Diffusion controlled encounter rates

Comparison with Diffusion Controlled Encounter Rates. 

As applied to proton transfers, the theory takes the form of equations 1.7 and 1.8, where kdiff is the diffusion-controlled encounter rate. AG is the free energy of activation defined by equation 1.8, AG° being the standard equilibrium free energy change and AG the intrinsic barrier , the free energy of activation when AG° = 0, a constant of the particular type of reaction. 

The Stem-Volmer equation (6.19) was derived above on the assumption that the fonna-tion of the encounter-complex which precedes quenching is not reversible. If we assume that only a fraction (y) of encounter-complexes result in reaction, while the rest dissociate before quenching, we obtain a modified Stem-Volmer equation (6.23), in which ko is the diffusion-controlled encounter rate ... 

The quantity kcat/Km is a rate constant that refers to the overall conversion of substrate into product. The ultimate limit to the value of k at/Km is therefore set by the rate constant for the initial formation of the ES complex. This rate cannot be faster than the diffusion-controlled encounter of an enzyme and its substrate, which is between 10 to 10 per mole per second. The quantity kcat/Km is sometimes called the specificity constant because it describes the specificity of an enzyme for competing substrates. As we shall see, it is a useful quantity for kinetic comparison of mutant proteins. 

Based upon their data and upon results in the literature, the authors concluded that hydrogenations using 24 or related species as catalyst precursor proceed in solution by mechanisms involving iridium(I)/(III) formal oxidation states. During the course of their discussions, the authors made the interesting observation that the rate of gas-phase collisions between the thermalized iridium organometallic ions and D2 under their experimental conditions in the oc-topole were similar to the rate of diffusion-controlled encounters between iridium species and D2 in solution. 

Separation of an ion-pair to free ions, with rate constant d, has been used as a clock for other reactions of the ion pair. Estimates of d can be obtained from the simple relationship between the association constant for ion-pair formation (Xas, equation (2)) and the rate constant for diffusion-controlled encounter... 

Table 4.5 shows that for some efficient enzymes, kcJKM may be as high as 3 X 108 s-1 M l. In these cases, the rate-determining step for this parameter, which is the apparent second-order rate constant for the reaction of free enzyme with free substrate, is close to the diffusion-controlled encounter of the enzyme and the substrate. Briggs-Haldane kinetics holds for these enzymes (Chapter 3, section B3). 

We can use the two hypothetical steps of section Clb i.e., that kcJKM be maximized and that KM be greater than [S], to set up criteria for judging the state of evolution of an enzyme whose function is to maximize rate. We recall from Chapter 3 that the maximum value of kcJKM is the rate constant for the diffusion-controlled encounter of the enzyme and substrate, and from Chapter 4 that this is about 108 to 109 s "1 M l. A perfectly evolved enzyme should have a kcJKM in the range of 108 to 109 s"1 and a KM greater than [S]. Using the data for kcJKM listed in Table 4.4 and the substrate concentrations and KM values mentioned in this chapter, it appears that carbonic anhydrase and triosephosphate isomerase are perfectly evolved for the maximization of rate, which agrees with the conclusions of W. J. Albery and J. R. Knowles on triosephosphate isomerase.5... 

The molecules M and Q can come into contact (within the sphere of action) through their random diffusional motion. The rate constant kD of diffusion controlled encounters is the upper limit for any bimolecular reaction. This must be multiplied by the probability of an encounter leading to reaction (quenching in the present case), and the luminescence quantum yield then follows the Stern-Volmer equation... 

The rates for many of the e aq reactions in Table II are very fast, exceeding 1010M-1 sec.-1, and therefore, may be limited by the rates of diffusion-controlled encounters. The equation from which the diffusion-limited rate constants may be calculated for ionic species is due to Debye... 

Suppose that the rate of formation of product (k c t) is much faster than the rate of dissociation of the ES complex (A . j). The value of k jyi then approaches k j. Thus, the ultimate limit on the value of k JK is set by A j, the rate of formation of the ES complex. This rate cannot be faster than the diffusion-controlled encounter of an enzyme and its substrate. Diffusion limits the value of j so that it cannot be higher than between 10 and 10 s l M-f Hence, the upper limit on k. JK jyj is between 10 and 10 s i M f... 

The rate-limiting step in proton transfer between electronegative atoms is either /c, the diffusion-controlled encounter of the acid-base pair for thermodynamically favourable transfers or the diffusion-controlled dissociation of the acid-base pair, for thermodynamically unfavourable transfers (Figure 1). The rate-limiting step is rarely the proton transfer within the encounter complex. Eigen plots are predominantly observed in reactions in which proton transfer occurs to or from heteroatoms in reactive intermediates. 

Chemical reaction kinetics proceeds on the (often implicit) assumption that the reaction mixture is ideally mixed, and does not consider the time needed for reacting species to encounter each other by diffusion. The encounter rate follows from the theory of Smoluchowski. It turns out that most reactions in fairly dilute solutions follow chemical kinetics, but that reactions in low-moisture foods may be diffusion controlled. In the Bodenstein approximation, the Smoluchowski theory is combined with a limitation caused by an activation free energy. Unfortunately, the theory contains several uncertainties and unwarranted presumptions. 

Temperature dependence studies were undertaken in order to evaluate the activation energy of the association process. The association rates obtained by monitoring the disappearance of the free probe absorption, X ax = 990 nm, yielded remarkably linear Arrhenius plots (Figure 6) with the activation energies of 2.4 0.2 kcal/mol, that is, the activation energy for diffusion of a small molecule in THF. Therefore, it can be concluded that the primaiy event of the association process is a barrierless, diffusion-controlled encounter between the probe molecule and an ion pair of the salt. The association rates for TBAF were consistently -20% higher than for the somewhat more bulky TBABr. The... 

Although the kinetics of binding may control the reaction if enzyme complementarity is too good and the mobility of the enzyme is too restricted, this is unlikely to be a general situation. Even if, because of weak binding, the rate of dissociation of an initially formed enzyme-substrate complex was fast, say 10 /sec, there is still plenty of time for conformational changes which may take place in less than 10 ° sec. This is compatible with those enzyme-catalysed reactions where the rate-limiting step is diffusion-controlled encounter of the enzyme and substrate. 

When the rate-limiting step of a reaction such as that of Eqn. 7 is the diffusion-controlled encounter of two reagents an enzyme may increase the rate simply by having the nucleophile and catalyst preassociated i.e. they do not have to diffuse through solution before reaction can occur. This situation occurs [3] in simple intermolecular reactions when the rate of breakdown of the intermediate to regenerate reactants is faster than the rate of separation of the intermediate and catalyst, 10 -10"/sec, and when proton transfer between the intermediate and catalyst is thermodynamically favourable. The formation of the intermediate must take place by a preliminary association of the reactants and catalyst (Eqn. 9 and Fig. 3). 

An example of a proton-transfer reaction occurring by the stepwise trapping mechanism (Ch. 7, Section 2.1) is the general base-catalysed aminolysis of benzyl-penicillin. Amines react with penicillin to form an unstable tetrahedral intermediate which may be trapped by a diffusion-controlled encounter with a strong base as shown in Eqn. 8. Diamines also undergo this reaction but at a much faster rate than monoamines of the same basicity which is attributed to intramolecular general base catalysis i.e. the second amino group acts as a proton acceptor (V). However, the effect of intramolecularity itself is small and the effective concentration (Ch. 1)... 

There is an upper limit for the value of 2/ M which is contingent upon the rate of diffusion-controlled encounters of enzyme and substrate molecules, i.e., the rate of product formation is no longer limited by the reaction rate but by the diffusion rate. In an aqueous solution, this limiting value lies between 10 and 10 molL s (compare Sect. 20.2). Enzymes such as catalase that exhibit a value of k 2jK of this order of magnitude are considered to be (almost) catalytically perfect because (almost) every contact between enzyme and substrate leads to a reaction. 

Determining the effect of solvent on the rate or course of a reaction can often provide insight into the reaction mechanism. One solvent property that may be important in extremely fast reactions is viscosity. If a reaction is encounter-controlled (also termed diffusion-controlled), the rate constant for the reaction is limited by the ability of the reacting species to reach other. For example, in aqueous solution, the second-order rate constant for the encoxmter of two species is ca. 10 °Lmol In such cases, changing fromalower viscosity... 

Diffusion-Controlled Encounter. Elementary bimolecular reaction mechanisms require diffiisional encovmter before the reaction. If the intrinsic kinetics are fast, and/or the viscosity of the solution is high, diffusion-controlled encounter may occur. In a homogeneous medium, a rate constant /jdiff can be evaluated which reflects the effective bulk-averaged rate constant associated with bimolecular encounters (45). Diffusional bimolecular encounter should be considered in the appropriate context. If Areact is the intrinsic bimolecular rate constant and djff is the differential rate constant defined above, then the observed rate constant for the bimolecular reaction is given by equation (11) (46). The limiting cases of this equation can be readily identified that is when the rate constant is very large, the observed rate constant corresponds to the diffusional rate constant. 

Bimolecular reactions involving radicals typically possess very low activation energies and very high intrinsic rate constants. Diffusion-controlled encounter can occur even in low viscosity solvents. Radicals are known to be involved in some classes of homogeneous catalytic reactions (47). 

The mechanism of Scheme 8 is compatible with this observation. At low concentrations of hydroxide ion the rate of collapse of the tetrahedral intermediate to reactants must be faster than its reaction with hydroxide ion ( -1 At2[OH ]) the observed rate constant is dependent upon the concentration of hydroxide ion with the diffusion-controlled step, being rate-limiting, The calculated pAT,-values for the protonated amine of the tetrahedral intermediates are well below that for water. Proton transfer from the tetrahedral intermediate to hydroxide ion is therefore in the thermodynamically favourable direction and it is to be expected that the rate-limiting step for this process is the diffusion-controlled encounter of the proton donor and acceptor. 

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