Velocity/substrate concentration curves

The fact that so many transport systems (cf. Figs. 1-3) display the same characteristic form for the dependence of velocity on substrate concentration as do enzyme systems strongly suggests that a formal analysis of transport kinetics along the lines of that of enzyme kinetics might be valuable. The simple Michaelis-Menten or hyperbolic velocity vs. substrate concentration curve for enzymes has traditionally been interpreted as arising from the combination between enzyme and substrate, with the subsequent breakdown of this complex to product and free enzyme. One writes... 

Enzyme kinetics velocity versus substrate concentration curve... 

Figure 30.2 shows an initial velocity versus substrate concentration curve. The reaction velocity (v) increases in proportion to increasing concentration of substrate [5] until aU the catalytic sites of the enzyme are working as fast as they can and maximum reaction velocity (V )... 

The general shape of a velocity versus substrate concentration curve is that of a rectangular h5q>erbola (Fig. 3.4). At low substrate concentrations, the rate of the reaction is proportional to substrate concentration. In... 

Figure 8.4. (a) Simulation of the effects of varying the HUl exponent (n) on the shape of the initial velocity versus substrate concentration curve for a coopantive enzyme. (b) Simulation of the effects of varying the Hill constant (it ) on the shape of the initial velocity versus substrate concentration curve for a coopraative enzyme. 

FIGURE 14.7 Substrate saturation curve for au euzyme-catalyzed reaction. The amount of enzyme is constant, and the velocity of the reaction is determined at various substrate concentrations. The reaction rate, v, as a function of [S] is described by a rectangular hyperbola. At very high [S], v= Fnax- That is, the velocity is limited only by conditions (temperature, pH, ionic strength) and by the amount of enzyme present becomes independent of [S]. Such a condition is termed zero-order kinetics. Under zero-order conditions, velocity is directly dependent on [enzyme]. The H9O molecule provides a rough guide to scale. The substrate is bound at the active site of the enzyme. 

These practical approaches are by no means mutually exclusive, and attempts should be made to combine as many of these as possible to improve ones ability to experimentally measure the K-pp of tight binding inhibitors. Thus one should always work at the lowest enzyme concentration possible, and drive the substrate concentration as high as possible, when dealing with competitive inhibitors. A long preincubation step should be used before activity measurements, or the progress curves should be fitted to Equation (6.2) so that accurate determinations of the steady state velocity at each inhibitor concentration can be obtained. Finally, the concentration-response data should be fitted to Morrison s quadratic equation to obtain good estimates of the value of Arfpp. 

As substrate is consumed, the substrate concentration falls and the reaction may get slower. As product is made, the reaction may slow down if the product is an inhibitor of the enzyme. Some enzymes are unstable and die as you re assaying them. All these things may cause the velocity to change with time. If the velocity is constant with time, the plot of product against time is a straight line however, if velocity changes with time (the slope changes with time), this plot is curved (Fig. 8-2). 

For a FIRST-ORDER REACTION, the velocity decreases as the concentration of substrate decreases as it is converted to product. As a result, a plot of substrate concentration against time is a curved line. 

Probably the most important variable to consider in defining optimal conditions or standard conditions is the initial substrate concentration. Most enzymes show a hyperbolic curve as relation between initial reaction velocity and substrate concentration, well known now as the Michaelis-Menten curve. With increasing substrate concentration (S) the velocity (o) rises asymptotically to a maximum value (V) (Fig. 3), according to the expression ... 

Fig. 3. Relation between substrate concentration and reaction velocity (Mi-chaelis-Menten curve), and reaction velocity and time.

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) ... 

Thereafter, and V ax values for substrate turnover are determined in the absence (controls) and presence of several concentrations of the inhibitor of interest. It is recommended that substrate turnover in the presence of at least four concentrations of inhibitor are examined, at concentrations between 1/3 x IC50 and 4 x IC50. Velocity data are then plotted versus substrate concentration, yielding a control plot and plots at each of the concentrations of inhibitor assessed. Hyperbolic curves are then fitted to data with the Michaelis-Menten equation, or with whichever variation of the Michaelis-Menten equation was found to describe control enzyme behavior most appropriately (see Section 4.1.4 etseq.). In this way, a pattern of changes in Km and Vmax> or both, should become apparent with changing inhibitor concentration. 

A straight line whose perpendicular distance from a curve becomes progressively smaller as the distance from the origin at [0,0] becomes greater. For example, in a plot of velocity versus [Substrate Concentration] for an enzyme-catalyzed reaction, the asymptote reaches the maximal velocity when the enzyme molecules become saturated with substrate. 

An enzyme is said to obey Michaelis-Menten kinetics, if a plot of the initial reaction rate (in which the substrate concentration is in great excess over the total enzyme concentration) versus substrate concentration(s) produces a hyperbolic curve. There should be no cooperativity apparent in the rate-saturation process, and the initial rate behavior should comply with the Michaelis-Menten equation, v = Emax[A]/(7 a + [A]), where v is the initial velocity, [A] is the initial substrate concentration, Umax is the maximum velocity, and is the dissociation constant for the substrate. A, binding to the free enzyme. The original formulation of the Michaelis-Menten treatment assumed a rapid pre-equilibrium of E and S with the central complex EX. However, the steady-state or Briggs-Haldane derivation yields an equation that is iso-... 

Fig. 18. Variation of initial rate constant VJPa for decarboxylation of 6-nitrobenzisoxazole-3-carboxylate as a function of substrate concentration under conditions of excess substrafe. Initial velocities were corrected for the spontaneous hydrolysis of substrate in absence of polymer. Polymer is (C2H4N)m (C 2H25)0 2Sm(C2H5), 75m. The curve was drawn according to (30). with nk2 = 0.458 sec-1, and KM = 8.59 x 10-5 M.

FIGURE 6-11 Effect of substrate concentration on the initial velocity of an enzyme-catalyzed reaction. V max is extrapolated from the plot, because V0 approaches but never quite reaches /max. The substrate concentration at which V0 is half maximal is Km, the Michaelis constant. The concentration of enzyme in an experiment such as this is generally so low that [S] >> [E] even when [S] is described as low or relatively low. The units shown are typical for enzyme-catalyzed reactions and are given only to help illustrate the meaning of V0 and [S]. (Note that the curve describes part of a rectangular hyperbola, with one asymptote at /max. If the curve were continued below [S] = 0, it would approach a vertical asymptote at [S] = — Km.)... 

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). 

For heterotropic allosteric enzymes, those whose modulators are metabolites other than the normal substrate, it is difficult to generalize about the shape of the substrate-saturation curve. An activator may cause the curve to become more nearly hyperbolic, with a decrease in Z0.5 but no change in Fmax, resulting in an increased reaction velocity at a fixed substrate concentration (V0 is higher for any value of [S] Fig. 6-29b, upper curve). 

FIGURE 15-35 Elasticity coefficient, e, of an enzyme with typical Michaelis-Menten kinetics. At substrate concentrations far below the Km, each increase in [S] produces a correspondingly large increase in the reaction velocity, v. For this region of the curve, the enzyme has an elasticity, e, of about 1.0. At [S] Km, increasing [S] has little effect on v s here is close to 0.0. 

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. 

Figure 9-2 shows a plot of velocity against substrate concentration as given by Eq. 9-15. The position of Vmax on the ordinate is marked, but it should be clear that the experimental velocity (v) can never attain Vmax unless [S] is very high relative to Km. The value of v approaches Vmax asymptotically. Since Km is defined as the value of [S] at which v = Vmax/ 2, its value can be estimated from Fig. 9-2. However, Km cannot be determined reliably because of the difficulty of establishing the value of Vmax from a plot of this type. Notice that the curve of Figure 9-2 is identical in form to the saturation curve for reversible binding shown in Fig. 7-1.

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