Enzyme hyperbolic kinetics

Mathematically, the Michaelis-Menten equation is the equation of a rectangular hyperbola. Sometimes you ll here reference to hyperbolic kinetics, this means it follows the Michaelis-Menten equation. A number of other names also imply that a particular enzyme obeys the Michaelis-Menten equation Michaelis-Menten behavior, saturation kinetics, and hyperbolic kinetics. 

It has been demonstrated that lin-log equations capture hyperbolic kinetics slightly better than either a linear approximation or the power-law approach [318, 320 322]. One reason for the improved performance is that for lin-log kinetics the elasticities (and kinetic orders) are not constant, but change with changing metabolite concentrations. For a monosubstrate reaction v(Sj, and omitting the dependence on the enzyme concentration, we obtain... 

Referring to an enzyme whose kinetic properties do not yield hyperbolic saturation curves in plots of the initial rate as a function of the substrate concentration. 

Hexokinase catalyzes the phosphorylation of glucose and fructose by ATP. However the Km for glucose is 0.13 mmol L-1, whereas that for fructose is 1.3 mmol L-1. Assume is the same for both glucose and fructose and the enzyme displays hyperbolic kinetics [Eq. (9.5)]. (a) Calculate the normalized initial velocity of the reaction (i.e., V(/Umax) for each substrate when [S]0 = 0.13, 1.3, and 13.0 mmol L-1. (b) For which substrate does hexokinase have the greater affinity ... 

In contrast with Michaelian enzymes, which have hyperbolic kinetics, allosteric enzymes, thanks to their sigmoidal kinetics, possess an enhanced sensitivity towards variations in the concentration of an effector or of the substrate. This is the reason why many enzymes that play an important role in the control of metabolism are of the allosteric type. 

Why do chymotrypsin and ATCase have different velocity curves Chymotrypsin and aspartate transcar-bamoylase exhibit different types of kinetics. Chymotrypsin is a nonallosteric enzyme and exhibits hyperbolic kinetics. ATCase is an allosteric enzyme. It has multiple subunits, and the binding of one molecule of substrate affects the binding of the next molecule of substrate. It exhibits sigmoidal kinetics. 

Allosteric enzymes display sigmoidal kinetics when rates are plotted versus substrate concentration. Michaelis-Menten enzymes exhibit hyperbolic kinetics. Allosteric enzymes usually have multiple subunits, and the binding of substrates or effector molecules to one subunit changes the binding behavior of the other subunits. 

Michaelis-Menten (M-M) hyperbolic kinetics (Eq. 13.7) is often assumed and directly applied in rate determination, particularly in early discovery when the enzyme kinetic behavior is usually unknown ... 

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 ... 

Since significant meaning is placed on these measured constants and parameters, it is important that they be determined accurately and unambiguously. It is also important that the reader or practitioner in the field of enzymology be able to assess if the measurement of these parameters is reliable. Furthermore, since enzyme behavior is often modeled as Michaelis-Menten (hyperbolic) kinetics, it seems reasonable that interpretations of observations should be made in the context of the Michaelis-Menten model. In some cases, alternative explanations for enzyme kinetic behavior may be appropriate and one may be inclined to select one interpretation over another (preferably based on a kinetic analysis, although too often this is done on intuition). 

To make an appropriate assessment of the pattern of inhibition, one need only compare the pattern of reaction velocity versus [S] observed relative to the pattern predicted from an application of the hyperbolic kinetics model. This requires making an estimate of V ax and from the data available. Transforming the original data to a Lineweaver-Burke plot (despite the aforementioned limitations) indicates that only four data points (at low [S]) can be used to estimate Vmax and Km (as 3.58 units and 0.48 mM, respectively. Fig. 14.10). The predicted (uninhibited) behavior of the enzyme activity can now be calculated by applying the rectangular hyperbola [Eq. (14.5)] (yielding the upper curve in Fig. 14.11), and it becomes clear that inhibition was obvious at [S] <1 mM. The degree of inhibition is expressed appropriately as the difference between observed and predicted activity at any [S] value, if one makes interpretations within the context of the Michaelis-Menten model. 

In this situation, two different polypeptide chains interact to form a new specific complex, whose biologic activity can be modified by ligands that bind to the precursor subunits. This type of regulation has been extensively studied for the aspartate transcarbamylase of E. coli (Gerhart, 1970). This enzyme catalyzed the initial step in the synthesis of cytidine nucleotides it is allosterically inhibited by CTP and shows positive cooperativity for substrate. It may be dissociated by mercurials into catalytic subunits, which are insensitive to CTP and which exhibit hyperbolic kinetics for substrate, and into regulatory subunits which bind CTP. 

FIGURE 15.8 Sigmoid versus [S] plot. The dotted line represents the hyperbolic plot characteristic of normal Michaelis-Men ten-type enzyme kinetics. 

Different from conventional chemical kinetics, the rates in biochemical reactions networks are usually saturable hyperbolic functions. For an increasing substrate concentration, the rate increases only up to a maximal rate Vm, determined by the turnover number fccat = k2 and the total amount of enzyme Ej. The turnover number ca( measures the number of catalytic events per seconds per enzyme, which can be more than 1000 substrate molecules per second for a large number of enzymes. The constant Km is a measure of the affinity of the enzyme for the substrate, and corresponds to the concentration of S at which the reaction rate equals half the maximal rate. For S most active sites are not occupied. For S >> Km, there is an excess of substrate, that is, the active sites of the enzymes are saturated with substrate. The ratio kc.AJ Km is a measure for the efficiency of an enzyme. In the extreme case, almost every collision between substrate and enzyme leads to product formation (low Km, high fccat). In this case the enzyme is limited by diffusion only, with an upper limit of cat /Km 108 — 109M. v 1. The ratio kc.MJKm can be used to test the rapid... 

The necessity of developing approximate kinetics is unclear. It is sometimes argued that uncertainties in precise enzyme mechanisms and kinetic parameters requires the use of approximate schemes. However, while kinetic parameters are indeed often unknown, the typical functional form of generic rate equations, namely a hyperbolic Michaelis Menten-type function, is widely accepted. Thus, rather than introducing ad hoc functions, approximate Michaelis Menten kinetics can be utilized an approach that is briefly elaborated below. 

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. 

A linear reciprocal transformation of a function of the form [f(a) = a/(l + a)], such that l/f(a) is plotted on the vertical axis and 1/a is plotted on the horizontal axis. In the case of one-substrate enzyme kinetics, the hyperbolic function is ... 

Competitive, 249, 123, 146, 190 [partial, 249, 124 progress curve equations for, 249, 176, 180 for three-substrate systems, 249, 133, 136] competitive-uncompetitive, 249, 138 concave-up hyperbolic, 249, 143 dead-end, 249, 124 [for bireactant kinetic mechanism determination, 249, 130-133 definition of kinetic constants, 249, 220-221 effects on enzyme progress curves, nonlinear regression analysis, 249, 71-72 inhibition constant evaluation, 249, 134-135 kinetic analysis with, 249, 123-143 one-substrate systems, 249, 124-126 unireactant systems, theory,... 

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-... 

Decarboxylase reaction Kinetic constants The optimum pH of the decarboxylase reaction was determined with the natural substrates of both enzymes, pyruvate (PDC) and benzoylformate (BFD). Both enzymes show a pH optimum at pH 6.0-6.5 for the decarboxylation reaction [4, 5] and investigation of the kinetic parameters gave hyperbolic v/[S] plots. The kinetic constants are given in Table 2.2.3.1. The catalytic activity of both enzymes increases with the temperature up to about 60 °C. From these data activation energies of 34 kj moT (PDC) and 38 kJ mol (BFD) were calculated using the Arrhenius equation [4, 6-8]. 

Additional information <2, 3> (<2> the concentration of glutamate which yields half-maximal activity is 33 mM for y-glutamyl kinase DHPr, and 37mM for y-glutamyl kinase w-l-, no typical Michealis-menten kinetics [2] <3> plots of the enzyme activity as a function of ATP concentration are non-hyperbolic [5]) [2, 5]... 

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 =...

It is important to distinguish between the Michaelis-Menten equation and the specific kinetic mechanism on which it was originally based. The equation describes the kinetic behavior of a great many enzymes, and all enzymes that exhibit a hyperbolic dependence of V0 on [S] are said to follow Michaelis-Menten kinetics. The practical rule that... 

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). 

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