Steady-state kinetics measurement methods

Steady state kinetic measurements on an enzyme usually give only two pieces of kinetic data, the KM value, which may or may not be the dissociation constant of the enzyme-substrate complex, and the kcM value, which may be a microscopic rate constant but may also be a combination of the rate constants for several steps. The kineticist does have a few tricks that may be used on occasion to detect intermediates and even measure individual rate constants, but these are not general and depend on mechanistic interpretations. (Some examples of these methods will be discussed in Chapter 7.) In order to measure the rate constants of the individual steps on the reaction pathway and detect transient intermediates, it is necessary to measure the rate of approach to the steady state. It is during the time period in which the steady state is set up that the individual rate constants may be observed. 

At the present time, two methods are in common use for the determination of time-resolved anisotropy parameters—the single-photon counting or pulse method 55-56 and the frequency-domain or phase fluorometric methods. 57 59) These are described elsewhere in this series. Recently, both of these techniques have undergone considerable development, and there are a number of commercially available instruments which include analysis software. The question of which technique would be better for the study of membranes is therefore difficult to answer. Certainly, however, the multifrequency phase instruments are now fully comparable with the time-domain instruments, a situation which was not the case only a few years ago. Time-resolved measurements are generally rather more difficult to perform and may take considerably longer than the steady-state anisotropy measurements, and this should be borne in mind when samples are unstable or if information of kinetics is required. It is therefore important to evaluate the need to take such measurements in studies of membranes. Steady-state instruments are of course much less expensive, and considerable information can be extracted, although polarization optics are not usually supplied as standard. 

Steady-State Kinetics, There are two electrochemical methods for determination of the steady-state rate of an electrochemical reaction at the mixed potential. In the first method (the intercept method) the rate is determined as the current coordinate of the intersection of the high overpotential polarization curves for the partial cathodic and anodic processes, measured from the rest potential. In the second method (the low-overpotential method) the rate is determined from the low-overpotential polarization data for partial cathodic and anodic processes, measured from the mixed potential. The first method was illustrated in Figures 8.3 and 8.4. The second method is discussed briefly here. Typical current—potential curves in the vicinity of the mixed potential for the electroless copper deposition (average of six trials) are shown in Figure 8.13. The rate of deposition may be calculated from these curves using the Le Roy equation (29,30) ... 

The fastest steps in an enzymatic process cannot be observed by conventional steady-state kinetic methods because the latter cannot be applied to reactions with half-times of less than about 10 s. Consequently, a variety of methods have been developed18 56-593 to measure rates in the range of 1 to 1013 s... 

Methods for measurement An introduction to pre - steady state kinetics... 

There are also nonthematic methods that allow the formation of acylenzymes under conditions where they are stable, so that they can be stored in a syringe in a stopped-flow spectrophotometer. For example, it is possible to synthesize certain nonspecific acylenzymes and store them at low pH.9 12 When they are restored to high pH, they are found to deacylate at the rate expected from the steady state kinetics. This approach has been extended to cover specific acylenzymes. When acyl-L-tryptophan derivatives are incubated with chymotrypsin at pH 3 to 4, the acylenzyme accumulates. The solution may then be pH-jumped by mixing it with a concentrated high-pH buffer in the stopped-flow spectrophotometer.1314 The deacylation rate has been measured by the proflavin displacement method and by using furylacrylolyl compounds. 

In contrast to the batch fermentation based methods of determining kinetic constants, the use of a continuous fermenter (Fig. 3.71) requires more experiments to be performed, but the analysis tends to be more straightforward. In essence, the experimental method involves setting up a continuous stirred-tank fermenter to grow the micro-organisms on a sterile feed of the required substrate. The feed flowrate is adjusted to the desired value which, of course, must produce a dilution rate below the critical value for washout, and the system is allowed to reach steady state. Careful measurements of the microbial density X, the substrate concentration S, and the flowrate F are made when a steady state has been achieved, and the operation is then repeated at a series of suitable dilution rates. 

It should be noted that this solution procedure requires the knowledge of elementary rate constants, klt k2, and k3. The elementary rate constants can be measured by the experimental techniques such as pre-steady-state kinetics and relaxation methods (Bailey and Ollis, pp. 111 -113, 1986), which are much more complicated compared to the methods to determine KM and rmax. Furthermore, the initial molar concentration of an enzyme should be known, which is also difficult to measure as explained earlier. However, a numerical solution with the elementary rate constants can provide a more precise picture of what is occurring during the enzyme reaction, as illustrated in the following example problem. 

There are several reasons to work with dilute solutions of enzyme. First, there is the obvious practical issue of conserving what is often a precious supply of enzyme that has been obtained with some labor and cost. Second, dilution can aid in eliminating unwanted interactions, thereby linearizing the rate vs enzyme curve as described above. Finally, it may be difficult to make measurements of initial rates in steady state unless the enzyme preparation is sufficiently dilute. If too much substrate is converted in the time required to make the measurement, then one must slow the reaction, and this is typically done by reducing the amount of enzyme in the assay. Transient kinetic methods typically require the use of concentrated enzyme solutions. Hence, these methods are seldom used until after basic understanding of the reaction mechanism has been obtained through steady-state kinetic methods, and critical tests can be designed to elucidate further the mechanism by transient kinetic methods. 

MALDI-TOF-MS has been demonstrated as a useful tool in pre-steady state kinetic research by Houston et al., who combined it off-line with quench-flow methods to follow the appearance of a protein tyrosine phosphatase (PTPase) reaction intermediate.12 Houston et al. were able to measure rate constants up to 30 s 1, with k2/k3 ratios up to approximately 15. The device described in this chapter extends this technique to measure rate constants approximately 5 times greater, with k2/k3 ratios approximately twice previously measurable values. MALDI-TOF-MS is typically conducted on a centimeter-scale conducting plate the digital microfluidic system employed is a square plate approximately 2 cm on each side, with 16 experimental units per chip, which can be placed directly inside a standard MALDI-TOF-MS plate that has been machined appropriately. By grounding the exposed wires on the otherwise insulated chip, charging is negligible. 

Houston et al,12 reported the capability to measure pre-steady state kinetics using rapid chemical quench-flow methods coupled with MALDI-TOF-MS. However, they reported difficulties in obtaining accurate pre-steady state... 

The kinetic analysis of an enzyme mechanism often begins by analysis in the steady state therefore, we first consider the conclusions that can be derived by steady-state analysis and examine how this information is used to design experiments to explore the enzyme reaction kinetics in the transient phase. It has often been stated that steady-state kinetic analysis cannot prove a reaction pathway, it can only eliminate alternate models from consideration (5). This is true because the data obtained in the steady state provide only indirect information to define the pathway. Because the steady-state parameters, kcat and K, are complex functions of all of the reactions occurring at the enzyme surface, individual reaction steps are buried within these terms and cannot be resolved. These limitations are overcome by examination of the reaction pathway by transient-state kinetic methods, wherein the enzyme is examined as a stoichiometric reactant, allowing individual steps in a pathway to be established by direct measurement. This is not to say that steady-state kinetic analysis is without merit rather, steady-state and transient-state kinetic studies complement one another and analysis in the steady state should be a prelude to the proper design and interpretation of experiments using transient-state kinetic methods. Two excellent chapters on steady-state methods have appeared in this series (6, 7) and they are highly recommended. 

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. 

This study has important lessons for enzyme kinetic analysis. The use of pH variation and examination of isotope elfects can be a powerful combination to explore the chemistry of enzyme-catalyzed reactions and to dissect the contributions of individual reaction steps to the net steady-state turnover (27). Examination of the effects of pH on each step of the reaction pathway could resolve the contributions of ionizable groups toward ground-state binding energy and transition-state stabilization. The use of isotope effects by transient-state kinetic methods is more limited than in the steady state due to the errors involved in comparing two rate measurements. In the steady state, the ratio method has allowed isotope effects of less than 1% to be measured accurately (8a, 58). By transient-state kinetics, one would require at least a 10-20% change in rate to demonstrate a convincing difference between two rate measurements in most instances. 

These two approaches require different apparatus. Steady-state kinetics usually requires only the apparatus of the routine biochemical laboratory. Most steady-state studies are spectrophotometric. If the reaction does not involve a convenient absorbance change, it may be necessary to use other methods of measurement, but often it is possible to obtain an absorbance change by using coupling enzymes or secondary reactants. Rapid-reaction studies require more sophisticated equipment for precise and rapid mixing of enzymes with reactant solutions and for synchronous rapid recording of the appropriate signal. 

The techniques for characterizing the kinetics of electrode reactions can be classified into steady-state and transient methods. The steady-state methods involve the measurement of the current-potential relationships at constant current (galvanoslatic control) or constant potential (potentiostatic control) conditions and measuring the response, which is either the potential or the current after a steady state is achieved. The non-steady-state methods involve the perturbation of the system from an equilibrium or a steady-state condition, and follow the response of the system as a function of time using current, potential, charge, impedance, or any other accessible property of the interface. Relaxation methods are a subclass of perturbation methods. 

Enzyme reaction intermediates can be characterized, in sub-second timescale, using the so-called pulsed flow method [35]. It employs a direct on-line interface between a rapid-mixing device and a ESI-MS system. It circumvents chemical quenching. By way of this strategy, it was possible to detect the intermediate of a reaction catalyzed by 5-enolpyruvoyl-shikimate-3-phosphate synthase [35]. The time-resolved ESI-MS method was also implemented in measurements of pre-steady-state kinetics of an enzymatic reaction involving Bacillus circulans xylanase [36]. The pre-steady-state kinetic parameters for the formation of the covalent intermediate in the mutant xylanase were determined. The MS results were in agreement with those obtained by stopped-flow ultraviolet-visible spectroscopy. In a later work, hydrolysis of p-nitrophenyl acetate by chymotrypsin was used as a model system [27]. The chymotrypsin-catalyzed hydrolysis follows the mechanism [27] ... 

A procedure, based on the Gauss-Newton method for non-linear regression, has been developed to obtain constants from the analysis of progress curve data. Rules are presented which greatly simplify the derivation of the necessary equations. Values for kinetic parameters from the method agreed well with those obtained from actual steady-state rate measurements. 

In constrast to the situation described above, the major disadvantages of using CSTR s for kinetic studies stem from the fact that they can be difficult to construct and operate and they consume considerably larger supplies of raw materials. This last factor can be particularly troublesome for academic laboratories. On the other hand, a CSTR system which is operated at steady state provides a method for direct measurement of reaction rate, particle nucleation etc. without data differentiation. The required mass balance measurements yield accurate values for the various rates. 

---