Michaelis functions

The expressions in parenthesis are termed Michaelis functions of pH. The inverse of such functions represent the molar fraction of the total enzyme species that is in a particular ionic form. As clearly seen, that fractions (e and c ) are dependent on pH (—log h+) and will rise and fall passing through a maximum, since the expression contains terms with h+ in the numerator and the denominator. On the contrary, expressing them in terms of e" and c or e + and c", that fractions will rise or fall monotonically with pH. The second assumption of the Michaelis-Davidsohn hypothesis is then consistent with the experimental behavior, since pH profile of any enzymatic reaction resembles the dependence of c" (and not c" or c" ) with pH. 

A typical amino add, without a charge in its side chain, has a pjRTa = 2 for the carboxylic group, and 9 for the a-amino group. Now, with the aid of Eqs. (174), a titration curve for a t)q)ical amino add can be constmcted, assuming liiat p/Ti = 2, and pKs = 9 (Fig. 8). Figure 8 shows that it is convenient to express the Michaelis functions in the reciprocal form from this, it is evident that the zwitteiion form predominates in the pH region between pA, and pifa-... 

Micha.elAdditions. The reaction of a bismaleimide with a functional nucleophile (diamine, bisthiol, etc) via the Michael addition reaction converts a BMI building block into a polymer. The non stoichiometric reaction of an aromatic diamine with a bismaleimide was used by Rhc )ne Poulenc to synthesize polyaminobismaleimides as shown in Figure 6 (31). 

Figure 11.1 A plot of the reaction rate as a function of the substrate concentration for an enzyme catalyzed reaction. Vmax is the maximal velocity. The Michaelis constant. Km, is the substrate concentration at half Vmax- The rate v is related to the substrate concentration, [S], by the Michaelis-Menten equation ...

Saturation kinetics are also called zero-order kinetics or Michaelis-Menten kinetics. The Michaelis-Menten equation is mainly used to characterize the interactions of enzymes and substrates, but it is also widely applied to characterize the elimination of chemical compounds from the body. The substrate concentration that produces half-maximal velocity of an enzymatic reaction, termed value or Michaelis constant, can be determined experimentally by graphing r/, as a function of substrate concentration, [S]. 

The operational model allows simulation of cellular response from receptor activation. In some cases, there may be cooperative effects in the stimulus-response cascades translating activation of receptor to tissue response. This can cause the resulting concentration-response curve to have a Hill coefficient different from unity. In general, there is a standard method for doing this namely, reexpressing the receptor occupancy and/or activation expression (defined by the particular molecular model of receptor function) in terms of the operational model with Hill coefficient not equal to unity. The operational model utilizes the concentration of response-producing receptor as the substrate for a Michaelis-Menten type of reaction, given as... 

Enzyme kinetics. Data for reactions that follow the Michaelis-Menten equation are sometimes analyzed by a plot of v,/tA]o versus l/[A]o. This treatment is known as an Eadie-Hofstee plot. Following the style of Fig. 4-7b, sketch this function and label its features. 

The Michaelis-Menten rate equation for enzyme reactions is typically written as the rate of formation of product (Eq. 19a). This equation implies that 1/Rate (where rate is the rate of formation of product) depends linearly on the inverse of the substrate concentration [S]. This relation allows KM to be determined. Derive this equation and sketch 1/Rate against 1/[S]. Label the axes, the y-intercept, and the slope with their corresponding functions. 

The functioning of enzymes produces phenomena driving the processes which impart life to an organic system. The principal source of information about an enzyme-catalyzed reaction has been from analyses of the changes produced in concentrations of substrates and products. These observations have led to the construction of models invoking intermediate complexes of ingredients with the enzyme. One example is the Michaelis-Menten model, postulating an... 

Kemp and Waters also found the oxidations of cyclohexanone and of mandelic, malonic and a-hydroxyisobutyric acids by Cr(VI) to be Mn(II)-catalysed. In these cases, as with oxalic acid, the [Cr(VI)] versus time plots are almost linear and the reaction becomes first order in substrate (or involves Michaelis-Menten kinetics), and, except at lowest catalyst concentrations, approximately first order in [Mn(II)]. Detailed examination of the initial rate of oxidation of a-hydroxyrobutyric acid as a function of oxidant concentration revealed, however, that the dependence is... 

On the other hand, the macrolides showed unusual enzymatic reactivity. Lipase PF-catalyzed polymerization of the macrolides proceeded much faster than that of 8-CL. The lipase-catalyzed polymerizability of lactones was quantitatively evaluated by Michaelis-Menten kinetics. For all monomers, linearity was observed in the Hanes-Woolf plot, indicating that the polymerization followed Michaehs-Menten kinetics. The V, (iaotone) and K,ax(iaotone)/ m(iaotone) values increased with the ring size of lactone, whereas the A (iactone) values scarcely changed. These data imply that the enzymatic polymerizability increased as a function of the ring size, and the large enzymatic polymerizability is governed mainly by the reachon rate hut not to the binding abilities, i.e., the reaction process of... 

CA = concentration of feed Cm = a constant (Michaeli s constant that is a function of the reaction and conditions)... 

Fig. 20.1. Correlation between the air/water partition coefficient, Kaw, determined from measurements of the surface pressure as a function of drug concentration (Gibbs adsorption isotherm) in buffer solution (50 mM Tris/HCI, containing 114 mM NaCI) at pH 8.0 and the inverse of the Michaelis Menten constant, Km obtained from phosphate release...

Figure 8 Plot of the initial rate of the enzyme-catalyzed oxidation of 1-phenylpropanol as a function of % ee. The solid line represents a fit of the data to the Michaelis-Menten formalism for competitive inhibition where [S] = [ -(60)] and [ ] = [ -(60)]. The total alcohol concentration was maintained constant at lOmM.100...

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. 

Fig. 1 a and b show how the Andrews and the Michaelis-Menten functions (with the optimal parameter values) respectively fit the experimental data... 

Cyclodextrins as catalysts and enzyme models It has long been known that cyclodextrins may act as elementary models for the catalytic behaviour of enzymes (Breslow, 1971). These hosts, with the assistance of their hydroxyl functions, may exhibit guest specificity, competitive inhibition, and Michaelis-Menten-type kinetics. All these are characteristics of enzyme-catalyzed reactions. 

An exponential function that describes the increase in product during a first-order reaction looks a lot like a hyperbola that is used to describe Michaelis-Menten enzyme kinetics. It s not. Don t get them confused. If you can t keep them separated in your mind, then just forget all that you ve read, jump ship now, and just figure out the Michaelis-Menten description of the velocity of enzyme-catalyzed reaction—it s more important to the beginning biochemistry student anyway. 

The effect of inhibition is to decrease rP relative to rP given by the Michaelis-Menten equation 10.2-9 (c, = 0). The extent of inhibition is a function of cr. 

The effect of non-participating ligands on the copper catalyzed autoxidation of cysteine was studied in the presence of glycylglycine-phosphate and catecholamines, (2-R-)H2C, (epinephrine, R = CH(OH)-CH2-NHCH3 norepinephrine, R = CH(OH)-CH2-NH2 dopamine, R = CH2-CH2-NH2 dopa, R = CH2-CH(COOH)-NH2) by Hanaki and co-workers (68,69). Typically, these reactions followed Michaelis-Menten kinetics and the autoxidation rate displayed a bell-shaped curve as a function of pH. The catecholamines had no kinetic effects under anaerobic conditions, but catalyzed the autoxidation of cysteine in the following order of efficiency epinephrine = norepinephrine > dopamine > dopa. The concentration and pH dependencies of the reaction rate were interpreted by assuming that the redox active species is the [L Cun(RS-)] ternary complex which is formed in a very fast reaction between CunL and cysteine. Thus, the autoxidation occurs at maximum rate when the conditions are optimal for the formation of this species. At relatively low pH, the ternary complex does not form in sufficient concentration. 

Bosma et al. [1] have proposed including the details of extracellular substrate transport in the calculation of whole-cell Michaelis-Menten kinetics. For the situation of a quasi-steady-state (i.e. when the transport flux and the rate of degradation of the substrate are equal) the consumption of substrate by a microorganism is represented as a function of the distant, and effectively unavailable, substrate concentration ca ... 

It is perhaps wise to begin by questioning the conceptual simplicity of the uptake process as described by equation (35) and the assumptions given in Section 6.1.2. As discussed above, the Michaelis constant, Km, is determined by steady-state methods and represents a complex function of many rate constants [114,186,281]. For example, in the presence of a diffusion boundary layer, the apparent Michaelis-Menten constant will be too large, due to the depletion of metal near the reactive surface [9,282,283], In this case, a modified flux equation, taking into account a diffusion boundary layer and a first-order carrier-mediated uptake can be taken into account by the Best equation [9] (see Chapter 4 for a discussion of the limitations) or by other similar derivations [282] ... 

Figure 7. The Michaelis Menten rate equation as a function of substrate concentration S (in arbitrary units). Parameters are Km 1 and Vm 1. A A linear plot. B A semilogarithmic plot. At a concentration S Km, the rate attains half its maximal value Vm.

Figure 10. A sigmoidal rate equation as a function of the substrate concentration S. Shown is the rate for n 1 (Michaelis Menten, dotted line), n 2 (dashed line), and n 4 (solid line). For increasing n, the rate equation is increasingly switch like. Parameters are Vm 1 and Ks 1.

Taking into account the typical functional form of Michaelis Menten kinetics (see Section III.C), the rate of consumption will usually increase with increasing... 

Similar to Eq. (67), the first reaction (incorporating the enzyme phosphofructo-kinase) exhibits a Hill-type inhibition by its substrate ATP [126]. The overall ATP utilization v3 (ATP) is modeled by a saturable Michaelis Menten function. The system is specified by five kinetic parameters (with Gx lumped into Vm ), the Hill coefficient n, and the total concentration, 4 / = [ATP] + [ADP]. Note that the model is not intended to capture biological realism, rather it serves as a paradigmatic example to identify dynamic behavior in metabolic pathways. 

For power-law functions the (scaled) elasticities do not depend on the substrate concentration, that is, unlike Michaelis Menten rate equations, power-law functions will not saturate for increasing substrate concentration. 

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. 

Most problems associated with approximate kinetics are avoided when Michaelis Menten-type rate equations are utilized. Though this choice sacrifices the possibility of analytical treatment, reversible Michaelis Menten-type equations are straightforwardly consistent with fundamental thermodynamic constraints, have intuitively interpretable parameters, are computationally no more demanding than logarithmic functions, and are well known to give an excellent account of biochemical kinetics. Consequently, Michaelis Menten-type kinetics are an obvious choice to translate large-scale metabolic networks into (approximate) dynamic models. It should also be emphasized that simplified Michaelis Menten kinetics are common in biochemical practice almost all rate equations discussed in Section III.C are simplified instances of more complicated rate functions. 

---