- Email: [email protected]
The exponential model for a regulatory enzyme: Its extension to describe the binding of two ligands
J. theor. Biol. (1978) 75, 97-114
The Exponential Model for a Regulatory Enzyme: Its Extension to Describe the Binding of Two Ligands STANLEY AIN~WORTH AND ROGER 3. GREGORY Department
of Biochemistry, University of Shefield, Shefield ,910 2TN, England (Received 26 April 1978)
The exponential model for a regulatory enzyme (Ainsworth, 1977u) is extended to deal explicitly with the presence in solution of a second ligand. This is achieved by introducing exponential interaction coefficients which respectively describe how the affinity of the free and bound forms of the protein for the ligand depend on its fractional saturation by the second ligand. The basic equations, so derived, are applied to binding experiments
where the ligands bind independently or competitively
and to rate
experiments where the ligands represent two substrates or one substrate and a modifier which may be either competitive or non-competitive in type. The conditions required to display linkage between the binding of the two
ligands are established and it is also shown that rate data may display a maximum as one ligand concentration is varied at a fixed concentration of the other. The equations that are derived are tested by application to
experimental data and the conditions that have to be met to justify such an application are discussed.
1. Introduction In a recent paper Ainsworth (1977a) described an exponential model for a regulatory enzyme which leads to relationships between the initial velocity of the catalysed reaction and the varied concentration of a substrate that are non-inflected or sigmoidal in type, without a maximum. The model assumes that the most relevant measure of protein configuration (itself determining the kinetic behaviour of the enzyme) is the apparent association constant, o+,, measured for the given fractional saturation of enzyme by the substrate under investigation. It was further assumed that the original state of the protein in solution, cl,,, is destabilized by an increment of energy, AG:, that is proportional to the fractional saturation, p, of the enzyme by substrate, so that formation of the configurational state, aP, can be represented by - AG,OlRT = kp = ln uP/cq,,
(1)
97
0022-5193/78/210097+18 4
$02.00/O
0
1978 Academic
Press Inc.
(London)
Ltd.
98
S. AINSWORTH
AND
R.
B.
GREGORY
where a,, is the binding constant at zero saturation. Equation (1) then gives ap=
( > -_-P l-p
-1 = a0 ekP A
which describes both positive and negative co-operativity with respect to the binding of A. Equation (2) can also be applied to rate data if p is equated with the ratio of the measured initial velocity Y to the maximum velocity Y established at a saturating concentration of A. It is an important aspect of the model that @Prefers to the whole assembly of enzyme molecules in solution and is therefore an average quantity: it does not refer to specific enzyme complexes. The purpose of this paper is to extend the model so as to deal explicitly with the presence in solution of a second ligand and to test this approach by applying it to experimental data. It is assumed throughout that the fractional saturation of the protein by any additional ligand is constant. 2. Theory
It will be evident from the introduction that the exponential model assumes that the substrate and enzyme reach a state of quasi-equilibrium with enzyme-substrate complexes which is not disturbed by product formation during the timeintervalrequiredto establish an initial velocity of reaction and further that non-hyperbolic velocity-substrate relationships arise solely from changes in the binding constants. This approach will be continued and, for simplicity, the two ligand-protein system will first be examined as a binding problem: the modifications that arise when rate data alone are available will then be treated separately. (A)
DEFINITION
OF BINDING
CONSTANTS
Consider an experiment in which the fractional saturation of the protein by A is determined in the presence of a second ligand, B, which binds independently. By analogy with equation (2), we have (3)
where aab is the binding constant for A which refers to specific values of pA and pB between O-l. Note, however, that the value of pB refers only to that fraction of the saturation by B which, through a contingent configurational change in the protein, exerts an effect on the binding constant for A. It is possible to imagine that a value for pB might be determined directly, say for binding at several sites, which would be irrelevant to the experiment under
THE
EXPONENTIAL
MODEL
FOR
A REGULATORY
ENZYME
99
consideration. In what follows, however, we shall assume that the measured and effective saturations by B are identical. By following equation (l), we can write aab = aOb exp &4bPAh (4) where kAb is a constant independent of pA but dependent on pB = b. Again, gob is the binding constant for A measured at zero saturation by A but at PB = b. If we now assume, in a manner consistent with the original model, that the configurational state, mob, is related to that of the wholly-unbound protein, aoo, by an energy term which represents the destabilizing effect of binding B, then - AG&/RT
= IOBPB
= In (aOb/aOO>
(5)
(&Bh)
(6)
and, in a similar fashion, we can write %b
=
%O
exp
where 10s and IIB are exponential interaction coefficients which respectively describe how the affinity of the A-free and A-bound forms of the protein for A depends on PB. From equation (4), kAb = In f+,-h sob (7) and hence, by introducing equations (5) and (6) k*b = tin alo
-In
(8)
aOO> + (llB-lOB)PBT
or k Ab
(9)
= kAo -I- (11 B - IoB)PB-
By combining
equations (4) and (9), we obtain a nb = C(0b W? -K&m+ (11B- LJPBIP,J or, by making use of equation (5) a ab = aOO
exP &?4o + (11, -
10B)pBbA
(10)
+ IOBP,)
(10
and by analogy Dab = Poe e*P Kkok,, +- (L -&o)PA]P~+IAoPA) (12) which corresponds term by term with equation (11) except that the constants relate to the binding of B in the presence of A. (B) THE BINDING
FUNCTION
FOR A IN THE PRESENCE
Equations (3) and (4) can be combined to give V AaOb exP (k&‘/v> pA = ? = 1 +Aaob eXp (k&/v)
*
OF B
(13)
Equation (13) is analogous to the Michaelis-Menten equation and its denominator, correspondingly, is a distribution function which gives the
100
S. AINSWORTH
AND
R.
B.
GREGORY
ratio of the A-free to A-bound fractions of the enzyme. It does not, however, identify individual terms with the relative concentrations of specific, known enzyme forms and, indeed, the dependence of a,,,, on pA and pB emphasizes that the identities of the enzyme-species which determine Q, change with saturation. The analogy with Michaelis-Menten kinetics is nevertheless a useful one and further analogies will be drawn in deriving equations to represent the behaviour displayed by two ligand-protein systems. (C) DETERMINATION
OF CONSTANTS
The constants in equation (11) [and (12)] can be determined by multiple regression on their logarithmic transformations if values of pA and pB are available: in this connection, note that LX,,,,is derived from a value of pA by equation (3). (D)
LINEAR
TRANSFORMATION
The logarithmic form of equation (4) can be used as the basis of a linear plot In a,, = f(pA)B = f(~/y)a for those ligand-protein systems in which pB is independent ofp,: when this condition applies, all the constants in equation (11) can be obtained by replotting the primary slopes and intercepts as functions of pe [equations (9) and (5)]. In practice, the linear plot In Q, = f(z~/& requires the definition of maximum saturation or its equivalent V, the maximum velocity, (Ainsworth, 19770) and we have found it more convenient to employ a computer program (Kinderlerer & Ainsworth, 1978) to estimate the constants which also provides the value of this parameter. The program establishes initial estimates of the constants by the log plot, and then refines them, according to equation (13), by an iterative procedure which treats iyOb,kAb and Y as disposable constants. (This procedure implies that measured fractional saturations are treated as values of 0 and can be altered by a constant proportion in relation to the value of V determined by the iterative procedure itself.) (E) RELATIONSHIP
BETWEEN
CONSTANTS
The relationship between the kA and I constants is given by equation (9) and when pB = 1, (14) km+& = kx.il +Io~ Equations (9) and (14) indicate that the same final configuration, as measured by a&, is obtained on saturation by A and B, irrespective of the order of saturation. It should be noted, however, that the value of pA, established in the absence of B, will only be the same as that finally obtained when IIB and IOB are zero. Equation (14) can be modified to employ p constants, but an
THE
EXPONENTIAL
MODEL
FOR
A REGULATORY
ENZYME
101
“equilibrium box” relationship between the two equations does not exist. For, in the normal equilibrium box, separate, identifiable protein species co-exist in solution whose relationships to one another are defined by known ligand concentrations and equilibrium constants. By contrast, the exponential model treats the whole assembly of protein molecules in solution as one, average configurational state with properties which change continuously with saturation and which are assessed differently by different ligands. The last point deserves emphasis. The free energy changes that are associated with changes in the binding constants of individual ligands relate only to those configurational events within the protein molecule which are relevant to the ligand concerned: they do not relate to the protein as a whole. There is thus no basis upon which to formulate an obligatory relationship between the c1and p constants. This conclusion may be derived more explicitly by considering Wyman’s (1964) linkage relationship for a protein which binds an equal number of A and B molecules :
(&),=(i2tJ,
(15)
The right hand side of equation (15) can be represented as (16)
By differentiation
of equation (13) (Ainsworth, 1977u) (17)
where Ixr, = IIB-&. Similarly, from equation (12) and bna = p&l dp, -= dp, where IAx = I, 1 - IAo.
The combination -=dp, din A
-pB)B
(ZAO+I”xPJP,(lPe) Cl -P& - PB)~%NI + ZAXPA~I ’
(W
of equations (17) and (18) now gives
PAI -P.&U - PIN,40 +IAXPB) .Cl - PAO - P~)UGO +kB~dlC1 - PA1 -p&h +hdI - lk4(1- P.4)PAl - PINA + ~AXP&JB +Ix,PA)l
which demonstrates, is true
(19) by arguments of symmetry, that when equation (15) (I.40 + ZAXPdB = uon +k3PA)A.
(20)
But equation (20) can only be true for all values of pA and ps when IA0 z IOB and when I AX z I,, E zero. The latter condition requires that the function
102
S.
AINSWORTH
AND
R.
B.
GREGORY
In z,~ =&A) should have a constant slope when pB takes different fixed values. The data discussed below suggests that this condition is not always met and therefore that the a and p constants are not obligatorily linked. (F) APPLICATION
TO DATA
FROM
A COMPETITION
By analogy with the Michaelis-Menten inhibition, we can write:
equation
for a competitive
AaGo pA = 1 +B&, Re-arrangement
(21)
+ Aa,o
of equation (21) now gives (22)
pA = v = l+Aa,,(l+B&,)-l exp (k&/V) which can be analysed according to equation (13) if cl,&1 f&&J’ is equated with czob.Equations (3) and (22) also show that plots of In a,, = f(pA) at different constant values of B, present a series of parallel lines which move to more negative values as B increases. It also follows that:
Data consistent with equation (22) are presented later in this paper. 3. Application to Rate Data The application of the equations developed above to rate data depend on the operational relationship of the two ligands to one another. Two alternatives will be considered. B ARE SUBSTRATES
(A) A AND
Here : V -V =
pApB
=
-.- A%b 1+./h,,
%b l-+-B/&,’
In this equation, u/[ V&,J(l + &&)] will only provide a measure of PA if the second term remains constant as A is varied. This can be proved by comparing the velocity of the reaction at a saturating ‘concentration of A with a direct measurement of the fractional saturation by B in the absence of A, for the equality
0 V
v
can only be true when I,, BII, = kb
= A-m
B&b l+Bthb
=
(P&z‘,
(25)
= IA, = zero as shown by the definition.
exP
{[~~B$(JA1-IAO)IPBflAO}.
(26)
THE
EXPONENTIAL
MODEL
FOR
A REGULATORY
ENZYME
103
In the absence of binding data only circumstantial evidence to the same effect can be provided. In the first place plots of In aa,, = f(z~[k’&, according to equation (4) and where V,,, = VB&,/(l + B&J, can only be linear when pe is independent of pA. The prevalence of rate data that can be represented in this way (Ainsworth, 1977a) suggests that pB often does remain constant at constant B, but deviations from linearity were observed and it is likely that they arise from a systematic change in pB as A is varied. A second line of evidence follows if (v/ v3A= co is found to be an hyperbolic function of B for equation (26) shows that one condition leading to this result is that k OBPI,, and IA0 are all zero, thereby indicating that pB is a constant independent of pA. Unfortunately, an alternative interpretation of the hyperbolic relationship can be suggested, namely, that koB = (IAl -I.& = zero but, if this were true and I,, # zero, linear replots based on equations (5) and (9) and the value of pB calculated from (u/v>,,, would not be expected. The last test can also be applied when (v/V),, co = f(B) is not hyperbolic, for then linear replots calculated on the basis of the assumption k,, # IAl = IA0 = zero and (v/V), =m = pe would again not be found if the assumption were false. When pB is not independent of pA analysis of the binding behaviour of A can only be undertaken on rate data collected at saturating concentrations of B. However, more progress is possible if either PA or pB can be separately determined in a binding experiment conducted under identical conditions. With one value available, the second can be determined by equation (24). The values of PA and pB can then be inserted into the logarithmic forms of equations (11) and (12) and the constants estimated by multiple regression. Note that equation (24) also gives rise to the possibility of a maximum in o as A is varied at constant B. Differentiation of the equation shows that the maximum arises when (27) Equation (27) can be combined with equations (17) and (18) to give d(pApzt)
_
PBZ-‘A(~
-
PA)[~
+
(I-
Pd(IAoP,
-
koBPB)l
dln A D where D is the denominator of equation (19). Equation (28) shows that a maximum occurs when (I--P,)-’
= tkoBPrI,aPA)
=
o
,
(2%
(29) that is to say when the right hand side of equation (29) is positive. The values of k,, and I,,-, required to meet this condition are readily determined and have been confirmed by computation of the relationship pApe = f(A).
104 (B)
S. AINSWORTH
AND
R.
B.
GREGORY
A IS A SUBSTRATE,
B AN ALLOSTERIC
EFFECTOR
The second alternative can be described by analogy with the MichaelisMenten equation for a partially non-competitive modifier (Ainsworth, 19773), that is: V v = PAp-PB)+YPBl.
(30)
In this equation, V represents the rate of product formation when the enzyme is completely saturated by A in the absence of B. A different rate is measured when the enzyme is also saturated by B and the change is defined by multiplying V by the constant y : inhibition is associated with values 0 < y < 1, and activation with values y > 1. In addition, however, the effector B influences the rate of reaction through its effect on CQ,[equation (ll)]. Thus a pure K system will arise when y = 1, and a pure V system when y # 1 and I,, and IoEl eq ual zero. (Monod, Wyman 8z Changeux, 1965). Equation (30) describes both these influences and its further analysis might proceed as follows. First, consider the case where y # 1: by substituting for pA and pB equation (30) gives: V v=
Now, with the definition saturation with A:
A%,
.- 1 + @hzb
IfAoc,,
(31)
l+B&,’
that V,,, = V(1 +yB@,,)/(l K,,V B&b pB = (yV-V) = lfb’
+B&),
we obtain on (32)
When evidence of the type discussed in the last section can be advanced to show that pB is independent of pa, we can write PA
=
$ *aPP
(33)
and the values given by equations (32) and (33) can be analysed by the use of equation (13). When /?rb is not independent of PA, further progress can only take place if either pA or pB can be separately determined by a binding experiment. With one value available, the second can be obtained through equation (30). Equation (31) also implies the existence of turning points but the conditions for these to be observed have not yet been examined. The second case arises when y = 1. Here equation (31) reverts to equation (13) and (as described in connection with that equation) its complete analysis depends upon the availability of independent values of pB.
THE
EXPONENTIAL
MODEL
FOR
A
REGULATORY
ENZYME
105
4. Tests of the Model (A)
OXYGEN
BINDING
BY
HEMOGLOBIN
IN
THE
PRESENCE
OF
THE
COMPETITOR,
2$DIPHOSPHOGLYCERATE
This well-known competitive system (Benesch, Benesch & Yu, 1968) is complicated by the fact that hemoglobin (Hb) is a tetramer, at high concentrations, which binds four molecules of oxygen (A) but only one molecule of 2,3-diphosphoglycerate (B). It is therefore necessary to modify the equations for a competition reaction, developed above, so that the individual binding of A and B relate to a single aggregation state of the protein. To standardize in terms of A-binding sites, the monomer equivalent of Hb4 and HbaB is calculated as
where K is the association constant for the monomer-tetramer But Hb = (Hb,/K)’ so that A4 = Hb (1 + B/?&.
equilibrium. (35)
Equation (21) can now be modified to read
AGO PA = (1+B/3,,)~+Ac~,,
(36)
which can be analysed according to equation (13) if aoo(l +BpOb)-h is equated with sob. Goodford, Norrington, Paterson & Wootton (1977) have recently published oxygen dissociation curves for concentrated human hemoglobin solutions in the presence and absence of 2,3-diphosphoglycerate. The actual data points (kindly supplied by the authors) were analysed by fitting equation (13) and the resulting constants sob, kAb and Y are given in Table 1. Examination of the table shows that the values of kAb are almost constant (ka = 2.625f0.094) and that sob becomes smaller as B is increased. These observations are consistent with a competition and it appears that the constants should be employed to test equation (36). However, because the concentration of hemoglobin is too high to permit the approximation B = B (total), it is more convenient to test a modified form of equation (36)
AQO PA = (1-p&“+Acc~,
(37)
106
S.
AINSWORTH
AND
R.
TABLE
1
B.
GREGORY
Oxygen binding by human hemoglobin. Constants obtained by fitting
equation
(13) with v set equal to the measuredfractional saturation of human hemoglobin by oxygen (A) in the presence ofjxed concentrations of 2,3-diphosphoglycerate (B). Values of pB are calculated as described in the text Goodford, Norrington, 0.1
V
1 a006 ho.036 0.219 &0@06 2.697 &0.081
Gb (kpka-I) Rb
PB
0.5
1.0
2.5
5.0
O-976 10.021 0.160 *oaO3 2.714 10.054 0.623
0.963 ho.034 0.141
1408 &to.018 o-105 &O+lOl 2.646 ho.038
0.985 kO.016 0.091 &O*OOl 2.632 kO.037
0.2
1.046 fO*026 0.204 &0.003 2,436 10.050 0.172
0
Paterson & Wootton (1977)
0.978
&O-O19 0.197 f0*003 2.546 10.051 0.321
f 0405 2-703
& 0,092 0.816
0.933
O-968
Benesch, Benesch h Yu (1968) 2,3-DPG hM>
V &lb
(mm
Hg - I)
k Ab PB
which eqUateS expression
0
o-2
1.107
iO.032 O-0989 jO.0062 2.423 10.153
0
Nob
1.083
0.4 1.119
0.6 1.110
f0.030
kO.027
kO.020
0.0526
0.0361 10.0019 2.231 f0.142
0.0274 &0aoo9 2.400
&0+034 2.354 50.160 0.884
with a,,(1 -pB)*
0.945
*to.090 0.964
O-8
1.0
l-106 &0*019 0.0244
1.105 10.021 0.0213
iOGOO8
10-0008
2.430
i oa90 0.973
2.513 &O.loo
0.979
and to calculate pB from the quadratic
pB = (C~+PB,+BE,I-C(~+PB,+BE,)~-~P~B~E~I~~~~~E~)-~, (38) where B, represents the total concentration of 2,3-diphosphoglycerate
and E0 the total concentration of hemoglobin expressed as tetramers (0-4x 10m3 M). p is the association constant for the binding of 2,3diphosphoglycerate and has the value 6600 M-I (Goodford et al., 1977). Figure 1 shows a plot of ln Mob = h aoo +t In (1 -pB) (3% according to these data: the straight line through the points has unit slope as required. The data for the diphosphoglycerate competition provided by Fig. 4 of Benesch, Benesch & Yu (1968) can also be analysed by the procedure
THE
EXPONENTIAL
MODEL
FOR A REGULATORY
ENZYME
107
1/4In(I-pJ -1.0
-04
I
I
-1.5 -
I
-2.5
FIG. 1. Dependence of ln sob on the fractional saturation of human hemoglobin by 2,3-diphosphoglycerate. 0 from data given in Fig. 4 of Benesch, Benesch & Yu (1968); 0 from the datum points illustrated in Fig. 3 of Goodford, Norrington, Paterson & Wootton (1977).
described earlier in this paper. As before kAb is almost constant (kAb = 2.390 $-O-085) and aObdecreases as B is increased (Table 1). In contrast, however, these data were collected at a lower concentration of hemoglobin (4.6 x 10e5 M tetramers), where fi was estimated to have the higher value of 4.8 x IO4 M-’ and they were interpreted (Benesch, Benesch & Yu, 1968) by assuming that the binding of two oxygen molecules was completely concerted. This interpretation is consistent with the linear relationship between in aOb and 3 In (1 -pB) shown in Fig. 1. Note that the line with unit slope passes through all but one of the data points. (B)
THE
REACTION
OF
GLYCOGEN
AND
GLUCOSE-l-PHOSPHATE
CATALYSED
BY
FROG MUSCLE PHOSPHORYLASE U
initial velocity data for the phosphorylase a catalysed reaction were obtained from optical enlargments of Fig. S(b) of Metzger, Glaser &
108
S.
AINSWORTH
AND
R.
B.
GREGORY
@ucose-i-phosphate],mM
FIG. 2. The initial velocity of the reaction catalysed by frog skeletal muscle phosphorylase a [Fig. 8(b) of Metzger, Glaser & Helmreich, 19681 with glycogen (expressed as glucose residues at the non-reducing end of the chain) 0, 0.1 mu; 0, 0.25 mM; A, 0.5 mu; A, 2.5 mu; 0, 5 mu. pH 6.8, 25”. The points represent initial velocities expressed as bmoles product formed min- l (mg enzyme)- I. The curves are calculated as described in the text. Insert: dependence of the maximum velocity calculated for saturation by giucosel-phosphate on the concentration of glycogen.
2
TABLE
Frog muscle phosphorylase a. Constants obtained by Jitting equation (13) with v equal to the initial velocity of the reaction catalysed by phosphorylase a when measured as a function of [glucose-l-phosphate], (A), in the presence ofjixed concentrations of glycogen, (B), (Metzger, Glaser & Helmreich, 1968). Other details as in Fig. 2 legend. The last four columns give the values used to obtain the interaction and binding constants quoted in the text Glycogen
V
a0b
b-M
(units of v)
(mM- I)
0‘10
7.92 ho.48 17.08 zto.41 21.41 *O-69 28.99 &l-O6 35.30 hl.53
0.0532 rt O+IO26 0.0422 ~09011 0.0594 &-o-O019 0.0719 rtO.0044 0.0886 100341
0.25 o-50 2.50 5~00
k
Ab
1,763 & 0.200 1.830 &0~096 1.333 zto.143 l-270 A-O.203 0.694 rtO.216
h.l am
In 011b
(ln mM-l)
(In mki-I)
PB
-2.940
-1.171
0.213
-3.165
-1.335
0.404
-2.824
-1.491
0.575
-2.632
-1.363
0.871
-2.423
- 1.729
0.931
THE
EXPONENTIAL
MODEL
FOR
A
REGULATORY
ENZYME
109
Helmreich (1968). These data, shown in Fig. 2, conform to equation (24) when A is the concentration of glucose-l-phosphate and B the concentration of glycogen expressed in terms of glucose residues at the non-reducing end of the glycogen chain. The data were analysed by fitting equation (13) and resulting constants are given in Table 2. The maximum velocities were found to be a hyperbolic function of B, as shown by the double reciprocal plot inserted in Fig. 2. The hyperbolic relationship suggests that PB might be independent of pA, supports the use of equation (13), and provides values of V = 38.12k2.97 umoles min-’ (mg enzyme)-‘, and PI0 = 2.706 AO.195 mM-l. The assumption of independence is tested by setting Boo = plo (i.e. IAl = la0 = zero), and by its use calculating the value of pB for each glycogen concentration that is given in Table 2. Plots of kAb, In aOb and In c+, are linear functions of pB, thus supporting the assumption that was made, and unweighted linear regressions provide the values (units in Table 2) kAo = 2-17&O-22, ZXB = -1*32$0.30, Ina,, = -3*30+0*16, I,, = O-86+0*22, ha,, = -1*13+0.14 and ZIB = -0*47&0*19. These values, together with that given above for Boo, were then employed with an iterative procedure to calculate pA and pB for the given concentrations of A and B. By substitution into the equation v = Vpp, calculated values of v were obtained which are shown as the curves drawn in Fig. 2. The agreement between the calculated and experimental values is comparable in quality to that which has been achieved in the analysis of two-substrate reactions, conforming to first degree rate equations, by the use of Cleland’s (1963, 1967) procedures (Heyde & Ainsworth, 1968). (C)
THE
INHIBITION
OF
RABBIT
MUSCLE
PYRUVATE
KINASE
BY
PHENYLALANINE
Data for this inhibition, in which A is the substrate ADP, and B the inhibitor, phenylalanine (the concentrations of phosphoenolpyruvate and Kf and Mg2+ ions were kept constant) are shown in Fig. 3. These data were analysed by fitting equation (13) and the resulting constants are given in Table 3. Examination of the table shows that the maximum velocity falls to zero as B is given increasing fixed values, from which it can be concluded that y = 0 [equation (30)]. The plot of pB = f(Z3) according to equation (32) is not hyperbolic, however, and one indication that ps is independent of PA is therefore lacking. In spite of this, plots of kAb, In CQ,,and In alb are found to be linear functions of pB. These linear relations provide circumstantial evidence that pB is independent of p A (I A1 = Z,, = zero) and when analysed by an unweighted linear regression give the values (units in Table 3) kAo = l-92+ 0.04, ZXB = - 1.01 &O-082, In LX,,,,= - 1*49+0*04, ZoB = O-830 &O-09; In alo = 0*420+0+48 and ZIB = -0.175&0*10. The values of
110
S.
AINSWORTH
AND
R.
B.
GREGORY
12e
IOC
7: V
5c
25
[ADP],mM
3. The inhibition of rabbit muscle pyruvate kinase by phenylalanine at 25”. The reaction mixtures contained phosphoenolpyruvate (O-2 mu), Mg2+ (0.1 mu), and K+ (100 mu) in Nethylmorpholine buffer (0.1 M, pH 7.4), with phenylalanine 0, 0 mu 0, O-5 mu; A, 1.0 mu; A, 2.5 mu; q , 5 mu; W, 10 mu. The points represent initial velocities expressed as p moles of pyruvate produced per minute per mg of enzyme corrected to a specific activity of 300 units mg-I. The curves are calculated as described in the text. TABLE 3 FIG.
Inhibition of rabbit muscle pyruvate kinase. Constants obtained by Jitting equation (13) with v equal to the initial velocity of the reaction catalysed by pyruvate kinase when measured as a function of [ADP], (A) in the presence of fixed concentrations of phenylalanine, (B). Other details as in Fig. 3 legend. The last four columns give the values used to obtain the interaction and binding constants quoted in the text Phenylalanine (mM> 0
0.5 1.0 2.5 5.0 10.0
Vor K,, when B # 0 &?& (units of v) 13363 13.16 119.97 A4.71 114.21 h4.51 75*19 c6*00 38.30 h2.59 16.81 zt1.63
0.231 -10-011 0.243 10.018 0.242 &to*015 0.318 &OGll 0.435 ,to4xo 0.426 ItO.
k*b 1.936 ho-124 1.886 kO.208 1.701 &-to.185 1442 10.425 1.184 A-to.444 1.135 10.612
- 1.464
0.471
0
-1.414
0.472
0.077
-1.417
0.284
0.160
-1.144
0.297
0.427
-0.833
0.350
0.702
-0.854
0.282
0.860
THE
EXPONENTIAL
MODEL
FOR
A REGULATORY
111
ENZYME
k,, = 1*67+ 0.27 and In ,!IoO = - 1*92j:O*lO were also estimated by using an appropriately modified form of equation (13) in which z)was equated with p,,. The constants thus obtained were then employed with an iterative procedure to calculated pA and pB for the given concentrations of A and B. By substitution into the equation v = Vp, (1 -pe), calculated values of D were obtained: these are shown as the curves in Fig. 3. The agreement between the calculated and experimental values is sufficiently good to suggest that the assumptions made in the analysis must be at least approximately true. (D)
THE
ACTIVATION
OF RABBIT
MUSCLE
PYRUVATE
KINASE
BY FRUCTOSE-1,6-
DIPHOSPHATE
Phillips & Ainsworth (1977) have demonstrated that the enhancement of pyruvate kinase activity brought about by fructose diphosphate is an hyperbolic function of the activator concentration which indicates an apparent binding constant of 0.80 mK1. Phillips & Ainsworth (1977) also showed that the same final velocity was achieved at high concentrations of the substrate ADP with or without addition of the activator. It appears on this basis that 4 Activation of rabbit muscle pyruvate kinase. Constants obtained by fitting equation (13) with v equal to the initial velocity of reaction catalysed bypyruvate kinase when measured as a function of [ADP], (A) in the presence of Jixed concentrations of fructose-1,6-diphosphate, (B). Other details as in Fig. 4 legend. The last four columns give the values used to obtain the interaction and binding constants quoted in the text TABLE
5.0
147.08 12.88 147.85 +2-83 156.59 +9*78 154.84 ztz4.83 158.85 h3.73
0.202 1OGO6 0.264 iO406 0.279 10.021 o-359 *0*017 0.502 10*021
1.813 &O-O80 1.411 &to.084 1 a308 10.275 1.218 &0*160 0.942 10.150
155.61 49.43
0.590 f0.076
0.859 10440
-1.601
0.213
0
-1.330
0.081
0.167
- 1.275
0.032
0.286
- 1.023
0.195
0444
-0.690
0.252
0.667
-0~528
0.331
0.800
112
S.
AINSWORTH
AND
R.
B.
GREGORY
the activation of pyruvate kinase conforms to equation (30) when y = 1. This conclusion is supported by additional data for which the constants given in Table 4 were obtained by fitting equation (13) with A identified as the substrate ADP, and B as the activator, fructose diphosphate. Examination of the table confirms that V is constant within the error of the estimates, demonstrating thereby that y = 1. Plots of kAb, In c& and ln aI,, are linear functions of pB (calculated on the basis of the binding constant given above), and unweighted linear regressions provide the values (units in Table 4)
5
FIG. 4. The activation of rabbit muscle pyruvate kinase with fructose-1,6-diphosphate 0, 0 mu; l , 0.25 mh%; A, 0.5 mu; A, 1.0 mu; 0, 2.5 mu; a, 5.0 mu. Other details as in Fig. 3. The points represent data, the curves are calculated as described in the text.
kA,, = l-69+0*06, IxB = -l-11+0*13, Ina,, = -1.60fO.03, IOB = 1.34 *O-06, In cllo = 0.09 kO.07, and IIB = 0.23 f0.14. Note that IIB is almost zero within the error of the estimate: this is to be expected for, if A and B
promote the same conformation of the enzyme as indicated by a common maximum velocity, then B cannot modify further the conformation that has already been achieved by the binding of A. The values of the A-binding constants, given above, together with that given for poo, have been employed with an iterative procedure to calculate pA for the given concentrations of A and B. These values are shown as the curves drawn in Fig. 4, where they can be compared with the experimental values calculated from the ratios (o/V)B.
THE
EXPONENTIAL
MODEL
FOR
A
REGULATORY
ENZYME
113
5. Discussion Two topics arise for discussion; first, the validity of the model itself and second, the results of its application to experimental data. With regard to the validity of the exponential model, it is quite clear, on the basis of previous trials (Ainsworth, 19774 and those conducted above, that its characteristic equation is extremely effective as a device for fitting sigmoidal or non-inflected curves. This, in itself, could be trivial for it is well known (see for example, Tanford, 1973) that any suitable equation with a sufficient number of constants could do the same thing. Because this is so, care must obviously be taken to ensure that equation (13) is applied to data only when the evidence suggests that the constraints of the model are being met. Such care has been taken in treating the experimental data introduced above but it remains a matter of concern that no justification was provided for the assumption, made both here and by Ainsworth (1977a), that the fractional saturation of the protein by additional ligands remains constant as the concentrations of those ligands that are being explicitly considered change their values. This is particularly worrying when velocity data for multisubstrate enzymes, such as pyruvate kinase, are being analysed. There is no doubt therefore that the model could be more rigorously tested by binding data than by the velocity data that are more freely available. At a more fundamental level, it should be re-iterated that the model was introduced as an interim procedure to exclude the individual processes that occur during saturation because they cannot be defined experimentally. But the processes remain and will finally require description. It is therefore to be noted that there is as yet no logical connection that can be drawn between the conformational states of a general model (Whitehead, 1970; Ainsworth, 1977b) and the average conformational state that is postulated here. In spite of these reservations, it must be acknowledged that the experimental data examined above are satisfactorily fitted by the model, that it succeeds in the task with fewer constants than are required by several alternative models that have been suggested and that it can provide values for the constants that are required. The second topic arises from the values of the constants that were obtained. It appears possible from these data that the binding constants for A might be influenced by pB whilst those for B remain independent of pA: in other words, that the binding behaviour of two ligands might not be linked reciprocally. The notion that a protein consists of independently flexible parts, to which this result leads, is an interesting one and one that can be easily modelled by a tangled coil of wire. If this can be confirmed, it suggests that the study of reciprocal binding effects might become a useful
114
S. AINSWORTH
AND
R.
means of studying the directed transmission structure.
B.
GREGORY
of strain within the protein
REFERENCES AINSWORTH, S. (1977u). J. theou. Biol. 68, 391. AINSWORTH, S. (1977b). Steady State Enzyme Kinetics, pp. 201-243, p. 53. London: Macmillan. BENESCH, R., BENESCH, R. E. & Yu, C. I. (1968). Proc. natn. Acud. Sci. U.S.A. 59, 526. CLELAND, W. W. (1963). Nature 198,463. CLELAND, W. W. (1967). Ado. Enzymology 29, 1. GOODFORD, P. J., NORRINGTON, F. E., PATERSON, R. A. & WOOTTON, R. (1977). J. Physiol. 273, 631. H&DE, E. & AINSWORTH, S. (1968). J. biol. Cbem. 243, 2413. KINLIERLEFSR, J. & AINSWORTH, S. (1978). Znt. J. Bio-med. Comp. In press. METZGER, B. E., GLASER, L. & HELMREICH, E. (1968). Biochem. 7,202l. MONOD, J., WYMAN, J. & CHANC+EUX, J.-P. (1965). J. mol. Biol. 12, 88. PHILLIPS, F. C. & AINSWORTH, S. (1977). Znt. J. Biochem. 8, 729. TANFORD, C. (1973). The Hydrophobic Effect, pp.32-3. New York: Wiley. WHITEHEAD, E. (1970). Prog. Biophys. mol. Biol. 21, 323. WYMAN, J. (1964). Adu. prot. Chem. 13, 223.
The Exponential Model for a Regulatory Enzyme: Its Extension to Describe the Binding of Two Ligands STANLEY AIN~WORTH AND ROGER 3. GREGORY Department
of Biochemistry, University of Shefield, Shefield ,910 2TN, England (Received 26 April 1978)
The exponential model for a regulatory enzyme (Ainsworth, 1977u) is extended to deal explicitly with the presence in solution of a second ligand. This is achieved by introducing exponential interaction coefficients which respectively describe how the affinity of the free and bound forms of the protein for the ligand depend on its fractional saturation by the second ligand. The basic equations, so derived, are applied to binding experiments
where the ligands bind independently or competitively
and to rate
experiments where the ligands represent two substrates or one substrate and a modifier which may be either competitive or non-competitive in type. The conditions required to display linkage between the binding of the two
ligands are established and it is also shown that rate data may display a maximum as one ligand concentration is varied at a fixed concentration of the other. The equations that are derived are tested by application to
experimental data and the conditions that have to be met to justify such an application are discussed.
1. Introduction In a recent paper Ainsworth (1977a) described an exponential model for a regulatory enzyme which leads to relationships between the initial velocity of the catalysed reaction and the varied concentration of a substrate that are non-inflected or sigmoidal in type, without a maximum. The model assumes that the most relevant measure of protein configuration (itself determining the kinetic behaviour of the enzyme) is the apparent association constant, o+,, measured for the given fractional saturation of enzyme by the substrate under investigation. It was further assumed that the original state of the protein in solution, cl,,, is destabilized by an increment of energy, AG:, that is proportional to the fractional saturation, p, of the enzyme by substrate, so that formation of the configurational state, aP, can be represented by - AG,OlRT = kp = ln uP/cq,,
(1)
97
0022-5193/78/210097+18 4
$02.00/O
0
1978 Academic
Press Inc.
(London)
Ltd.
98
S. AINSWORTH
AND
R.
B.
GREGORY
where a,, is the binding constant at zero saturation. Equation (1) then gives ap=
( > -_-P l-p
-1 = a0 ekP A
which describes both positive and negative co-operativity with respect to the binding of A. Equation (2) can also be applied to rate data if p is equated with the ratio of the measured initial velocity Y to the maximum velocity Y established at a saturating concentration of A. It is an important aspect of the model that @Prefers to the whole assembly of enzyme molecules in solution and is therefore an average quantity: it does not refer to specific enzyme complexes. The purpose of this paper is to extend the model so as to deal explicitly with the presence in solution of a second ligand and to test this approach by applying it to experimental data. It is assumed throughout that the fractional saturation of the protein by any additional ligand is constant. 2. Theory
It will be evident from the introduction that the exponential model assumes that the substrate and enzyme reach a state of quasi-equilibrium with enzyme-substrate complexes which is not disturbed by product formation during the timeintervalrequiredto establish an initial velocity of reaction and further that non-hyperbolic velocity-substrate relationships arise solely from changes in the binding constants. This approach will be continued and, for simplicity, the two ligand-protein system will first be examined as a binding problem: the modifications that arise when rate data alone are available will then be treated separately. (A)
DEFINITION
OF BINDING
CONSTANTS
Consider an experiment in which the fractional saturation of the protein by A is determined in the presence of a second ligand, B, which binds independently. By analogy with equation (2), we have (3)
where aab is the binding constant for A which refers to specific values of pA and pB between O-l. Note, however, that the value of pB refers only to that fraction of the saturation by B which, through a contingent configurational change in the protein, exerts an effect on the binding constant for A. It is possible to imagine that a value for pB might be determined directly, say for binding at several sites, which would be irrelevant to the experiment under
THE
EXPONENTIAL
MODEL
FOR
A REGULATORY
ENZYME
99
consideration. In what follows, however, we shall assume that the measured and effective saturations by B are identical. By following equation (l), we can write aab = aOb exp &4bPAh (4) where kAb is a constant independent of pA but dependent on pB = b. Again, gob is the binding constant for A measured at zero saturation by A but at PB = b. If we now assume, in a manner consistent with the original model, that the configurational state, mob, is related to that of the wholly-unbound protein, aoo, by an energy term which represents the destabilizing effect of binding B, then - AG&/RT
= IOBPB
= In (aOb/aOO>
(5)
(&Bh)
(6)
and, in a similar fashion, we can write %b
=
%O
exp
where 10s and IIB are exponential interaction coefficients which respectively describe how the affinity of the A-free and A-bound forms of the protein for A depends on PB. From equation (4), kAb = In f+,-h sob (7) and hence, by introducing equations (5) and (6) k*b = tin alo
-In
(8)
aOO> + (llB-lOB)PBT
or k Ab
(9)
= kAo -I- (11 B - IoB)PB-
By combining
equations (4) and (9), we obtain a nb = C(0b W? -K&m+ (11B- LJPBIP,J or, by making use of equation (5) a ab = aOO
exP &?4o + (11, -
10B)pBbA
(10)
+ IOBP,)
(10
and by analogy Dab = Poe e*P Kkok,, +- (L -&o)PA]P~+IAoPA) (12) which corresponds term by term with equation (11) except that the constants relate to the binding of B in the presence of A. (B) THE BINDING
FUNCTION
FOR A IN THE PRESENCE
Equations (3) and (4) can be combined to give V AaOb exP (k&‘/v> pA = ? = 1 +Aaob eXp (k&/v)
*
OF B
(13)
Equation (13) is analogous to the Michaelis-Menten equation and its denominator, correspondingly, is a distribution function which gives the
100
S. AINSWORTH
AND
R.
B.
GREGORY
ratio of the A-free to A-bound fractions of the enzyme. It does not, however, identify individual terms with the relative concentrations of specific, known enzyme forms and, indeed, the dependence of a,,,, on pA and pB emphasizes that the identities of the enzyme-species which determine Q, change with saturation. The analogy with Michaelis-Menten kinetics is nevertheless a useful one and further analogies will be drawn in deriving equations to represent the behaviour displayed by two ligand-protein systems. (C) DETERMINATION
OF CONSTANTS
The constants in equation (11) [and (12)] can be determined by multiple regression on their logarithmic transformations if values of pA and pB are available: in this connection, note that LX,,,,is derived from a value of pA by equation (3). (D)
LINEAR
TRANSFORMATION
The logarithmic form of equation (4) can be used as the basis of a linear plot In a,, = f(pA)B = f(~/y)a for those ligand-protein systems in which pB is independent ofp,: when this condition applies, all the constants in equation (11) can be obtained by replotting the primary slopes and intercepts as functions of pe [equations (9) and (5)]. In practice, the linear plot In Q, = f(z~/& requires the definition of maximum saturation or its equivalent V, the maximum velocity, (Ainsworth, 19770) and we have found it more convenient to employ a computer program (Kinderlerer & Ainsworth, 1978) to estimate the constants which also provides the value of this parameter. The program establishes initial estimates of the constants by the log plot, and then refines them, according to equation (13), by an iterative procedure which treats iyOb,kAb and Y as disposable constants. (This procedure implies that measured fractional saturations are treated as values of 0 and can be altered by a constant proportion in relation to the value of V determined by the iterative procedure itself.) (E) RELATIONSHIP
BETWEEN
CONSTANTS
The relationship between the kA and I constants is given by equation (9) and when pB = 1, (14) km+& = kx.il +Io~ Equations (9) and (14) indicate that the same final configuration, as measured by a&, is obtained on saturation by A and B, irrespective of the order of saturation. It should be noted, however, that the value of pA, established in the absence of B, will only be the same as that finally obtained when IIB and IOB are zero. Equation (14) can be modified to employ p constants, but an
THE
EXPONENTIAL
MODEL
FOR
A REGULATORY
ENZYME
101
“equilibrium box” relationship between the two equations does not exist. For, in the normal equilibrium box, separate, identifiable protein species co-exist in solution whose relationships to one another are defined by known ligand concentrations and equilibrium constants. By contrast, the exponential model treats the whole assembly of protein molecules in solution as one, average configurational state with properties which change continuously with saturation and which are assessed differently by different ligands. The last point deserves emphasis. The free energy changes that are associated with changes in the binding constants of individual ligands relate only to those configurational events within the protein molecule which are relevant to the ligand concerned: they do not relate to the protein as a whole. There is thus no basis upon which to formulate an obligatory relationship between the c1and p constants. This conclusion may be derived more explicitly by considering Wyman’s (1964) linkage relationship for a protein which binds an equal number of A and B molecules :
(&),=(i2tJ,
(15)
The right hand side of equation (15) can be represented as (16)
By differentiation
of equation (13) (Ainsworth, 1977u) (17)
where Ixr, = IIB-&. Similarly, from equation (12) and bna = p&l dp, -= dp, where IAx = I, 1 - IAo.
The combination -=dp, din A
-pB)B
(ZAO+I”xPJP,(lPe) Cl -P& - PB)~%NI + ZAXPA~I ’
(W
of equations (17) and (18) now gives
PAI -P.&U - PIN,40 +IAXPB) .Cl - PAO - P~)UGO +kB~dlC1 - PA1 -p&h +hdI - lk4(1- P.4)PAl - PINA + ~AXP&JB +Ix,PA)l
which demonstrates, is true
(19) by arguments of symmetry, that when equation (15) (I.40 + ZAXPdB = uon +k3PA)A.
(20)
But equation (20) can only be true for all values of pA and ps when IA0 z IOB and when I AX z I,, E zero. The latter condition requires that the function
102
S.
AINSWORTH
AND
R.
B.
GREGORY
In z,~ =&A) should have a constant slope when pB takes different fixed values. The data discussed below suggests that this condition is not always met and therefore that the a and p constants are not obligatorily linked. (F) APPLICATION
TO DATA
FROM
A COMPETITION
By analogy with the Michaelis-Menten inhibition, we can write:
equation
for a competitive
AaGo pA = 1 +B&, Re-arrangement
(21)
+ Aa,o
of equation (21) now gives (22)
pA = v = l+Aa,,(l+B&,)-l exp (k&/V) which can be analysed according to equation (13) if cl,&1 f&&J’ is equated with czob.Equations (3) and (22) also show that plots of In a,, = f(pA) at different constant values of B, present a series of parallel lines which move to more negative values as B increases. It also follows that:
Data consistent with equation (22) are presented later in this paper. 3. Application to Rate Data The application of the equations developed above to rate data depend on the operational relationship of the two ligands to one another. Two alternatives will be considered. B ARE SUBSTRATES
(A) A AND
Here : V -V =
pApB
=
-.- A%b 1+./h,,
%b l-+-B/&,’
In this equation, u/[ V&,J(l + &&)] will only provide a measure of PA if the second term remains constant as A is varied. This can be proved by comparing the velocity of the reaction at a saturating ‘concentration of A with a direct measurement of the fractional saturation by B in the absence of A, for the equality
0 V
v
can only be true when I,, BII, = kb
= A-m
B&b l+Bthb
=
(P&z‘,
(25)
= IA, = zero as shown by the definition.
exP
{[~~B$(JA1-IAO)IPBflAO}.
(26)
THE
EXPONENTIAL
MODEL
FOR
A REGULATORY
ENZYME
103
In the absence of binding data only circumstantial evidence to the same effect can be provided. In the first place plots of In aa,, = f(z~[k’&, according to equation (4) and where V,,, = VB&,/(l + B&J, can only be linear when pe is independent of pA. The prevalence of rate data that can be represented in this way (Ainsworth, 1977a) suggests that pB often does remain constant at constant B, but deviations from linearity were observed and it is likely that they arise from a systematic change in pB as A is varied. A second line of evidence follows if (v/ v3A= co is found to be an hyperbolic function of B for equation (26) shows that one condition leading to this result is that k OBPI,, and IA0 are all zero, thereby indicating that pB is a constant independent of pA. Unfortunately, an alternative interpretation of the hyperbolic relationship can be suggested, namely, that koB = (IAl -I.& = zero but, if this were true and I,, # zero, linear replots based on equations (5) and (9) and the value of pB calculated from (u/v>,,, would not be expected. The last test can also be applied when (v/V),, co = f(B) is not hyperbolic, for then linear replots calculated on the basis of the assumption k,, # IAl = IA0 = zero and (v/V), =m = pe would again not be found if the assumption were false. When pB is not independent of pA analysis of the binding behaviour of A can only be undertaken on rate data collected at saturating concentrations of B. However, more progress is possible if either PA or pB can be separately determined in a binding experiment conducted under identical conditions. With one value available, the second can be determined by equation (24). The values of PA and pB can then be inserted into the logarithmic forms of equations (11) and (12) and the constants estimated by multiple regression. Note that equation (24) also gives rise to the possibility of a maximum in o as A is varied at constant B. Differentiation of the equation shows that the maximum arises when (27) Equation (27) can be combined with equations (17) and (18) to give d(pApzt)
_
PBZ-‘A(~
-
PA)[~
+
(I-
Pd(IAoP,
-
koBPB)l
dln A D where D is the denominator of equation (19). Equation (28) shows that a maximum occurs when (I--P,)-’
= tkoBPrI,aPA)
=
o
,
(2%
(29) that is to say when the right hand side of equation (29) is positive. The values of k,, and I,,-, required to meet this condition are readily determined and have been confirmed by computation of the relationship pApe = f(A).
104 (B)
S. AINSWORTH
AND
R.
B.
GREGORY
A IS A SUBSTRATE,
B AN ALLOSTERIC
EFFECTOR
The second alternative can be described by analogy with the MichaelisMenten equation for a partially non-competitive modifier (Ainsworth, 19773), that is: V v = PAp-PB)+YPBl.
(30)
In this equation, V represents the rate of product formation when the enzyme is completely saturated by A in the absence of B. A different rate is measured when the enzyme is also saturated by B and the change is defined by multiplying V by the constant y : inhibition is associated with values 0 < y < 1, and activation with values y > 1. In addition, however, the effector B influences the rate of reaction through its effect on CQ,[equation (ll)]. Thus a pure K system will arise when y = 1, and a pure V system when y # 1 and I,, and IoEl eq ual zero. (Monod, Wyman 8z Changeux, 1965). Equation (30) describes both these influences and its further analysis might proceed as follows. First, consider the case where y # 1: by substituting for pA and pB equation (30) gives: V v=
Now, with the definition saturation with A:
A%,
.- 1 + @hzb
IfAoc,,
(31)
l+B&,’
that V,,, = V(1 +yB@,,)/(l K,,V B&b pB = (yV-V) = lfb’
+B&),
we obtain on (32)
When evidence of the type discussed in the last section can be advanced to show that pB is independent of pa, we can write PA
=
$ *aPP
(33)
and the values given by equations (32) and (33) can be analysed by the use of equation (13). When /?rb is not independent of PA, further progress can only take place if either pA or pB can be separately determined by a binding experiment. With one value available, the second can be obtained through equation (30). Equation (31) also implies the existence of turning points but the conditions for these to be observed have not yet been examined. The second case arises when y = 1. Here equation (31) reverts to equation (13) and (as described in connection with that equation) its complete analysis depends upon the availability of independent values of pB.
THE
EXPONENTIAL
MODEL
FOR
A
REGULATORY
ENZYME
105
4. Tests of the Model (A)
OXYGEN
BINDING
BY
HEMOGLOBIN
IN
THE
PRESENCE
OF
THE
COMPETITOR,
2$DIPHOSPHOGLYCERATE
This well-known competitive system (Benesch, Benesch & Yu, 1968) is complicated by the fact that hemoglobin (Hb) is a tetramer, at high concentrations, which binds four molecules of oxygen (A) but only one molecule of 2,3-diphosphoglycerate (B). It is therefore necessary to modify the equations for a competition reaction, developed above, so that the individual binding of A and B relate to a single aggregation state of the protein. To standardize in terms of A-binding sites, the monomer equivalent of Hb4 and HbaB is calculated as
where K is the association constant for the monomer-tetramer But Hb = (Hb,/K)’ so that A4 = Hb (1 + B/?&.
equilibrium. (35)
Equation (21) can now be modified to read
AGO PA = (1+B/3,,)~+Ac~,,
(36)
which can be analysed according to equation (13) if aoo(l +BpOb)-h is equated with sob. Goodford, Norrington, Paterson & Wootton (1977) have recently published oxygen dissociation curves for concentrated human hemoglobin solutions in the presence and absence of 2,3-diphosphoglycerate. The actual data points (kindly supplied by the authors) were analysed by fitting equation (13) and the resulting constants sob, kAb and Y are given in Table 1. Examination of the table shows that the values of kAb are almost constant (ka = 2.625f0.094) and that sob becomes smaller as B is increased. These observations are consistent with a competition and it appears that the constants should be employed to test equation (36). However, because the concentration of hemoglobin is too high to permit the approximation B = B (total), it is more convenient to test a modified form of equation (36)
AQO PA = (1-p&“+Acc~,
(37)
106
S.
AINSWORTH
AND
R.
TABLE
1
B.
GREGORY
Oxygen binding by human hemoglobin. Constants obtained by fitting
equation
(13) with v set equal to the measuredfractional saturation of human hemoglobin by oxygen (A) in the presence ofjxed concentrations of 2,3-diphosphoglycerate (B). Values of pB are calculated as described in the text Goodford, Norrington, 0.1
V
1 a006 ho.036 0.219 &0@06 2.697 &0.081
Gb (kpka-I) Rb
PB
0.5
1.0
2.5
5.0
O-976 10.021 0.160 *oaO3 2.714 10.054 0.623
0.963 ho.034 0.141
1408 &to.018 o-105 &O+lOl 2.646 ho.038
0.985 kO.016 0.091 &O*OOl 2.632 kO.037
0.2
1.046 fO*026 0.204 &0.003 2,436 10.050 0.172
0
Paterson & Wootton (1977)
0.978
&O-O19 0.197 f0*003 2.546 10.051 0.321
f 0405 2-703
& 0,092 0.816
0.933
O-968
Benesch, Benesch h Yu (1968) 2,3-DPG hM>
V &lb
(mm
Hg - I)
k Ab PB
which eqUateS expression
0
o-2
1.107
iO.032 O-0989 jO.0062 2.423 10.153
0
Nob
1.083
0.4 1.119
0.6 1.110
f0.030
kO.027
kO.020
0.0526
0.0361 10.0019 2.231 f0.142
0.0274 &0aoo9 2.400
&0+034 2.354 50.160 0.884
with a,,(1 -pB)*
0.945
*to.090 0.964
O-8
1.0
l-106 &0*019 0.0244
1.105 10.021 0.0213
iOGOO8
10-0008
2.430
i oa90 0.973
2.513 &O.loo
0.979
and to calculate pB from the quadratic
pB = (C~+PB,+BE,I-C(~+PB,+BE,)~-~P~B~E~I~~~~~E~)-~, (38) where B, represents the total concentration of 2,3-diphosphoglycerate
and E0 the total concentration of hemoglobin expressed as tetramers (0-4x 10m3 M). p is the association constant for the binding of 2,3diphosphoglycerate and has the value 6600 M-I (Goodford et al., 1977). Figure 1 shows a plot of ln Mob = h aoo +t In (1 -pB) (3% according to these data: the straight line through the points has unit slope as required. The data for the diphosphoglycerate competition provided by Fig. 4 of Benesch, Benesch & Yu (1968) can also be analysed by the procedure
THE
EXPONENTIAL
MODEL
FOR A REGULATORY
ENZYME
107
1/4In(I-pJ -1.0
-04
I
I
-1.5 -
I
-2.5
FIG. 1. Dependence of ln sob on the fractional saturation of human hemoglobin by 2,3-diphosphoglycerate. 0 from data given in Fig. 4 of Benesch, Benesch & Yu (1968); 0 from the datum points illustrated in Fig. 3 of Goodford, Norrington, Paterson & Wootton (1977).
described earlier in this paper. As before kAb is almost constant (kAb = 2.390 $-O-085) and aObdecreases as B is increased (Table 1). In contrast, however, these data were collected at a lower concentration of hemoglobin (4.6 x 10e5 M tetramers), where fi was estimated to have the higher value of 4.8 x IO4 M-’ and they were interpreted (Benesch, Benesch & Yu, 1968) by assuming that the binding of two oxygen molecules was completely concerted. This interpretation is consistent with the linear relationship between in aOb and 3 In (1 -pB) shown in Fig. 1. Note that the line with unit slope passes through all but one of the data points. (B)
THE
REACTION
OF
GLYCOGEN
AND
GLUCOSE-l-PHOSPHATE
CATALYSED
BY
FROG MUSCLE PHOSPHORYLASE U
initial velocity data for the phosphorylase a catalysed reaction were obtained from optical enlargments of Fig. S(b) of Metzger, Glaser &
108
S.
AINSWORTH
AND
R.
B.
GREGORY
@ucose-i-phosphate],mM
FIG. 2. The initial velocity of the reaction catalysed by frog skeletal muscle phosphorylase a [Fig. 8(b) of Metzger, Glaser & Helmreich, 19681 with glycogen (expressed as glucose residues at the non-reducing end of the chain) 0, 0.1 mu; 0, 0.25 mM; A, 0.5 mu; A, 2.5 mu; 0, 5 mu. pH 6.8, 25”. The points represent initial velocities expressed as bmoles product formed min- l (mg enzyme)- I. The curves are calculated as described in the text. Insert: dependence of the maximum velocity calculated for saturation by giucosel-phosphate on the concentration of glycogen.
2
TABLE
Frog muscle phosphorylase a. Constants obtained by Jitting equation (13) with v equal to the initial velocity of the reaction catalysed by phosphorylase a when measured as a function of [glucose-l-phosphate], (A), in the presence ofjixed concentrations of glycogen, (B), (Metzger, Glaser & Helmreich, 1968). Other details as in Fig. 2 legend. The last four columns give the values used to obtain the interaction and binding constants quoted in the text Glycogen
V
a0b
b-M
(units of v)
(mM- I)
0‘10
7.92 ho.48 17.08 zto.41 21.41 *O-69 28.99 &l-O6 35.30 hl.53
0.0532 rt O+IO26 0.0422 ~09011 0.0594 &-o-O019 0.0719 rtO.0044 0.0886 100341
0.25 o-50 2.50 5~00
k
Ab
1,763 & 0.200 1.830 &0~096 1.333 zto.143 l-270 A-O.203 0.694 rtO.216
h.l am
In 011b
(ln mM-l)
(In mki-I)
PB
-2.940
-1.171
0.213
-3.165
-1.335
0.404
-2.824
-1.491
0.575
-2.632
-1.363
0.871
-2.423
- 1.729
0.931
THE
EXPONENTIAL
MODEL
FOR
A
REGULATORY
ENZYME
109
Helmreich (1968). These data, shown in Fig. 2, conform to equation (24) when A is the concentration of glucose-l-phosphate and B the concentration of glycogen expressed in terms of glucose residues at the non-reducing end of the glycogen chain. The data were analysed by fitting equation (13) and resulting constants are given in Table 2. The maximum velocities were found to be a hyperbolic function of B, as shown by the double reciprocal plot inserted in Fig. 2. The hyperbolic relationship suggests that PB might be independent of pA, supports the use of equation (13), and provides values of V = 38.12k2.97 umoles min-’ (mg enzyme)-‘, and PI0 = 2.706 AO.195 mM-l. The assumption of independence is tested by setting Boo = plo (i.e. IAl = la0 = zero), and by its use calculating the value of pB for each glycogen concentration that is given in Table 2. Plots of kAb, In aOb and In c+, are linear functions of pB, thus supporting the assumption that was made, and unweighted linear regressions provide the values (units in Table 2) kAo = 2-17&O-22, ZXB = -1*32$0.30, Ina,, = -3*30+0*16, I,, = O-86+0*22, ha,, = -1*13+0.14 and ZIB = -0*47&0*19. These values, together with that given above for Boo, were then employed with an iterative procedure to calculate pA and pB for the given concentrations of A and B. By substitution into the equation v = Vpp, calculated values of v were obtained which are shown as the curves drawn in Fig. 2. The agreement between the calculated and experimental values is comparable in quality to that which has been achieved in the analysis of two-substrate reactions, conforming to first degree rate equations, by the use of Cleland’s (1963, 1967) procedures (Heyde & Ainsworth, 1968). (C)
THE
INHIBITION
OF
RABBIT
MUSCLE
PYRUVATE
KINASE
BY
PHENYLALANINE
Data for this inhibition, in which A is the substrate ADP, and B the inhibitor, phenylalanine (the concentrations of phosphoenolpyruvate and Kf and Mg2+ ions were kept constant) are shown in Fig. 3. These data were analysed by fitting equation (13) and the resulting constants are given in Table 3. Examination of the table shows that the maximum velocity falls to zero as B is given increasing fixed values, from which it can be concluded that y = 0 [equation (30)]. The plot of pB = f(Z3) according to equation (32) is not hyperbolic, however, and one indication that ps is independent of PA is therefore lacking. In spite of this, plots of kAb, In CQ,,and In alb are found to be linear functions of pB. These linear relations provide circumstantial evidence that pB is independent of p A (I A1 = Z,, = zero) and when analysed by an unweighted linear regression give the values (units in Table 3) kAo = l-92+ 0.04, ZXB = - 1.01 &O-082, In LX,,,,= - 1*49+0*04, ZoB = O-830 &O-09; In alo = 0*420+0+48 and ZIB = -0.175&0*10. The values of
110
S.
AINSWORTH
AND
R.
B.
GREGORY
12e
IOC
7: V
5c
25
[ADP],mM
3. The inhibition of rabbit muscle pyruvate kinase by phenylalanine at 25”. The reaction mixtures contained phosphoenolpyruvate (O-2 mu), Mg2+ (0.1 mu), and K+ (100 mu) in Nethylmorpholine buffer (0.1 M, pH 7.4), with phenylalanine 0, 0 mu 0, O-5 mu; A, 1.0 mu; A, 2.5 mu; q , 5 mu; W, 10 mu. The points represent initial velocities expressed as p moles of pyruvate produced per minute per mg of enzyme corrected to a specific activity of 300 units mg-I. The curves are calculated as described in the text. TABLE 3 FIG.
Inhibition of rabbit muscle pyruvate kinase. Constants obtained by Jitting equation (13) with v equal to the initial velocity of the reaction catalysed by pyruvate kinase when measured as a function of [ADP], (A) in the presence of fixed concentrations of phenylalanine, (B). Other details as in Fig. 3 legend. The last four columns give the values used to obtain the interaction and binding constants quoted in the text Phenylalanine (mM> 0
0.5 1.0 2.5 5.0 10.0
Vor K,, when B # 0 &?& (units of v) 13363 13.16 119.97 A4.71 114.21 h4.51 75*19 c6*00 38.30 h2.59 16.81 zt1.63
0.231 -10-011 0.243 10.018 0.242 &to*015 0.318 &OGll 0.435 ,to4xo 0.426 ItO.
k*b 1.936 ho-124 1.886 kO.208 1.701 &-to.185 1442 10.425 1.184 A-to.444 1.135 10.612
- 1.464
0.471
0
-1.414
0.472
0.077
-1.417
0.284
0.160
-1.144
0.297
0.427
-0.833
0.350
0.702
-0.854
0.282
0.860
THE
EXPONENTIAL
MODEL
FOR
A REGULATORY
111
ENZYME
k,, = 1*67+ 0.27 and In ,!IoO = - 1*92j:O*lO were also estimated by using an appropriately modified form of equation (13) in which z)was equated with p,,. The constants thus obtained were then employed with an iterative procedure to calculated pA and pB for the given concentrations of A and B. By substitution into the equation v = Vp, (1 -pe), calculated values of D were obtained: these are shown as the curves in Fig. 3. The agreement between the calculated and experimental values is sufficiently good to suggest that the assumptions made in the analysis must be at least approximately true. (D)
THE
ACTIVATION
OF RABBIT
MUSCLE
PYRUVATE
KINASE
BY FRUCTOSE-1,6-
DIPHOSPHATE
Phillips & Ainsworth (1977) have demonstrated that the enhancement of pyruvate kinase activity brought about by fructose diphosphate is an hyperbolic function of the activator concentration which indicates an apparent binding constant of 0.80 mK1. Phillips & Ainsworth (1977) also showed that the same final velocity was achieved at high concentrations of the substrate ADP with or without addition of the activator. It appears on this basis that 4 Activation of rabbit muscle pyruvate kinase. Constants obtained by fitting equation (13) with v equal to the initial velocity of reaction catalysed bypyruvate kinase when measured as a function of [ADP], (A) in the presence of Jixed concentrations of fructose-1,6-diphosphate, (B). Other details as in Fig. 4 legend. The last four columns give the values used to obtain the interaction and binding constants quoted in the text TABLE
5.0
147.08 12.88 147.85 +2-83 156.59 +9*78 154.84 ztz4.83 158.85 h3.73
0.202 1OGO6 0.264 iO406 0.279 10.021 o-359 *0*017 0.502 10*021
1.813 &O-O80 1.411 &to.084 1 a308 10.275 1.218 &0*160 0.942 10.150
155.61 49.43
0.590 f0.076
0.859 10440
-1.601
0.213
0
-1.330
0.081
0.167
- 1.275
0.032
0.286
- 1.023
0.195
0444
-0.690
0.252
0.667
-0~528
0.331
0.800
112
S.
AINSWORTH
AND
R.
B.
GREGORY
the activation of pyruvate kinase conforms to equation (30) when y = 1. This conclusion is supported by additional data for which the constants given in Table 4 were obtained by fitting equation (13) with A identified as the substrate ADP, and B as the activator, fructose diphosphate. Examination of the table confirms that V is constant within the error of the estimates, demonstrating thereby that y = 1. Plots of kAb, In c& and ln aI,, are linear functions of pB (calculated on the basis of the binding constant given above), and unweighted linear regressions provide the values (units in Table 4)
5
FIG. 4. The activation of rabbit muscle pyruvate kinase with fructose-1,6-diphosphate 0, 0 mu; l , 0.25 mh%; A, 0.5 mu; A, 1.0 mu; 0, 2.5 mu; a, 5.0 mu. Other details as in Fig. 3. The points represent data, the curves are calculated as described in the text.
kA,, = l-69+0*06, IxB = -l-11+0*13, Ina,, = -1.60fO.03, IOB = 1.34 *O-06, In cllo = 0.09 kO.07, and IIB = 0.23 f0.14. Note that IIB is almost zero within the error of the estimate: this is to be expected for, if A and B
promote the same conformation of the enzyme as indicated by a common maximum velocity, then B cannot modify further the conformation that has already been achieved by the binding of A. The values of the A-binding constants, given above, together with that given for poo, have been employed with an iterative procedure to calculate pA for the given concentrations of A and B. These values are shown as the curves drawn in Fig. 4, where they can be compared with the experimental values calculated from the ratios (o/V)B.
THE
EXPONENTIAL
MODEL
FOR
A
REGULATORY
ENZYME
113
5. Discussion Two topics arise for discussion; first, the validity of the model itself and second, the results of its application to experimental data. With regard to the validity of the exponential model, it is quite clear, on the basis of previous trials (Ainsworth, 19774 and those conducted above, that its characteristic equation is extremely effective as a device for fitting sigmoidal or non-inflected curves. This, in itself, could be trivial for it is well known (see for example, Tanford, 1973) that any suitable equation with a sufficient number of constants could do the same thing. Because this is so, care must obviously be taken to ensure that equation (13) is applied to data only when the evidence suggests that the constraints of the model are being met. Such care has been taken in treating the experimental data introduced above but it remains a matter of concern that no justification was provided for the assumption, made both here and by Ainsworth (1977a), that the fractional saturation of the protein by additional ligands remains constant as the concentrations of those ligands that are being explicitly considered change their values. This is particularly worrying when velocity data for multisubstrate enzymes, such as pyruvate kinase, are being analysed. There is no doubt therefore that the model could be more rigorously tested by binding data than by the velocity data that are more freely available. At a more fundamental level, it should be re-iterated that the model was introduced as an interim procedure to exclude the individual processes that occur during saturation because they cannot be defined experimentally. But the processes remain and will finally require description. It is therefore to be noted that there is as yet no logical connection that can be drawn between the conformational states of a general model (Whitehead, 1970; Ainsworth, 1977b) and the average conformational state that is postulated here. In spite of these reservations, it must be acknowledged that the experimental data examined above are satisfactorily fitted by the model, that it succeeds in the task with fewer constants than are required by several alternative models that have been suggested and that it can provide values for the constants that are required. The second topic arises from the values of the constants that were obtained. It appears possible from these data that the binding constants for A might be influenced by pB whilst those for B remain independent of pA: in other words, that the binding behaviour of two ligands might not be linked reciprocally. The notion that a protein consists of independently flexible parts, to which this result leads, is an interesting one and one that can be easily modelled by a tangled coil of wire. If this can be confirmed, it suggests that the study of reciprocal binding effects might become a useful
114
S. AINSWORTH
AND
R.
means of studying the directed transmission structure.
B.
GREGORY
of strain within the protein
REFERENCES AINSWORTH, S. (1977u). J. theou. Biol. 68, 391. AINSWORTH, S. (1977b). Steady State Enzyme Kinetics, pp. 201-243, p. 53. London: Macmillan. BENESCH, R., BENESCH, R. E. & Yu, C. I. (1968). Proc. natn. Acud. Sci. U.S.A. 59, 526. CLELAND, W. W. (1963). Nature 198,463. CLELAND, W. W. (1967). Ado. Enzymology 29, 1. GOODFORD, P. J., NORRINGTON, F. E., PATERSON, R. A. & WOOTTON, R. (1977). J. Physiol. 273, 631. H&DE, E. & AINSWORTH, S. (1968). J. biol. Cbem. 243, 2413. KINLIERLEFSR, J. & AINSWORTH, S. (1978). Znt. J. Bio-med. Comp. In press. METZGER, B. E., GLASER, L. & HELMREICH, E. (1968). Biochem. 7,202l. MONOD, J., WYMAN, J. & CHANC+EUX, J.-P. (1965). J. mol. Biol. 12, 88. PHILLIPS, F. C. & AINSWORTH, S. (1977). Znt. J. Biochem. 8, 729. TANFORD, C. (1973). The Hydrophobic Effect, pp.32-3. New York: Wiley. WHITEHEAD, E. (1970). Prog. Biophys. mol. Biol. 21, 323. WYMAN, J. (1964). Adu. prot. Chem. 13, 223.