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The theoretical analysis of kinetic behaviour of “hysteretic” allosteric enzymes III. Dissociating and associating enzyme systems in which the rate of installation of equilibrium between the oligomeric forms is comparable to that of enzymatic reaction
J. theor. Biol. (1976) 60, 287-299
The Theoretical Anaiysis of Kinetic Behaviour of “Hysteretic” Allosteric Enzymes III. Dissociating and Associating Enzyme Systems in which the Rate of Installation of Equilibrium between the Oligomeric For&s is Comparable to that of Enzymatic Reaction B. I. KURGANOV, A. I. DOROZHKO, Z. S. KAGAN AND V. A. YAKOVLEV The All-Union
Vitamin Research Institute, Moscow, U.S.S.R. (Received 23 December 1974)
The theoretical analysis of the shape of product (Pr) accumulation versus time (t) plots is represented for the slowly associating enzyme systems 2p s P (where P is an enzyme oligomer reversibly dissociating into two identical halves p) and M * Ma 2 M3 z . . . (where M is the monomer having two association sites overlapping with the active sites). It is assumed that the rate of installation of equilibrium between oligomeric forms is comparable to that of enzymatic reaction and that the complexes of oligomerit forms with the substrate are in fast equilibrium with free components. It is shown that the specific property of kinetic behaviour of the slowly dissociating enzyme systems is revealed by the dependence of r on enzyme concentration (z is the intercept of the linear part of the kinetic curve of Ipr] versus t with time axis).
1. Introduction A “hysteresis” of the allosteric enzyme may be explained partly by a relatively low rate of installation of equilibrium between enzyme oligomeric forms having different molecular weights and kinetic parameters. A shift of equilibrium between enzyme oligomeric forms in the course of enzymatic reaction may accelerate an enzymatic process and give rise to a lag-period in the timedependent product accumulation curves. On the other hand, it may slow down an enzymatic reaction with appearance of burst in the kinetic curves. The most simple explanation of an equilibrium shift between enzyme oligomerit forms may be the change of enzyme concentration
when the reaction
is launched by the enzyme. For example, the burst in kinetics of enzymatic reaction catalyzed by a-N-acetylgalactosaminidase (EC 3.2.1.49) from beef liver when the reaction is started by addition of small quantity of the enzyme 287
288
B.
I.
KURGANOV
ET
AL.
to the substrate solution may be due to dissociation of active enzyme oligomer upon diluting? to give inactive subunits (Wang & Weissmann, 1971). Evidently, the most interesting cases are those when an equilibrium shift between oligomeric enzyme forms in the course of enzymatic reaction is due to associating or dissociating substrate effect (or allosteric effector effects) which appears upon substrate addition to the enzyme. “Biodegradative” L-threonine dehydratase (EC 4.2.1.16) from Clostridium tetanomorphum (Vanquickenbourne & Phillips, 1968), a-N-acetylgalactosaminidase from beef liver (Wang & Weissmann, 1971), phosphofructokinase (EC 2.7.1.11) from rabbit skeletal muscles (Hofer, 1971), pyruvate kinase (EC 2.7.1.40) from BacilIus licheniformis (Tuominen & Bernlohr, 1971), phosphoenolpyruvate carboxylase (EC 4.1.1.31) from Escherichia coli (Wohl & Marcus, 1972) and anthranylate synthase (EC 4.1.3.27) from Bacillus sibtilis (Kane, Holmes, Smiley & Jensen, 1973) are examples of dissociating allosteric enzymes in which an appearance of lag-period in the time dependent product accumulation curves is due to an association of inactive (or less active) subunits into active oligomer under the influence of substrate. Dissociation of inactive associates of the enzyme under the influence of substrate may produce the lag-period in the time dependent product accumulation plot. Examples of such associating enzyme systems may be glycogen-phosphorylase A (EC 2.4.1 .I) from frog skeletal muscles (Metzger, Glaser and Helmreich, 1968), phosphorylase kinase (EC 2.7.1.38) from rabbit skeletal muscles (Kim & Graves, 1973) and polyoxyethylene ester of N-benzoylhistidine, a polymeric catalyst which may be considered as a model of associating allosteric enzyme (Topchieva, Solov’eva, Kurganov & Kabanov, 1972). “Biodegradative” threonine dehydratase of Escherichia coli offers an excellent example of the hysteretic allosteric enzyme systems whose activity is determined by the enzyme concentration (Dunne & Wood, 1975). The present paper is concerned with the pattern of product [Pr] accumulation versus time (t) curves for associating enzymic systems in which the rate of installation of equilibrium between oligomeric forms is comparable to that of enzymic reaction. 2. Dissociating
Enzyme System 2p + P
The system P F? 2p is evidently the most simple dissociating enzyme system. P is an enzyme oligomer able to dissociate reversibly into two identical halves p. In general the form p may include several subunits which are bound essentially stronger then p halves in oligomer P. t In this experiment the change of enzyme dissociation was due not only to the changing of enzyme concentration but to changing of the temperature as well (from 0” to 38T).
HYSTERETIC
PROPERTIES
OF
ALLOSTERIC
Let us assume that the rate of installation of equilibrium and P is comparable to that of enzymic reaction: f+oo
ENZYMES
289
between forms
2p P P k-cm
p
(1)
where k+,,,, is the rate constant for association of subunits p and k-,, is the rate constant of oligomer P dissociation. Let us assume that (1) an enzymatic reaction involves one substrate and is irreversible, (2) the substrate concentration is constant (S), (3) the reaction is not inhibited by the product, (4) the sites binding the substrate are equivalent and independent in each oligomeric form and finally (5) the complexes PSi and PS, are at rapid equilibrium with free components. With the last two conditions the concentrations of complexes psi and PSI are connected by the following relation with concentrations of free oligomeric forms : [psi]
=
(n’2)!
cs),
i!(n/2 - i)! ( Ks >
Cpsi] = i!(:A
*[p]
ij, (e)‘[P]
(0 < i < n/2) (0 < i < n)
where Kk and Kg are the microscopic dissociation constants of substrate complexes with forms p and P respectively. (It is assumed that the forms p and P contain n/2 and n sites binding the substrate respectively). Let k+, and k-, be the rate constants of mutual transformation of oligomeric forms which are in the complex with substrate: k+, PSx+PS, +r PSx+, k-xy
(4)
(0 < x < n/2 and 0 < y < n/2). In equilibrium (4) the form PS,,, is an oligomer one part of which contains x bound substrate molecules and the other part involves y bound substrate molecules (i.e. the form PS,,, may be written as pS;pS,). Let us choose two cycles:
and
under condition that in the left cycle x substrate molecules are bound by one half of the oligomer, while in the right cycle other half of the oligomer binds y
290
B.
I.
KURGANOV
ET
AL
substrate molecules. The conditions of detailed balance for these cycles yield the following relation for the rate constants k+xy and k-,, and k,,, and k-,,: k +.Ty =a--~ li +oo I($ x+p _--. k-0, ( r;c; > k-x,
(6)
where c( = 1 at x = J and c( = 2 at x # y. Let y be the portion of enzyme present in p form: i=n/2 izo
[Psi1
(7) y
=
-mo
where [El0 is total molar enzyme concentration per form p. Let us assume that an enzymatic activity is measured in the following manner. An enzyme solution (of certain concentration [El,) is preincubated until the equilibrium between oligomeric forms is installed. The portion of p form prior to the substrate addition (and thus at an initial moment of reaction) is (Kurganov, 1967, 1968): yo = -741So
(8)
where z. = Jl + SK,[E],, K. is association constant in the absence of substrate: K, = [P]/[p]’ = k+oo/k-oo. If the substrate shifts an equilibrium between oligomeric forms p and P (i.e. K; # Kg) the portion of form p will change with the time after launching the reaction and approach limiting value after sufficiently large time, yr: yf = 2/(1 -tzf)
(9)
where zI. = &+8rQC,[E]~; q = [( I+ [S],/l;;)/( I -t [S]o/rC’,)]“. The change of portion of form p with the time (t) is describedby the following relation : (Zf + l)[(zo + zr) -(z. - zf) .exp ( - GzJ t)] Y = Y, -__ CZf+l)(zo+zf)+(zfl)(z,--zf).exp( -Gzft)
(IO)
where
and k-,, = km,,. Tt should be noted that in expression for G the rate constants k-,, for which 2 < (x+y) d (n-2) are the “concealed” parameters (the term proposed by Kobosev, 1959), i.e. they cannot be found experimentally from G versus [Slo plot. Let us give an illustrative example of dissociating system: dimer (D) F? tetramer (T). The tetramer (which may be denoted as D. D) gives two types of complexes TS, one of which (D . DS,) may dissociateinto D+ DS, with rate constant k-,, and the other (DS*DS)
HYSTERETIC
PROPERTIES
OF
ALLOSTERIC
291
ENZYMES
into DS + DS with rate constant k- i r. Using experimental plot G versus [S]e one may estimate only the sum of rate constants (4k-,, +2kmoz) instead of individual values, in other words the rate constants ksoz and k-ll are “concealed” parameters. It should be noted that for the system 2D # T the rate constants kwol, k-,, and k-,, are “explicit” parameters and could be determined from G versus [S],, plot. In this view it is reasonable to introduce the effective rate constants k-, which are explicit parameters and related to k-,, rate constants by the expression (i = x+y):
The expression of type (10) for the portion of form p versus time is valid when an enzymatic activity of dissociating system 2p * P is measured in the presence of allosteric effector (F) on condition that the latter affects enzymatic activity only by shifting the equilibrium 2p s P (i.e. the effector does not change specific enzymic activity of forms p and P). Assuming that the forms p and P contain n/2 and n sites binding the effector, respectively, the parameter 9 defining zf has the form: (13) and the relation for G:
(1+ t-Slo/W(1+ CFloPW
(14)
’
In these relations Kk and Kk are the microscopic dissociation constants for effector complexes with forms p and P respectively and kmij are the effective rate constants for dissociation of oligomeric form PSiFj into two halves. (A)
THE CASE OF INACTIVE
p FORM
The pattern of time-dependent product accumulation curve is determined by ratio of specific enzymatic activities of forms p(q) and P(a2). For the system 2p it P in which the p form is inactive (a, = 0) the dependence of product [Pr] concentration on time t has the form: a2(zf--
cprl= -G-
O2 0
t-
a2 __
2GvKo
(z,+~)(z~~z~)-(z~-~)(z,--~).~~P(-G~~~)
2z,(zo
+ 1)
>
(15)
When q > 1 (the summary influence of substrate and allosteric effector shifts
292
B.
I.
KURGANOV
ET
AL.
an equilibrium 2p it P towards formation of active form P) the rate of enzymatic reaction enhances during the process, i.e. the time-dependent curve of product accumulation has a lag-period. When q < 1 (the summary effect of substrate and allosteric effector shifts an equilibrium 2p + P towards formation of inactive form p) the rate of enzymatic reaction is diminished during the process, i.e. the kinetic curve has an initial burst. Let us introduce the dimensionless product concentration :
LPI = y The dimensionless product concentration
[Pr]. [II]
(16)
versus dimensionless time Gt
I.0
XII
Gf 1
I
(b)
/
0.01 -
U-II 0~005-
-5 FIG. 1. The dependences dimensionless time Gt for inactive at 11= 10 (a) and calculated by the equation
0
5
Gf of dimensionless product concentration [II] = 2G&[Pr]/az on the slowly dissociating enzyme system 2p d P with p form 0.1 (b) and varied dimensionless enzyme concentration KOIElo (17).
HYSTERETIC
PROPERTIES
OF
ALLOSTERIC
ENZYMES
293
for system 2p # P at a, = 0 has the form:
(z,+1)(~~+~~)-(~~-11)(z~--~).ex~[:-z,W>l . c1,j + 1) > The plots of [II] versus Gt calculated by means of this equation for q = 10 and 0.1 and varied dimensionless enzyme concentration &,[E],, are shown in Fig. 1. The portion of the time axis limited by the continuation of a 2zxz,
o-
-4
, -6
I -4
f
I -2
1
I 0
,
I 2
I
I
IqKoIEI,
FIG. 2. The dcpendences of value Gr on dimensionless enzyme concentration KOIEIO in logarithmic co-ordinates for the slowly dissociating enzyme system 2p F? P with inactive p form at varied q [(a) corresponds to q > 1 and (b) conesponds to v < 11. The dependences of Gs on &[E], are calculated by the equation (18). Gt is a portion of time axis limited by continuation of a linear asymptote of m versus Gt at Gt + co.
294
B.
I.
KURGANOV
ET
AL.
linear asymptote in [II] versus Gt plot for Gt -+ a3 is equal to:
The dependences of CT calculated by this relation (for ye> 1) and -Gr (for Q < 1) upon dimensionless enzyme concentration K,,[E], for different values of q are shown in Fig. 2. The absolute value Gz decreases with increasing enzyme concentration and the tangent of the slope for the plot in logarithmic co-ordinates being close to - 1 at the sufficiently high enzyme concentration. A decrease of lag-period, T, for time-dependent product accumulation versus time curves at an increase of enzyme concentration has been observed for phosphoenolpyruvate carboxylase (EC 4.1.1.3 1) from Escherichiu coli (Wohl
& Marcus,
1972) [Fig.
3(a)].
The lag-period
for this enzyme
association of inactive dimers to give active tetramer
is due to
under influence of
30 .5 E 20 i
0
\
IO
\ \
0
_~
_A-.--L-.-I20 40
0.F
60
L--~---L~--i-~-3 ‘,
CElo,/Lgim
.A-I 1 ,I
1-6
I.8
jag [El,
FIG. 3. The dependence of duration of lag-period 7 (in the ~LILLC‘Sof product accumulation in the course of enzymatic reaction catalyzed by phosphoenolpyruvate carboxylase from E. co/i) on enzyme concentration in the (7; [El,; co-ordinates (a) (Wohl & Marcus, 1972) and in the {log 7; log [El,} co-ordinates (b). The reaction was initiated by addition of substrate (phosphoenolpyruvale, 12 111M) to the enzyme solution containing 0.1 M 2-mercaptoethanol and 10 mM asparate (inhibitor). 0.1 M Tris-HCI buffer, pH 8.5, 25°C. FIG. 4. The dependences of value Gr on dimensionless in the logarithmic co-ordinates for the slowly dissociating inactive P form, at varied q [(a) corresponds to r] < 1 and dependences of CT on &[E], are calculated by the equation
enzyme concentration K,[E]o enzymic system 2p ti P with (b) corresponds to rl > 11. The (20).
I (‘S-)
T.B.
601
20
B. I. KURGANOV
296
ET AL.
substrate. Figure 3(b) shows the z versus [E-Jo plot in logarithmic coordinates. The slope is equal to - 1. Such slope should be observed in log z wms iog [El0 plot for the system 2p s P with p form being inactive at sufficiently high enzyme concentrations. THE
CASE
WHEN
P FORM
IS INACTIVE
When ~1~= 0 (the P form is inactive) the dependence of dimensionless product concentration [l-l] = 2G@,[Pr]/ a, on dimensionlesstime Gt has the form :
(19) The portion of time axis limited by continuation of a linear asymptote of [II] versus Gt plot at Gt -+ 00 is equal to: ln _ ~~~~&I+ 1) (20) { (z,+ l)(z,+~/) i When 11< 1 the value GT is positive (the lag-period is observed in the timedependent product accumulation plot). When 9 > 1 the value CT is negative (initial burst is observed in the kinetical curve). The absolute value of Gt versus K,[Elo plots have maxima, with the tangent of the slope closing to unity at sufficiently low enzyme concentration and approaching -0.5 at high K,,[E], in logarithmic co-ordinates (Fig. 4). Figure 5 shows the plots of dimensionless product concentration [II] versus dimensionlesstime Gt for ye= 0; 1 and 10, calculated by the equation (19). The dimensionlessenzyme concentration K, [El0 was chosento illustrate the fact that the absolute magnitude of Gz passesthrough a maximum with increasing enzyme concentration. Gz = -2 (“/-I)
3. Associating Enzymic System M # M, d M, * . . . Let us discussthe pattern of time-dependent product accumulation plots for the slowly associatingenzyme systemsin which protein association affords linear infinite molecules (M P M, z$ M, it *se) with monomer M having two association sites overlapping with the active sites. Let us assumethat a FIG. 5. The dependences of dimensionless product concentration [II] = fG+&[Pr]/a,
on
dimensionless time Gt for the slowlydissociating enzymesystem2p 7-rP with inactiveP form calculatedby the equation(19), at q = 0.1 (a) and (b) and 10 (c) andvaried dimensionless enzyme concentration &[E], [(a) and (c) correspond to &fElo < 1 and (b) corresponds to &[E], 2 11.
I
-
I
I
298
B.
I.
KURGANOV
ET
AL.
half quantity of active sites in the monomer is sterically screened by each neighbour upon associate formation and the association constants in each step M + Mi- 1 F? Mj are identicalt :
EMJ
(21)
K” = [M][M,-,I’ Let k,, and k-, be the rate constants gomer Mi respectively : M+M,-,
of formation k +0 or-? M;. k-o
and breakdown
of oli-
(221
It has been shown (Kurganov, 1974) that the dependence of specific activity on enzyme concentration for a similar enzyme system M ti M, ti M 3 P . . . has the same pattern as in the case of 2p $ P system with inactive P form. Thus, as one should have expected in the linearly associating enzyme system M&M,pM,*..* with steric screening of active sites of monomer the dependence of [Pr] on t and that of z on [El0 have the same pattern as in 2p P P system with a, = 0. In other words the equations (19) and (20) hold for the system discussed. However one should make the following substitutions : G = k-,, r] = l/(1 + [S]o/ty)“, z. = V’ 1+4K,[E], and z/. = x,: I +49Z
where v is the current rate of enzymatic reaction and u/. is the rate at the stationary part of kinetical curve. The apparent association constant qKo may be determined from u/ versus [El0 plot (Kurganov, 1974). In these co-ordinates the tangent of the slope is equal to the rate constant k-,. t Such association is observed, for example, in glutamate dehydrogenase (EC 1.4.1.3) from beef liver (Dessen & Pantaloni, 1969; Chun, Kim, Stanley & Ackers, 1969; Chun & Kim, 1969; Reisler, Pouyet & Eisenberg, 1970; Markau, Schneider & Sund, 1971; Kempfle, Mosebach & Winkler, 1972).
HYSTERETIC
PROPERTIES
OF
ALLOSTERIC
ENZYMES
299
4. Conclusion The kinetics of the slowly associating enzyme systems is specified by dependence of z on enzyme concentration. If a variation of enzyme concentration produces the change of r, it means that the changes in degree of enzyme oligomer dissociation in the course of enzymatic reaction under the influence of substrate (or allosteric effecters) take place. One should bear in mind that generally the “hysteresis” of allosteric enzyme may be due to the shift of equilibrium between oligomeric forms and as a result of relatively slow conformation changes in the oligomeric forms themselves. In cases when variation of enzyme concentration gives the curves of time-dependent product accumulation having r equal to zero one may conclude that the enzyme “hysteresis” is solely a result of change in the degree of enzyme oligomer dissociation under the effect of substrate (or allosteric effecters). To conclude it should be noted that our expressions for r versus [E-Jo plots in the associating enzyme systems 2p $ P and M @ M2 $ M, $ * * * hold as well in the case of unequivalent or interacting active sites binding the substrate. In such case however, the q versus [S], and G versus [S], plots have more complex pattern. REFERENCES CHUN, P. W. & KIM, S. J. (1969). Biochemistry 8, 1633. CHIJN, P. W., KIM, S. J., STANLEY, C. A. & ACKERS, G. K. (1969). Biochemistry DEN, P. & PANTAL~NI, D. (1969). Eur. J. Biochem. 8,292. D~~JNE, C. P. & WOOD, W. A. (1975). Curr. Top. Cell. Regui. 9,65. HOFER, H. W. (1971). Hoppe-Seyler’s, Z. physiol. Chem. 352,997. KANE, J. F., HOLMES, W. M., SMILEY, K. L. &JENSEN, R. A. (1973). J. Bucteriof. KEMPFLE, M., MOSEBACH, K.-O. & WINKLER, H. (1972). FEBSLett. 19, 301. KIM, G. & GRAVES, D. J. (1973). Biochemistry 12,20!90. Koaozw. N. I. (1959). Zh. Fiz. Khim. 33. 1002. KURGAN&, B. I: (1987). Molec. Biol. 1, i7. KURGANOV; B. I. (1968). Molec. Biof. 2,430. KURGANOV. B. I. (1974). Molec. Biol. 8. 525. KURGANOV: B. I. & Y~KOVLEV, V.‘A. 6973). Molec. Biol. 7,429. MARKAU, k., SCHNEIDER, J. & SUNJJ,fi. (1971). Eur. J. Biochem. 24, 393. METZGER. B. E.. GLASER. L. & HELMREICH, E. (1968). Biochemistry 7.2021. RELSLER, k., PO&T, J. k EISENBERG, H. (i97Oj. Biochemistry 9, 3095. TOPCHIEVA, I. N., SOLOV’EVA, A. B., KURGANOV, B. I. & KABANOV, V. A. (1972).
molek. Soedin. 14A, 825. TUOMIN~N, F. W. & BERNLOHR, R. W. (1971). J. bioi. Chem. 246,1733. VANQUICKENBORNE, A. & PHILLIPS, A. T. (1968). J. biol. Chem. 243, 1312. WANG, C.-T. & WEISSMANN, B. (1971). Biochemistry 10,1068. WOHL, R. G. & MARCUS, G. (1972). J. biol. Chem. 247, 5785.
8, 1625.
113,224.
Vysoko-
The Theoretical Anaiysis of Kinetic Behaviour of “Hysteretic” Allosteric Enzymes III. Dissociating and Associating Enzyme Systems in which the Rate of Installation of Equilibrium between the Oligomeric For&s is Comparable to that of Enzymatic Reaction B. I. KURGANOV, A. I. DOROZHKO, Z. S. KAGAN AND V. A. YAKOVLEV The All-Union
Vitamin Research Institute, Moscow, U.S.S.R. (Received 23 December 1974)
The theoretical analysis of the shape of product (Pr) accumulation versus time (t) plots is represented for the slowly associating enzyme systems 2p s P (where P is an enzyme oligomer reversibly dissociating into two identical halves p) and M * Ma 2 M3 z . . . (where M is the monomer having two association sites overlapping with the active sites). It is assumed that the rate of installation of equilibrium between oligomeric forms is comparable to that of enzymatic reaction and that the complexes of oligomerit forms with the substrate are in fast equilibrium with free components. It is shown that the specific property of kinetic behaviour of the slowly dissociating enzyme systems is revealed by the dependence of r on enzyme concentration (z is the intercept of the linear part of the kinetic curve of Ipr] versus t with time axis).
1. Introduction A “hysteresis” of the allosteric enzyme may be explained partly by a relatively low rate of installation of equilibrium between enzyme oligomeric forms having different molecular weights and kinetic parameters. A shift of equilibrium between enzyme oligomeric forms in the course of enzymatic reaction may accelerate an enzymatic process and give rise to a lag-period in the timedependent product accumulation curves. On the other hand, it may slow down an enzymatic reaction with appearance of burst in the kinetic curves. The most simple explanation of an equilibrium shift between enzyme oligomerit forms may be the change of enzyme concentration
when the reaction
is launched by the enzyme. For example, the burst in kinetics of enzymatic reaction catalyzed by a-N-acetylgalactosaminidase (EC 3.2.1.49) from beef liver when the reaction is started by addition of small quantity of the enzyme 287
288
B.
I.
KURGANOV
ET
AL.
to the substrate solution may be due to dissociation of active enzyme oligomer upon diluting? to give inactive subunits (Wang & Weissmann, 1971). Evidently, the most interesting cases are those when an equilibrium shift between oligomeric enzyme forms in the course of enzymatic reaction is due to associating or dissociating substrate effect (or allosteric effector effects) which appears upon substrate addition to the enzyme. “Biodegradative” L-threonine dehydratase (EC 4.2.1.16) from Clostridium tetanomorphum (Vanquickenbourne & Phillips, 1968), a-N-acetylgalactosaminidase from beef liver (Wang & Weissmann, 1971), phosphofructokinase (EC 2.7.1.11) from rabbit skeletal muscles (Hofer, 1971), pyruvate kinase (EC 2.7.1.40) from BacilIus licheniformis (Tuominen & Bernlohr, 1971), phosphoenolpyruvate carboxylase (EC 4.1.1.31) from Escherichia coli (Wohl & Marcus, 1972) and anthranylate synthase (EC 4.1.3.27) from Bacillus sibtilis (Kane, Holmes, Smiley & Jensen, 1973) are examples of dissociating allosteric enzymes in which an appearance of lag-period in the time dependent product accumulation curves is due to an association of inactive (or less active) subunits into active oligomer under the influence of substrate. Dissociation of inactive associates of the enzyme under the influence of substrate may produce the lag-period in the time dependent product accumulation plot. Examples of such associating enzyme systems may be glycogen-phosphorylase A (EC 2.4.1 .I) from frog skeletal muscles (Metzger, Glaser and Helmreich, 1968), phosphorylase kinase (EC 2.7.1.38) from rabbit skeletal muscles (Kim & Graves, 1973) and polyoxyethylene ester of N-benzoylhistidine, a polymeric catalyst which may be considered as a model of associating allosteric enzyme (Topchieva, Solov’eva, Kurganov & Kabanov, 1972). “Biodegradative” threonine dehydratase of Escherichia coli offers an excellent example of the hysteretic allosteric enzyme systems whose activity is determined by the enzyme concentration (Dunne & Wood, 1975). The present paper is concerned with the pattern of product [Pr] accumulation versus time (t) curves for associating enzymic systems in which the rate of installation of equilibrium between oligomeric forms is comparable to that of enzymic reaction. 2. Dissociating
Enzyme System 2p + P
The system P F? 2p is evidently the most simple dissociating enzyme system. P is an enzyme oligomer able to dissociate reversibly into two identical halves p. In general the form p may include several subunits which are bound essentially stronger then p halves in oligomer P. t In this experiment the change of enzyme dissociation was due not only to the changing of enzyme concentration but to changing of the temperature as well (from 0” to 38T).
HYSTERETIC
PROPERTIES
OF
ALLOSTERIC
Let us assume that the rate of installation of equilibrium and P is comparable to that of enzymic reaction: f+oo
ENZYMES
289
between forms
2p P P k-cm
p
(1)
where k+,,,, is the rate constant for association of subunits p and k-,, is the rate constant of oligomer P dissociation. Let us assume that (1) an enzymatic reaction involves one substrate and is irreversible, (2) the substrate concentration is constant (S), (3) the reaction is not inhibited by the product, (4) the sites binding the substrate are equivalent and independent in each oligomeric form and finally (5) the complexes PSi and PS, are at rapid equilibrium with free components. With the last two conditions the concentrations of complexes psi and PSI are connected by the following relation with concentrations of free oligomeric forms : [psi]
=
(n’2)!
cs),
i!(n/2 - i)! ( Ks >
Cpsi] = i!(:A
*[p]
ij, (e)‘[P]
(0 < i < n/2) (0 < i < n)
where Kk and Kg are the microscopic dissociation constants of substrate complexes with forms p and P respectively. (It is assumed that the forms p and P contain n/2 and n sites binding the substrate respectively). Let k+, and k-, be the rate constants of mutual transformation of oligomeric forms which are in the complex with substrate: k+, PSx+PS, +r PSx+, k-xy
(4)
(0 < x < n/2 and 0 < y < n/2). In equilibrium (4) the form PS,,, is an oligomer one part of which contains x bound substrate molecules and the other part involves y bound substrate molecules (i.e. the form PS,,, may be written as pS;pS,). Let us choose two cycles:
and
under condition that in the left cycle x substrate molecules are bound by one half of the oligomer, while in the right cycle other half of the oligomer binds y
290
B.
I.
KURGANOV
ET
AL
substrate molecules. The conditions of detailed balance for these cycles yield the following relation for the rate constants k+xy and k-,, and k,,, and k-,,: k +.Ty =a--~ li +oo I($ x+p _--. k-0, ( r;c; > k-x,
(6)
where c( = 1 at x = J and c( = 2 at x # y. Let y be the portion of enzyme present in p form: i=n/2 izo
[Psi1
(7) y
=
-mo
where [El0 is total molar enzyme concentration per form p. Let us assume that an enzymatic activity is measured in the following manner. An enzyme solution (of certain concentration [El,) is preincubated until the equilibrium between oligomeric forms is installed. The portion of p form prior to the substrate addition (and thus at an initial moment of reaction) is (Kurganov, 1967, 1968): yo = -741So
(8)
where z. = Jl + SK,[E],, K. is association constant in the absence of substrate: K, = [P]/[p]’ = k+oo/k-oo. If the substrate shifts an equilibrium between oligomeric forms p and P (i.e. K; # Kg) the portion of form p will change with the time after launching the reaction and approach limiting value after sufficiently large time, yr: yf = 2/(1 -tzf)
(9)
where zI. = &+8rQC,[E]~; q = [( I+ [S],/l;;)/( I -t [S]o/rC’,)]“. The change of portion of form p with the time (t) is describedby the following relation : (Zf + l)[(zo + zr) -(z. - zf) .exp ( - GzJ t)] Y = Y, -__ CZf+l)(zo+zf)+(zfl)(z,--zf).exp( -Gzft)
(IO)
where
and k-,, = km,,. Tt should be noted that in expression for G the rate constants k-,, for which 2 < (x+y) d (n-2) are the “concealed” parameters (the term proposed by Kobosev, 1959), i.e. they cannot be found experimentally from G versus [Slo plot. Let us give an illustrative example of dissociating system: dimer (D) F? tetramer (T). The tetramer (which may be denoted as D. D) gives two types of complexes TS, one of which (D . DS,) may dissociateinto D+ DS, with rate constant k-,, and the other (DS*DS)
HYSTERETIC
PROPERTIES
OF
ALLOSTERIC
291
ENZYMES
into DS + DS with rate constant k- i r. Using experimental plot G versus [S]e one may estimate only the sum of rate constants (4k-,, +2kmoz) instead of individual values, in other words the rate constants ksoz and k-ll are “concealed” parameters. It should be noted that for the system 2D # T the rate constants kwol, k-,, and k-,, are “explicit” parameters and could be determined from G versus [S],, plot. In this view it is reasonable to introduce the effective rate constants k-, which are explicit parameters and related to k-,, rate constants by the expression (i = x+y):
The expression of type (10) for the portion of form p versus time is valid when an enzymatic activity of dissociating system 2p * P is measured in the presence of allosteric effector (F) on condition that the latter affects enzymatic activity only by shifting the equilibrium 2p s P (i.e. the effector does not change specific enzymic activity of forms p and P). Assuming that the forms p and P contain n/2 and n sites binding the effector, respectively, the parameter 9 defining zf has the form: (13) and the relation for G:
(1+ t-Slo/W(1+ CFloPW
(14)
’
In these relations Kk and Kk are the microscopic dissociation constants for effector complexes with forms p and P respectively and kmij are the effective rate constants for dissociation of oligomeric form PSiFj into two halves. (A)
THE CASE OF INACTIVE
p FORM
The pattern of time-dependent product accumulation curve is determined by ratio of specific enzymatic activities of forms p(q) and P(a2). For the system 2p it P in which the p form is inactive (a, = 0) the dependence of product [Pr] concentration on time t has the form: a2(zf--
cprl= -G-
O2 0
t-
a2 __
2GvKo
(z,+~)(z~~z~)-(z~-~)(z,--~).~~P(-G~~~)
2z,(zo
+ 1)
>
(15)
When q > 1 (the summary influence of substrate and allosteric effector shifts
292
B.
I.
KURGANOV
ET
AL.
an equilibrium 2p it P towards formation of active form P) the rate of enzymatic reaction enhances during the process, i.e. the time-dependent curve of product accumulation has a lag-period. When q < 1 (the summary effect of substrate and allosteric effector shifts an equilibrium 2p + P towards formation of inactive form p) the rate of enzymatic reaction is diminished during the process, i.e. the kinetic curve has an initial burst. Let us introduce the dimensionless product concentration :
LPI = y The dimensionless product concentration
[Pr]. [II]
(16)
versus dimensionless time Gt
I.0
XII
Gf 1
I
(b)
/
0.01 -
U-II 0~005-
-5 FIG. 1. The dependences dimensionless time Gt for inactive at 11= 10 (a) and calculated by the equation
0
5
Gf of dimensionless product concentration [II] = 2G&[Pr]/az on the slowly dissociating enzyme system 2p d P with p form 0.1 (b) and varied dimensionless enzyme concentration KOIElo (17).
HYSTERETIC
PROPERTIES
OF
ALLOSTERIC
ENZYMES
293
for system 2p # P at a, = 0 has the form:
(z,+1)(~~+~~)-(~~-11)(z~--~).ex~[:-z,W>l . c1,j + 1) > The plots of [II] versus Gt calculated by means of this equation for q = 10 and 0.1 and varied dimensionless enzyme concentration &,[E],, are shown in Fig. 1. The portion of the time axis limited by the continuation of a 2zxz,
o-
-4
, -6
I -4
f
I -2
1
I 0
,
I 2
I
I
IqKoIEI,
FIG. 2. The dcpendences of value Gr on dimensionless enzyme concentration KOIEIO in logarithmic co-ordinates for the slowly dissociating enzyme system 2p F? P with inactive p form at varied q [(a) corresponds to q > 1 and (b) conesponds to v < 11. The dependences of Gs on &[E], are calculated by the equation (18). Gt is a portion of time axis limited by continuation of a linear asymptote of m versus Gt at Gt + co.
294
B.
I.
KURGANOV
ET
AL.
linear asymptote in [II] versus Gt plot for Gt -+ a3 is equal to:
The dependences of CT calculated by this relation (for ye> 1) and -Gr (for Q < 1) upon dimensionless enzyme concentration K,,[E], for different values of q are shown in Fig. 2. The absolute value Gz decreases with increasing enzyme concentration and the tangent of the slope for the plot in logarithmic co-ordinates being close to - 1 at the sufficiently high enzyme concentration. A decrease of lag-period, T, for time-dependent product accumulation versus time curves at an increase of enzyme concentration has been observed for phosphoenolpyruvate carboxylase (EC 4.1.1.3 1) from Escherichiu coli (Wohl
& Marcus,
1972) [Fig.
3(a)].
The lag-period
for this enzyme
association of inactive dimers to give active tetramer
is due to
under influence of
30 .5 E 20 i
0
\
IO
\ \
0
_~
_A-.--L-.-I20 40
0.F
60
L--~---L~--i-~-3 ‘,
CElo,/Lgim
.A-I 1 ,I
1-6
I.8
jag [El,
FIG. 3. The dependence of duration of lag-period 7 (in the ~LILLC‘Sof product accumulation in the course of enzymatic reaction catalyzed by phosphoenolpyruvate carboxylase from E. co/i) on enzyme concentration in the (7; [El,; co-ordinates (a) (Wohl & Marcus, 1972) and in the {log 7; log [El,} co-ordinates (b). The reaction was initiated by addition of substrate (phosphoenolpyruvale, 12 111M) to the enzyme solution containing 0.1 M 2-mercaptoethanol and 10 mM asparate (inhibitor). 0.1 M Tris-HCI buffer, pH 8.5, 25°C. FIG. 4. The dependences of value Gr on dimensionless in the logarithmic co-ordinates for the slowly dissociating inactive P form, at varied q [(a) corresponds to r] < 1 and dependences of CT on &[E], are calculated by the equation
enzyme concentration K,[E]o enzymic system 2p ti P with (b) corresponds to rl > 11. The (20).
I (‘S-)
T.B.
601
20
B. I. KURGANOV
296
ET AL.
substrate. Figure 3(b) shows the z versus [E-Jo plot in logarithmic coordinates. The slope is equal to - 1. Such slope should be observed in log z wms iog [El0 plot for the system 2p s P with p form being inactive at sufficiently high enzyme concentrations. THE
CASE
WHEN
P FORM
IS INACTIVE
When ~1~= 0 (the P form is inactive) the dependence of dimensionless product concentration [l-l] = 2G@,[Pr]/ a, on dimensionlesstime Gt has the form :
(19) The portion of time axis limited by continuation of a linear asymptote of [II] versus Gt plot at Gt -+ 00 is equal to: ln _ ~~~~&I+ 1) (20) { (z,+ l)(z,+~/) i When 11< 1 the value GT is positive (the lag-period is observed in the timedependent product accumulation plot). When 9 > 1 the value CT is negative (initial burst is observed in the kinetical curve). The absolute value of Gt versus K,[Elo plots have maxima, with the tangent of the slope closing to unity at sufficiently low enzyme concentration and approaching -0.5 at high K,,[E], in logarithmic co-ordinates (Fig. 4). Figure 5 shows the plots of dimensionless product concentration [II] versus dimensionlesstime Gt for ye= 0; 1 and 10, calculated by the equation (19). The dimensionlessenzyme concentration K, [El0 was chosento illustrate the fact that the absolute magnitude of Gz passesthrough a maximum with increasing enzyme concentration. Gz = -2 (“/-I)
3. Associating Enzymic System M # M, d M, * . . . Let us discussthe pattern of time-dependent product accumulation plots for the slowly associatingenzyme systemsin which protein association affords linear infinite molecules (M P M, z$ M, it *se) with monomer M having two association sites overlapping with the active sites. Let us assumethat a FIG. 5. The dependences of dimensionless product concentration [II] = fG+&[Pr]/a,
on
dimensionless time Gt for the slowlydissociating enzymesystem2p 7-rP with inactiveP form calculatedby the equation(19), at q = 0.1 (a) and (b) and 10 (c) andvaried dimensionless enzyme concentration &[E], [(a) and (c) correspond to &fElo < 1 and (b) corresponds to &[E], 2 11.
I
-
I
I
298
B.
I.
KURGANOV
ET
AL.
half quantity of active sites in the monomer is sterically screened by each neighbour upon associate formation and the association constants in each step M + Mi- 1 F? Mj are identicalt :
EMJ
(21)
K” = [M][M,-,I’ Let k,, and k-, be the rate constants gomer Mi respectively : M+M,-,
of formation k +0 or-? M;. k-o
and breakdown
of oli-
(221
It has been shown (Kurganov, 1974) that the dependence of specific activity on enzyme concentration for a similar enzyme system M ti M, ti M 3 P . . . has the same pattern as in the case of 2p $ P system with inactive P form. Thus, as one should have expected in the linearly associating enzyme system M&M,pM,*..* with steric screening of active sites of monomer the dependence of [Pr] on t and that of z on [El0 have the same pattern as in 2p P P system with a, = 0. In other words the equations (19) and (20) hold for the system discussed. However one should make the following substitutions : G = k-,, r] = l/(1 + [S]o/ty)“, z. = V’ 1+4K,[E], and z/. = x,: I +49Z
where v is the current rate of enzymatic reaction and u/. is the rate at the stationary part of kinetical curve. The apparent association constant qKo may be determined from u/ versus [El0 plot (Kurganov, 1974). In these co-ordinates the tangent of the slope is equal to the rate constant k-,. t Such association is observed, for example, in glutamate dehydrogenase (EC 1.4.1.3) from beef liver (Dessen & Pantaloni, 1969; Chun, Kim, Stanley & Ackers, 1969; Chun & Kim, 1969; Reisler, Pouyet & Eisenberg, 1970; Markau, Schneider & Sund, 1971; Kempfle, Mosebach & Winkler, 1972).
HYSTERETIC
PROPERTIES
OF
ALLOSTERIC
ENZYMES
299
4. Conclusion The kinetics of the slowly associating enzyme systems is specified by dependence of z on enzyme concentration. If a variation of enzyme concentration produces the change of r, it means that the changes in degree of enzyme oligomer dissociation in the course of enzymatic reaction under the influence of substrate (or allosteric effecters) take place. One should bear in mind that generally the “hysteresis” of allosteric enzyme may be due to the shift of equilibrium between oligomeric forms and as a result of relatively slow conformation changes in the oligomeric forms themselves. In cases when variation of enzyme concentration gives the curves of time-dependent product accumulation having r equal to zero one may conclude that the enzyme “hysteresis” is solely a result of change in the degree of enzyme oligomer dissociation under the effect of substrate (or allosteric effecters). To conclude it should be noted that our expressions for r versus [E-Jo plots in the associating enzyme systems 2p $ P and M @ M2 $ M, $ * * * hold as well in the case of unequivalent or interacting active sites binding the substrate. In such case however, the q versus [S], and G versus [S], plots have more complex pattern. REFERENCES CHUN, P. W. & KIM, S. J. (1969). Biochemistry 8, 1633. CHIJN, P. W., KIM, S. J., STANLEY, C. A. & ACKERS, G. K. (1969). Biochemistry DEN, P. & PANTAL~NI, D. (1969). Eur. J. Biochem. 8,292. D~~JNE, C. P. & WOOD, W. A. (1975). Curr. Top. Cell. Regui. 9,65. HOFER, H. W. (1971). Hoppe-Seyler’s, Z. physiol. Chem. 352,997. KANE, J. F., HOLMES, W. M., SMILEY, K. L. &JENSEN, R. A. (1973). J. Bucteriof. KEMPFLE, M., MOSEBACH, K.-O. & WINKLER, H. (1972). FEBSLett. 19, 301. KIM, G. & GRAVES, D. J. (1973). Biochemistry 12,20!90. Koaozw. N. I. (1959). Zh. Fiz. Khim. 33. 1002. KURGAN&, B. I: (1987). Molec. Biol. 1, i7. KURGANOV; B. I. (1968). Molec. Biof. 2,430. KURGANOV. B. I. (1974). Molec. Biol. 8. 525. KURGANOV: B. I. & Y~KOVLEV, V.‘A. 6973). Molec. Biol. 7,429. MARKAU, k., SCHNEIDER, J. & SUNJJ,fi. (1971). Eur. J. Biochem. 24, 393. METZGER. B. E.. GLASER. L. & HELMREICH, E. (1968). Biochemistry 7.2021. RELSLER, k., PO&T, J. k EISENBERG, H. (i97Oj. Biochemistry 9, 3095. TOPCHIEVA, I. N., SOLOV’EVA, A. B., KURGANOV, B. I. & KABANOV, V. A. (1972).
molek. Soedin. 14A, 825. TUOMIN~N, F. W. & BERNLOHR, R. W. (1971). J. bioi. Chem. 246,1733. VANQUICKENBORNE, A. & PHILLIPS, A. T. (1968). J. biol. Chem. 243, 1312. WANG, C.-T. & WEISSMANN, B. (1971). Biochemistry 10,1068. WOHL, R. G. & MARCUS, G. (1972). J. biol. Chem. 247, 5785.
8, 1625.
113,224.
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