Sigmoidal velocity curves

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 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). 

In the absence of activators AMP aminohydrolase from brain (149), erythrocytes (143, 150), muscle (145), and liver (128) gave sigmoid curves for velocity vs. AMP concentration which were hyperbolic after the addition of monovalent cations, adenine nucleotides, or a combination of monovalent cations and adenine nucleotides. For the rabbit muscle enzyme (145), addition of K+, ADP, or ATP produced normal hyperbolic saturation curves for AMP as represented by a change in the Hill slope nH from 2.2 to 1.1 Fmax remained the same. The soluble erythrocyte enzyme and the calf brain enzyme required the presence of both monovalent cations and ATP before saturation curves became hyperbolic. In contrast, the bound human erythrocyte membrane enzyme did not exhibit sigmoid saturation curves and K+ activation was not affected by ATP (142). 

Figure 4-51 shows the effect of different interaction factors on the velocity curve of an allosteric tetramer. As the interaction factors decrease (i.e., as the cooperativity increases), the curves become more sigmoidal and [S]o.s decreases. 

Fipire 4- 4 According to the concerted-symmetry model, an allosteric inhibitor binds preferendaliy to the T form. This causes the velocity curve to become more sigmoidal tvith a higher [S]gj. An allosteric activator mimics the substrate by binding preferentially to the R form. As a result, the velocity curve becomes less sigmoidal (hyperbolic at saturating activator) and fS]oi decreases. These observations can also be explained in terms of the sequential interaction model. 

Why do chymotrypsin and ATCase have different velocity curves Chymotrypsin and aspartate transcar-bamoylase exhibit different types of kinetics. Chymotrypsin is a nonallosteric enzyme and exhibits hyperbolic kinetics. ATCase is an allosteric enzyme. It has multiple subunits, and the binding of one molecule of substrate affects the binding of the next molecule of substrate. It exhibits sigmoidal kinetics. 

The Km value of human amidotransferase for PP-ribose-P is 0.25 mM [47]. The intracellular concentrations of PP-ribose-P are 0.005 to 0.05 mfl (erythrocytes and fibroblasts) [49-51]. Thus, intracellular concentrations of PP-ribose-P are in the low range of the substrate-velocity curve, and factors which affect their intracellular, perhaps chiefly intrahepatic, concentrations will have a corresponding influence upon purine biosynthesis de novo. The amidotransferase reaction displays sigmoidal kinetics with respect to PP-ribose-P in the presence of nucleotide regulators such as A1 P and GMP [47]. Small increases in concentrations of PP-ribose-P may therefore affect purine biosynthesis disproportionately. Indeed, in isolated systems such as Ehrlich ascites cells [52] or cultured human fibroblasts [49,53], manipulations of the concentrations of PP-ribose-P profoundly affect the rate of the first steps of purine biosynthesis. 

A lower value of the nonexclusive binding coefficient is associated with a higher cooperativity, and therefore sigmoidicity, of the velocity curves. A lower value of this coefficient implies a decreased affinity of the T state for substrate relative to the R state. If the enzyme in the T state does not bind substrate kj = oo), c = 0. 

Flo. 16. Hyperbolic and sigmoidal inhibition curves. The velocity of the inhibited reaction (v is expressed in percent of the velocity of the uninhibited reaction (v) and is plotted against inhibitor concentration. 

Typical decay curves for various ionic masses are shown in Figure 3b, where the effect of differing thermal velocity is clearly evident. Neglect of such decay characteristics can lead to serious discrimination effects. Note that all the experimental curves have the sigmoid shape... 

Cooperative enzymes show sigmoid or sigmoidal kinetics because the dependence of the initial velocity on the concentration of the substrate is not Michaelis-Menten-like but gives a sigmoid curve (Fig. 8-7). 

If an approximate Km value for the enzyme-substrate combination of interest is known, a full-scale kinetic assay may be done immediately. However, often an approximate value is not known and it is necessary first to do a range finding or suck and see preliminary assay. For such an assay, a concentrated substrate solution is prepared and tenfold serial dilutions of the substrate are made so that a range of substrate concentrations is available within which the experimenter is confident the Km value lies. Initial velocities are determined at each substrate concentration, and data may he plotted either hyperholically (as V versus [S]) or with [S] values expressed as logio values. In the latter case, a sigmoidal curve is fitted to data with a three parameter logistic equation (O Eq. 4) ... 

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. 

Regulatory enzymes are usually identified by the deviation of their kinetics from Michaelis-Menten kinetics plots of velocity versus substrate concentration can be a sigmoidal curve or a modified hyperbola [Fig. 9-7(o)]. If these curves are plotted in the double-reciprocal (Lineweaver-Burk) form, nonlinear graphs are obtained [Fig. 9-7(6)]. 

In this experiment a sigmoid dependence of reaction velocity on substrate concentration was interpreted as evidence for cooperativity of binding of the substrate to the enzyme. What kinds of experiments would be necessary to confirm this proposal Describe other possible situations, especially in a crude enzyme preparation, that would result in such a sigmoid curve that is, what kinds of artifacts might lead you to conclude falsely that a sigmoid curve results from cooperativity in substrate binding ... 

With a constant gas velocity, the mass transfer coefficient kg does not greatly depend on temperature, so that NA will be practically constant for a certain value of Ca and at high temperatures. Accordingly, the left-hand term of Equation 4.10, i.e. the chemical heat production rate per unit external area depends on Tt in the manner indicated in Figure 4.3, curves la and lb. The equation of these sigmoid curves is given by... 

Figure 4 51 Effect of the interaction factors on the sigmoidicity and [SJti.5 of a four-site enzyme. Curve A a = = c = 0.05. Curve B a = b c = 0.1. Curve C a = b = c 0.2. Curves A, B, and C were calculated using the fuU velocity equation. Curves A B and C were calculated from the corresponding Hill equation (i.e., only the terms corresponding to E and ES4 are taken into account).

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. 

Sigmoidal curves are generated by plots of the velocity versus the substrate concentration. 

D. Without ADP, the curve is sigmoidal, so Michaelis-Menten kinetics are not exhibited. Vm is the same at all ADP concentrations shown. The substrate concentration at V2 Vm decreases as the ADP concentration increases therefore, ADP decreases the Km, activating the enzyme. (The velocity is higher at lower substrate concentrations in the presence of ADP.)... 

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