
since we find the two relaxation times with only one substrata. Therefore the conservation conditions become somewhat changed relative to the published results. Here three different cases yielding two relaxation times are considered. It will be shown that from the concentration dependence of the second--longer-relaxation time it is possible to decide between the different mechanisms. The following three mechanisms are considered here: (I) A binding reaction of the NADH-ligand L to the enzyme E is followed b y a monomolecular interconversion of the EL-complex k~2
k23
E + L ~2~ E L ~32 ~ EL*,
(10)
where E = enzyme, L = ligand, E L = enzyme-ligand complex before conformation change, EL* = enzyme-ligand complex after conformation change and the k~3 are the rate constants in the different steps. The other two cases must be considered owing to the fact that experimental data (Huang and Frieden, 1969; Pantaloni and Dessen, 1969; Koberstein and Sund, 1973) show two N A D H binding sites on each of the six subunits which build up the G]uDH molecule (Eisenberg and Tomkins, 1968; Mosebach and Kempfle, 1969). Thus we have to treat: (II) Two binding sites completely independent from another--we can treat this case formally like two different enzymes competing for only one substrata: E I + L ~ EL1, ~2~
E2+L ~
/¢3~
E~L,
(11)
where El, E~ are the different binding sites, ElL, E2L the different enzymeligand complexes and kij are again the rate constants for the isolated steps. (III) Binding of the N A D H proceeds in a consecutive manner, the second molecule can only be bound when the first has been taken up. Here we are able to consider all cases of any kind of cooperativity: E+L
/c12
~- EL,
/~21
EL+L
k23
~- ELg,
]¢32
(12)
where E L is the enzyme molecule with one molecule of N A D H bound, this takes up another N A D H molecule leading to EL~. Cooperativity appears in different values of the rate constants. The derivation of the expressions for the relaxation times and their concentration dependence is given in a more detailed fashion for the first case; for the two others we give the results only. After this, all results are compared with each other,leading to the decision asto which of the three mechanisms is present in our experimental case.
302
MANFI~ED KEMPFLE
M e c h a n i s m I . Let us calculate the zi for model (I):
E+L
]/~12
k23
k21
k32
~ E L ~ EL*,
(10)
with /c21 [E][L] K21 - ]c12 - [EL---~'
/ca2 [EL] Ks2 - ]c23 - [EL*]"
We can derive four possible differential equations for the rate equations but only two of these are independent. We choose dE = l c 1 9 E ' L - k21C dt
---
dC* dt
= k 2 3 E ' L - /~a2C*
(13)
We see that if there is any coupling of the two steps both terms are no longer independent from one another. Now we have the same procedure as in the case of the single step mechanism already treated. The conservation relations are E o = E+C+C*
= R,+C+C*,
L o = L+C+C*
= L+C+C*,
(14)
where E 0 and L 0 represent the total "weighed-in" concentrations of enzyme and substrate respectively. Again using the method developed in the equations before, the rate expressions in (13) can be linearized b y introducing timeindependent and time-dependent terms for the instantaneous concentrations. (~S "~ -- 5 E L - - (~EL*,
5 L "~ -- 5 . E L - - 5 E L * .
(15)
Introducing for the concentration changes 58 = 5L = X 1 and 5EL*-----X2 with the conservation condition (~E-~ (~EL-]- S E L * .-~ O,
(16)
5 E L ~- -- 5 E - - 5EL* ~-- -- (Xl"Jr X2).
(17)
we get The rate equations after substitution and linearization read dS~ dt
dX1 - ~ 1 ( X 1 + X2) + k12(2, + L)XI, dt
d6EL*
dX2
dt
dt
- k 3 2 X 2 - k~3(X1 + X2).
RELAXATION
SPECTRA
OF TWO-STEP
MECHANISMS
303
Further rearrangement leads to dX1 dt
= X 1 = - (k2~ + k ~ ( ] £ + L ) ) X ~ -
k~X~,
dX2 dt
-
X2 =
- k ~ a X 1 - (k28 + k39.)X2.
(18)
This we can generalize and write X 1 = a ~ l X l + cq~X~,
~ 2 ~--- ~21X1~- g22X2,
(19)
with ~1~ =
- (k2~ + k ~ ( / ~
+ L)).
~12 ----- --/~21, ~21 ----- - - ~ 2 3 ,
a~2 =
(19a)
- (k,.a + k3~.).
It should be noted that (19) is valid for all two-step mechanisms but the a 0 are different in the different reaction types. To determine the expressions for the two relaxation times characterizing mechanism (I) we have to solve the two simultaneous differential equations (19) for X1 and Xg.. The solutions for X1 and X2 are X1 = A1 e - k i t + A 2 e-k2 t,
X2 = B1 e - k i t + B 2 e-k2 t,
(20)
where kl and k2 are the reciprocals of the two relaxations times zl and z2 and A1, A2, B2 and B1 are constants. B u t here it is of interest only to obtain the relaxation times. We obtain solutions of the form shown in (20) if kl and k2 are chosen to satisfy the following determinantal equation ~11 - - k
Ct12
a9,1
ag.,. - k
-~-- 0
(20)
or ]c2 - (all + a22)/c + (a11~22- area21) = 0. This yields the exact expressions for the two relaxation times
=
TI,II
kl,2 ----- -
2
Jr --
( ~ i i ~ 2 2 - - a12~21)-
(22)
From this we get the solutions in final form when the substitutions for the ~ j are made from (19a). This yields the two quantities 1
1 4
=
--(~11+~22)
= k12(~+L)+k21÷k23+ka2
(23)
304
MANFRED KEMI:)FLE
and 1
1 (~ii~22-
. . . . I~I TII
~2i~12)
= k12(E' + L ) ( k 2 a + k32) + k21]~32 •
(24)
I f we plot both expressions vs (E + L) we obtain two straight lines: the slopes y i e l d k12 a n d k12(]~23 + k32) .respectively and the intercepts ]c21 + ]C23 + k32 and k~ika~ respectively. From this all four rate constants can be determined d i r e c t l y : k12 -- slope of (23); ]c2i = intercept of (23)- (slope of (24))/(slope of (23)); k32 = (intercept of (24)/(k21); k2a = intercept of (23)-(k21+k32). Experimentally the two relaxation times have often very different values • 2 >> ~1 o r 1/,1 >> 1/,2. ( I n o u r case z2 is about fifty times larger than *i!) This leads to a great simplification because then (0(11+C(22) 2 >> 4((Xll~22-al~.a2x) is valid. The square root term in (22) m a y be expanded using the binomial theorem. With only the first two terms we have l
-
( a l l + (~22)2 _+ [(~11 "4- (~22)2 -- 2((~11a22 -- (X12~21)]
k~,2 =
"CI,II
(25)
2 ( a l l + a22)
this leads to 1/zi = a l l + ~22
( + ) sign of the term in (25),
1/¢2 = (¢¢11922 - 0¢120¢21)/(c¢11 + a22)
(--) sign of the term in (25).
(26)
I f we assume further that the first step will be the fast one, this means 0~11 ~ ~22
(27)
and we get 1/Vl---- (~11,
l / z 2 = a22--(C(12~21)/(1/Z1).
(28)
This leads to the following expressions for the relaxation times of mechanisms (I):
1 -
-
=
k21+ k12(~ + L),
T1
] z2
E+L = k32 + k28
K2i+E+L
•
(29)
From plots of the 1/z's versus the concentrations it is possible to determine all four rate constants. This mechanism (I) was treated extensively here because the basic equations will later be used again, but only (23), (24) and (29) of this treatment deal with mechanism (I). Now we can handle mechanisms (II) and (III) quite easily.
RELAXATION SPECTRA OF TWO-STEP MECItANISMS
305
Mechanism (II) kl2
k23
E I + L ~ ElL,
E u + L ~ E2L.
(11)
This leads to the following differential equations for the first and second steps : dE1
--
dt dE 2
-
dt
kl2E1L + k21C1,
k23E2L + k32C2.
-
(30)
The relations among the concentration changes are E°
=
El+C1
L°
=
L+CI+C2
=
E2+C2,
(31)
given b y
~E1 = (~L
=
- - (~EIL,
(~E2 =
--~EtL--~EsL
- - (~E2L,
~- ~E~'4-~E 2.
(32)
This leads to
dgE1 d~ -
(kls(E1 + L) + k~l)SE1 - klsE1.5E2,
dfE2
(k23Eu)SE~- (k23(E2 + L) + k82)5~2.
d--/- -
(33)
From this we obtain the following coefficients of the matrix (19), 0~11 =
-
-
(]Cl2(El-{- L) + ]621),
~12 = -]c12E1~ ~z21 =
~22 =
--/C28~2, -
(k23(E2 + L ) + k32).
(34)
The solution .leads to the relaxation times given in 1
4
•1
1
=
- ( a l l + a22)
T2
= (1c12(~1+ L) + kSl} + {ksa(E2 + L) + ka2},
1 1 • i
"~2
CZII(Z22- (212~21
(]C12(~ 1 + L ) + ]~21} " {]~23(E2 -]- L) + ]¢32} - ks~klsE1Es.
(35)
306
1KANFREDKEMPFLE
Again we have 1/Zl > l/z2 ~ all > a22 and we obtain 1 -- k12(/~1-4- L ) + k21,
1
k23 1 L +
•2
[
~2 ] E1 I + k a ~ 1 + - L + K21 _1
(36)
Mechanism (III) kt2
E+L
k23
EL+L ~EL2,
~EL,
~21
k31
(12)
leads to the following ~wo differential equations: dE
dt
-
k l 2 E ' L + k21C', C' = [EL],
dEL2 = k2aC'L-k82[C"], C"=
dt
[EL2].
(37)
The relations among the concentration changes are from E o = E+C'+C", L o = L+C'+2C", ~EL =
--~E--(~EL 2,
5 L = 5 E - - ( ~ E L 2,
Introducing this into (37) yields, if 0 ' = [EL], d6E
dt
--
dSEL2 dt
(kl2(E 4- L) -4-k21)bE - (k21 - kl2E) • gEL2, -t
(k2a(L - C ))5E -- (k2a(L + C') + ka2)@L2,
(39)
leading to these coefficients of the m a t r i x
all = - (k12(N' + L) + k ~ ) , a12 = -- (1¢21--ki2E) a2i =
- (kza(L-
az2 =
- (k2a(L + 0") + ka2).
C ' ) ),
(40)
RELAXAT~O~q
S P ] ~ C T I ~ A O:F T W O - S T E P
MEC~IA~NISMS
307
From experimental knowledge of the two widely separated relaxation times (l/z1 >> l/z2) we have to introduce (211 >> (2~. which results in 1
1 --
z2
- (211 = /~12(E+L)+/c21, g12(221 ~_~ ( 2 2 2 - - ~
1/~1
]c~a[(L(C' ÷ L) ÷ 2B} ÷ 2K210'] E+L+K21
= ]ca2+
(41)
These three expressions for the second relaxation time are quite complicated. Furthermore the concentration terms E, L and C' respectively refer to the concentrations of free enzyme, coenzyme and correspondent complex present at equilibrium. Calculation of these quantities from the known total "weighed-in" concentrations requires knowledge of the equilibrium constants K21 = k21/k12 and K32 = k32/k23. But in the case of the GluDH. N A D H complex the values for K,j given in literature vary very great]y (Bayley and Radda, 1966 ; Yielding and Holt, 1967; Malcolm, 1972; Krause et al., 1974) due to different experimental conditions. It is therefore not possible to calculate E, L and C' in this case. B u t there are two possible ways to solve this problem. The first is to use the relationship between the equilibrium concentrations and the total "weighedin" concentrations. But these expressions are fairly complicated (for example the book of Czerlinsky (1966) is recommended) so that this procedure is not given here. But in principle it may be possible to express these quantities for the relaxation times b y the total reagent concentrations. The other method is to use a large excess of one rectant such that the sum of E and L m a y be equated with the total reagent in excess, either by E o or L o. This w a y is gone in our further treatment. I t can be shown that if the first step is very fast in all three cases the first relaxation times are identical. Therefore only the second will be treated further. I t can be seen that it is possible to decide from the concentration dependence of this time what mechanism is present. (a) E0 >> L0 Mechanism (I): (29) leads to 1 --
ze
E0 =
/ c 3 2 + ]~23"
K21 + E °"
(42)
Mechanism (II): (36) with E ° ~ E ° m E ° yields 1 E0 z2 = k32 + k23. K21 K21 + E e"
(43)
308
iVIA17FI%E D KEMPFLE
and Mechanism (III): (41) gives
1 z2
-- ]c32 + 1c23
2(LE + K210') K21 + E °
(44)
The numerator is the expression for equilibrium E L = K210 thus we have 1 "C2
E o -
k32 +/c23- 4L
K21 + E 0"
(45)
In the first two eases these simplifications lead to straight lines if we plot 1/~2 versus EO/(K21+ E 0) but with different slopes: slope of (42) = /c23,
slope of (43) = /c23.K21
whereas the intercept ks2 remains identical. Therefore we cannot discriminate mechanism (I) and (II) from this procedure but from amplitude considerations we are able to distinguish these mechanisms (Eigen, personal communication):
•,x k a
E.O
Eo +
Figure 2. E 0 >> L 0 In (I) the amplitude of the second relaxation time becomes larger and larger up to a constant value with increasing enzyme concentration due to increasing and saturating amounts of C and C* complexes. In (II) the amplitude of the second relaxation time must vanish with increasing concentration due to the saturation of the binding sites. The mechanism (III) shows a very interesting behaviour. Since we have to consider in (45) the concentration of the free ligand it becomes evident t h a t this concentration/~, goes through a maximum and at high enzyme concentrations it becomes zero. Thus we have the shape of the second relaxation time of mechanism (III) shown in Figure 2. From this plot we can only see the concentration dependence but we can h a r d l y calculate k23. From these results we can already conclude t h a t in our experimental ease only mechanism (II) is valid. Because the plot vs E°/(E ° +K2I) is a straight line and the amplitudes of the relaxation process go to zero.
R E L A X A T I O N SPECTRA OF TWO-STEP MECHANISMS
309
But in general we can introduce a further simplification: (b) L 0 >> E 0 # O. Mechanism (I) leads to 1
Z0
-*2 -=
(46)
] c 3 2 + K 2 1 + L °"
Mechanism (II) leads to 1
(47)
= k39,+ k~aL °
T2
and mechanism (III) to
1 •2
"~2
(L0)2 =
k82 + k23
(4s)
K21 + L0"
[L° ] >> [E°]
~
/
fi°:> mechanism[I) ~; ~.7/_I...~-'TJ--m--e c~a hni-'= k 3 2 * k23
k3z
i --
,
[L°]
m=
F i g u r e 3. Concentration dependence of t h e second reciprocal r e l a x a t i o n t i m e of t h e three two-step b i a 0 m g m e c h a n i s m s as ,tiseussed in t h e t e x t , w i t h ligand c o n c e n t r a t i o n L 0 in excess over t h e e n z y m e concentration. This plot d e m o n s t r a t e s t h a t it is possible f r o m it to distinguish the different reaction mechanisms
These three plots show different behaviour if we plot 1/z2 vs the concentration in excess L 0 (Figure 3), so that we can see what mechanism is present. Mechanism (I) (46) levels off at high concentrations resulting in a straight line independent from concentration with a constant value k3~+k2~; thus it is possible to determine these values. Mechanism (II) (47) results in a straight line with a slope ke~ and an intercept k32. Mechanism (III) (48) at last has at low concentrations the slope zero but it increases and results in a straight line at high concentrations (L0 >> K~I) parallel to this of mechanism (II) with the slope k23. Thus we see that it is not only possible to determine the rate constants for any known reaction mechanism b u t possible too to distinguish between unknown mechanisms. I t is known that steady-state experiments allow 'only
310
MANFI~ED KEMPFLE
the measurement of the ratios of these single step rate constants (the equilibrium constants) or the overall value of coupled reactions (not the isolated single steps). Thus relaxation kinetic experiments are a powerful tool in d i s c r i m i n a t i n g d i f f e r e n t p o s s i b l e m e c h a n i s m s (here t h e t w o s t e p m e c h a n i s m s ) and in determining their rate constants. An EMB0
s h o r t - t e r m f e l l o w s h i p is g l a d l y a c k n o w l e d g e d .
LITERATURE Bayley, P. M. and G. K. Radda. 1966. "Conformational changes and the regulation of glutamate dehydrogenase a c t i v i t y . " Biochem. J . 98, 105-111. Czerlinsky, G. 1966. Chemical Relaxation. New York: Dekker. Eigen, M. 1957. "Determination of general and specific ionic interactions in solution." Discuss. Farad. Soc., 24, 25-36. - - - - - - and L. deMaeyer. 1963. "Very rapid reactions in solution. Relaxation Methods." I n Techniques of organic chemistry, Vol. 8, p a r t 2, pp. 895-1054 (Eds. S. L. Fries, E. S. Lewis, and A. Weissberger). New York: Interscience. • 1967. "Kinetics of reaction control and information transfer in enzymes and nucleic acids." I a Nobel Symposium No. 5, pp. 333-369 (Ed. S. Claesson). Stockholm: Almquist and Wiksell. Eisenberg, H. and G. M. Tomkins. 1968. "Molecular weight of the subunits, oligomeric and associated forms of bovine liver glutamate dehydrogenase." J. Melee. Biol., 31, 37-49. Hammes, G. G. 1968. "Relaxation spectrometry of biological systems." I n Advances in P~otein Chemistry, Vol. 23, pp. 1-58. (Eds. C. B. Anfmsen, J. T. Edsall, M. L. Anson, and F. M. Richards). New York: Academic Press. Huang, Ch. Y. and C. Frieden. 1969. " R a t e s of GDP-induced and GTP-induced depolimerization of glutamate dehydrogenase: A possible factor in metabolic regulation." Prec. Natn. Acad. Sci. U.S.A., 64, 338-344. Kempfle, M., K.-O. Mosebach and I-I. Winkler. 1974. "Relaxation kinetic studies of coenzyme binding to glutamate dehydrogenase." Ninth JFEBS-Meeting, Budapest 1974, Abstr. No. s2e4. Koberstein, D. and H. Sund. 1973. "Studies of glutamate dehydrogenase. The influence of ADP, GTP and L-glutamate on the binding of the reduced eoenzyme to beef-liver glutamate dehydrogenase." Ear. J. Biochem., 36, 545-552. Krause, J., M. Bdhner and t{. Sund. 1974. "Studies of glutamate dehydrogenase. The binding of I~ADR and N A D P H to beef-llver glutamate dehydrogenase." Eur. J. Biochem., 41, 593-602. Malcolm, A. D. B. 1972. "Coenzyme binding to glutamate dehydrogenase. A s t u d y b y relaxation kinetics." Eur. J. Bioehem., 27, 453-461. Mosebach, K.-O. and M. Kempfle. 1969. "Das Molekulargewieht der Glutamatdehydrogenase und ihrer Untercinheiten." Z. Naturforsch., 24b, 580-583. Pantalonl, D. and P. Dessen. 1969. " G l u t a m a t e d~shydrogenase. Fixation des coenzymes N A D et N A D P et d'autres nucl@otides d@rives de l'adenosine-5'-phosphate." Eur. J. Biochem., l l , 510-519. Yapel, A. F. and 1%. Lumry. 1971. "A practical guide to the temperature-jump method for measuring the rate of fast reactions." I n Methods of biochemical analysis, Vol. 20, pp. 169-350 (Ed. D. Glick). New York: Interscience. Yielding, 14. L. and B. Holt. 1967. "Binding b y glutamate dehydrogenase of reduced diphosphopyridine nucleotide." J . Biol. Chem. 242, 1079-1082.
I~ECE£VED 5-24-75
OF
MATHEMATICAL
BIOLOGY
VOLUME 39, 1977
R E L A X A T I O N SPECTRA OF TWO-STEP MECHANISMS AS MEASURED BY THE N A D t t - B I N D I N G TO GLUTAMATE D E H Y D R O G E N A S E
[]MANFRED KEMPFLE
Physikalisch-chemische Abtcilang des Physiologisch-ehemisehen Instituts der Universitgt, Nussallee 11, 5300 Bonn, Germany
Temperature-jump relaxation experiments performed with glutamate dehydrogenase and the reduced coenzyme nicotinamide adenosine dinueleotide show two clearly separated relaxation times. Three of the simplest and most plausible mechanisms attributable to two relaxation times are treated here theoretically and the concentration dependence of the relaxation times is determined in each case with different experimental conditions. This makes it possible to distinguish between the mechanisms and also to determine the rate constants of the different reaction steps.
Relaxation kinetic experiments are not only a powerful tool in determining the single rate constants of chemical reaction steps but also in deciding between different possible reaction mechanisms. This paper deals with the second case: it will be shown from the concentration dependence of the measured relaxation times we are able to conclude which reaction mechanism is present. For those who are not familiar with relaxation kinetics a very brief introduction is given. For more detailed information they should refer to recent publications (Eigen and DeMaeyer, 1963; Czerlinsky, 1966; Hammes, 1968; Yapel and Lumry, 1971). The concept of relaxation in a physical process relates to the time delay between a sudden change in external conditions and the readjustment of the system in equilibrium to the change. Almost any chemical equilibrium can be 297
298
MANF~ED KEMPFLE
perturbed rapidly by stepwise or periodical changes of external parameters according to the van t'Hoff equation
(31nK~ /
(3 In K'~ ,
lnK
=
/O l n K \ ...,
where K represents the equilibrium constant of the reaction, T the temperature, p the pressure, and E the electric field strength. The following internal re-equilibration (the "relaxation process") is now characterized by a time constant z, the so-called relaxation time, and can be monitored by suitable techniques. I f this chemical re-equilibration process includes several reactions, there is a spectrum of time constants (the relaxation spectrum) which can be expressed in terms of the rate constants by welldefined mathematical transformations (Eigen, 1957). The analysis of the relaxation phenomena becomes fairly simple if the perturbation is kept sma]l to admit the linearization of the rate equations in terms of deviations of the concentrations from their reference values (deviations not larger than 10%). The one-step equilibrium reaction between an enzyme E and a ligand L is a useful introduction to the general principle in the derivation of the expression for the relaxation times, because of its simplicity:
~,2 E+L ~EL,
K =-
/~2,
k~: k12
-
[E][L] [EL]
(1)
where [E], [L], [EL] are the equilibrium concentrations and k:2, k2: are the rate constants for the forward (from left to right in (1)) and for the backward reaction step respectively. As a consequence of a rapidly applied perturbation the system is suddenly produced in a nonequilibrium state. From it the system relaxes to a new equilibrium. The expression for the occurring single relaxation time of this mechanism is obtained by analyzing the time-dependence of system (1) as it shifts to achieve its new equilibrium conditions. Let us assume that the perturbation has been carried out by a temperature jump and that the equilibrium constant K for the reaction (1) is dependent on temperature (due to the van t'Hoff equation AH 0 ¢ 0). It is immaterial whether AH ° is positive or negative, since the mathematical expression for the relaxation time is independent of the direction in which the equilibrium shifts after the perturbation. Using the law of mass action (the general principle of reaction kinetics) we can describe the rate equations for the equilibrium dE dt
dL dt
d[EL] dt
-
k:2E.L-k~:C,
C = EL.
(2)
RELAXATION If we
SPECTRA
OF TWO-STEP
carry out only small perturbations,
MECHANISMS
the instantaneous
concentrations
E, L, and EL may be expressed as the sum of the time-independent concentration and a small time-dependent concentration term: E = E-F6F,,L = L-t-6L, EL = EL+sEL,
299
C -= EL.
equilibrium (3)
E, L, and C are the equilibrium concentrations at the higher T-jump temperature. From the conservation expressions Eo=
E+C
= E+C,
LO = L + C
= L+C,
(4)
where E ° and L ° represent the initial total concentrations of E and L in our system, we obtain the relation between the small time-dependent concentrations 6~, 6L, 6~L by substituting (3) into (4):
E+6~+C+6EL = L+JL+C+JEL
E+C,
-= L q - C .
Therefore, 6E-{-6EL -~ 0;
6L-{-6EL --~ O.
(5)
From this we can express the 6 z , 6L, 5~L by a single term 6 : 6 = 6~ = 6L = -- 6EL. With this from (3) we get E = ETS,
L = L-FS, C = C-6.
(6)
Substituting this again into the rate equations (2) gives d(E + 6) dt
k12(E + 6)(L + 6) - k 2 1 ( C - 6).
-
(7)
Expanding and rearranging results in d6 - ~l-t = kl~E.L - k21C + [k~(E + L) + ~.fl. 6 + k~2- 63. The first two terms (not containing 6) cancel out since they are equal to each other (see (1)). Now the great mathematical simplicity of the method is due to the small value of 6~' which can then be neglected. This procedure results in a single first-order differential equation d6 -
d--t = [ k ~ 2 ( E + L ) + k z ~ ] . 6
1 = k.6 = -.6.~
(8)
The coefficient k of 6 is the reciprocal relaxation time given by 1
k = - = kl~(E+L) +kzl.
(0)
300
M A N F I ~ E D KEIVfPFLE
This is a concentration-dependent term for 1/z allowing the separate determination of the rate constants k12 and k21 of system (1). A plot of the reciprocals of the measured relaxation times 1/z vs the sum E + L of the correspondent concentrations of the reaction partners yields the recombination rate constant k12 from the slope of the straight line and the dissociation rate constant k21 from the intercept with the ordinate (Figure 1). But since it is not always possible to calculate the equilibrium concentrations E + L, the relaxation times m a y also be expressed in terms of the total "weighed in" concentrations. However these terms are much more complicated (Czerlinsky 1966).
1 1;
E.L
k12 ~ EL k21
1 -_ k21 " k12 ( [E]f + [C]f) ~-K d = k2.__[ k12
(
[~]f ° [tYlf)
F i g u r e 1. L i n e a r relationship b e t w e e n t h e first reciprocal relaxation t i m e and the s u m of t h e free concentrations of the reactants at equilibrium. This plot allows t h e d e t e r m i n a t i o n of t h e rate constants for t h e first .(fast) isolated step a n d t h u s t h e calculation of the dissociation (or association) constant
After this simple example and the calculation of its relaxation time we change to a much more complicated system. T-jump relaxation experiments of the reduced nicotineamide-adenine-dinucleotide (NADH) binding to glutamate dehydrogenase (GIuDH) from beef liver show two clearly separated relaxation times when we measure the fluorescence change of the coenzyme due to binding (Kempfle et al., 1974). In general, each relaxation time corresponds to at least one reaction step (whereas the opposite m a y not be valid--see Eigen, 1967). Therefore we have to consider at least a two-step mechanism in our case. In recent publications such two-step reactions have been treated (Czerlinsky, 1966 ; Hammes, 1968; Yapel and Lumry, 1971) and the concentration dependence of the relaxation times in those cases was shown. Our case is different from those of recent publications,
RELAXATION
SPECTRA OF TWO-STEP MECHANISMS
301
since we find the two relaxation times with only one substrata. Therefore the conservation conditions become somewhat changed relative to the published results. Here three different cases yielding two relaxation times are considered. It will be shown that from the concentration dependence of the second--longer-relaxation time it is possible to decide between the different mechanisms. The following three mechanisms are considered here: (I) A binding reaction of the NADH-ligand L to the enzyme E is followed b y a monomolecular interconversion of the EL-complex k~2
k23
E + L ~2~ E L ~32 ~ EL*,
(10)
where E = enzyme, L = ligand, E L = enzyme-ligand complex before conformation change, EL* = enzyme-ligand complex after conformation change and the k~3 are the rate constants in the different steps. The other two cases must be considered owing to the fact that experimental data (Huang and Frieden, 1969; Pantaloni and Dessen, 1969; Koberstein and Sund, 1973) show two N A D H binding sites on each of the six subunits which build up the G]uDH molecule (Eisenberg and Tomkins, 1968; Mosebach and Kempfle, 1969). Thus we have to treat: (II) Two binding sites completely independent from another--we can treat this case formally like two different enzymes competing for only one substrata: E I + L ~ EL1, ~2~
E2+L ~
/¢3~
E~L,
(11)
where El, E~ are the different binding sites, ElL, E2L the different enzymeligand complexes and kij are again the rate constants for the isolated steps. (III) Binding of the N A D H proceeds in a consecutive manner, the second molecule can only be bound when the first has been taken up. Here we are able to consider all cases of any kind of cooperativity: E+L
/c12
~- EL,
/~21
EL+L
k23
~- ELg,
]¢32
(12)
where E L is the enzyme molecule with one molecule of N A D H bound, this takes up another N A D H molecule leading to EL~. Cooperativity appears in different values of the rate constants. The derivation of the expressions for the relaxation times and their concentration dependence is given in a more detailed fashion for the first case; for the two others we give the results only. After this, all results are compared with each other,leading to the decision asto which of the three mechanisms is present in our experimental case.
302
MANFI~ED KEMPFLE
M e c h a n i s m I . Let us calculate the zi for model (I):
E+L
]/~12
k23
k21
k32
~ E L ~ EL*,
(10)
with /c21 [E][L] K21 - ]c12 - [EL---~'
/ca2 [EL] Ks2 - ]c23 - [EL*]"
We can derive four possible differential equations for the rate equations but only two of these are independent. We choose dE = l c 1 9 E ' L - k21C dt
---
dC* dt
= k 2 3 E ' L - /~a2C*
(13)
We see that if there is any coupling of the two steps both terms are no longer independent from one another. Now we have the same procedure as in the case of the single step mechanism already treated. The conservation relations are E o = E+C+C*
= R,+C+C*,
L o = L+C+C*
= L+C+C*,
(14)
where E 0 and L 0 represent the total "weighed-in" concentrations of enzyme and substrate respectively. Again using the method developed in the equations before, the rate expressions in (13) can be linearized b y introducing timeindependent and time-dependent terms for the instantaneous concentrations. (~S "~ -- 5 E L - - (~EL*,
5 L "~ -- 5 . E L - - 5 E L * .
(15)
Introducing for the concentration changes 58 = 5L = X 1 and 5EL*-----X2 with the conservation condition (~E-~ (~EL-]- S E L * .-~ O,
(16)
5 E L ~- -- 5 E - - 5EL* ~-- -- (Xl"Jr X2).
(17)
we get The rate equations after substitution and linearization read dS~ dt
dX1 - ~ 1 ( X 1 + X2) + k12(2, + L)XI, dt
d6EL*
dX2
dt
dt
- k 3 2 X 2 - k~3(X1 + X2).
RELAXATION
SPECTRA
OF TWO-STEP
MECHANISMS
303
Further rearrangement leads to dX1 dt
= X 1 = - (k2~ + k ~ ( ] £ + L ) ) X ~ -
k~X~,
dX2 dt
-
X2 =
- k ~ a X 1 - (k28 + k39.)X2.
(18)
This we can generalize and write X 1 = a ~ l X l + cq~X~,
~ 2 ~--- ~21X1~- g22X2,
(19)
with ~1~ =
- (k2~ + k ~ ( / ~
+ L)).
~12 ----- --/~21, ~21 ----- - - ~ 2 3 ,
a~2 =
(19a)
- (k,.a + k3~.).
It should be noted that (19) is valid for all two-step mechanisms but the a 0 are different in the different reaction types. To determine the expressions for the two relaxation times characterizing mechanism (I) we have to solve the two simultaneous differential equations (19) for X1 and Xg.. The solutions for X1 and X2 are X1 = A1 e - k i t + A 2 e-k2 t,
X2 = B1 e - k i t + B 2 e-k2 t,
(20)
where kl and k2 are the reciprocals of the two relaxations times zl and z2 and A1, A2, B2 and B1 are constants. B u t here it is of interest only to obtain the relaxation times. We obtain solutions of the form shown in (20) if kl and k2 are chosen to satisfy the following determinantal equation ~11 - - k
Ct12
a9,1
ag.,. - k
-~-- 0
(20)
or ]c2 - (all + a22)/c + (a11~22- area21) = 0. This yields the exact expressions for the two relaxation times
=
TI,II
kl,2 ----- -
2
Jr --
( ~ i i ~ 2 2 - - a12~21)-
(22)
From this we get the solutions in final form when the substitutions for the ~ j are made from (19a). This yields the two quantities 1
1 4
=
--(~11+~22)
= k12(~+L)+k21÷k23+ka2
(23)
304
MANFRED KEMI:)FLE
and 1
1 (~ii~22-
. . . . I~I TII
~2i~12)
= k12(E' + L ) ( k 2 a + k32) + k21]~32 •
(24)
I f we plot both expressions vs (E + L) we obtain two straight lines: the slopes y i e l d k12 a n d k12(]~23 + k32) .respectively and the intercepts ]c21 + ]C23 + k32 and k~ika~ respectively. From this all four rate constants can be determined d i r e c t l y : k12 -- slope of (23); ]c2i = intercept of (23)- (slope of (24))/(slope of (23)); k32 = (intercept of (24)/(k21); k2a = intercept of (23)-(k21+k32). Experimentally the two relaxation times have often very different values • 2 >> ~1 o r 1/,1 >> 1/,2. ( I n o u r case z2 is about fifty times larger than *i!) This leads to a great simplification because then (0(11+C(22) 2 >> 4((Xll~22-al~.a2x) is valid. The square root term in (22) m a y be expanded using the binomial theorem. With only the first two terms we have l
-
( a l l + (~22)2 _+ [(~11 "4- (~22)2 -- 2((~11a22 -- (X12~21)]
k~,2 =
"CI,II
(25)
2 ( a l l + a22)
this leads to 1/zi = a l l + ~22
( + ) sign of the term in (25),
1/¢2 = (¢¢11922 - 0¢120¢21)/(c¢11 + a22)
(--) sign of the term in (25).
(26)
I f we assume further that the first step will be the fast one, this means 0~11 ~ ~22
(27)
and we get 1/Vl---- (~11,
l / z 2 = a22--(C(12~21)/(1/Z1).
(28)
This leads to the following expressions for the relaxation times of mechanisms (I):
1 -
-
=
k21+ k12(~ + L),
T1
] z2
E+L = k32 + k28
K2i+E+L
•
(29)
From plots of the 1/z's versus the concentrations it is possible to determine all four rate constants. This mechanism (I) was treated extensively here because the basic equations will later be used again, but only (23), (24) and (29) of this treatment deal with mechanism (I). Now we can handle mechanisms (II) and (III) quite easily.
RELAXATION SPECTRA OF TWO-STEP MECItANISMS
305
Mechanism (II) kl2
k23
E I + L ~ ElL,
E u + L ~ E2L.
(11)
This leads to the following differential equations for the first and second steps : dE1
--
dt dE 2
-
dt
kl2E1L + k21C1,
k23E2L + k32C2.
-
(30)
The relations among the concentration changes are E°
=
El+C1
L°
=
L+CI+C2
=
E2+C2,
(31)
given b y
~E1 = (~L
=
- - (~EIL,
(~E2 =
--~EtL--~EsL
- - (~E2L,
~- ~E~'4-~E 2.
(32)
This leads to
dgE1 d~ -
(kls(E1 + L) + k~l)SE1 - klsE1.5E2,
dfE2
(k23Eu)SE~- (k23(E2 + L) + k82)5~2.
d--/- -
(33)
From this we obtain the following coefficients of the matrix (19), 0~11 =
-
-
(]Cl2(El-{- L) + ]621),
~12 = -]c12E1~ ~z21 =
~22 =
--/C28~2, -
(k23(E2 + L ) + k32).
(34)
The solution .leads to the relaxation times given in 1
4
•1
1
=
- ( a l l + a22)
T2
= (1c12(~1+ L) + kSl} + {ksa(E2 + L) + ka2},
1 1 • i
"~2
CZII(Z22- (212~21
(]C12(~ 1 + L ) + ]~21} " {]~23(E2 -]- L) + ]¢32} - ks~klsE1Es.
(35)
306
1KANFREDKEMPFLE
Again we have 1/Zl > l/z2 ~ all > a22 and we obtain 1 -- k12(/~1-4- L ) + k21,
1
k23 1 L +
•2
[
~2 ] E1 I + k a ~ 1 + - L + K21 _1
(36)
Mechanism (III) kt2
E+L
k23
EL+L ~EL2,
~EL,
~21
k31
(12)
leads to the following ~wo differential equations: dE
dt
-
k l 2 E ' L + k21C', C' = [EL],
dEL2 = k2aC'L-k82[C"], C"=
dt
[EL2].
(37)
The relations among the concentration changes are from E o = E+C'+C", L o = L+C'+2C", ~EL =
--~E--(~EL 2,
5 L = 5 E - - ( ~ E L 2,
Introducing this into (37) yields, if 0 ' = [EL], d6E
dt
--
dSEL2 dt
(kl2(E 4- L) -4-k21)bE - (k21 - kl2E) • gEL2, -t
(k2a(L - C ))5E -- (k2a(L + C') + ka2)@L2,
(39)
leading to these coefficients of the m a t r i x
all = - (k12(N' + L) + k ~ ) , a12 = -- (1¢21--ki2E) a2i =
- (kza(L-
az2 =
- (k2a(L + 0") + ka2).
C ' ) ),
(40)
RELAXAT~O~q
S P ] ~ C T I ~ A O:F T W O - S T E P
MEC~IA~NISMS
307
From experimental knowledge of the two widely separated relaxation times (l/z1 >> l/z2) we have to introduce (211 >> (2~. which results in 1
1 --
z2
- (211 = /~12(E+L)+/c21, g12(221 ~_~ ( 2 2 2 - - ~
1/~1
]c~a[(L(C' ÷ L) ÷ 2B} ÷ 2K210'] E+L+K21
= ]ca2+
(41)
These three expressions for the second relaxation time are quite complicated. Furthermore the concentration terms E, L and C' respectively refer to the concentrations of free enzyme, coenzyme and correspondent complex present at equilibrium. Calculation of these quantities from the known total "weighed-in" concentrations requires knowledge of the equilibrium constants K21 = k21/k12 and K32 = k32/k23. But in the case of the GluDH. N A D H complex the values for K,j given in literature vary very great]y (Bayley and Radda, 1966 ; Yielding and Holt, 1967; Malcolm, 1972; Krause et al., 1974) due to different experimental conditions. It is therefore not possible to calculate E, L and C' in this case. B u t there are two possible ways to solve this problem. The first is to use the relationship between the equilibrium concentrations and the total "weighedin" concentrations. But these expressions are fairly complicated (for example the book of Czerlinsky (1966) is recommended) so that this procedure is not given here. But in principle it may be possible to express these quantities for the relaxation times b y the total reagent concentrations. The other method is to use a large excess of one rectant such that the sum of E and L m a y be equated with the total reagent in excess, either by E o or L o. This w a y is gone in our further treatment. I t can be shown that if the first step is very fast in all three cases the first relaxation times are identical. Therefore only the second will be treated further. I t can be seen that it is possible to decide from the concentration dependence of this time what mechanism is present. (a) E0 >> L0 Mechanism (I): (29) leads to 1 --
ze
E0 =
/ c 3 2 + ]~23"
K21 + E °"
(42)
Mechanism (II): (36) with E ° ~ E ° m E ° yields 1 E0 z2 = k32 + k23. K21 K21 + E e"
(43)
308
iVIA17FI%E D KEMPFLE
and Mechanism (III): (41) gives
1 z2
-- ]c32 + 1c23
2(LE + K210') K21 + E °
(44)
The numerator is the expression for equilibrium E L = K210 thus we have 1 "C2
E o -
k32 +/c23- 4L
K21 + E 0"
(45)
In the first two eases these simplifications lead to straight lines if we plot 1/~2 versus EO/(K21+ E 0) but with different slopes: slope of (42) = /c23,
slope of (43) = /c23.K21
whereas the intercept ks2 remains identical. Therefore we cannot discriminate mechanism (I) and (II) from this procedure but from amplitude considerations we are able to distinguish these mechanisms (Eigen, personal communication):
•,x k a
E.O
Eo +
Figure 2. E 0 >> L 0 In (I) the amplitude of the second relaxation time becomes larger and larger up to a constant value with increasing enzyme concentration due to increasing and saturating amounts of C and C* complexes. In (II) the amplitude of the second relaxation time must vanish with increasing concentration due to the saturation of the binding sites. The mechanism (III) shows a very interesting behaviour. Since we have to consider in (45) the concentration of the free ligand it becomes evident t h a t this concentration/~, goes through a maximum and at high enzyme concentrations it becomes zero. Thus we have the shape of the second relaxation time of mechanism (III) shown in Figure 2. From this plot we can only see the concentration dependence but we can h a r d l y calculate k23. From these results we can already conclude t h a t in our experimental ease only mechanism (II) is valid. Because the plot vs E°/(E ° +K2I) is a straight line and the amplitudes of the relaxation process go to zero.
R E L A X A T I O N SPECTRA OF TWO-STEP MECHANISMS
309
But in general we can introduce a further simplification: (b) L 0 >> E 0 # O. Mechanism (I) leads to 1
Z0
-*2 -=
(46)
] c 3 2 + K 2 1 + L °"
Mechanism (II) leads to 1
(47)
= k39,+ k~aL °
T2
and mechanism (III) to
1 •2
"~2
(L0)2 =
k82 + k23
(4s)
K21 + L0"
[L° ] >> [E°]
~
/
fi°:> mechanism[I) ~; ~.7/_I...~-'TJ--m--e c~a hni-'= k 3 2 * k23
k3z
i --
,
[L°]
m=
F i g u r e 3. Concentration dependence of t h e second reciprocal r e l a x a t i o n t i m e of t h e three two-step b i a 0 m g m e c h a n i s m s as ,tiseussed in t h e t e x t , w i t h ligand c o n c e n t r a t i o n L 0 in excess over t h e e n z y m e concentration. This plot d e m o n s t r a t e s t h a t it is possible f r o m it to distinguish the different reaction mechanisms
These three plots show different behaviour if we plot 1/z2 vs the concentration in excess L 0 (Figure 3), so that we can see what mechanism is present. Mechanism (I) (46) levels off at high concentrations resulting in a straight line independent from concentration with a constant value k3~+k2~; thus it is possible to determine these values. Mechanism (II) (47) results in a straight line with a slope ke~ and an intercept k32. Mechanism (III) (48) at last has at low concentrations the slope zero but it increases and results in a straight line at high concentrations (L0 >> K~I) parallel to this of mechanism (II) with the slope k23. Thus we see that it is not only possible to determine the rate constants for any known reaction mechanism b u t possible too to distinguish between unknown mechanisms. I t is known that steady-state experiments allow 'only
310
MANFI~ED KEMPFLE
the measurement of the ratios of these single step rate constants (the equilibrium constants) or the overall value of coupled reactions (not the isolated single steps). Thus relaxation kinetic experiments are a powerful tool in d i s c r i m i n a t i n g d i f f e r e n t p o s s i b l e m e c h a n i s m s (here t h e t w o s t e p m e c h a n i s m s ) and in determining their rate constants. An EMB0
s h o r t - t e r m f e l l o w s h i p is g l a d l y a c k n o w l e d g e d .
LITERATURE Bayley, P. M. and G. K. Radda. 1966. "Conformational changes and the regulation of glutamate dehydrogenase a c t i v i t y . " Biochem. J . 98, 105-111. Czerlinsky, G. 1966. Chemical Relaxation. New York: Dekker. Eigen, M. 1957. "Determination of general and specific ionic interactions in solution." Discuss. Farad. Soc., 24, 25-36. - - - - - - and L. deMaeyer. 1963. "Very rapid reactions in solution. Relaxation Methods." I n Techniques of organic chemistry, Vol. 8, p a r t 2, pp. 895-1054 (Eds. S. L. Fries, E. S. Lewis, and A. Weissberger). New York: Interscience. • 1967. "Kinetics of reaction control and information transfer in enzymes and nucleic acids." I a Nobel Symposium No. 5, pp. 333-369 (Ed. S. Claesson). Stockholm: Almquist and Wiksell. Eisenberg, H. and G. M. Tomkins. 1968. "Molecular weight of the subunits, oligomeric and associated forms of bovine liver glutamate dehydrogenase." J. Melee. Biol., 31, 37-49. Hammes, G. G. 1968. "Relaxation spectrometry of biological systems." I n Advances in P~otein Chemistry, Vol. 23, pp. 1-58. (Eds. C. B. Anfmsen, J. T. Edsall, M. L. Anson, and F. M. Richards). New York: Academic Press. Huang, Ch. Y. and C. Frieden. 1969. " R a t e s of GDP-induced and GTP-induced depolimerization of glutamate dehydrogenase: A possible factor in metabolic regulation." Prec. Natn. Acad. Sci. U.S.A., 64, 338-344. Kempfle, M., K.-O. Mosebach and I-I. Winkler. 1974. "Relaxation kinetic studies of coenzyme binding to glutamate dehydrogenase." Ninth JFEBS-Meeting, Budapest 1974, Abstr. No. s2e4. Koberstein, D. and H. Sund. 1973. "Studies of glutamate dehydrogenase. The influence of ADP, GTP and L-glutamate on the binding of the reduced eoenzyme to beef-liver glutamate dehydrogenase." Ear. J. Biochem., 36, 545-552. Krause, J., M. Bdhner and t{. Sund. 1974. "Studies of glutamate dehydrogenase. The binding of I~ADR and N A D P H to beef-llver glutamate dehydrogenase." Eur. J. Biochem., 41, 593-602. Malcolm, A. D. B. 1972. "Coenzyme binding to glutamate dehydrogenase. A s t u d y b y relaxation kinetics." Eur. J. Bioehem., 27, 453-461. Mosebach, K.-O. and M. Kempfle. 1969. "Das Molekulargewieht der Glutamatdehydrogenase und ihrer Untercinheiten." Z. Naturforsch., 24b, 580-583. Pantalonl, D. and P. Dessen. 1969. " G l u t a m a t e d~shydrogenase. Fixation des coenzymes N A D et N A D P et d'autres nucl@otides d@rives de l'adenosine-5'-phosphate." Eur. J. Biochem., l l , 510-519. Yapel, A. F. and 1%. Lumry. 1971. "A practical guide to the temperature-jump method for measuring the rate of fast reactions." I n Methods of biochemical analysis, Vol. 20, pp. 169-350 (Ed. D. Glick). New York: Interscience. Yielding, 14. L. and B. Holt. 1967. "Binding b y glutamate dehydrogenase of reduced diphosphopyridine nucleotide." J . Biol. Chem. 242, 1079-1082.
I~ECE£VED 5-24-75