Biochem. J. (1 977) 167, 479-482 Printed in Great Britain
479
A Model for the Allosteric Regulation of pH-Sensitive Enzymes By JULIAN S. SHINDLER and KEITH F. TIPTON* Department ofBiochemistry, University ofCambridge, Tennis Court Road, Cambridge CB2 1 Q W, U.K. (Received 13 June 1977) 1. In an enzyme that has two independent binding sites for a ligand, any inhibitor that binds solely to the free enzyme will give rise to positive co-operativity. 2. A model is considered for the allosteric control of enzymes by effectors in which their effects are mediated by ligand-induced perturbations ofthe ionization constants ofa group or groups involved in the binding of substrate to the active site. 3. The model described offers a plausible explanation for the observation that the sigmoidal initial-rate curves reported for some regulatory enzymes are not expressed at all pH values where the enzyme is catalytically active. In an enzyme that has two independent binding sites for a ligand, any inhibitor that binds solely to the free enzyme will give rise to positive co-operativity. This is because the first ligand to bind has to displace a bound inhibitor, which lowers the affinity for the first ligand molecule to bind relative to the second. This makes the binding positively co-operative by a degree that is controlled by the extent of saturation with the inhibitor. Several well-characterized regulatory enzymes do not exhibit sigmoid kinetics throughout their catalytic pH range, and it is often the case that their maximal catalytic potential is not expressed at physiological pH values (Gerhardt & Pardee, 1963; Bloxham & Lardy, 1974). Consequently, not only do these pH-sensitive enzymes function at a fraction of their actual efficiency but, assuming that in vivo such enzymes operate at, say, half-way down one limb of their pH-activity profile, they are extremely susceptible to fluctuations of intracellular pH (Trivedi & Danforth, 1966). This may be contrasted with the situation at the pH optimum, where the percentage change in velocity for a correspondingly small change of pH is much less. The model described here attempts to examine and explain these effects in terms of ligandinduced changes of the pK. values of one or more groups at the enzyme substrate-binding site.
However, a ligand (X) that binds only to the free will increase the dissociation constant for the first substrate molecules to bind by a factor of (1 + (where K, is the dissociation constant for the EX complex) without affecting the binding of the second substrate molecules. Thus the equation will become: KA[S] + [S]2 enzyme
KA2 (1 + []) +2KA[S] + [S]2
which according to the analysis of Ferdinand (1966)
ES
(a)
KA
SES
E
KA
KB
SE
Theory
Consider the simple model shown in Scheme l(a) in which the binding sites for substrate have equal affinities (KA = KB). Such a mechanism can only give rise to a hyperbolic binding curve, since the fractional saturation (FE) is given by: 2(KA[S] + [S]2) [ES] + [SE] + 2[SES]
2[Etotal]
(b)
ES
'EX =
KA+[S] * Present address: Department of Biochemistry, Trinity College, Dublin 2, Ireland. Vol. 167
X
~
~
KBl ~
~
SES
E
KA
KB
2(KA2+2KA[S]+[S]2)
[S]
KA
Ki~
SE
Scheme
1.
Simple model for homotropic effects
(a) With no inhibitor present; (b) with a competitive inhibitor, X, present. When X represents a H+ ion, K; will be replaced by K., the ionization constant.
J. S. SHINDLER AND K. F. TIPTON
480
will give rise to a sigmoid saturation curve when
[X]/Ki > 1.
A somewhat more general case, in which KA . KB, will give rise to the equation:
(KA+KB)[S]+2[S]2
(1) 2{KAKB + (KA +KB)[S] + [S]2} Seeking the condition for sigmoidicity, we find that (KA2+K2)
479
A Model for the Allosteric Regulation of pH-Sensitive Enzymes By JULIAN S. SHINDLER and KEITH F. TIPTON* Department ofBiochemistry, University ofCambridge, Tennis Court Road, Cambridge CB2 1 Q W, U.K. (Received 13 June 1977) 1. In an enzyme that has two independent binding sites for a ligand, any inhibitor that binds solely to the free enzyme will give rise to positive co-operativity. 2. A model is considered for the allosteric control of enzymes by effectors in which their effects are mediated by ligand-induced perturbations ofthe ionization constants ofa group or groups involved in the binding of substrate to the active site. 3. The model described offers a plausible explanation for the observation that the sigmoidal initial-rate curves reported for some regulatory enzymes are not expressed at all pH values where the enzyme is catalytically active. In an enzyme that has two independent binding sites for a ligand, any inhibitor that binds solely to the free enzyme will give rise to positive co-operativity. This is because the first ligand to bind has to displace a bound inhibitor, which lowers the affinity for the first ligand molecule to bind relative to the second. This makes the binding positively co-operative by a degree that is controlled by the extent of saturation with the inhibitor. Several well-characterized regulatory enzymes do not exhibit sigmoid kinetics throughout their catalytic pH range, and it is often the case that their maximal catalytic potential is not expressed at physiological pH values (Gerhardt & Pardee, 1963; Bloxham & Lardy, 1974). Consequently, not only do these pH-sensitive enzymes function at a fraction of their actual efficiency but, assuming that in vivo such enzymes operate at, say, half-way down one limb of their pH-activity profile, they are extremely susceptible to fluctuations of intracellular pH (Trivedi & Danforth, 1966). This may be contrasted with the situation at the pH optimum, where the percentage change in velocity for a correspondingly small change of pH is much less. The model described here attempts to examine and explain these effects in terms of ligandinduced changes of the pK. values of one or more groups at the enzyme substrate-binding site.
However, a ligand (X) that binds only to the free will increase the dissociation constant for the first substrate molecules to bind by a factor of (1 + (where K, is the dissociation constant for the EX complex) without affecting the binding of the second substrate molecules. Thus the equation will become: KA[S] + [S]2 enzyme
KA2 (1 + []) +2KA[S] + [S]2
which according to the analysis of Ferdinand (1966)
ES
(a)
KA
SES
E
KA
KB
SE
Theory
Consider the simple model shown in Scheme l(a) in which the binding sites for substrate have equal affinities (KA = KB). Such a mechanism can only give rise to a hyperbolic binding curve, since the fractional saturation (FE) is given by: 2(KA[S] + [S]2) [ES] + [SE] + 2[SES]
2[Etotal]
(b)
ES
'EX =
KA+[S] * Present address: Department of Biochemistry, Trinity College, Dublin 2, Ireland. Vol. 167
X
~
~
KBl ~
~
SES
E
KA
KB
2(KA2+2KA[S]+[S]2)
[S]
KA
Ki~
SE
Scheme
1.
Simple model for homotropic effects
(a) With no inhibitor present; (b) with a competitive inhibitor, X, present. When X represents a H+ ion, K; will be replaced by K., the ionization constant.
J. S. SHINDLER AND K. F. TIPTON
480
will give rise to a sigmoid saturation curve when
[X]/Ki > 1.
A somewhat more general case, in which KA . KB, will give rise to the equation:
(KA+KB)[S]+2[S]2
(1) 2{KAKB + (KA +KB)[S] + [S]2} Seeking the condition for sigmoidicity, we find that (KA2+K2)
