Hyperbolic curves

This is shown graphically by the (Vm, p, T) surface given in Figure 1.3. The three variables p, Vm, and T exist together only on this surface. Cutting through the surface with isothermal (constant T) planes generates hyperbolic curves... 

The logarithmic form of Eq. (25) is a hyperbolic curve, just like Eq. (24), with an apparent pK associated with the pH at half-slope positions (cf Fig. 3.4, solid-line curves). 

Usually, one plots the initial rate V against the initial amount X, which produces a hyperbolic curve, such as shown in Fig. 39.17a. The rate and amount at time 0 are larger than those at any later time. Hence, the effect of experimental error and of possible deviation from the proposed model are minimal when the initial values are used. The Michaelis-Menten equation can be linearized by taking reciprocals on both sides of eq. (39.114) (Section 8.2.13), which leads to the so-called Lineweaver-Burk form ... 

After the usual corrections for analyte impurity, potential drop in the solution and volume increase during titration, the experimental results were in perfect agreement with the theoretical hyperbolic curve. 

The Roche investigators were able to relate their measured fluxes to human absorption values with a hyperbolic curve, much like that indicated in Caco-2... 

These results show that if the relationship between the concentration of an agonist and the proportion of receptors that it occupies is measured directly (e.g., using a radioligand binding method), the outcome should be a simple hyperbolic curve. Although the curve is describable by the Hill-Langmuir equation, the dissociation equilibrium constant for the binding will be not KA but Ke, which is determined by both E and KA. 

Experimental values of H, obtained from equation (4.46) and plotted against the rate of flow of mobile phase for a given solute and set of conditions, produce a hyperbolic curve showing an optimum flow rate for maximum efficiency (Figure 4.15). The position of the maximum varies with the solute, and a family of curves can be derived for the components of a mixture. The most efficient flow rate for a particular sample is, therefore, a matter of compromise. The equation also indicates that the highest... 

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 ... 

Velocity data may be plotted in any one of a number of ways to illustrate the relation between v and [S], and plotting procedures are detailed later in this chapter. However, the Michaelis-Menten equation is an equation for a rectangular hyperbola, and plotting v versus [S] yields a hyperbolic curve, in the absence of cooperative or other unusual behaviors (O Figure 4-3). 

Under many circumstances, the behavior of a simple unireactant enzyme system cannot be described by the Michaelis-Menten equation, although a v versus [S] plot is still hyperbolic and can be described by a modified version of the equation. For example, as will be discussed later, when enzyme activity is measured in the presence of a competitive inhibitor, hyperbolic curve fitting with the Michaelis-Menten equation yields a perfectly acceptable hyperbola, but with a value for Km which is apparently different from that in the control curve O Figure 4-7). Of course, neither the affinity of the substrate for the active site nor the turnover number for that substrate is actually altered by the presence of a competitive... 

Hyperbolic curve fits to control enzymatic data and to data obtained in the presence of a competitive inhibitor. Curve fitting to the Michaelis-Menten equation results in two different values for Km- However, Km does not, in actuality, change, and the value in the presence of inhibitor (15 uiM) is an apparent value. Fitting with the correct equation, that for turnover in the presence of a competitive inhibitor ( Eq. 5), results in plots identical in appearance to those obtained with the Michaelis-Menten equation. However, nonlinear regression now reveals that Km remains constant at 5 ulM and that [l]/Ki = 2.5 with knowledge of [/], calculation of K is straightforward... 

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. 

With knowledge of iCi, a and p for a reversible inhibitor, along with Km and Vmax values for the substrate of interest, it is possible to fit hyperbolic curves to multiple data sets by applying the appropriate inhibitor equation. It is a relatively straightforward task to enter such equations into graphing programs such as... 

If there is a nonlinear dependence of kapp on [Lq] then the reaction cannot be a simple bimolecular process. However, the extended equation described inO Section 2.6.1 will give rise to a hyperbolic curve as shown in Figure 10-5b, where Ki is given by k i/ki, and kapp is described by ... 

Figure 3.7 First and zero order components of a hyperbolic curve of enzyme activity against substrate concentration. For the first order, rate is directly proportional to concentration at zero order, rate is independent or almost independent of substrate concentration.

A hyperbolic curve results when v is plotted against the concentration of S, where [S] indicates concentration of S and a and b are constants. Two questions arise ... 

The values for these constants can be obtained from the hyperbolic curve when v is plotted against the concentration of S this method is not very accurate unless calculated using a computer program but the most commonly used... 

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-... 

The dose-dependent saturation of a constant amount of receptor is described by sigmoid or hyperbolic curve shape. If the measuring signal is plotted against ligand concentration, the curves show a minimal signal (blank, lower plateau) and an upper plateau (Bmax)-The equation for the hyperbola is... 

This is the equation for the hyperbolic curve shown in Fig. Hemoglobin with its four subunits has more complex behaviour it approximately follows the equation... 

Even in intact animals or patients, responses to low doses of a drug usually increase in direct proportion to dose. As doses increase, however, the response increment diminishes finally, doses may be reached at which no further increase in response can be achieved. In idealized or in vitro systems, the relation between drug concentration and effect is described by a hyperbolic curve (Figure 2-1A) according to the following equation ... 

Fig. 2. Hyperbolic curves of the effectiveness of the seme alternative rewards, available at the same times as in Fig. 1. Note that predicted preference changes between time A and time B. As delays become minuscule, effectiveness becomes enormous, making objective size unimportant.

Fig. 4. Two consumption patterns of reward over time (solid lines), and the aggregate value of all remaining moments in each, discounted in hyperbolic curves (broken lines) (A) Rapid, intense consumption (B) more gradual consumption (C) the two patterns compared.

Fig. 5. Hyperbolic curves of the effectiveness of three rewards A, if each reward is an exclusive alternative to each other reward and B, if the earliest reward predudes the middle one and the middle one precludes the last one, but the earliest one does not preclude the last one. The summed effect of the earliest and last rewards is depicted by the dashed curve. Note that the curve from the middle reward dominates that from each other reward at some time before the first reward is available in contingency A, but in contingency never rises above the summed curve from the first and last rewards during the time before the first reward is available.

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). 

When v0 is plotted against [S], it is not always possible to determine when Vmax has been achieved, because of the gradual upward slope of the hyperbolic curve at high substrate concentrations. However, if 1A/0 is plotted versus 1/[S], a straight line is obtained (Figure 5.11). This plot, the Lineweaver-Burke plot (also called a double-reciprocal plot) can be used to calculate Km and Vmax> as well as to determine the mechanism of action of enzyme inhibitors. 

---