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The theoretical analysis of kinetic behaviour of “hysteretic” allosteric enzymes V. Relaxation kinetics of dissociating enzyme systems
I. theor. Biol. (1982) 98,73-90
The Theoretical Analysis of Kinetic Behaviour of “Hysteretic” Allosteric Enzymes V. Relaxation Kinetics of Dissociating Enzyme Systems S. V. KLINOVAND
The Ail-Union
B. I. KURGANOV
Vitamin Research Institute, Moscow, U.S.S.R.
(Received 17 September 1981 and in revised form 10 March 1982) The expressionsfor relaxation time as a function of enzyme and specific ligand concentration are deducedfor dissociatingenzyme system2peP (P is enzyme oligomer which is able to dissociatereversibly forming two identical halvesp), It is assumedthat ligand binding sitesare equivalent and independent in each oligomeric enzyme form and the equilibrium betweenoligomericforms developsrather slowly in comparisonwith the rate of the bindingof the ligand. The kineticsof relaxation of the dissociating enzyme system2peP with progressivechangeof the rate constants for associationof oligomeric form p hasbeen analysedin graphic form. The situationswhen one of the oligomeric enzyme forms is not able to bind the ligand are also considered.The principles of the analysisof relaxation kinetics of dissociatingenzyme systems2peP are discussed.
1. Introduction In a previous communication (Kurganov, 1977) we discussedthe kinetics of equilibration of the dissociating enzyme system 2p~P (P is an enzyme oligomer which is able to dissociate reversibly, forming two identical halves
p) after changes in storage conditions or after addition of allosteric ligand. The methods of estimation of the rate constant for association of the halves p and of the rate constant for dissociation of the oligomer P suggested in this article are applicable for relatively slow processesof equilibration when
the kinetics of the dissociation-association process may be followed by the registration of a change in specific enzyme activity. The fast processes may be studied using, for example, relaxation methods (Pecht & Rigler, 1977). The work of Kirschner et al. (1966) may serve as an excellent illustration of the application of relaxation methods for elucidation of the mechanism of specific ligands binding by oligomeric enzymes. Kirschner et al. investigated the binding of NAD’ to yeast D-glyceraldehyde-3-phosphate dehydrogenase with the temperature-jump method. The enzyme molecule contains four coenzyme binding sites and at certain conditions (e.g. at pH 8.5 73 0022-5193/82/170073+18
$03.00/O
@ 1982AcademicPress
Inc. (London)
Ltd.
74
S.
V.
KLINOV
AND
B.
I.
KURGANOV
and 40°C) the interactions between binding sites are revealed in the Sshaped character of the saturation function. Kinetic measurements led to the detection of three distinct relaxation processes. Two fast relaxation processes were assigned respectively to coenzyme binding to two different conformational states of D-glyceraldehyde-3-phosphate dehydrogenase. The slowest relaxation process was related to isomerization of the enzyme. The experimental data were quantitatively analysed in terms of the concerted model of allosteric interactions (Monod, Wyman & Changeux, 1965). These authors adduced the theoretical expressions for the dependences of relaxation times on concentration of free binding sites of the enzyme and free ligand concentration. In the present communication we have analysed the relaxation kinetics of dissociating enzyme system 2p~ P in the presence of a specific ligand.
2. Relaxation The equilibration
Kinetics
of a dissociating enzyme system of the type k+m
2p*P k-oo is discussed, where P is an enzyme oligomer which is able to dissociate reversibly forming two identical halves p, k+oo is the second-order rate constant for association and keoo is the first-order rate constant for dissociation. Let us assume that (1) oligomeric forms P and p contain n and n/2 specific ligand (L) binding sites respectively; (2) ligand binding sites are equivalent and independent in each oligomeric form; (3) the steps of Iigand binding, (pLi-i + L$ pLi and PLi-i + Ls PLi) are in rapid equilibrium; (4) the total ligand concentration [L10 exceeds greatly the total enzyme concentration [El,-,; (5) the microscopic complexes of oligomeric enzyme formed with ligand are able to undergo reversible dissociation-association: k
pL, + pL, *
k-x,
L,PL,,
(2)
where pL, and pL, are microscopic complexes of oligomeric form p with x and y molecules of the ligand respectively, L,PL, is a microscopic complex of oligomeric form P with the ligand where one part of the oligomer contains x bound ligand molecules and the other part contains y bound ligand molecules, and k+xy and k-,,are microscopic rate constants for association and dissociation respectively.
KINETICS
OF
DISSOCIATING
ENZYME
SYSTEMS
75
With such assumptions the concentrations of microscopic complexes of oligomeric forms p and P with the ligand are given by the relationships
[PLI = [PI (rLl”IKk )“, [L,PL,l=
Plm.-lo/K:
Ix+‘,
(05x
(n/2;
(3) OCry In/2)
(4)
where [p] and [P] are concentrations of free oligomeric enzyme forms p and P respectively, Kk and K[ are microscopic dissociation constants for the complexes of p and P forms with the ligand respectively. The conditions of detailed balance yield the following relationship for microscopic rate constants k,,, and k-,, and microscopic rate constants k+,,,, and kQo:
The kinetic equation for the process (2) has the following form: -- ;
$g
G2 [PLXI) n/2
n/2
= C C (k+xrChCL2 x=0 y=o
[pLApL,l-
bC/2C,2
[LxPL, I).
(6)
Taking into account the equations (3), (4) and (5) the following expression for the relaxation time (T) can be obtained:
(7) where K. = k+oo/k-oo is the constant of association of halves p and [p] is the equilibrium concentration of free oligomeric form p. It should be noted that the rate constants k,,, and k-,, for which 2 I (x + y ) 5 (n - 2) are the “concealed” parameters, i.e. they cannot be found from the experimental dependence of r on [LIO and [p]. This is a result of uncertainty in distribution of ligand molecules between two parts of oligomer P. Therefore it is reasonable to introduce the effective rate constants for dissociation (k-i) and for association (k+i) which are explicit parameters and related to k-,, and k+,, rate constants by the expressions (i = x + y )
(8) Chk+i = nf nf CX,/2C$2k+,y. x=0 y=o
(9)
76
S.
The relationship
between k+i and km, is identical to relation (5):
V.
KLINOV
AND
B.
I.
KURGANOV
k+i 7tJ=Kod where cp= KQKL. The equation of material
balance
42 [Elo
= x;.
n/2 C”,,,
(10)
[@x1
+ 2
n/2
c
c C”,,,C’,,2 x=0y=o
(where [El0 is the total enzyme concentration form p) can be solved with respect to [p]:
[ULI
(11)
calculated on the oligomeric
4Ko[p] = (dl + 8K[Elo - 1) ( (;lT;;;,$);:2
7
(12)
where K is the apparent association constant
1+ [Lb/K:
K = K”
(
1 +[L]iJK;
>.
(13)
Taking into account equations (8) and (12) one can transform the expression (7) to the following form: t= k-d1 +8K(E]o.
(14)
In the expression (14), k- represents the apparent rate constant for dissociation of oligomer P into two halves p in the presence of the ligand: k-c i. C’;k-i(g)i/(
I +$$)n,
(15)
The following equation is also valid for relaxation time:
~=$JI+~K[EI,, where k, is the apparent rate constant for association of oligomeric p in the presence of the ligand:
(16)
forms
(17)
KINETICS
OF
DISSOCIATING
ENZYME
SYSTEMS
77
The linear anamorphoses connecting the values of r and [El0 may be obtained from equation (14): $=
k? +8k’K[E]o,
1 ,=Bk’K+k$p [Elo~
(19)
These anamorphoses have the same form as those obtained for relaxation of the dissociating enzyme system 2p *P in the absence of the ligand: L= 72
1
Elo~
kTOo +Bk&K,[E],,, 2=8k?ooKo+k?m[E10,
1
(22)
T2 - - 1 - 8K,,[E]Or’. k&
The following linear anamorphoses correspond to equation (16): (24) -- 1
[Elope-
8k:: K
k: 1 K*[Elo’
(25)
2
T* = KF-8K[E]o~2. +
3. The Principles
of Analysis of Relaxation
Kinetics
The dependence of relaxation time on ligand concentration for dissociating enzyme system 2p C$ P are generally rather complex and cannot be used for estimation of the values of kinetic constants. However, if the steps of ligand binding (pLi-1 + L s pLi and PLi-1 + LSPLi) are in rapid equilibrium the dependences of relaxation time on total enzyme concentration obtained at fixed concentrations of the ligand have simple form and can be transformed to linear relationships. Using linear anamorphoses (18), (19) or (20) allows one to determine the apparent association constant
78
S.
V.
KLINOV
AND
B.
I.
KURGANOV
(K) and the apparent rate constant for dissociation of oligomer P(k..) at definite fixed concentration of the ligand. If the values of K and km are determined at various concentrations of the ligand, the microscopic dissociation constants for complexes of oligomeric forms with the ligand (KL and Kf respectively) and the effective rate constants for association of halves p(k+i) and for dissociation of oligomer P(kei) may be calculated from the quantitative analysis of the dependences of K and k- on [L& based on using theoretical expressions (13) and (15). If we use the linear anamorphoses (24)-(26) the following analysis of the dependences of the apparent constants K and k, on ligand concentration with the aid of equations (13) and (17) allows us to determine the parameters Kt, K[, KO, k-i and k,,. It should be noted that above expressions for dependence of relaxation time r on total enzyme concentration [El0 remain valid also for the situation when the ligand binding sites in oligomeric forms p and P are non-equivalent or interacting. However, in this case complication of the dependences of K, k- and k, on [LIO will take place. It is evident that the relaxation spectrum of the dissociating enzyme system under discussion must include not only the slow relaxation process corresponding to the change in oligomeric state of the enzyme but additional faster relaxation processes which are due to ligand binding to oligomeric enzyme forms. If ligand binding sites in each oligomeric enzyme form are equivalent and independent, kinetically significant species are free ligand binding sites in oligomeric forms p and P and their complexes with the ligand. According to Eigen (1967), the system consisting of n kinetically significant species has (n - 1) relaxation times. Therefore dissociating enzyme system 2ptiP will have two relaxation times when ligand binding sites of oligomeric forms p and P are kinetically identical and three relaxation times when ligand binding sites of oligomeric enzyme forms are kinetically non-identical (see, for example, Kirschner et al., 1971). This evident conclusion has not yet received an experimental corroboration. It is worth noting that Tai & Kegeles (1975) observed a single relaxation time for Helix pomatia a-haemocyanin (the dissociating protein system being of the type 2pgP) in the presence of the specific ligand (Ca” ions). This circumstance may testify to the reality of simplifying assumptions used by us in the theoretical analysis. 4. Particular
Cases
We have assumed above that oligomeric form p contains n/2 equivalent and independent ligand binding sites. In addition we assume that association
KINETICS
OF
DISSOCIATING
ENZYME
SYSTEMS
79
site of oligomeric form p consists of n/2 association subsites which are equivalent and independent. The association of two halves p to give oligomer P results in the formation of pair-contacts between all association subsites of halves p. We assume also that each protomer contains one ligand binding site and one association subsite. Two possible states of the protomer must be taken into account: the state (A) in which the ligand binding site of the protomer is not occupied by the ligand and the state (B) in which the ligand binding site of the protomer is occupied by the ligand. The oligomeric form P will contain three types of contacts between protomers: AA, AB and BB. If microscopic complex L,PL, has m contacts BB, 1 contacts AB and (n/2-m -I) contacts AA respectively (where m and 1 are integer numbers), the activation free energy for its dissociation can be written (see, for example, Ricard, Mouttet & Nari, 1974) as AG!,,
= m AG&
+ IAG!ae
+ (n/2 -m - l)AGTaA,
(27)
where AG&, AGfAB and AG! AA are the activation free energies for the dissociation of contacts BB, AB and AA respectively. The following relationship is fulfilled in the absence of the ligand: AG!,,o = (n/2)AG!aA.
(28)
If the association subsites are equivalent and independent, we can assume that the change in activation free energy for the dissociation of the contact between two protomers after ligand binding to one of the protomers remains invariable whether the other protomer is occupied by the ligand or not: AG& Equation
- AGtAB = AG:Ar, - AGTA/,.
(27) can be simplified AG!,,
(29)
to the following form:
= AGfoo +i(AG!*,
- AGtAA),
(30)
wherei=2m+l. The expression for the rate constant for dissociation of microscopic complex L,PL, which has m contacts BB and 1 contacts AB can be written in terms of the absolute rate theory as k-,,=(gexp(
-AGt,,/RT
) = k-,,,,ai,
(31)
where k-oo = (kBT/h)
exp {-AG!,JRT}
(32)
AB-A@AA)/RT).
(33)
and a = exp {-(AG!
80
S. V.
KLINOV
AND
B.
I. KURGANOV
ks, h, R and T are the Boltzmann constant, the Planck constant, the gas constant and the absolute temperature, respectively. It should be noted that different microscopic complexes L,PL, may have the same number of contacts BB and AB and the same their spatial disposition but differ in location of states A and B in contacts AB: state B (or A) may belong to one or other half of oligomer P. Using the known combinatoric relationship (see, for example, Ezhov, Skorochod & Yadrenko, 1977)
ck+, =
i
cic”,-i
(34)
one can transform equation (8) in the following way: k-i = k-ooa i
or
k-i/k-(i-1,
= OZ.
(35) The analogous expression for k+i can be obtained with the aid of relation (10): or k+i/k+,i-1, = CY(P. k+i = k+oo(ap)’ (36) The denominators of the corresponding geometric progressions (a or crcp) characterize the influence of the ligand on the rates of dissociation of oligomer P and association of halves p respectively. the expressions for apparent rate constants for dissociation of oligomer P and for association of halves p acquire the following form: k- = k-o,,
1+ (Y[L]O/K[ ” 1+ [L]lJ/K[ > ’
(37)
With such assumptions the change of the dependence of T on [El0 at variation of ligand concentration is determined by the values of parameters CYand cp. Parameter cy determines the character of the dependence of kon [L&,, parameter cp determines the character of the dependence of K on [L],, and the product of parameters cx and Q determines the character of the dependence of k+ on [L]o. The theoretical dependences of T on [El0 calculated at various fixed values of concentration of the ligand and some values of parameters cu and Q are given below for illustration (for II = 4). It is reasonable to introduce the dimensionless ligand concentration lo = [L],/Kt. The expressions for K, k- and k+ are rewritten as (39)
KiNETICS
OF
DISSOCIATING
ENZYME
SYSTEMS
81
n
k-=
k-00 i’
~ l+cpf, +aplo)
’
(41)
k, = k+oo(%Jn.
(A)
THE
CASE
(40)
WHEN
cp = 1
If the oligomeric forms p and P are characterized by identical affinity with respect to the ligand (rp = l), the linear anamorphoses constructed in coordinates {l/(koO~)*; Ko[Elo} at various fixed values of ligand concentration must have a common point of intersection on the abscissa (Fig. 1).
4
;:
(b)
I, = 0 /
K, IElo
FIG. 1. The theoretical dependences of relaxation time T on enzyme concentration [E]a represented in the coordinates {~/(/c-~~T)~; KOIE] 0} at n = 4, cp= 1 and various fixed values of dimensionless ligand concentration I0 = [L&/K;. (a) a =: 1.3 and (b) CI= 0.7.
82
S.
V.
KLINOV
AND
B.
I.
KURGANOV
Increasing ligand concentration results in a decrease in the value of relaxation time when LY> 1 [Fig. l(a)]. On the other hand, if cr < 1 increasing ligand concentration will result in an increase in the value of relaxation time [Fig. l(b)]. (B)
THE
CASE
WHEN
LY =
1
If the rate constant for dissociation of oligomer P does not depend on the degree of its saturation by the ligand (ar = 1) the linear anamorphoses constructed in coordinates {l/(k.+,~)~; &[Eb} at various fixed values of ligand concentration must have a common point of intersection on the ordinate (Fig. 2). Increasing ligand concentration results in a decrease in (0) 8-
4-
FIG. 2. The theoretical dependences of the reciprocal value of the square of the dimensionless relaxation time k-,,o~ on dimensionless enzyme concentration KOIE10 at n = 4, Q = 1 and various fixed values of dimensionless ligand concentration lo= [L],,/Kt. (a) cp=‘0.6 and (b) cp= 2.0.
KINETICS
OF
DISSOCIATING
ENZYME
SYSTEMS
83
the value of the relaxation time when cp< 1 [Fig. 2(a)]. On the other hand, if cp> 1 increasing ligand concentration will result in an increase in the value of the relaxation time [Fig. 2(b)]. (C)
THE
CASE
WHEN
&
= 1
If the values of parameters (Y and (o obey the condition (Y’C+Y = 1 the linear anamorphcses constructed in coordinates {l/(k-00r)2; &[E&} at f0 = 0 and l,, + co have the same slope (Fig. 3). Increasing ligand concentration results in a decrease in the value of the relaxation time when (Y> 1 [Fig. 3(a)]. On the other hand, if (Y< 1 increasing ligand concentration results in an increase in the value of the relaxation itime [Fig. 3(b)].
15 -
2 ;:
(b) 2-
FIG. 3. The theoretical dependences of l/(k-Oo~)2 on K&Z], at n = 4, a*~ = 1 and various fixed values of dimensionless ligand concentration lo = [L&/K;. (a) (I = 1.3 and (b) a = 0.7.
84
~.V.KLIN~V (D)
AND THE
B.I.KuRGANOV
CASE
WHEN
crcp=l
If the rate constant for association of halves p does not depend on the degree of its saturation by the ligand (acp = 1) the linear anamorphoses constructed at various fixed values of ligand concentration do not have a common point of intersection (Fig. 41, but all points of intersection are
N ;: 8
-02
0
02
O-4
06
FIG. 4. The theoretical dependences of l/(k-oor)2 on &[E& fixed values of dimensionless ligand concentration lo = [L]o/Kt.
at n = 4, ap = 1 and various (a) a = 1.3 and (b) a = 0.7.
disposed in the left lower quadrant. Increasing ligand concentration results in a decrease in the value of the relaxation time when cy> 1 [Fig. 4(a)]. On the other hand if (Y< 1 increasing ligand concentration results in an increase of the value of the relaxation time [Fig. 4(b)].
KINETICS (E)
THE
CASE
OF WHEN
DISSOCIATING
THE
ENZYME
VALUES
OF
ARBITRARY
BUT
85
SYSTEMS
PARAMETERS
(Y
AND
q~
ARE
FINITE
In this case increasing ligand concentration results in a decrease or an increase in the value of relaxation time when the value of parameter (Y is greater or less than unity respectively. The values of intercepts on abscissa and ordinate axes have definite limits when [L]” + 0 or [L],, + CO. (F)
THE
DETERMINATION
OF
THE
PARAMETERS
The primary analysis of relaxation kinetics with the aid of linear relationships (18)-(20) allows us to construct the dependences of K and kp (or k,) on ligand concentration. The following analysis of these dependences using theoretical equations (39)-(41) enables us to estimate parameters K,, Kt, K[, kdoo and kcoo. We shall henceforth assume that parameter n is known (it can be found by an independent method). If the experimental dependence of K on [L-k, allows the association constant to be determined at very low concentrations of the ligand (Kd), the dissociation constants K[ and Kfl may be calculated by means of the linear relationship e-11
Klo
-
= &vKJKo. L
(42)
I
If, however, the experimental data enable the apparent association constant to be calculated at sufficiently a high concentration of the ligand (Kli,), then in order to find KL and K{ one may utilize the linear relationship (43) When it is not possible to determine the limiting values of the association constant (when [L&+ 0 and [L10 +a~), the difference method may be adopted as follows: the experimental data are presented in the form of a plot of fi against [LlO. A pair of points are chosen on this plot, whose abscissae differ by the quantity A. We designate the ordinates of these points by @ and sl/Ka respectively. It can then readily be shown that the following linear relationship holds, enabling the constant Kh and the ratio of constants e/K: to be found:
This plot may be constructed various values of [Llo and A. Knowing Kt
86
S.
and e/K:,
V.
KLINOV
AND
B.
I.
KURGANOV
it is possible to calculate K,, from the equation
Finally, the constant KII. is calculated from the values of %JKL and K. so obtained. If one of the dissociation constants is determined in an independent way (for example from the dependence of k- on [L],J the other dissociation constant and parameter K. may be calculated using the following linear relationships:
~(~+[L]~/K~)=~+(~/K~)[L]~ (if Kt
(461
is known) and
1+DAoIKLYz
1 + --hLlo
+‘i?i,
(47)
3’&Kt
(if Kf: is known). The determination of the parameters from the dependence of k- on ligand concentration in the accordance with theoretical equation (37) is fulfilled in an analogous manner. If experimental data enable the rate constant to be determined at very low concentrations of the ligand (k-d, parameters (Yand K[ may be calculated by means of the linear relationship a-1
[Llo
=g-+QiZi.
(48) L
If, however, the experimental data enable the apparent rate constant to be calculated at sufficiently high concentrations of the ligand (k-),i,, then in order to find parameters (Y and K[ one may use the linear relationship
When it is not possible to determine the limiting values of the rate constant k- (when [Llo+ 0 and [Llo + co) the difference method may be ado ted. The experimental data are presented in the form of a plot of ti k- against [Llo. A pair of points are chosen of this plot, whose abscissae differ by the uantity A. We designate the ordinates of these points by 9% and tp kea, respectively. The following linear relationship holds, enabling the constant KL and the value of ~/cxG to be found:
A viy~=~+avi&
K;:
1
AK ( [L1o+ic&c
).
(50)
KINETICS
Knowing equation
K[
OF
DISSOCIATING
and l/a(f/k-
ENZYME
SYSTEMS
00, it is possible to calculate
87
kmoo from the
Finally, parameter cy is calculated from the values of l/,G and kpoo. If the dissociation constant K[ is determined by independent way (for example, from the dependence of K on [LlO) the rate constant for dissociation of oligomer P(k-& and parameter (Y may be calculated using the following linear relationship:
1 : WI0 = &---oo+atG ,,[Llo. K;) L
q
(52)
In definite cases the dependences of apparent rate constant k, on ligand concentration may be convenient for determination of some parameters of the dissociating enzyme system (namely parameters k+oo, Kt and acp) using the theoretical equation (38). The linear relationships (48)-(52) are applicable for this aim after replacing k-, kmoo, Kt and cr by k+, k+oo, K; and (~9 respectively. 5. The Limiting Consider the limiting to bind the ligand.
(A) OLIGOMERIC
FORM
Situations
situations when one of oligomeric
P DOES
NOT
BIND
THE
LIGAND
forms is not able
(Kfl. + 00, cp + 0)
The association of halves p may result in steric screening of ligand binding sites. In this case the ligand interacts exclusively with p form (cp= 0) and therefore induces the dissociation of oligomeric form P. The saturating concentrations of the ligand provoke the complete dissociation of P form into its two halves. When oligomeric form P is not able to bind the ligand the dependence of T on [E]o is described by equations (14) and (16) in which the apparent association constant K and the apparent rate constants for dissociation kand for association k+ are expressed in the following way: (53) k- = keoo
(54)
88
S. V.
KLINOV
AND
B.
I.
KURGANOV
(55) The linear anamorphoses constructed in co-ordinates { 1 /(k -OOr)2; KOIE],} at various fixed values of ligand concentration have the common point of intersection which is disposed on ordinate axis (Fig. 5). Increasing ligand concentration results in an increase in the value of relaxation time. When lo + co the linear anamorphosis becomes parallel to abscissa axis.
-2
0
2
4
6
FIG. 5. The theoretical dependences of ~/(/c-~~T)* on Ko[E]o at various fixed values of dimensionless ligand concentration I,J=[L]~/K~ in the case when the oligomeric enzyme form P is not able to bind the ligand (n = 4). (B)
OLIGOMERIC
FORM
p DOES
NOT
BIND
THE
LIGAND
(K;
-, a,
Cp + 0;))
One may now discuss the situation when the ligand interacts exclusively with P form (cp+ co) and, as a result of this, displaces the equilibrium 2p$ P towards the formation of the oligomer P. Such a situation is possible when ligand binding site is created in the region of the contact of halves p in the oligomer P. The apparent association constant K and the apparent rate constants for dissociation (k-) and for association (k,) determining the character of dependence of T on [El0 are expressed in the following way: K =&(l+[L]o/K:)“,
(56)
k_=k_oo/(l+[L],/K~)“,
(57)
k+ = k+oo.
(58)
The linear anamorphoses constructed in coordinates {l/(k-oar)*; Ko[E]o} at various fixed values of ligand concentration do not have common point
KINETICS
OF
DISSOCIATING
ENZYME
SYSTEMS
89
of intersection (Fig. 6), but all points of intersection are found in the left lower quadrant. Increasing ligand concentration results in an increase in the value of relaxation time. When [L&,/K; + 00 the linear anamorphosis fuses with abscissa axis.
FIG. 6. The theoretical dependences of l/(k-,,,~)’ on &[E], at various fixed values of dimensionless ligand concentration [L]JK; in the case when the oligomeric enzyme form p is not able to bind the ligand (n = 4).
6. Conclusion
The principles of analysis of relaxation kinetics of dissociating enzyme systems 2peP in the presence of specific ligand elaborated in this paper supplement the general principles of analysis of kinetic behaviour of dissociating enzyme systems (Kurganov, 1973,1977,1978). Whether we study any properties of dissociating enzyme system at equilibrium conditions or the kinetics of attainment of equilibrium between oligomeric forms in the presence of specific ligands, the analysis must be started from the dependences of definite characteristics of dissociating enzyme system (the degree of dissociation, specific enzyme activity, the degree of saturation by the ligand, relaxation time and so on) on enzyme concentration obtained at varoius fixed values of ligand concentration. REFERENCES M. (1967). In Fast Reactions and Primary Processes in Chemical Kinetics (Claesson, S., ed.), New York: Wiley-Interscience, D. 333. EZHOV, I. I., SKOROCHO~, A. V. & YA~RENKO, M. I. (1977). Elements of Combinatorics, Moscow: Nauka, p. 40. EIGEN,
90
S. V. KLINOV
AND B. 1. KURGANOV
KIRSCHNER, K., EIGEN, M., BITYMAN, R. & VOIGT, B. (1966). Pm. natn. Acad. Sci. U.S.A. 56, 1661. KIRSCHNER, K., GALLEGO, E., SCHUSTER, I. & GOODALL, D. (1971). J. mol. Biol. 58,29. KLJRGANOV, B. I. (1973). Acra biol. med. germ. 31, 181. KURGANOV, B. I. (1977). J. theor. Biol. 68,521. KURGANOV, B. I. (1978). Allosteric Enzymes, Moscow: Nauka. MONOD, J., WYMAN, J. & CHANGEUX, J.-P. (1965). J. mol. Biol. 12, 88. PECHT, I. & RIGLER, R. (eds) (1977). Chemical Relaxation in Molecular Biology, Berlin,
Heidelberg, New York: Springer Verlag. RICARD, J., MOUTTET, TAI, M.-S. & KEGELES,
C. & NARI, J. (1974). Eur. J. Biochem. G. (1975). Biophys. Chem. 3, 307.
41,479.
APPENDIX
Nomenclature
Elo k +00 k-m k+i, k-i k+ k-
k +XY k-x, Ko, K Kt, KII. L lo n P [PI, [PI
P t a 7 cp
and Notation
the total molar enzyme concentration calculated on the oligomeric form p the rate constantfor associationof two halvesp into the oligomer P the rate constantfor dissociationof oligomer P to two halvesp the effective rate constantsfor associationof two p and for dissociation of oligomer P respectiveiy the apparent rate constant for associationof halves p into oligomer P in the presenceof a specificligand the apparent rate constant for dissociationof oligomer P to two halves p in the presenceof a specificligand the rate constantfor associationof oligomericforms p containingbound ligand (pL, + pL, + L,PL, ) the rate constantfor dissociationof oligomerP onehalf of which contains x bound ligand moleculesand the other half contains y bound ligand molecules(L,PL, + pL, + pL,) the true and apparent (in the presenceof a specific ligand) association constantsfor the equiibrium 2p z P the microscopicdissociationconstantsfor the complexesof p and P forms with a ligand a specificligand [L&,/K L (the dimensionless ligand concentration) the number of ligand binding sitesin oligomeric form P the oligomeric enzyme form which is formed at reversible dissociation of enzyme oligomer P the current and equilibrium concentrationsof oligomericform p enzyme oligomer which is able to dissociatereversibly forming two identical halvesp time k-ilk-o-u relaxation time K;/K;:
The Theoretical Analysis of Kinetic Behaviour of “Hysteretic” Allosteric Enzymes V. Relaxation Kinetics of Dissociating Enzyme Systems S. V. KLINOVAND
The Ail-Union
B. I. KURGANOV
Vitamin Research Institute, Moscow, U.S.S.R.
(Received 17 September 1981 and in revised form 10 March 1982) The expressionsfor relaxation time as a function of enzyme and specific ligand concentration are deducedfor dissociatingenzyme system2peP (P is enzyme oligomer which is able to dissociatereversibly forming two identical halvesp), It is assumedthat ligand binding sitesare equivalent and independent in each oligomeric enzyme form and the equilibrium betweenoligomericforms developsrather slowly in comparisonwith the rate of the bindingof the ligand. The kineticsof relaxation of the dissociating enzyme system2peP with progressivechangeof the rate constants for associationof oligomeric form p hasbeen analysedin graphic form. The situationswhen one of the oligomeric enzyme forms is not able to bind the ligand are also considered.The principles of the analysisof relaxation kinetics of dissociatingenzyme systems2peP are discussed.
1. Introduction In a previous communication (Kurganov, 1977) we discussedthe kinetics of equilibration of the dissociating enzyme system 2p~P (P is an enzyme oligomer which is able to dissociate reversibly, forming two identical halves
p) after changes in storage conditions or after addition of allosteric ligand. The methods of estimation of the rate constant for association of the halves p and of the rate constant for dissociation of the oligomer P suggested in this article are applicable for relatively slow processesof equilibration when
the kinetics of the dissociation-association process may be followed by the registration of a change in specific enzyme activity. The fast processes may be studied using, for example, relaxation methods (Pecht & Rigler, 1977). The work of Kirschner et al. (1966) may serve as an excellent illustration of the application of relaxation methods for elucidation of the mechanism of specific ligands binding by oligomeric enzymes. Kirschner et al. investigated the binding of NAD’ to yeast D-glyceraldehyde-3-phosphate dehydrogenase with the temperature-jump method. The enzyme molecule contains four coenzyme binding sites and at certain conditions (e.g. at pH 8.5 73 0022-5193/82/170073+18
$03.00/O
@ 1982AcademicPress
Inc. (London)
Ltd.
74
S.
V.
KLINOV
AND
B.
I.
KURGANOV
and 40°C) the interactions between binding sites are revealed in the Sshaped character of the saturation function. Kinetic measurements led to the detection of three distinct relaxation processes. Two fast relaxation processes were assigned respectively to coenzyme binding to two different conformational states of D-glyceraldehyde-3-phosphate dehydrogenase. The slowest relaxation process was related to isomerization of the enzyme. The experimental data were quantitatively analysed in terms of the concerted model of allosteric interactions (Monod, Wyman & Changeux, 1965). These authors adduced the theoretical expressions for the dependences of relaxation times on concentration of free binding sites of the enzyme and free ligand concentration. In the present communication we have analysed the relaxation kinetics of dissociating enzyme system 2p~ P in the presence of a specific ligand.
2. Relaxation The equilibration
Kinetics
of a dissociating enzyme system of the type k+m
2p*P k-oo is discussed, where P is an enzyme oligomer which is able to dissociate reversibly forming two identical halves p, k+oo is the second-order rate constant for association and keoo is the first-order rate constant for dissociation. Let us assume that (1) oligomeric forms P and p contain n and n/2 specific ligand (L) binding sites respectively; (2) ligand binding sites are equivalent and independent in each oligomeric form; (3) the steps of Iigand binding, (pLi-i + L$ pLi and PLi-i + Ls PLi) are in rapid equilibrium; (4) the total ligand concentration [L10 exceeds greatly the total enzyme concentration [El,-,; (5) the microscopic complexes of oligomeric enzyme formed with ligand are able to undergo reversible dissociation-association: k
pL, + pL, *
k-x,
L,PL,,
(2)
where pL, and pL, are microscopic complexes of oligomeric form p with x and y molecules of the ligand respectively, L,PL, is a microscopic complex of oligomeric form P with the ligand where one part of the oligomer contains x bound ligand molecules and the other part contains y bound ligand molecules, and k+xy and k-,,are microscopic rate constants for association and dissociation respectively.
KINETICS
OF
DISSOCIATING
ENZYME
SYSTEMS
75
With such assumptions the concentrations of microscopic complexes of oligomeric forms p and P with the ligand are given by the relationships
[PLI = [PI (rLl”IKk )“, [L,PL,l=
Plm.-lo/K:
Ix+‘,
(05x
(n/2;
(3) OCry In/2)
(4)
where [p] and [P] are concentrations of free oligomeric enzyme forms p and P respectively, Kk and K[ are microscopic dissociation constants for the complexes of p and P forms with the ligand respectively. The conditions of detailed balance yield the following relationship for microscopic rate constants k,,, and k-,, and microscopic rate constants k+,,,, and kQo:
The kinetic equation for the process (2) has the following form: -- ;
$g
G2 [PLXI) n/2
n/2
= C C (k+xrChCL2 x=0 y=o
[pLApL,l-
bC/2C,2
[LxPL, I).
(6)
Taking into account the equations (3), (4) and (5) the following expression for the relaxation time (T) can be obtained:
(7) where K. = k+oo/k-oo is the constant of association of halves p and [p] is the equilibrium concentration of free oligomeric form p. It should be noted that the rate constants k,,, and k-,, for which 2 I (x + y ) 5 (n - 2) are the “concealed” parameters, i.e. they cannot be found from the experimental dependence of r on [LIO and [p]. This is a result of uncertainty in distribution of ligand molecules between two parts of oligomer P. Therefore it is reasonable to introduce the effective rate constants for dissociation (k-i) and for association (k+i) which are explicit parameters and related to k-,, and k+,, rate constants by the expressions (i = x + y )
(8) Chk+i = nf nf CX,/2C$2k+,y. x=0 y=o
(9)
76
S.
The relationship
between k+i and km, is identical to relation (5):
V.
KLINOV
AND
B.
I.
KURGANOV
k+i 7tJ=Kod where cp= KQKL. The equation of material
balance
42 [Elo
= x;.
n/2 C”,,,
(10)
[@x1
+ 2
n/2
c
c C”,,,C’,,2 x=0y=o
(where [El0 is the total enzyme concentration form p) can be solved with respect to [p]:
[ULI
(11)
calculated on the oligomeric
4Ko[p] = (dl + 8K[Elo - 1) ( (;lT;;;,$);:2
7
(12)
where K is the apparent association constant
1+ [Lb/K:
K = K”
(
1 +[L]iJK;
>.
(13)
Taking into account equations (8) and (12) one can transform the expression (7) to the following form: t= k-d1 +8K(E]o.
(14)
In the expression (14), k- represents the apparent rate constant for dissociation of oligomer P into two halves p in the presence of the ligand: k-c i. C’;k-i(g)i/(
I +$$)n,
(15)
The following equation is also valid for relaxation time:
~=$JI+~K[EI,, where k, is the apparent rate constant for association of oligomeric p in the presence of the ligand:
(16)
forms
(17)
KINETICS
OF
DISSOCIATING
ENZYME
SYSTEMS
77
The linear anamorphoses connecting the values of r and [El0 may be obtained from equation (14): $=
k? +8k’K[E]o,
1 ,=Bk’K+k$p [Elo~
(19)
These anamorphoses have the same form as those obtained for relaxation of the dissociating enzyme system 2p *P in the absence of the ligand: L= 72
1
Elo~
kTOo +Bk&K,[E],,, 2=8k?ooKo+k?m[E10,
1
(22)
T2 - - 1 - 8K,,[E]Or’. k&
The following linear anamorphoses correspond to equation (16): (24) -- 1
[Elope-
8k:: K
k: 1 K*[Elo’
(25)
2
T* = KF-8K[E]o~2. +
3. The Principles
of Analysis of Relaxation
Kinetics
The dependence of relaxation time on ligand concentration for dissociating enzyme system 2p C$ P are generally rather complex and cannot be used for estimation of the values of kinetic constants. However, if the steps of ligand binding (pLi-1 + L s pLi and PLi-1 + LSPLi) are in rapid equilibrium the dependences of relaxation time on total enzyme concentration obtained at fixed concentrations of the ligand have simple form and can be transformed to linear relationships. Using linear anamorphoses (18), (19) or (20) allows one to determine the apparent association constant
78
S.
V.
KLINOV
AND
B.
I.
KURGANOV
(K) and the apparent rate constant for dissociation of oligomer P(k..) at definite fixed concentration of the ligand. If the values of K and km are determined at various concentrations of the ligand, the microscopic dissociation constants for complexes of oligomeric forms with the ligand (KL and Kf respectively) and the effective rate constants for association of halves p(k+i) and for dissociation of oligomer P(kei) may be calculated from the quantitative analysis of the dependences of K and k- on [L& based on using theoretical expressions (13) and (15). If we use the linear anamorphoses (24)-(26) the following analysis of the dependences of the apparent constants K and k, on ligand concentration with the aid of equations (13) and (17) allows us to determine the parameters Kt, K[, KO, k-i and k,,. It should be noted that above expressions for dependence of relaxation time r on total enzyme concentration [El0 remain valid also for the situation when the ligand binding sites in oligomeric forms p and P are non-equivalent or interacting. However, in this case complication of the dependences of K, k- and k, on [LIO will take place. It is evident that the relaxation spectrum of the dissociating enzyme system under discussion must include not only the slow relaxation process corresponding to the change in oligomeric state of the enzyme but additional faster relaxation processes which are due to ligand binding to oligomeric enzyme forms. If ligand binding sites in each oligomeric enzyme form are equivalent and independent, kinetically significant species are free ligand binding sites in oligomeric forms p and P and their complexes with the ligand. According to Eigen (1967), the system consisting of n kinetically significant species has (n - 1) relaxation times. Therefore dissociating enzyme system 2ptiP will have two relaxation times when ligand binding sites of oligomeric forms p and P are kinetically identical and three relaxation times when ligand binding sites of oligomeric enzyme forms are kinetically non-identical (see, for example, Kirschner et al., 1971). This evident conclusion has not yet received an experimental corroboration. It is worth noting that Tai & Kegeles (1975) observed a single relaxation time for Helix pomatia a-haemocyanin (the dissociating protein system being of the type 2pgP) in the presence of the specific ligand (Ca” ions). This circumstance may testify to the reality of simplifying assumptions used by us in the theoretical analysis. 4. Particular
Cases
We have assumed above that oligomeric form p contains n/2 equivalent and independent ligand binding sites. In addition we assume that association
KINETICS
OF
DISSOCIATING
ENZYME
SYSTEMS
79
site of oligomeric form p consists of n/2 association subsites which are equivalent and independent. The association of two halves p to give oligomer P results in the formation of pair-contacts between all association subsites of halves p. We assume also that each protomer contains one ligand binding site and one association subsite. Two possible states of the protomer must be taken into account: the state (A) in which the ligand binding site of the protomer is not occupied by the ligand and the state (B) in which the ligand binding site of the protomer is occupied by the ligand. The oligomeric form P will contain three types of contacts between protomers: AA, AB and BB. If microscopic complex L,PL, has m contacts BB, 1 contacts AB and (n/2-m -I) contacts AA respectively (where m and 1 are integer numbers), the activation free energy for its dissociation can be written (see, for example, Ricard, Mouttet & Nari, 1974) as AG!,,
= m AG&
+ IAG!ae
+ (n/2 -m - l)AGTaA,
(27)
where AG&, AGfAB and AG! AA are the activation free energies for the dissociation of contacts BB, AB and AA respectively. The following relationship is fulfilled in the absence of the ligand: AG!,,o = (n/2)AG!aA.
(28)
If the association subsites are equivalent and independent, we can assume that the change in activation free energy for the dissociation of the contact between two protomers after ligand binding to one of the protomers remains invariable whether the other protomer is occupied by the ligand or not: AG& Equation
- AGtAB = AG:Ar, - AGTA/,.
(27) can be simplified AG!,,
(29)
to the following form:
= AGfoo +i(AG!*,
- AGtAA),
(30)
wherei=2m+l. The expression for the rate constant for dissociation of microscopic complex L,PL, which has m contacts BB and 1 contacts AB can be written in terms of the absolute rate theory as k-,,=(gexp(
-AGt,,/RT
) = k-,,,,ai,
(31)
where k-oo = (kBT/h)
exp {-AG!,JRT}
(32)
AB-A@AA)/RT).
(33)
and a = exp {-(AG!
80
S. V.
KLINOV
AND
B.
I. KURGANOV
ks, h, R and T are the Boltzmann constant, the Planck constant, the gas constant and the absolute temperature, respectively. It should be noted that different microscopic complexes L,PL, may have the same number of contacts BB and AB and the same their spatial disposition but differ in location of states A and B in contacts AB: state B (or A) may belong to one or other half of oligomer P. Using the known combinatoric relationship (see, for example, Ezhov, Skorochod & Yadrenko, 1977)
ck+, =
i
cic”,-i
(34)
one can transform equation (8) in the following way: k-i = k-ooa i
or
k-i/k-(i-1,
= OZ.
(35) The analogous expression for k+i can be obtained with the aid of relation (10): or k+i/k+,i-1, = CY(P. k+i = k+oo(ap)’ (36) The denominators of the corresponding geometric progressions (a or crcp) characterize the influence of the ligand on the rates of dissociation of oligomer P and association of halves p respectively. the expressions for apparent rate constants for dissociation of oligomer P and for association of halves p acquire the following form: k- = k-o,,
1+ (Y[L]O/K[ ” 1+ [L]lJ/K[ > ’
(37)
With such assumptions the change of the dependence of T on [El0 at variation of ligand concentration is determined by the values of parameters CYand cp. Parameter cy determines the character of the dependence of kon [L&,, parameter cp determines the character of the dependence of K on [L],, and the product of parameters cx and Q determines the character of the dependence of k+ on [L]o. The theoretical dependences of T on [El0 calculated at various fixed values of concentration of the ligand and some values of parameters cu and Q are given below for illustration (for II = 4). It is reasonable to introduce the dimensionless ligand concentration lo = [L],/Kt. The expressions for K, k- and k+ are rewritten as (39)
KiNETICS
OF
DISSOCIATING
ENZYME
SYSTEMS
81
n
k-=
k-00 i’
~ l+cpf, +aplo)
’
(41)
k, = k+oo(%Jn.
(A)
THE
CASE
(40)
WHEN
cp = 1
If the oligomeric forms p and P are characterized by identical affinity with respect to the ligand (rp = l), the linear anamorphoses constructed in coordinates {l/(koO~)*; Ko[Elo} at various fixed values of ligand concentration must have a common point of intersection on the abscissa (Fig. 1).
4
;:
(b)
I, = 0 /
K, IElo
FIG. 1. The theoretical dependences of relaxation time T on enzyme concentration [E]a represented in the coordinates {~/(/c-~~T)~; KOIE] 0} at n = 4, cp= 1 and various fixed values of dimensionless ligand concentration I0 = [L&/K;. (a) a =: 1.3 and (b) CI= 0.7.
82
S.
V.
KLINOV
AND
B.
I.
KURGANOV
Increasing ligand concentration results in a decrease in the value of relaxation time when LY> 1 [Fig. l(a)]. On the other hand, if cr < 1 increasing ligand concentration will result in an increase in the value of relaxation time [Fig. l(b)]. (B)
THE
CASE
WHEN
LY =
1
If the rate constant for dissociation of oligomer P does not depend on the degree of its saturation by the ligand (ar = 1) the linear anamorphoses constructed in coordinates {l/(k.+,~)~; &[Eb} at various fixed values of ligand concentration must have a common point of intersection on the ordinate (Fig. 2). Increasing ligand concentration results in a decrease in (0) 8-
4-
FIG. 2. The theoretical dependences of the reciprocal value of the square of the dimensionless relaxation time k-,,o~ on dimensionless enzyme concentration KOIE10 at n = 4, Q = 1 and various fixed values of dimensionless ligand concentration lo= [L],,/Kt. (a) cp=‘0.6 and (b) cp= 2.0.
KINETICS
OF
DISSOCIATING
ENZYME
SYSTEMS
83
the value of the relaxation time when cp< 1 [Fig. 2(a)]. On the other hand, if cp> 1 increasing ligand concentration will result in an increase in the value of the relaxation time [Fig. 2(b)]. (C)
THE
CASE
WHEN
&
= 1
If the values of parameters (Y and (o obey the condition (Y’C+Y = 1 the linear anamorphcses constructed in coordinates {l/(k-00r)2; &[E&} at f0 = 0 and l,, + co have the same slope (Fig. 3). Increasing ligand concentration results in a decrease in the value of the relaxation time when (Y> 1 [Fig. 3(a)]. On the other hand, if (Y< 1 increasing ligand concentration results in an increase in the value of the relaxation itime [Fig. 3(b)].
15 -
2 ;:
(b) 2-
FIG. 3. The theoretical dependences of l/(k-Oo~)2 on K&Z], at n = 4, a*~ = 1 and various fixed values of dimensionless ligand concentration lo = [L&/K;. (a) (I = 1.3 and (b) a = 0.7.
84
~.V.KLIN~V (D)
AND THE
B.I.KuRGANOV
CASE
WHEN
crcp=l
If the rate constant for association of halves p does not depend on the degree of its saturation by the ligand (acp = 1) the linear anamorphoses constructed at various fixed values of ligand concentration do not have a common point of intersection (Fig. 41, but all points of intersection are
N ;: 8
-02
0
02
O-4
06
FIG. 4. The theoretical dependences of l/(k-oor)2 on &[E& fixed values of dimensionless ligand concentration lo = [L]o/Kt.
at n = 4, ap = 1 and various (a) a = 1.3 and (b) a = 0.7.
disposed in the left lower quadrant. Increasing ligand concentration results in a decrease in the value of the relaxation time when cy> 1 [Fig. 4(a)]. On the other hand if (Y< 1 increasing ligand concentration results in an increase of the value of the relaxation time [Fig. 4(b)].
KINETICS (E)
THE
CASE
OF WHEN
DISSOCIATING
THE
ENZYME
VALUES
OF
ARBITRARY
BUT
85
SYSTEMS
PARAMETERS
(Y
AND
q~
ARE
FINITE
In this case increasing ligand concentration results in a decrease or an increase in the value of relaxation time when the value of parameter (Y is greater or less than unity respectively. The values of intercepts on abscissa and ordinate axes have definite limits when [L]” + 0 or [L],, + CO. (F)
THE
DETERMINATION
OF
THE
PARAMETERS
The primary analysis of relaxation kinetics with the aid of linear relationships (18)-(20) allows us to construct the dependences of K and kp (or k,) on ligand concentration. The following analysis of these dependences using theoretical equations (39)-(41) enables us to estimate parameters K,, Kt, K[, kdoo and kcoo. We shall henceforth assume that parameter n is known (it can be found by an independent method). If the experimental dependence of K on [L-k, allows the association constant to be determined at very low concentrations of the ligand (Kd), the dissociation constants K[ and Kfl may be calculated by means of the linear relationship e-11
Klo
-
= &vKJKo. L
(42)
I
If, however, the experimental data enable the apparent association constant to be calculated at sufficiently a high concentration of the ligand (Kli,), then in order to find KL and K{ one may utilize the linear relationship (43) When it is not possible to determine the limiting values of the association constant (when [L&+ 0 and [L10 +a~), the difference method may be adopted as follows: the experimental data are presented in the form of a plot of fi against [LlO. A pair of points are chosen on this plot, whose abscissae differ by the quantity A. We designate the ordinates of these points by @ and sl/Ka respectively. It can then readily be shown that the following linear relationship holds, enabling the constant Kh and the ratio of constants e/K: to be found:
This plot may be constructed various values of [Llo and A. Knowing Kt
86
S.
and e/K:,
V.
KLINOV
AND
B.
I.
KURGANOV
it is possible to calculate K,, from the equation
Finally, the constant KII. is calculated from the values of %JKL and K. so obtained. If one of the dissociation constants is determined in an independent way (for example from the dependence of k- on [L],J the other dissociation constant and parameter K. may be calculated using the following linear relationships:
~(~+[L]~/K~)=~+(~/K~)[L]~ (if Kt
(461
is known) and
1+DAoIKLYz
1 + --hLlo
+‘i?i,
(47)
3’&Kt
(if Kf: is known). The determination of the parameters from the dependence of k- on ligand concentration in the accordance with theoretical equation (37) is fulfilled in an analogous manner. If experimental data enable the rate constant to be determined at very low concentrations of the ligand (k-d, parameters (Yand K[ may be calculated by means of the linear relationship a-1
[Llo
=g-+QiZi.
(48) L
If, however, the experimental data enable the apparent rate constant to be calculated at sufficiently high concentrations of the ligand (k-),i,, then in order to find parameters (Y and K[ one may use the linear relationship
When it is not possible to determine the limiting values of the rate constant k- (when [Llo+ 0 and [Llo + co) the difference method may be ado ted. The experimental data are presented in the form of a plot of ti k- against [Llo. A pair of points are chosen of this plot, whose abscissae differ by the uantity A. We designate the ordinates of these points by 9% and tp kea, respectively. The following linear relationship holds, enabling the constant KL and the value of ~/cxG to be found:
A viy~=~+avi&
K;:
1
AK ( [L1o+ic&c
).
(50)
KINETICS
Knowing equation
K[
OF
DISSOCIATING
and l/a(f/k-
ENZYME
SYSTEMS
00, it is possible to calculate
87
kmoo from the
Finally, parameter cy is calculated from the values of l/,G and kpoo. If the dissociation constant K[ is determined by independent way (for example, from the dependence of K on [LlO) the rate constant for dissociation of oligomer P(k-& and parameter (Y may be calculated using the following linear relationship:
1 : WI0 = &---oo+atG ,,[Llo. K;) L
q
(52)
In definite cases the dependences of apparent rate constant k, on ligand concentration may be convenient for determination of some parameters of the dissociating enzyme system (namely parameters k+oo, Kt and acp) using the theoretical equation (38). The linear relationships (48)-(52) are applicable for this aim after replacing k-, kmoo, Kt and cr by k+, k+oo, K; and (~9 respectively. 5. The Limiting Consider the limiting to bind the ligand.
(A) OLIGOMERIC
FORM
Situations
situations when one of oligomeric
P DOES
NOT
BIND
THE
LIGAND
forms is not able
(Kfl. + 00, cp + 0)
The association of halves p may result in steric screening of ligand binding sites. In this case the ligand interacts exclusively with p form (cp= 0) and therefore induces the dissociation of oligomeric form P. The saturating concentrations of the ligand provoke the complete dissociation of P form into its two halves. When oligomeric form P is not able to bind the ligand the dependence of T on [E]o is described by equations (14) and (16) in which the apparent association constant K and the apparent rate constants for dissociation kand for association k+ are expressed in the following way: (53) k- = keoo
(54)
88
S. V.
KLINOV
AND
B.
I.
KURGANOV
(55) The linear anamorphoses constructed in co-ordinates { 1 /(k -OOr)2; KOIE],} at various fixed values of ligand concentration have the common point of intersection which is disposed on ordinate axis (Fig. 5). Increasing ligand concentration results in an increase in the value of relaxation time. When lo + co the linear anamorphosis becomes parallel to abscissa axis.
-2
0
2
4
6
FIG. 5. The theoretical dependences of ~/(/c-~~T)* on Ko[E]o at various fixed values of dimensionless ligand concentration I,J=[L]~/K~ in the case when the oligomeric enzyme form P is not able to bind the ligand (n = 4). (B)
OLIGOMERIC
FORM
p DOES
NOT
BIND
THE
LIGAND
(K;
-, a,
Cp + 0;))
One may now discuss the situation when the ligand interacts exclusively with P form (cp+ co) and, as a result of this, displaces the equilibrium 2p$ P towards the formation of the oligomer P. Such a situation is possible when ligand binding site is created in the region of the contact of halves p in the oligomer P. The apparent association constant K and the apparent rate constants for dissociation (k-) and for association (k,) determining the character of dependence of T on [El0 are expressed in the following way: K =&(l+[L]o/K:)“,
(56)
k_=k_oo/(l+[L],/K~)“,
(57)
k+ = k+oo.
(58)
The linear anamorphoses constructed in coordinates {l/(k-oar)*; Ko[E]o} at various fixed values of ligand concentration do not have common point
KINETICS
OF
DISSOCIATING
ENZYME
SYSTEMS
89
of intersection (Fig. 6), but all points of intersection are found in the left lower quadrant. Increasing ligand concentration results in an increase in the value of relaxation time. When [L&,/K; + 00 the linear anamorphosis fuses with abscissa axis.
FIG. 6. The theoretical dependences of l/(k-,,,~)’ on &[E], at various fixed values of dimensionless ligand concentration [L]JK; in the case when the oligomeric enzyme form p is not able to bind the ligand (n = 4).
6. Conclusion
The principles of analysis of relaxation kinetics of dissociating enzyme systems 2peP in the presence of specific ligand elaborated in this paper supplement the general principles of analysis of kinetic behaviour of dissociating enzyme systems (Kurganov, 1973,1977,1978). Whether we study any properties of dissociating enzyme system at equilibrium conditions or the kinetics of attainment of equilibrium between oligomeric forms in the presence of specific ligands, the analysis must be started from the dependences of definite characteristics of dissociating enzyme system (the degree of dissociation, specific enzyme activity, the degree of saturation by the ligand, relaxation time and so on) on enzyme concentration obtained at varoius fixed values of ligand concentration. REFERENCES M. (1967). In Fast Reactions and Primary Processes in Chemical Kinetics (Claesson, S., ed.), New York: Wiley-Interscience, D. 333. EZHOV, I. I., SKOROCHO~, A. V. & YA~RENKO, M. I. (1977). Elements of Combinatorics, Moscow: Nauka, p. 40. EIGEN,
90
S. V. KLINOV
AND B. 1. KURGANOV
KIRSCHNER, K., EIGEN, M., BITYMAN, R. & VOIGT, B. (1966). Pm. natn. Acad. Sci. U.S.A. 56, 1661. KIRSCHNER, K., GALLEGO, E., SCHUSTER, I. & GOODALL, D. (1971). J. mol. Biol. 58,29. KLJRGANOV, B. I. (1973). Acra biol. med. germ. 31, 181. KURGANOV, B. I. (1977). J. theor. Biol. 68,521. KURGANOV, B. I. (1978). Allosteric Enzymes, Moscow: Nauka. MONOD, J., WYMAN, J. & CHANGEUX, J.-P. (1965). J. mol. Biol. 12, 88. PECHT, I. & RIGLER, R. (eds) (1977). Chemical Relaxation in Molecular Biology, Berlin,
Heidelberg, New York: Springer Verlag. RICARD, J., MOUTTET, TAI, M.-S. & KEGELES,
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41,479.
APPENDIX
Nomenclature
Elo k +00 k-m k+i, k-i k+ k-
k +XY k-x, Ko, K Kt, KII. L lo n P [PI, [PI
P t a 7 cp
and Notation
the total molar enzyme concentration calculated on the oligomeric form p the rate constantfor associationof two halvesp into the oligomer P the rate constantfor dissociationof oligomer P to two halvesp the effective rate constantsfor associationof two p and for dissociation of oligomer P respectiveiy the apparent rate constant for associationof halves p into oligomer P in the presenceof a specificligand the apparent rate constant for dissociationof oligomer P to two halves p in the presenceof a specificligand the rate constantfor associationof oligomericforms p containingbound ligand (pL, + pL, + L,PL, ) the rate constantfor dissociationof oligomerP onehalf of which contains x bound ligand moleculesand the other half contains y bound ligand molecules(L,PL, + pL, + pL,) the true and apparent (in the presenceof a specific ligand) association constantsfor the equiibrium 2p z P the microscopicdissociationconstantsfor the complexesof p and P forms with a ligand a specificligand [L&,/K L (the dimensionless ligand concentration) the number of ligand binding sitesin oligomeric form P the oligomeric enzyme form which is formed at reversible dissociation of enzyme oligomer P the current and equilibrium concentrationsof oligomericform p enzyme oligomer which is able to dissociatereversibly forming two identical halvesp time k-ilk-o-u relaxation time K;/K;: