Hyperbolic relationship

Assuming a hyperbolic relationship between response and the amount of agonist-receptor complex, response is defined as... 

Though this looks complicated, it still predicts a simple hyperbolic relationship (as with the Hill-Langmuir equation see Figure 1.1 and the accompanying text) between agonist concentration and the proportion of receptors in the state (AR G ) that leads to a response. If a very large concentration of A is applied, so that all the receptors are occupied, the value of pAR.G. asymptotes to ... 

A plot of the initial reaction rate, v, as a function of the substrate concentration [S], shows a hyperbolic relationship (Figure 4). As the [S] becomes very large and the enzyme is saturated with the substrate, the reaction rate will not increase indefinitely but, for a fixed amount of [E], it reaches a plateau at a limiting value named the maximal velocity (vmax). This behavior can be explained using the equilibrium model of Michaelis-Menten (1913) or the steady-state model of Briggs and Haldane (1926). The first one is based on the assumption that the rate of breakdown of the ES complex to yield the product is much slower that the dissociation of ES. This means that k2 tj. 

The relationship between substrate concentration ([S]) and reaction velocity (v, equivalent to the degree of binding of substrate to the active site) is, in the absence of cooperativity, usually hyperbolic in nature, with binding behavior complying with the law of mass action. However, the equation describing the hyperbolic relationship between v and [S] can be simple or complex, depending on the enzyme, the identity of the substrate, and the reaction conditions. Quantitative analyses of these v versus [S] relationships are referred to as enzyme kinetics. 

Figure 22 displays the time-integrated mean of mass flux as function of standard volume flux of primary air for all the three wood fuels, respectively. As indicated by Figure 22, the time-integrated mean of mass flux of conversion gas exhibits a hyperbolic relationship with the volume flux of primary air. In the low range of volume fluxes the conversion gas rate increases up to a maximum. After the maximum point is passed, the mass flux of conversion gas decreases due to convective cooling of the conversion reaction. 

What is the mechanism of catalysis that accounts for the hyperbolic relationship ... 

The law of mass action describes the hyperbolic relationship between binding (B) and ligand concentration (c). This relationship is characterized by the drug s affinity (1/Itotal number of binding sites per unit of weight of membrane homogenate. 

The hyperbolic relationship be tween plasma concentration and effect explains why the time course of the effect, unlike that of the plasma concentration, cannot be described in terms of a simple exponential function. A half-life can be given for the processes of drug absorption and elimination, hence for the change in plasma levels, but generally not for the onset or decline of the effect... 

K,y can then be obtained from a plot of v against [D] if the receptor concentration is constant. This is, of course, the same as the direct plot of enzyme activity shown in every biochemistry textbook. As with all hyperbolic relationships, there are several drawbacks to this technique many data points are needed at the beginning of the curve, at low [D] values, where accuracy is limited. Also, determination of the maximum effect is almost impossible, since we are dealing with an asymptotic curve. 

Fig. 7.90. Illustration of the hyperbolic relationship that exists between the square root of the transition time and the 000-6111 density.

Figure 17.16 Relationships of biodegradation rate, v, to substrate concentration, [/], when Michaelis-Menten enzyme kinetics is appropriate (a) when plotted as hyperbolic relationship (Eq. 17-79 in text), or (b) when plotted as inverse equation, Vv =...

Substrate concentration is yet another variable that must be clearly defined. The hyperbolic relationship between substrate concentration ([S ) and reaction velocity, for simple enzyme-based systems, is well known (Figure C1.1.1). At very low substrate concentrations ([S] ATm), there is a linear first-order dependence of reaction velocity on substrate concentration. At very high substrate concentrations ([S] A m), the reaction velocity is essentially independent of substrate concentration. Reaction velocities at intermediate substrate concentrations ([S] A"m) are mixed-order with respect to the concentration of substrate. If an assay is based on initial velocity measurements, then the defined substrate concentration may fall within any of these ranges and still provide a quantitative estimate of total enzyme activity (see Equation Cl. 1.5). The essential point is that a single substrate concentration must be used for all calibration and test-sample assays. In most cases, assays are designed such that [S] A m, where small deviations in substrate concentration will have a minimal effect on reaction rate, and where accurate initial velocity measurements are typically easier to obtain. 

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. 

Similarly, Figure 6b summarizes conductivity results. In contrast with pH, only conductivity measured in the first fractions (up to L/S 0.5 for BA and L/S 2 for 2SL) was of the same order of magnitude as that observed in the prerequisite study (Tab. 5). Moreover, conductivity measured in BA leachates, as well as in 2SL leachates, depicted a hyperbolic relationship with L/S ratio and showed marked... 

It should be noted that the mechanism depicted in Scheme 1 is the simplest that is consistent with mechanism-based inhibition. The mechanism for a given inhibitor and enzyme may be considerably more complex due to (a) multiple intermediates [e.g., MIC formation often involves four or more intermediates (29)], (b) detectable metabolite that may be produced from more than one intermediate, and (c) the fact that enzyme-inhibitor complex may produce a metabolite that is mechanistically unrelated to the inactivation pathway. Events such as these will necessitate alternate definitions for Z inact, Kh and r in terms of the microrate constants of the appropriate model. The hyperbolic relationship between rate of inactivation and inhibitor concentration will, however, remain, unless nonhyperbolic kinetics characterize this interaction. Silverman discussed this possibility from the perspective of an allosteric interaction between inhibitor and enzyme (5). Nonhyperbolic kinetics has been observed for the interaction of several drugs with members of the CYPs (30). 

Gaertner and Dhurjati (1993) used the initial concentration of base medium (B) in an attempt to handle the absence of information concerning the limiting substrate (Equation 45). The base medium corresponded to a DMEM formulation without GLC, GLN, NaCl, and NaHCOj Different concentrations of this base were tested. The solution showed a hyperbolic relationship between the specific growth rate and the basal medium concentration, independently of the effective limiting component. 

A close inspection of the Caco-2 data in Fig. 4.9a suggests a hyperbolic relationship, with log Papp values leveling off at about —5. Similar log Papp curve shapes have been observed elsewhere [41], and are thought to arise as a consequence of the unstirred water layer phenomenon. 

In eqns.(3.21) and (3.22) i denotes the solute and n and n +1 the preceding and the following n-alkane, respectively (see figure 2.2). It is seen that there is a hyperbolic relationship between the retention index and the temperature Although over small sections of the hyperbola a linear approximation is often used, this is not a sound basis for temperature optimization, especially not since a straight line can easily be obtained by plotting In (k/T) vs. 1/T(eqn.3.10). 

Fig. 9-1 The hyperbolic relationship between initial velocity (u0) and initial substrate concentration ([S]0) of an enzyme-catalyzed reaction.

Figure 8-19. Idealized hyperbolic relationship between the photosynthetic photon flux incident on the upper leaf surface and the net C02 uptake rate for a C3 plant. The intercept on the ordinate (y-axis) indicates the net COz flux by respiration in the dark (-1 pmol m-2 s 1), the intercept on the dashed line indicates the light compensation point (a PPF of 15 pmol m 2s l), the essentially linear initial slope (37co2 ppf) indicates the quantum yield (Eq. 4.16) for photosynthesis [(5 - 0 pmol m 2 s l)/(115 -15 pmol m-2 s l) = 0.05 mol C02/mol PPF], and the maximum Jco2reached asymptotically at high PPF indicates the light-saturated net C02 uptake rate (about 12 (xrnol m-2 s l often designated AmaK or Amax). Here the quantum yield is based on incident photons, but more appropriately it should be based on absorbed photons.

Light is a key environmental factor having an impact on nearly aU phytoplankton physiological processes N uptake is no different. Nutrient uptake versus irradiance curves often show a hyperbolic relationship that have been mathematically fit by... 

Vroax. We are not dealing ivith a linear relationship but, instead, a hyperbolic relationship. 

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