Hill-shaped curves

The bell-shaped curve is of particular interest in the analysis of how structural changes in a protein affect L. The Hill constants of a wide series of modified and mutant hemoglobins fit such a curve (Figure 10.9).19... 

The hidden units of a radial basis function network are not the same as used for a multilayer perception, and the weights between input and hidden layer have different meanings. Transfer functions typically used include the Gaussian function, spline functions and various quadratic functions they all are smooth functions, which taper off as distance from a center point increases. In two dimensions, the Gaussian is the well-known bell-shaped curve in three dimensions it forms a hill. 

Analogously, a plot of log[y/(l -y)] vs. log[B], known as the Hill plot, where y is the degree of saturation (r/m), is linear with unit slope in the absence of cooperativity (Eq. 5), whereas the presence of cooperativity is evidenced by two lines of unit slope connected by an S-shaped curve. The value of the slope... 

A large family of enzymes that deviate from hyperbolic kinetics (Michaelis) is the allosteric enzymes. These enzymes contain two or more topologically distinct binding sites that interact functionally with each other. Most commonly, sigmoidal or S-shaped curves are obtained, being indicative of positive substrate cooperativity. The reaction rate for these enzymes can be calculated by the Hill equation ... 

The intercept on the adsorption axis, and also the value of c, diminishes as the amount of retained nonane increases (Table 4.7). The very high value of c (>10 ) for the starting material could in principle be explained by adsorption either in micropores or on active sites such as exposed Ti cations produced by dehydration but, as shown in earlier work, the latter kind of adsorption would result in isotherms of quite different shape, and can be ruled out. The negative intercept obtained with the 25°C-outgassed sample (Fig. 4.14 curve (D)) is a mathematical consequence of the reduced adsorption at low relative pressure which in expressed in the low c-value (c = 13). It is most probably accounted for by the presence of adsorbed nonane on the external surface which was not removed at 25°C but only at I50°C. (The Frenkel-Halsey-Hill exponent (p. 90) for the multilayer region of the 25°C-outgassed sample was only 1 -9 as compared with 2-61 for the standard rutile, and 2-38 for the 150°C-outgassed sample). 

Where large samples of reactant are used and/or where C02 withdrawal is not rapid or complete, the rates of calcite decomposition can be controlled by the rate of heat transfer [748] or C02 removal [749], Draper [748] has shown that the shapes of a—time curves can be altered by varying the reactant geometry and supply of heat to the reactant mass. Under the conditions used, heat flow, rather than product escape, was identified as rate-limiting. Using large ( 100 g) samples, Hills [749] concluded that the reaction rate was controlled by both the diffusion of heat to the interface and C02 from it. The proposed models were consistent with independently measured values of the transport parameters [750—752] whether these results are transfenable to small samples is questionable. 

Double torsion test specimens take the form of rectangular plates with a sharp groove cut down the centre to eliminate crack shape corrections. An initiating notch is cut into one end of each specimen (Hill Wilson, 1988) and the specimen is then tested on two parallel rollers. A load is applied at a constant rate across the slot by two small balls. In essence the test piece is subjected to a four-point bend test and the crack is propagated along the groove. The crack front is found to be curved. 

The shape of curve is modulated by varying the Hill coefficient to fit the observed behavior. Sensitivity of a drug is how concentration translates into effect as described by the shape coefficient or the Hill coefficient at a value of 1, it is a classical parabola. Eg is... 

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. 

Figure 19.7. Competition curves for two compounds versus a known radioligand. (Top) These data represent the competition of two compounds with a known radioligand (in this case a radioligand that labels the dopamine D receptor, a member of the G protein-coupled superfamily). It is important to note not only the left-right difference between Compound A and Compound B, but also the difference in the shape of their competition curves. (Bottom) A Hill plot [based on Eq. (19.20)] of the competition curves shown in the top figure provides two pieces of data. First, the slopes of the lines are different (Compound A = -1.0 Compound B = -0.6), which has important mechanistic meaning that is discussed in Section 19.3.4b. Hill plots also allow more precise estimation of IC50s. By definition, at 50% inhibition, the Hill coefficient is 0. As shown, one can estimate the IC50 for each compound from this plot.

FIGURE 6.21.3 Plant species in a new area uncovered as a glacier receded. The shape of the curve is reminiscent of an exponential shape. (From Molles, M.C. Jr., Ecology Concepts and Applications, McGraw-Hill, New York, 1999. With permisssion.)... 

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. 

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