Rapid pre-equilibrium

If in the mechanism of Equation (4.1) the first-order rate constant k+2 is much smaller than the first-order constant k-, then the association reaction E + S = ES occurs on a much faster timescale than the transformation ES — E + P. Hence, the first reaction may be assumed to achieve a rapid equilibrium 

The timescale for the association step E + S = ESis (/c+i [S + k-i) l (which is even smaller than 1/k-i), while the timescale for the transformation ES E + P is l/k+2. Thus, ifk-i k+2, the association step equilibrates rapidly compared to the rate of product formation. 

Under the assumption of rapid pre-equilibrium, the rate of product formation is then 

In this case the Michaelis-Menten parameters are Km = k-i/k+i, which is the dissociation constant, and Vmax = k+2E0. Here we have arrived at the familiar Michaelis-Menten result without making the assumption that the substrate concentration [S] is in excess of the enzyme concentration. Recall that the quasi-steady 

Kinetics involving rapid pre-equilibrium steps finds numerous applications both within and beyond the study of enzyme kinetics. Other important examples are the theory of proton-deuterium exchange kinetics of a protein [169] and gene activation involving DNA looping [186], Because of its central importance in biological kinetics, let us provide a more complete mathematical treatment of the problem in a short digression. 

---

During the next fifty years the interest in derivatives of divalent carbon was mainly confined to methylene (CHg) and substituted methylenes obtained by decomposition of the corresponding diazo compounds this phase has been fully reviewed by Huisgen. The first convincing evidence for the formation of dichlorocarbene from chloroform was presented by Hine in 1950. Kinetic studies of the basic hydrolysis of chloroform in aqueous dioxane led to the suggestion that the rate-determining step was loss of chloride ion from the tri-chloromethyl anion which is formed in a rapid pre-equilibrium with hydroxide ions ... 

Michaelis—Menten mechanism A model of enzyme catalysis in which the enzyme and its substrate reach a rapid pre-equilibrium with the bound substrate-enzyme complex. 

The simplest overall interpretation of these data is in terms of a rate-determining dissociation. Entropies of activation are positive and the solvent-dependence for a better leaving group (Cl) is less marked than for a worse one (Br) in the case of reaction (38) . For the dimeric carbonyls, [M(CO)4X]2, bridge-breaking, essentially the same dissociation, could result in a rapid pre-equilibrium. If this were followed by a second dissociative step, then the kinetics could be first-order (as for Mn), while a rate-determining entry of L could produce second-order kinetics (as for Re). 

Preliminary results have been reported of oxidation of cyclobutanol by the Cr(VI)-V(IV) couple to 4-hydroxybutyraldehyde. This proceeds at the same rate as the oxidation of V(IV) by Cr(VI) and cannot involve attack of Cr(V) upon the alcohol, for this oxidation state is formed in a rapid pre-equilibrium, but rather attack by Cr(IV), viz. 

A mechanistic study of acid and metal ion (Ni2+, Cu2+, Zn2+) promoted hydrolysis of [N-(2-carboxyphenyl)iminodiacetate](picolinato)chromate (III) indicated parallel H+- or M2+-dependent and -independent pathways. Solvent isotope effects indicate that the H+-dependent path involves rapid pre-equilibrium protonation followed by rate-limiting ring opening. Similarly, the M2+-dependent path involves rate-determining Cr-0 bond breaking in a rapidly formed binuclear intermediate. The relative catalytic efficiencies of the three metal ions reflect the Irving-Williams stability order (88). 

The reaction was second order in acid and first order in substrate, so both rearrangements and the disproportionation reaction proceed via the doubly-protonated hydrazobenzene intermediate formed in a rapid pre-equilibrium step. The nitrogen and carbon-13 kinetic isotope effects were measured to learn whether the slow step of each reaction was concerted or stepwise. The nitrogen and carbon-13 kinetic isotope effects were measured using whole-molecule isotope ratio mass spectrometry of the trifluoroacetyl derivatives of the amine products and by isotope ratio mass spectrometry on the nitrogen and carbon dioxide gases produced from the products. The carbon-12/carbon-14 isotope... 

Mo2(H20)8+ is one of only three dimeric aqua ions with no bridging ligands (Hgf and Rh are the others). The structure 5 is eclipsed with 6-bond formation and quadruple metal-metal bonding. A rapid pre-equilibrium involving substitution of anion into the axial (end) waters. 

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-... 

In the rapid pre-equilibrium the high-energy cis-isomer is formed, and double hydrogen atom transfer takes place in the last step. It is likely that the value KIE = 3.3 will be typical of other hydrogenations by diazene392. 

Criteria for determining whether an intermediate is formed in simple reactions previously thought to occur in one step have been suggested.125 A mechanism that requires the formation of an intermediate ion-dipole complex in a rapid pre-equilibrium step before the rate-determining substitution is proposed for the XN2 reaction between 4-nitrophenoxide ion and methyl iodide. 

Since there is a rapid pre-equilibrium (Alsub, Fig. 3 A) established in the system, one could write ... 

The physical significance of the rate law is that consumption of alkene occurs by two pathways. One involves a rapid pre-equilibrium between 7.9 and dihydrogen to give 7.10. This is followed by a reaction with the alkene that eventually leads to the formation of the alkane. The other involves a similar equilibrium between 7.9 and alkene to give an alkene complex. This alkene complex then reacts with dihydrogen. In Fig. 7.3 for simplicity only the forward reaction of the former equilibrium has been shown. 

The flux expression in Equation (4.16) displays the canonical Michaelis-Menten hyperbolic dependence on substrate concentration [S], We have shown that this dependence can be obtained from either rapid pre-equilibration or the assumption that [S] [E]. The rapid pre-equilibrium approximation was the basis of Michaelis and Menten s original 1913 work on the subject [140], In 1925 Briggs and Haldane [24] introduced the quasi-steady approximation, which follows from [S] 2> [E], (In his text on enzyme kinetics [35], Cornish-Bowden provides a brief historical account of the development of this famous equation, including outlines of the contributions of Henri [80, 81], Van Slyke and Cullen [203], and others, as well as those of Michaelis and Menten, and Briggs and Haldane.)... 

A form for the flux more general than that of Equation (7.6) is obtained if we do not invoke the rapid pre-equilibrium assumption [108] ... 

The halogenation of ketones is also general acid catalysed. The mechanism usually consists of a rapid pre-equilibrium protonation of the carbonyl group followed by a slow proton transfer from carbon to the base catalyst [41]. The enol thus produced reacts rapidly with halogen. The overall mechanism is similar to mechanism (7) described earlier and the observed rate coefficient is a product of the equilibrium constant for protonation of the carbonyl group and the rate coefficient for the proton transfer from carbon, and therefore does not refer to a single proton transfer step. 

At low concentrations of BH+ such that k2 > k j [BH+], the observed first-order rate coefficient for conditions where buffer is present in excess over reactant is kx [B]. The rate of reaction is determined by a slow proton removal by base from carbon. At high concentrations of BH+, the observed first-order rate coefficient is (k1fe2/k-i)[B]/[BH+]. In this case, if the reaction is carried out in aqueous solution, the rate of reaction depends upon the hydroxide ion concentration and is independent of the buffer concentration at a fixed buffer ratio (specific base catalysis). The mechanism under these conditions consists of rapid pre-equilibrium formation of a carbanion followed by a slow step. Over the whole range of buffer concentration the first-order rate coefficient (M,hs) measured at fixed buffer ratio first increases (/ bs = kl [B]) with buffer concentration but reaches a limiting value (kohs = (ki k2 /k-i) [B] /[BH+]). This change in mechanism has been observed for a limited number of reactions [58]. Reactions (38) [58(a)] and (39) [58(b)] occurring in ethanol and reaction (40) [58(c)] in aqueous... 

Conversely, when both rates are fast, the preceding chemical reaction acts as a rapid pre-equilibrium, and (A) = K(Z). The electrochemical data are then formally identical to that of the virtual Z + ne B electron transfer, with a formal potential... 

Most metal ion-promoted reactions in which labile metal ions are employed involve a reaction scheme of the type shown in equations (2)-(5). There is a rapid pre-equilibrium formation of... 

Kinetic studies suggest that the mechanism involves removal of the N-bonded proton by OH in a rapid pre-equilibrium, followed by the ratedetermining, dissociative decomposition of the resultant conjugate base.168... 

An Id mechanism in which an ion-pair precursor complex is formed in a rapid pre-equilibrium step, equation (4.12), followed by a rate determining loss of H20 and entry of Y- from the outer-sphere of the complex. 

Most metal complex reactions are more complex however, and the kso does not reflect a simple bimolecular collision step. Our reaction between [Co(en)3]2+ and [Co(ox)2en] is an example of such a complex reaction. Recall from the stoichiometric measurements in Experiment 5.4 that this reaction yields two products [Co(en)3]3+ (80%) and [Co(en)2ox]+ (20%), indicating that the reaction proceeds by at least two different pathways. Further, the reaction involves a rapid pre-equilibrium step during which the two reactants are brought together. This complexity is embedded in the second-order rate constant. As an exercise, follow steps 1 -7 below to derive the second-order rate constant. 

---