Enzyme kinetics diffusion-limited

Equations 2.26 and 2.27 carmot be solved analytically except for a series of limiting cases considered by Bartlett and Pratt [147,192]. Since fine control of film thickness and organization can be achieved with LbL self-assembled enzyme polyelectrolyte multilayers, these different cases of the kinetic case-diagram for amperometric enzyme electrodes could be tested [147]. For the enzyme multilayer with entrapped mediator in the mediator-limited kinetics (enzyme-mediator reaction rate-determining step), two kinetic cases deserve consideration in this system in both cases I and II, there is no substrate dependence since the kinetics are mediator limited and the current is potential dependent, since the mediator concentration is potential dependent. Since diffusion is fast as compared to enzyme kinetics, mediator and substrate are both approximately at their bulk concentrations throughout the film in case I. The current is first order in both mediator and enzyme concentration and k, the enzyme reoxidation rate. It increases linearly with film thickness since there is no... 

Comparison with similar parameters obtained from reactions with free pyridoxamine indicated that IFABP-PX60 catalyzed transamination some 200 times more efficiently. Analysis of the specific kinetic constants kcat and KM indicated that the observed rate acceleration was due mostly to an increase in substrate binding (50-fold), with a smaller effect on the maximal rate (4-fold). While this is an impressive result, the absolute magnitude of kcat/Ku (0.02 s 1 m 1) makes it clear that this catalyst is still quite primitive compared to natural enzyme systems that occasionally operate with catalytic efficiencies near the diffusion limit. 

Increase of the load of enzyme of a highly active enzyme shifts the kinetically controlled enzyme electrode to diffusional limitation. Since diffusion-limiting effects are associated with the immobilization, they may be manipulated by the procedure of coupling an enzyme to the electrode. 

Fig. 10.6. Change of the rate hmiting step in t5rrosinase carbon paste electrodes, (a) ind (b) show the flow rate dependence of steady-state currents for unmediated (a) emd mediated (b) enzyme electrodes. Applied potential — 0.05 V vs. Ag/AgCl for ( ) 1 p.M phenol ind ( ) 0.5 mM ferrocyanide as control of diffusion limited response. Error bars show the standard deviation of six electrodes. The cheinge from kinetic to dififtisional control in mediated electrodes results in higher sensitivity and improved operational stability as demonstrated in (c) where (1) represents the FIA response of mediated electrodes and (2) unmediated with consecutive 20 /rl injections of 10 p,M phenol in a thin-layer cell. Applied potential - 0.05 V vs. Ag/AgCl, mobile phase 0.25 M phosphate buffer pH 6.0 ind flow rate 0.7 ml min . ...

In the liquid phase diffusion to the catalyst may become the limiting step. Diffusion limitations provides an upper bound to the observed reaction rate (see Chapter 8). It appears that some enzyme catalytic reactions are so fast, e.g. carbonic anhydrase or acetyl cholesterase, that they exhibit this phenomenon. Catalysis under such condition is called "kinetic perfection". 

It can be seen that this sequence of reactions is simply an expression in kinetic terms of the mechanism shown in Tig. 17 which was derived for the soluble enzyme. Whereas in the soluble system fca was the rate limiting step, kr (and perhaps fcs) is rate limiting in the bound system. The bound system is therefore diffusion-limited. The data indicate that in the intact microsome, NADH-cytochrome 65 reductase is turning over at about 60% of the maximum velocity sp). 

The problem of crystal reactivity and diffusion limitations has been considered in detail by Makinen and Fink [170]. They provide a simple treatment for crystals approximated as a plane sheet of material which leads to the definition of a limiting crystal thickness below which kinetic measurements of second-order rate constants are not affected by rate-limiting diffusion processes. For papain [172], ribonuclease A [173] and deoxyhaemoglobin [174], where the crystal thicknesses are comparable to the critical crystal thickness, reactivities are the same in the crystal and solution. In the case of glycogen phosphorylase b Kasvinsky and Madsen [175] demonstrated that the values for both substrates, glucose 1-phosphate (37 + 8mM) and malto-heptaose (176 + 20 mM), were the same in the crystal and solution. The 10-100-fold reduction in rate, despite the fact that crystal thickness was only twice the critical thickness, may be attributable partly to the allosteric nature of this enzyme and partly to the fact that the large substrate maltoheptaose (molecular weight, 1152) may not obey the simple diffusion rules in the crystal. 

The substrate concentration at which deviations from the analytically usable linear measuring range occur depends on the extent of diffusion limitation. According to the Michaelis-Men-ten equation, under kinetic control a linear dependence may only be expected for substrate concentrations below Km- Under diffusion control the decrease of substrate concentration in the enzyme layer caused by slow substrate diffusion results in an extended linear range. It has to be considered, however, that for two-substrate reactions deviations from linearity may also be produced by cosubstrate consumption. 

Owing to the excess of enzyme in the membrane a diffusion limited enzyme sensor has a higher functional stability than a kinetically controlled one. With the former, 2000-10 000 measurements per enzyme membrane can be performed, while kinetically controlled sensors typically permit only 200-500 measurements. 

Finally, we note that in a biochemically versatile world, enzyme kinetics may often reach the limit of the possible as defined by the rate of diffusion of the substrate. In other words, it is the rule rather than the exception that enzyme kinetics for a limiting substrate become so effective in nature that the rate of transformation of the substrate is limited by the rate of diffusion to the cell surface as much as by the enzyme reactions themselves. The kinetics of such a situation have been worked out by Pasciak and Gavis (1974). The net result is a rate equation of the form... 

Feedback. When an oxidoreductase enzyme is immobilized at the specimen surface, a redox mediator present in solution may be recycled by the diffusion-limited electrochemical process at the tip and electron exchange with the enzyme active site as described in Sec. I.C. The mass transport rate is defined by the tip radius and height of the tip above the specimen. The tip current depends on the mass transport rate and the enzyme kinetics. Kinetic information may therefore be obtained from the dependence of tip current on height, i.e., an approach curve. When the mediator is fed... 

Time response. In most situations enzyme kinetics have very little effect on the response time of enzyme-based biosensors. From the analysis given above, it is clear that one should operate these devices under conditions where the analyte concentration within the sensor is much less than Km- For sensors which are in the membrane diffusion limiting regime (section 7.3.1.1 above), the response characteristics of the membrane material will be governing. These depend on the thickness of the membrane and the diffusivity of the analyte in the membrane material. An approximate estimate of the membrane lag time is... 

MnSOD catalyzes the dismutation of H02 into dioxygen and hydrogen peroxide. As a redox enzyme, it shuttles between the Mn" and the Mn111 oxidation states (77). This process has been studied, and the enzyme has been shown to catalyze this reaction at a rate of 1-2.2 X 109 M s-1, which is at the diffusion limit (52). A representation of the proposed catalytic cycle is shown in Scheme 3 (78). Several studies on MnSOD kinetics have been published (78-82). Early studies on the reaction of MnSOD from Escherichia coli and Bacillus thermophilus indicated evidence for a four-step process involving two fast and... 

The dynamics of substrate conversion therefore depend on enzyme kinetics as well as on mass transport conditions. Diffusion through the membrane matrix and within the flowing solution, play the most important roles in transport mechanisrris. Since the flow is laminar in most cases, substrate and product transport resistances through the dense layer are exceedingly small relative to diffusional resistances in the flowing solution. The rate-limiting step in sub-... 

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