Sigmoidal curves, allosteric enzymes

A plot of VQ against [S] for an allosteric enzyme gives a sigmoidal-shaped curve. Allosteric enzymes often have more than one active site which co-operatively bind substrate molecules, such that the binding of substrate at one active site induces a conformational change in the enzyme that alters the affinity of the other active sites for substrate. Allosteric enzymes are often multi-subunit proteins, with an active site on each subunit. In addition, allosteric enzymes may be controlled by effector molecules (activators or inhibitors) that bind to a site other than the active site and alter the rate of enzyme activity. Aspartate transcarbamoylase is an allosteric enzyme that catalyzes the committed step in pyrimidine biosynthesis. This enzyme consists of six catalytic subunits each with an active site and six regulatory subunits to which the allosteric effectors cytosine triphosphate (CTP) and ATP bind. Aspartate transcarbamoylase is feedback-inhibited by the end-product of the pathway, CTP, which acts as an allosteric inhibitor. In contrast, ATP an intermediate earlier in the pathway, acts as an allosteric activator. 

In contrast to the kinetics of isosteric (normal) enzymes, allosteric enzymes such as ACTase have sigmoidal (S-shaped) substrate saturation curves (see p. 92). In allosteric systems, the enzyme s af nity to the substrate is not constant, but depends on the substrate concentration [A]. Instead of the Michaelis constant Km (see p. 92), the substrate concentration at half-maximal rate ([AJo.s) is given. The sigmoidal character of the curve is described by the Hill coef cient h. In isosteric systems, h = 1, and h increases with increasing sigmoid icity. 

Allosteric enzymes show relationships between V0 and [S] that differ from Michaelis-Menten kinetics. They do exhibit saturation with the substrate when [S] is sufficiently high, but for some allosteric enzymes, plots of V0 versus [S] (Fig. 6-29) produce a sigmoid saturation curve, rather than the hyperbolic curve typical of non-regulatory enzymes. On the sigmoid saturation curve we can find a value of [S] at which V0 is half-maximal, but we cannot refer to it with the designation Km, because the enzyme does not follow the hyperbolic Michaelis-Menten relationship. Instead, the symbol [S]0 e or K0,5 is often used to represent the substrate concentration giving half-maximal velocity of the reaction catalyzed by an allosteric enzyme (Fig. 6-29). 

Other heterotropic allosteric enzymes respond to an activator by an increase in Fmax with little change in if0i5 (Fig. 6-29c). A negative modulator (an inhibitor) may produce a more sigmoid substrate-saturation curve,... 

FIGURE 6-29 Substrate-activity curves for representative allosteric enzymes. Three examples of complex responses of allosteric enzymes to their modulators, (a) The sigmoid curve of a homotropic enzyme, in which the substrate also serves as a positive (stimulatory) modulator, or activator. Note the resemblance to the oxygen-saturation curve of hemoglobin (see Fig. 5-12). (b) The effects of a positive modulator (+) and a negative modulator (—) on an allosteric enzyme in which K0 5 is altered without a change in Zmax. The central curve shows the substrate-activity relationship without a modulator, (c) A less common type of modulation, in which Vmax is altered and /C0.sis nearly constant. 

Hyperbolic shape of the enzyme kinetics curve Most enzymes show Michaelis-Menten kinetics (see p. 58), in which the plot of initial reaction velocity, v0, against substrate concentration [S], is hyperbolic (similar in shape to that of the oxygen-dissociation curve of myoglobin, see p. 29). In contrast, allosteric enzymes frequently show a sigmoidal curve (see p. 62) that is similar in shape to the oxygen-dissociation curve of hemoglobin (see p. 29). 

Shapes of the kinetics curves for simple and allosteric enzymes Enzymes following Michaelis-Menten kinetics show hyperbolic curves when the initial reaction velocity (v0) of the reaction is plotted against substrate concentration. In contrast, allosteric enzymes generally show sigmoidal curves. 

Allosteric enzymes do not follow the Michaelis-Menten kinetic relationships between substrate concentration Fmax and Km because their kinetic behaviour is greatly altered by variations in the concentration of the allosteric modulator. Generally, homotrophic enzymes show sigmoidal behaviour with reference to the substrate concentration, rather than the rectangular hyperbolae shown in classical Michaelis-Menten kinetics. Thus, to increase the rate of reaction from 10 per cent to 90 per cent of maximum requires an 81-fold increase in substrate concentration, as shown in Fig. 5.34a. Positive cooperativity is the term used to describe the substrate concentration-activity curve which is sigmoidal an increase in the rate from 10 to 90 per cent requires only a nine-fold increase in substrate concentration (Fig. 5.346). Negative cooperativity is used to describe the flattening of the plot (Fig. 5.34c) and requires requires over 6000-fold increase to increase the rate from 10 to 90 per cent of maximum rate. 

There are exceptions to Michaelis-Menten behaviour. For example allosteric enzymes which instead of a hyperbolic curve in a Lversus [S] graph yield a sigmoidal plot (the behaviour is rather like non-catalytic allosteric proteins, such as haemoglobin, Section 2.5. This type of curve can indicate cooperative binding of the substrate to the enzyme. We have discussed cooperativity in Section 1.5 (see also Section 10.4.3). In addition, regulatory molecules can further alter the activity of allosteric enzymes. 

Not all enzymes afford a hyperbolic relationship between their rate and substrate concentration. Sigmoidal curves are common and indicate either cooperativity in a multienzyme complex or involvement of an allosteric site on the enzyme (Figure 4.10). These types of curves can be fit by introducing exponents on [S] and Km in Equation 4.11. 

Aspartate transcarbamoylase (aspartate carbamoyltransferase ATCase), a key enzyme in pyrimidine biosynthesis (see Topic FI), provides a good example of allosteric regulation. ATCase catalyzes the formation of N-carbamoylaspar-tate from aspartate and carbamoyl phosphate, and is the committed step in pyrimidine biosynthesis (Fig. 2). The binding of the two substrates aspartate and carbamoyl phosphate is cooperative, as shown by the sigmoidal curve of V0 against substrate concentration (Fig. 3). 

Fig. 9-10 Behavior of an MWC allosteric enzyme in the presence of positive and negative heterotropic effectors. The activator term, y, in Eq. (9.62) causes the curve to become more hyperbolic, whereas the inhibitor term (j3) renders it more sigmoidal. The curves were constructed using Eq. (9.62) with L = 1,000 and n - 4.

Figure 10.12. Basis for the Sigmoidal Curve. The generation of the sigmoidal curve by the property of cooperativity can be understood by imagining an allosteric enzyme as a mixture of two Michaelis-Menten enzymes, one with a high value of n, that corresponds to the T state and another with a low value of that corresponds to the R state. As the concentration of substrate is increased, the equilibrium shifts from the T state to the R state, which results in a steep rise in activity with respect to substrate concentration.

Figure 4-50 The sequential interaction model of allosteric enzymes. As each site is occupied, the subunit carrying the site undergoes a change from the A conformation to the B conformation. As a result, new interactions between subunits are established and the affinities of the vacant sites change. K represents a dissociation constant. Thus, if the affinities of vacant sites increase, a, b, and c (the interaction factors) are <1 and we observe positive cooperativity (a sigmoidal velocity curve). The sequential interaction model also provides for, negative cooperativity (a, b, and c are > ). (o ) Dimer model. The two ways of arranging S to form a singly-occupied species is shown, (fe) Tetramer model. For simplicity, only one arrangement of each occupied species is shown.

Allosteric enzymes bind activators or inhibitors at sites other than the active site. Plots of the velocity versus substrate concentration for allosteric enzymes produce curves that are sigmoidal. 

The sigmoidal binding curve displayed by many allosteric enzymes resembles the curve for the cooperative binding of 02 to hemoglobin. 

Allosteric enzymes show a sigmoid curve relating initial velocity to substrate concentration (Fig. 8.11). The sigmoid curve starts very slowly because the enzyme has low affinity for substrate. However, binding of one molecule of substrate increases the affinity of the enzyme for subsequent substrate molecules and the curve turns upward and the maximum velocity is soon reached. This is an example of positive cooperativity. 

Cooperative enzymes show sigmoidal curves of reaction rate as a function of substrate concentration. Allosteric regulators may change the Cm ax of such enzymes and/or change the cooperativity. 

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