For the prediction and optimisation of the equilibrium conversion in biphasic catalysed reactions, the equilibrium constant of the desired reaction and the partition coefficients of all reactants have to be known. Within this contribution we have examined the alcohol dehydrogenase-catalysed reduction of several linear and aromatic ketones in biphasic reaction media with respect to equilibrium conversion. In this example, the equilibrium constant can be expressed in terms of differences in oxidation-reduction potentials DE 0 . However, for a large variety of organic compounds, these data are quite rare in the literature. To overcome this lack of data, we have utilised methods of computational chemistry to calculate data for the Gibbs free energy DG R leading to the equilibrium constants of a homologous series of linear ketones. To obtain comparable data for the reduction of substituted acetophenone derivatives, the Hammett relation leads to the necessary equilibrium constants. Furthermore, we compare the equilibrium conversions of a set of cofactor regeneration methods for the alcohol dehydrogenase-catalysed reductions. These results lead to a time-saving experimental design for the enantioselective reduction of 2-octanone to (R)-2-octanol on a preparative scale utilising biphasic reaction conditions.

Using the organic solvents acetonitrile and 1,4-dioxane as water-miscible additives for the alcohol dehydrogenase (ADH)-catalyzed reduction of butan-2-one, we investigated the influence of the solvents on enzyme reaction behavior and enantioselectivity. The NADP(+)-dependent (R)-selective ADH from Lactobacillus brevis (ADH-LB) was chosen as biocatalyst. For cofactor regeneration, the substrate-coupled approach using propan-2-ol as co-substrate was applied. Acetonitrile and 1,4-dioxane were tested from mole fraction 0.015 up to 0.1. Initial rate experiments revealed a complex kinetic behavior with enzyme activation caused by the substrate butan-2-one, and increasing K(M) values with increasing solvent concentration. Furthermore, these experiments showed an enhancement of the enantioselectivity for (R)-butan-2-ol from 37% enantiomeric excess (ee) in pure phosphate buffer up to 43% ee in the presence of 0.1 mol fraction acetonitrile. Finally, the influence of the co-solvents on water activity of the reaction mixture and on enzyme stability was investigated.

Asymptotic expansions for large deviation probabilities are used to approximate the cumulative distribution functions of noncentral generalized chi-square distributions, preferably in the far tails. The basic idea of how to deal with the tail probabilities consists in first rewriting these probabilities as large parameter values of the Laplace transform of a suitably defined function f k ; second making a series expansion of this function, and third applying a certain modification of Watson's lemma. The function f k is deduced by applying a geometric representation formula for spherical measures to the multivariate domain of large deviations under consideration. At the so-called dominating point, the largest main curvature of the boundary of this domain tends to one as the large deviation parameter approaches infinity. Therefore, the dominating point degenerates asymptotically. For this reason the recent multivariate asymptotic expansion for large deviations in Breitung and Richter (1996, J. Multivariate Anal. 58, 1 20) does not apply. Assuming a suitably parametrized expansion for the inverse g~& 1 of the negative logarithm of the density-generating function, we derive a series expansion for the function f k . Note that low-order coefficients from the expansion of g~& 1 influence practically all coefficients in the expansion of the tail probabilities. As an application, classification probabilities when using the quadratic discriminant function are discussed. Academic PressAMS 1991 subject classifications: 41A60, 41A63, 60F10, 62E17, 62E20, 62H30.

Abstract. A geometric approach to asymptotic expansions for large-deviation probabilities, developed for the Gaussian law by Breitung and Richter [J. Multivariate Anal., 58, 1-20 (1996)], will be extended in the present paper to the class of spherical measures by utilizing their common geometric properties. This approach consists of rewriting the probabilities under consideration as large parameter values of the Laplace transfoma of a suitably defined function, expanding this function in a power series, and then applying Watson's lemma. A geometric representation of the Laplace transform allows one to combine the global and local properties of both the underlying measure and the large-deviation domain. A special new type of difficulty is to be dealt with because the so-called dominating points of the large-deviation domain degenerate asymptotically. As is shown in Richter and Schumacher (in print), the typical statistical applications of large-deviation theory lead to such situations. In the present paper, consideration is restricted to a certain two-dimensional domain of large-deviations having asymptotically degenerating dominating points. The key assumption is a parametrized expansion for the inverse ~-1 of the negative logarithm of the density-generating function of the two-dimensional spherical law under consideration,

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