Initial rate behavior

Cleland described the following rules for reversible inhibition patterns observed in double-reciprocal plots of initial rate behavior. 

An enzyme is said to obey Michaelis-Menten kinetics, if a plot of the initial reaction rate (in which the substrate concentration is in great excess over the total enzyme concentration) versus substrate concentration(s) produces a hyperbolic curve. There should be no cooperativity apparent in the rate-saturation process, and the initial rate behavior should comply with the Michaelis-Menten equation, v = Emax[A]/(7 a + [A]), where v is the initial velocity, [A] is the initial substrate concentration, Umax is the maximum velocity, and is the dissociation constant for the substrate. A, binding to the free enzyme. The original formulation of the Michaelis-Menten treatment assumed a rapid pre-equilibrium of E and S with the central complex EX. However, the steady-state or Briggs-Haldane derivation yields an equation that is iso-... 

Figure 2. Analysis of the pH-dependence of the key parameters influencing the initial rate behavior of an enzyme containing two ionizable groups that determine catalytic activity.

Kinetic Considerations. The reaction kinetics are masked by a desorption process as shown below and are further complicated by rate deactivation. The independence of the 400-sec rate on reactant mole ratio is not indicative of zero-order kinetics but results because of the nature of the particular kinetic, desorption, and rate decay relationships under these conditions. It would not be expected to be more generally observed under widely varying conditions. The initial rate behavior is considered more indicative of the intrinsic kinetics of the system and is consistent with a model involving competitive adsorption between the two reactants with the olefin being more strongly adsorbed. Such kinetic behavior is consistent with that reported by Venuto (16). Kinetic analysis depends on the assumption that quasi-steady state behavior holds for the rate during rate decay and that the exponential decay extrapolation is valid as time approaches zero. Detailed quantification of the intrinsic kinetics was not attempted in this work. 

Kinetic studies also provide other evidence for the SnI mechanism. One technique used F NMR to follow the solvolysis of trifluoroacetyl esters. If this mechanism operates essentially as shown on page 393, the rate should be the same for a given substrate under a given set of conditions, regardless of the identity of the nucleophile or its concentration. In one experiment that demonstrates this, benzhy-dryl chloride (Ph2CHCl) was treated in SO2 with the nucleophiles fluoride ion, pyridine, and triethylamine at several concentrations of each nucleophile. In each case, the initial rate of the reaction was approximately the same when corrections were made for the salt effect. The same type of behavior has been shown in a number of other cases, even when the reagents are as different in their nucleophilicities (see p. 438) as H2O and OH . 

The initial rate of the reaction can be determined from the early linear portion of the plot of [C] vs. n. (Note that there is often abrief adjustment period before the system settles into normal kinetic behavior. This brief adjustment period should not be included in the initial linear period.) This is illustrated in Figure 8.3, where the initial slope is found to be 4.83 ingredients/iteration (ingr/itn). Based on Eq. (8.2), the value of the rate constant k for this system is 4.83/(500)2 2.0 X 10 ... 

Equations and describe how concentration changes with time when only a single reactant is involved. However, most reactions involve concentration changes for more than one species. Although it is possible to develop equations relating concentration and time for such reactions, such equations are more complicated and more difficult to interpret than the equations that involve just one reactant. Fortunately, it is often possible to simplify the experimental behavior of a reaction. We describe two experimental methods that accomplish this, the isolation method and the method of initial rates. 

A second way to simplify the behavior of a reaction is the method of initial rates. In this method, we measure the rate at the very beginning of the reaction for different concentrations. A set of experiments is done, changing only one initial concentration each time. Instead of measuring the concentration at many different times during the reaction, we make just one measurement for each set of concentrations. The reaction orders can be evaluated from the relationships between the changes in concentration and the changes in initial rates. We illustrate how this works using a gas-phase reaction of H2 with NO 2H2(g) -b 2NO(g) N2(g) + 2H2 0(g)... 

In thermal polymerization where the rate of initiation may also vary with composition, an abnormal cross initiation rate may introduce a further contribution to nonadditive behavior. The only system investigated quantitatively is styrene-methyl methacrylate, rates of thermal copolymerization of which were measured by Walling. The rate ratios appearing in Eq. (26) are known for this system from studies on the individual monomers, from copolymer composition studies, and from the copolymerization rate at fixed initiation rate. Hence a single measurement of the thermal copolymerization rate yields a value for Ri. Knowing hm and ki22 from the thermal initiation rates for either monomer alone (Chap. IV), the bimolecular cross initiation rate constant kii2 may be calculated. At 60°C it was found to be 2.8 times that... 

Under normal circumstances, when a material reacts, its initial rate of disappearance is high, but the rate decreases progressively as the reactant is consumed. In an autocatalytic reaction, on the other hand, the initial rate is relatively slow, because little or no product is present. The rate increases to a maximum as products are formed and then decreases to a low value as reactants are consumed or equilibrium is achieved. If there are no product species present in the initial reaction mixture, autocatalytic reactions exhibit the type of behavior shown in Figure 9.13a. If the product species that is catalytic is present in the original reaction mixture, the type of behavior that the system will exhibit is shown in Figure 9.13b. 

From Figure 4 (a) and (b) it is seen that especially the initial rate of adsorption (adsorption within the first 5 min of the experiment) as well as the adsorption isotherm at pH=6 have a near linear behavior. The situation is less clear at pH=3, where the initial rate increases much slower with initial gold concentration and the adsorption isotherm shows non-linear behavior. This shows furthermore, that pH=6 is more favorable for the adsorption of more gold from solution. 

While there have been several studies on the synthesis of block copolymers and on the molecular weight evolution during solution as well as bulk polymerizations (initiated by iniferters), there have been only a few studies of the rate behavior and kinetic parameters of bulk polymerizations initiated by iniferters. In this paper, the kinetics and rate behavior of a two-component initiation system that produces an in situ living radical polymerization are discussed. Also, a model that incorporates the effect of diffusion limitations on the kinetic constants is proposed and used to enhance understanding of the living radical polymerization mechanism. 

The rate behavior is modeled using kinetic expressions based on elementary reactions of the species involved. Generation of radicals can occur through five different initiation mechanisms. First, species such as DMPA or TED can generate either two carbon radicals or two DTC radicals. If XDT-like initiators are considered, one carbon radical and one DTC radical are generated upon photolysis, and a similar reaction for reinitiation of DTC-terminated polymer chains exists. Lastly, initiation of polymer chains by DTC radicals should be included for completeness. These reactions can be summarized as ... 

As the enzyme itself is usually the focus of interest, information on the behavior of that enzyme can be obtained by incubating the enzyme with a suitable substrate under appropriate conditions. A suitable substrate in this context is one which can be quantified by an available detection system (often absorbance or fluorescence spectroscopy, radiometry or electrochemistry), or one which yields a product that is similarly detectable. In addition, if separation of substrate from product is necessary before quantification (for example, in radioisotopic assays), this should be readily achievable. It is preferable, although not always possible, to measure the appearance of product, rather than the disappearance of substrate, because a zero baseline is theoretically possible in the former case, improving sensitivity and resolution. Even if a product (or substrate) is not directly amenable to an available detection method, it maybe possible to derivatize the product with a chemical species to form a detectable adduct, or to subject a product to a second enzymatic step (known as a coupled assay, discussed further later) to yield a detectable product. But, regardless of whether substrate, product, or an adduct of either is measured, the parameter we are interested in determining is the initial rate of change of concentration, which is determined from the initial slope of a concentration versus time plot. 

The major advantage associated with continuous assays is that the initial rate of product formation can be determined with complete confidence, and any unusual behavior of the enzyme would be immediately apparent. The major disadvantage is a question of throughput an instrument such as a platereader would remain dedicated to the reading of a single plate for the duration of the enzyme-substrate incubation period, compared with an equivalent discontinuous assay where an entire plate may be measured in a... 

Perhaps of first concern in determining the overall design of a particular assay is the actual method used for product identification (or for substrate depletion) per unit time. Many different methods have been utilized (e.g., radiometric, spectrophotometric, fluorometric, pH-stat, polarimetric, etc.) No matter which method is used, the product has to be clearly identified (or substrate, if substrate depletion is being measured). With stopped-time assays, it may be necessary to separate product(s) from substrate(s) prior to determination of the amounts of the metabolite(s) present (as well as demonstration that product(s) and substrate(s) are truly separated). If so, the investigator should be able to demonstrate that the assay procedure clearly measures true initial rates (see below). Closely related to these issues are concerns about purity (See Substrate Purity Enzyme Purity Water Purity, etc.) and stability (See Substrate Stability Enzyme Stability, etc.. If the components of the assay mixture are not stable over the time course of the experiment (or, if certain side reactions occur), then corrections have to be made in analyzing the rate behavior. 

One of the basic assumptions in kinetic studies of an enzyme-catalyzed reaction is that true initial rates are being measured. In such cases, a plot of the product concentration versus time must yield a straight line. (This behavior is only observed when the substrate is at or near its initial (or, r = 0) concentration. As time increases, product accumulation and substrate depletion will result in a curvature of this progress curve hence, the reaction velocity at these later times would be correspondingly lower.)... 

The initial rate assumption is one of the most powerful and widely used assumptions in the kinetic characterization of enzyme action. The proper choice of reaction conditions that satisfy the initial rate assumption is itself a challenge, but once conditions are established for initial rate measurements, the kinetic treatment of an enzyme s rate behavior becomes much more tractablek In reporting initial rate data, investigators would be well advised to provide the following information ... 

Figure 2. Illustration of the importance of the choice of reaction conditions on the determination of initial velocity. Shown are four conditions applied to examine the rate behavior of Escherichia coli NAD+-dependent coenzyme A-linked aldehyde dehydrogenase (Reaction NAD+ + CoA-SH + Acetaldehyde = NADH + Acetyl-S-CoA + H+). All assay mixtures contained enzyme, 0.5 mM NAD+, 8 /jlW CoA-SFI, 16 mM acetaldehyde, and 22.5 mM Tris buffer at pFI 8.1. (a) Time-course observed when enzyme was added to the standard assay (b) time-course observed when enzyme was added to standard assay augmented with 10 mM 2-mercaptoethanol (c) time-course observed when enzyme was first preincubated for 15 min with 8 /jlW CoA-SH, 16 mM acetaldehyde, 10 mM 2-mercaptoethanol, and 22.5 mM Tris buffer at pH 8.1, and the reaction was initiated by addition of NAD+ (d) time-course observed when enzyme was preincubated with lOmM 2-mercaptoethanol for 15 min andthen addedtostandard assay augmented with 10 mM 2-mercaptoethanol. The data are most compatible with the idea that the enzyme has an active-site thiol group that must be reduced to express full catalytic activity during assay.

In enzyme-catalyzed kinetics, one must necessarily deal with the behavior of a multistep reaction scheme. For initial rate enzyme processes, one typically deals with collections of rate constants which appear in the form of the maximal velocity Um (shortened to U) or the specificity constant VJK (shortened to VIK). Accordingly, enzyme kineticists will use °V and °V/K as an easy way to indicate the respective isotope effects [(Um)H/(Um)D ] and [(VJK )u/(VJK )b], respectively. 

The quotient of rate constants obtained in steady-state treatments of enzyme behavior to define a substrate s interaction with an enzyme. While the Michaelis constant (with overall units of molarity) is a rate parameter, it is not itself a rate constant. Likewise, the Michaelis constant often is only a rough gauge of an enzyme s affinity for a substrate. 2. Historically, the term Michaelis constant referred to the true dissociation constant for the enzyme-substrate binary complex, and this parameter was obtained in the Michaelis-Menten rapid-equilibrium treatment of a one-substrate enzyme-catalyzed reaction. In this case, the Michaelis constant is usually symbolized by Ks. 3. The value equal to the concentration of substrate at which the initial rate, v, is one-half the maximum velocity (Lmax) of the enzyme-catalyzed reaction under steady state conditions. 

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