Steady-state conditions, enzyme kinetics

The intermediate reacts sufficiently rapidly to be on the reaction pathway. These criteria require that pre-steady state kinetics be used at some stage in order to measure the relevant formation and decomposition rate constants of the intermediate. But the rapid reaction measurements are not sufficient by themselves, since the rate constants must be shown to be consistent with the activity of the enzyme under steady state conditions. Hence the power, and the necessity, of combining the two approaches. 

Equations (2.10) and (2.12) are identical except for the substitution of the equilibrium dissociation constant Ks in Equation (2.10) by the kinetic constant Ku in Equation (2.12). This substitution is necessary because in the steady state treatment, rapid equilibrium assumptions no longer holds. A detailed description of the meaning of Ku, in terms of specific rate constants can be found in the texts by Copeland (2000) and Fersht (1999) and elsewhere. For our purposes it suffices to say that while Ku is not a true equilibrium constant, it can nevertheless be viewed as a measure of the relative affinity of the ES encounter complex under steady state conditions. Thus in all of the equations presented in this chapter we must substitute Ku for Ks when dealing with steady state measurements of enzyme reactions. 

In this chapter we have seen that enzymatic catalysis is initiated by the reversible interactions of a substrate molecule with the active site of the enzyme to form a non-covalent binary complex. The chemical transformation of the substrate to the product molecule occurs within the context of the enzyme active site subsequent to initial complex formation. We saw that the enormous rate enhancements for enzyme-catalyzed reactions are the result of specific mechanisms that enzymes use to achieve large reductions in the energy of activation associated with attainment of the reaction transition state structure. Stabilization of the reaction transition state in the context of the enzymatic reaction is the key contributor to both enzymatic rate enhancement and substrate specificity. We described several chemical strategies by which enzymes achieve this transition state stabilization. We also saw in this chapter that enzyme reactions are most commonly studied by following the kinetics of these reactions under steady state conditions. We defined three kinetic constants—kai KM, and kcJKM—that can be used to define the efficiency of enzymatic catalysis, and each reports on different portions of the enzymatic reaction pathway. Perturbations... 

In steady-state kinetic studies, the total concentration of the enzyme should be much less than the concentration of the substrate(s), product(s), and effector(s) typically, by at least a thousandfold. When this condition is not true, the steady-state condition will not be valid and other methods, such as global analysis, have to be utilized to analyze the kinetic data. 

Except for very simple systems, initial rate experiments of enzyme-catalyzed reactions are typically run in which the initial velocity is measured at a number of substrate concentrations while keeping all of the other components of the reaction mixture constant. The set of experiments is run again a number of times (typically, at least five) in which the concentration of one of those other components of the reaction mixture has been changed. When the initial rate data is plotted in a linear format (for example, in a double-reciprocal plot, 1/v vx. 1/[S]), a series of lines are obtained, each associated with a different concentration of the other component (for example, another substrate in a multisubstrate reaction, one of the products, an inhibitor or other effector, etc.). The slopes of each of these lines are replotted as a function of the concentration of the other component (e.g., slope vx. [other substrate] in a multisubstrate reaction slope vx. 1/[inhibitor] in an inhibition study etc.). Similar replots may be made with the vertical intercepts of the primary plots. The new slopes, vertical intercepts, and horizontal intercepts of these replots can provide estimates of the kinetic parameters for the system under study. In addition, linearity (or lack of) is a good check on whether the experimental protocols have valid steady-state conditions. Nonlinearity in replot data can often indicate cooperative events, slow binding steps, multiple binding, etc. 

Although the use of pre-steady state kinetics is undoubtedly superior as a means of analyzing the chemical mechanisms of enzyme catalysis (Chapters 4 and 7), steady state kinetics is more important for the understanding of metabolism, since it measures the catalytic activity of an enzyme in the steady state conditions in the cell. 

Before we can discuss the measurement of active-site concentration, we need to consider the kinetics of the substrate reaction. The majority of kinetic studies of enzymes are carried out on systems described by Scheme 11.16 where all terms have their usual meanings and where the intermediates have come to a steady-state concentration otherwise, studies of the kinetics of the pre-steady-state conditions usually require the use of specialist, fast reaction, equipment. The Michaelis-Menten equation, Equation 11.12, where all terms again have their usual meanings, can be derived from Scheme 11.16 when the system has reached a steady state at this point the values of [ES] and [P] are still very much less than that of [S] ... 

In the same vein and under dimensionally restricted conditions, the description of the Michaelis-Menten mechanism can be governed by power-law kinetics with kinetic orders with respect to substrate and enzyme given by noninteger powers. Under quasi-steady-state conditions, Savageau [25] defined a fractal Michaelis constant and introduced the fractal rate law. The behavior of this fractal rate law is decidedly different from the traditional Michaelis-Menten rate law ... 

Since the concentrations of all the intermediate states are constant under steady state conditions, all of these states can, at least formally, be incorporated into a single kinetic intermediate state. It follows that under steady state conditions, kinetic data can provide no information about the existence and kinetic properties of intermediate enzyme-substrate complexes. An understanding of the mechanism of an enzyme catalysed reaction needs information about these intermediate states, which is therefore usually obtained from kinetic studies before steady state has been established, usually by rapid reaction methods. Comprehensive coverage of the techniques and methods of analysis of pre-steady state kinetics is beyond the scope of this chapter, but we discuss here methods for analysing simple exponential processes. Two approaches are used. In the first, the observed signal S(t) is fitted to an exponential function of the following form ... 

Kinetic analysis of TD-62 under steady-state conditions with ferricyanide as electron acceptor showed that kcat and Km values for L-lactate (165 sec and 0.96 mM, at 25°C, 10 mM Tris-HCl, pH 7.5, / = 0.10 M NaCl) were not dramatically different from the values determined for wild-type enzyme (200 sec and 0.49 mM, conditions as above) 144). However, the behavior of TD-62 under assay conditions was unusual in that the rate of lactate oxidation decreased long before either L-lactate or ferricyanide became depleted. The decrease in reaction rate occurred as a biphasic process, leading eventually to complete loss of activity. This deactivation of TD-62 was only observed under turnover conditions. Deactivation was independent of ferricyanide and was also observed when cytochrome c was used as electron acceptor (150). 

Initial experiments involved the use of hydroquinones, ascorbate and dihydroxy fumarate as substrates for the enzyme [120], Radicals (e.g., 5 and 6) were detected under steady-state conditions using a flow system to minimize substrate depletion. The narrow line spectra (Fig. 6a,b) of the radical anions are identical to those generated in chemical systems, indicating that the radicals are free in solution rather than associated with the enzyme. If it is assumed that the reaction proceeds by one-electron oxidation of the substrate and that the product radicals decay by self-reaction (e.g., disproportionation), kinetic analysis predicts that the steady-state radical... 

Most measurements of glycosidase kinetics are carried out under steady-state conditions. Substrate is in large excess over enzyme and the reaction is monitored on a time-scale that is long compared with the reciprocals of the rate constants for individual molecular events, so that changes in the concentrations of various liganded and unliganded forms of the enzyme can be set to zero. If only one substrate is involved and the active sites are independent, eqn. (5.1), the Michaelis-Menten equation, holds ... 

In glutamate mutase [43], the forward and reverse steady-state deuterium (kn/ko of 3.9 forward and 6.3 reverse) and tritium kn/kj of 21 forward and 19 reverse) kinetic isotope effects are both suppressed. However large deuterium isotope effects of 28 and 35 in the forward and reverse directions respectively have been observed for cob(ii)alamin formation under pre-steady-state conditions. These large kinetic isotope effects suggest that quantum mechanical tunneling also dominates this enzyme reaction. 

Knowing that the enzyme stability is sufficiently high at pH 3.75 (50% deactivation after 150 h), water was chosen as the reaction medium and the reaction was performed at room temperature. This selection of reaction conditions was followed by a detailed kinetic analysis of the system, the investigation of the reactor kinetics and the simulation of steady state conditions in continuous experiments155, 60]. 

Pb being product concentration in the permeate. Once steady state conditions are attained, all enzyme is virtually confined within region B at a concentration Eb=N/Vb. Therefore, enzyme kinetics in region B can be experimentally determined through Equation 32. For details on the initial conditions and integration of the set of Equations 30 and 31 readers are referred to Reference 30. 

Under steady-state conditions, the cleavage of this substrate obeys Michaelis-Menten kinetics with a Km of 20 p,M and a fecat of 77 s h The initial phase of the reaction was examined by using the stopped-flow method. This technique permits the rapid mixing of enzyme and substrate and allows almost instantaneous monitoring of the reaction. At the beginning of the reaction, this method revealed a burst phase during which the... 

The theoiy developed in the preceding sections has, for the sake of simplicity, dealt with enzyme mechanisms that involve only one substrate. For sinple kinetic models, pH dependent rate equations, such as (14.16), (14.21) and (14.27), may be derived by simple algebraic manipulation. If, however, steady-state conditions are... 

Multi-substrate enzymes (see) catalyse reactions of two or more substrates. Such enzymes can form a number of different complexes (known as enzyme species) with one or both substrates and/or products. The order in which these species are formed may be random or ordered. Cleland s short notation (see) is a convenient way of representing the possibilities. The kinetics of such reactions become extremely complicated enzyme networks (see Enzyme graphs) provide a means of sununarizing them. To evaluate the kinetic data for such systems, one must resort to a computer. Furthermore, the information gained from steady-state experiments may not be sufficient. A number of methods of very rapid measiu ement have been used to investigate the pre-steady-state condition of reactions, including stopped flow, temperature jump and flash methods. 

Rat equation in Enzyme kinetics (see), an equation expressing the rate of a reaction in terms of rate constants and the concentrations of enzyme spedes, substrate and product. When it is assumed that steady state conditions obtain, the Michaelis-Menten equation (see) is a suitable approximation. R.e. are represented graphically (see Enzyme graph) they may be derived by the King-Altman method (see). 

The modeling of the biosensors utilizing ntm-Michaelis-Menten kinetics of the enzymes action revealed a muMstate possibility at the steady-state conditions [13]. Multi-steady state may have many interesting consequences for the stability of the biosensor response. It can generate oscillations of the concentration and the biosensor response if the negligible perturbation of the enzyme activity or the mass transport occurs. 

The complexity of the kinetics of both reactions indicates that some disturbances of the steady-state conditions prevailing in the freshly prepared particles have taken place during storage. These could be due to a variety of factors, such as an increased availability of precursors for one particular step, the impairment of a rate-determining step, or the acceleration of a reverse reaction, such as, for example, would occur by the activation of a proteolytic enzyme. The sharp increase in the initial rate of reac-... 

The catalytic function of an enzyme is described by enzyme kinetics usually determined under steady-state conditions. A steady state refers to a complete balance of a particular quantity between its rate of formation and its rate of disappearance. In steady-state enzyme kinetics, the concentrations of enzyme-bound intermediates are meant to be in a steady state. On mixing an enzyme with a large excess of substrates, there is an initial period, known as a presteady state, during which the concentrations of the intermediates build up to a maximal level under the reaction conditions. Then the reaction rate changes relatively slowly with time and the intermediates are considered to be at steady-state concentrations. Note that the steady state is an approximation because the substrate is gradually depleted during the course of reaction. Therefore, steady-state kinetic measurements should be performed in a relatively short time interval over which the... 

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