Steady-state kinetic parameters

In this chapter we described the thermodynamics of enzyme-inhibitor interactions and defined three potential modes of reversible binding of inhibitors to enzyme molecules. Competitive inhibitors bind to the free enzyme form in direct competition with substrate molecules. Noncompetitive inhibitors bind to both the free enzyme and to the ES complex or subsequent enzyme forms that are populated during catalysis. Uncompetitive inhibitors bind exclusively to the ES complex or to subsequent enzyme forms. We saw that one can distinguish among these inhibition modes by their effects on the apparent values of the steady state kinetic parameters Umax, Km, and VmdX/KM. We further saw that for bisubstrate reactions, the inhibition modality depends on the reaction mechanism used by the enzyme. Finally, we described how one may use the dissociation constant for inhibition (Kh o.K or both) to best evaluate the relative affinity of different inhibitors for ones target enzyme, and thus drive compound optimization through medicinal chemistry efforts. 

The reduction of 7,8-dihydrofolate (H2F) to 5,6,7,8-tetrahydrofolate (H4F) has been analyzed extensively14 26-30 and a kinetic scheme for E. Coli DHFR was proposed in which the steady-state kinetic parameters as well as the full time course kinetics under a variety of substrate concentrations and pHs were determined. From these studies, the pKa of Asp27 is 6.5 in the ternary complex between the enzyme, the cofactor NADPH and the substrate dihydrofolate. The second observation is that, contrary to earlier results,27 the rate determining step involves dissociation of the product from the enzyme, rather than hydride ion transfer from the cofactor to the substrate. 

Table 3 Steady-State Kinetic Parameters of Dihydrogen Peroxide Decomposition by Mn Catalases...

Fig. 9 2 Lineweaver-Burk graphical procedure for determining the two steady-state kinetic parameters in the Michaelis-Menten equation.

To describe completely the effects of pH changes on enzyme catalysis is an almost impossible task. Many of the amino acid side chains in an enzyme are ionizable, but in environments with polarities different from that of the free solution, their pKa s (Chap. 3) will probably be significantly altered. However, experimentally, it is a simple matter to determine values of steady-state kinetic parameters (Km, Kmax) of an enzyme for various pH conditions. 

Wolthers, Kirsten R., Schimerlik, Michael I. (2002) Neuronal nitric oxide synthase substrate and solvent kinetic isotope effects on the steady-state kinetic parameters for the reduction of 2, 6-dichloroindophenol and cytochrome C3+, Biochemistry 41, 196-204. 

There have been k12/k13 KIEs measured on C02 release (1.033) by this enzyme which indicate that the ratio kjk5 is not very different from unity, i.e. that transimination and decarboxylation are both partially rate-limiting147. Based on a comparison of a variety of KIEs, as well as steady-state kinetic parameters Vmax and VmaJKM for the pyruvyl-dependent and pyridoxal-dependent decarboxylases, no obvious reasons could be found why nature would preferentially select one pathway over the other. 

Brandt DA, Barnett LB, Alberty RA. The temperature dependence of the steady state kinetic parameters of the fumarase reaction. J. Am. Chem. Soc. 1963 85 2204-2209. 

Table 12.5-7. Steady-state kinetic parameters for the hydrolysis of Boc-Xaa-OCp by trypsin according to Thormann et ai.[198].

The two kinetic constants, and itcat. are most often misinterpreted as the substrate dissociation constant and the rate of the chemical reaction, respectively. However, this is not always the case, and /Cm can be greater than, less than, or equal to the true substrate dissociation constant, K. The steady-state kinetic parameters only provide information sufficient to describe a minimal kinetic scheme. In terms of measurable steady-state parameters, a reaction sequence must be reduced to a minimal mechanism (Scheme I),... 

Although steady-state kinetic methods cannot establish the complete enzyme reaction mechanism, they do provide the basis for designing the more direct experiments to establish the reaction sequence. The magnitude of kcm will establish the time over which a single enzyme turnover must be examined for example, a reaction occurring at 60 sec will complete a single turnover in approximately 70 msec (six half-lives). The term kcJKm allows calculation of the concentration of substrate (or enzyme if in excess over substrate) that is required to saturate the rate of substrate binding relative to the rate of the chemical reaction or product release. In addition, the steady-state kinetic parameters define the properties of the enzyme under multiple turnovers, and one must make sure that the kinetic properties measured in the first turnover mimic the steady-state kinetic parameters. Thus, steady-state and transient-state kinetic methods complement one another and both need to be applied to solve an enzyme reaction pathway. 

Table 1 Steady-state kinetic parameters for CTP formation cataiyzed by wiid-type CTP synthetase and the LI 09A CTP synthetase mutant using various nitrogen sources...

For determination of steady-state kinetic parameters of X2-X6 substrates, 0.5-ml reaction mixtures contained varied substrate concentrations (0.9-13 mM) in 100 mM succinate-NaOH, pH 5.3 at 25°C. For pH studies of SXA-catalyzed hydrolysis of X2 and X3, buffers of constant ionic strength (7=0.3 M), adjusted with NaCl, were used as indicated (replacing 100 mM succinate-NaOH, pH 5.3) 100 mM succinate-NaOH (pH 4.3-6), 100 mM sodium phosphate (pH 6-8), and 30 mM sodium pyrophosphate (pH 8-9.2). Before (time= 0 min) and after (time=0.5-2 min) initiating reactions with enzyme (7 xl SXA in 20 mM sodium phosphate, pH 7.0), 100-(xl aliquots of reaction mixtures were removed and quenched with an equal volume of 0.2 M sodium phosphate pH 11.3 at 0°C (so that quenched mixtures were pH 10.5-11) and diluted by adding 1 mM sodium phosphate, pH 10.5-11 at 0°C as necessary (typically 200-800 (il added to 200 pi quenched samples) to adjust concentrations of reactants and products to fall within the linear range of standard curves. Samples were kept on wet ice or the HPLC autosampler at 5°C until analyzed by HPLC. Initial rates, calculated from linear regressions of the [D-xylose] produced vs time, were fitted to Eq. 1 to determine steady-state kinetic parameters. Parameter, kcai. is expressed in moles of substrate hydrolyzed per second per mole enzyme active sites (protomers) thus, for substrate X2, the [D-xylose] produced was divided by two to provide the [X2] hydrolyzed, whereas for X3-X6, the [D-xylose] produced was taken as the concentration of substrate hydrolyzed. 

For D-xylose inhibition of SXAotatyzed hydrolysis of 4NPX, 1-ml reactions contained varied concentrations (0.2-7 mM) of 4NPX and varied concentrations (0, 20, 60, and 150 mM) of D-xylose in 100 mM succinate-NaOH, pH 5.3 at 25°C. Reactions were initialed by adding enzyme (7 pi SXA in 20 mM sodium phosphate, pH 7.0), and reaction progress was monitored continuously for 0.3 min at 380 nm to determine initial rates (fitted to lines). For determination of steady-state kinetic parameters, initial rates were fitted to Eq. 2 (competitive inhibition) and Eq. 3 (noncompetitive inhibition). 

Table 1 Steady-state kinetic parameters of SXA acting on xylooligosaccharides .

Fig. 2 pH dqiendence of steady-state kinetic parameters for 1,4-3-D-xylobiose hydrolysis catalyzed by SXA at 25°C. Initial rates were determined from reactions in buffers of constant ionic strength (7==0.3 M) and kinetic parameters were determined by fitting initial-rate data to Eq. 1 SEs ( ) are indicated, a kcat vs pH. The curve was generated by fitting values vs pH to Eq. 5 pXa=7.57 0.03, pH-independent ka = 2 2. b kcJKm vs pH. The curve was generated by fitting k lK values vs pH to Eq. 6 pXai=4.86 0.06,... 

Unfortunately, the form of Equation (8.53) is a little way off the form of the Michaelis-Menten equation. For this reason, the King-Altman approach is usually supplemented by an approach developed by Cleland. The Cleland approach seeks to group kinetic rate constants together into numbers (num), coefficients (Coef) and constants (const) that themselves can be collectively defined as experimental steady-state kinetic parameters equivalent to fccat> Umax and fCm of the original Michaelis-Menten equation. After such substitutions, the result is that equations may be algebraically manipulated to reproduce the form of the Michaelis-Menten equation (8.8). Use of the Cleland approach is illustrated as follows. 

Second, we will need to define steady state kinetic parameters as follows ... 

A critical evaluation of this explanation of the pH dependence for enzymatic reactions reveals that this mechanism is only one relatively simple type consistent with the data. For example, if the steps E -h S X, S -h EH2 XH2, etc., were included, exactly the same rate law would result, although the interpretation of the data would be considerably changed [5]. Other types of explanations for the pH dependence of steady-state kinetic parameters have emerged from time to time but have not gained wide acceptance. 

An excellent study has been made of the temperature dependence of the steady-state kinetic parameters of the fumarase reaction [6]. By studying the temperature dependence over a wide range of pH, the apparent activation energies and standard-enthalpy changes associated with the pH-independent steady-state parameters, the lower bounds of the rate constants, and the ionization constants of the groups at the active site were obtained. The results are summarized in Table 9-1. In this case the temperature dependence of all parameters appears quite normal. The standard-enthalpy changes of... 

Table 1. Steady-state kinetic parameters for COT and CPT-II. The data, taken from Nic a Bhaird et was obtained using purified enzymes in the absence of bovine serum albumin.

Nucleotide Incorporation Assay Modification Determining Steady-State Kinetic Parameters... 

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