Enzymes curvature

The hallmark of slow binding inhibition is that the degree of inhibition at a fixed concentration of compound will vary over time, as equilibrium is slowly established between the free and enzyme-bound forms of the compound. Often the establishment of enzyme-inhibitor equilibrium is manifested over the time course of the enzyme activity assay, and this leads to a curvature of the reaction progress curve over a time scale where the uninhibited reaction progress curve is linear. We saw... 

Figure 7.6 Double reciprocal plot for a tight binding competitive enzyme inhibitor, demonstrating the curvature of such plots. The dashed lines represent an attempt to fit the data at lower substrate concentrations to linear equations. This highlights how double reciprocal plots for tight binding inhibitors can be misleading, especially when data are collected only over a limited range of substrate concentrations.

As noted by the original authors (Dorovska et al., 1972), and cited by Fersht (1985), there is an excellent linear correlation between log/ccat/KM and the Hansch hydrophobicity parameters (v) of the side chains (Fig. 9, A), except for the two branched side chains (valine and isoleucine residues). However, since the ku values for the esters do vary somewhat (Table A6.8), the values of pKrs do not correlate as strongly with ir (Fig. 9, B). Moreover, the plot shows distinct curvature which probably indicates the onset of a saturation effect due to the physical limits of the Sj binding pocket, adjacent to the enzyme s active site. Still, the points for valine and isoleucine deviate below the others, suggesting that the pocket has a relatively narrow opening. 

The Bronsted relationship can be strictly accurate only over a certain range of acid and base strengths. When has diffusion-controlled values, which of course cannot be exceeded, the linear plot of log k/ y vs log must level off to a zero slope, that is a = 0. As well as being reported, although rarely, in simple metal complexes, the resultant curvature in the Bronsted plot is also shown by the zinc enzyme carbonic anhydrase (Chap. 8. Zn(II)). In... 

In standard kinetic studies, the initial velocity (v) should be directly proportional to the total enzyme concentration. Indeed, this plot should be one of the first items analyzed in the kinetic characterization of an enzyme. The enzyme concentration range in this plot should span from below to above the concentration to be used in the inital rate studies (and, ideally, the line should go through the origin). When curvature is observed in such plots, there are several potential explanations . ... 

A toxic impurity present in the reaction mixture (but not in the stock enzyme solution) may result in upward curvature at high concentrations of enzyme, a larger amount of enzyme remains active. 

A dissociable activator (or coenzyme) present in the enzyme solution may result in upward curvature as the enzyme concentration increases, so also does the concentration of the activator. 

Upward curvature may occur when the enzyme undergoes self-association or aggregation into a more active form such a process is favored at elevated concentrations. 

If the enzyme under consideration is assayed via a coupling enzyme(s) system, downward curvature in the plot will be observed when the primary enzyme fails to limit the observed velocity. In such instances, it would be necessary to increase the amount of coupling enzymes such that they remain in excess under aU conditions. This problem is often encountered in manometric assays, where the rate of diffusion of gas into the liquid begins to limit the enzyme reaction rate. 

One of the basic assumptions in kinetic studies of an enzyme-catalyzed reaction is that true initial rates are being measured. In such cases, a plot of the product concentration versus time must yield a straight line. (This behavior is only observed when the substrate is at or near its initial (or, r = 0) concentration. As time increases, product accumulation and substrate depletion will result in a curvature of this progress curve hence, the reaction velocity at these later times would be correspondingly lower.)... 

A graphical procedure for analyzing effects of two competitive inhibitors at the active site of an enzyme Vq/vi is plotted as a function of ([Ii] + [I2]) where Vq is the initial-rate velocity in the absence of an inhibitor(s), Vi is the velocity in the presence of the inhibitor(s), and [ii] and [I2] are the two inhibitor concentrations . If there are no interactions between the two inhibitors, then a straight line will be obtained if there is an interaction at the active site, curvature will be observed. See Yonetani-Theorell Treatment... 

NPs have physical dimensions close to cell membrane receptors and other biomolecules. This opens a new scenario to be explored. For example, the small dimensions are responsible for an enhanced penetrability of the cell membrane, since endocytosis is favored. The interaction with proteins is also affected by dimensions differences in surface curvature influence the ability of proteins to interact with surface functionalities, and consequently possible differences in conformational modification may occur.13 This is particularly relevant in the case of enzymes, where activity may depend on conformation. 

Notice that the line curves down from the right in Fig. 9-20. This is identical to the effect seen in negative cooperativity [see Fig. 9-19(c)[. In other words, this curvature is not diagnostic of control enzymes but can arise simply from a mixture of isoenzymes. 

In many situations, quadratic terms are important. This allows curvature, and is one way of obtaining a maximum or minimum. Most chemical reactions have an optimum performance at a particular pH, for example. Almost all enzymic reactions work in this way. Quadratic terms balance out the linear terms. 

Thus the initial phase of the [P] as a function of t is an increasing function with negative curvature. However, for certain enzyme kinetics, [P] as a function of t has a positive curvature, a lag, in its initial phase. This phenomenon is known as hysteresis, first discovered by Carl Frieden [60],... 

A biologic reason for the abundance of nonlamellar lipids in membranes is that they possess the ability to modulate the activities of membrane proteins (15, 16). It has been recognized that membranes exist in a state of curvature frustration, which may be sufficiently large to have significant effect on certain protein conformations (17). Many examples show that the lipid bilayer elastic curvature stress indeed couples to conformational changes of membrane proteins (15, 18, 19). Protein kinase C is one such example of an enzyme activated by lipids that exhibit a propensity for nonlamellar phase formation (20). The activity of Ca " -ATPase from sarcoplasmic reticulum membranes also strongly correlates with the occurrence of nonbilayer lipids in the membrane and increases with the increase of their amount. It is noteworthy that the protein activity does not depend on the chemical structure of the lipids but only on their phase propensity thus specific binding interactions are ruled out. The list of proteins with activities that depend on the phase properties... 

The Henri-Michaelis-Menten equation describes the curve obtained when initial velocity is plotted versus substrate concentration. The curve shown in Figure 4-7 is a right rectangular hyperbola with limits of and - K . The curvature is fixed regardless of the values of and V mxx- Consequently, the ratio of substrate concentrations for any two fractions of Vj m is constant for all enzymes that obey Henri-Michaelis-Menten kinetics. For example, the ratio of substrate required for 90% of Vmat to the substrate required for... 

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