A Method for Fast Diagonalization of a 2x2 or 3x3 Real Symmetric Matrix

A method is presented for fast diagonalization of a 2x2 or 3x3 real symmetric matrix, that is determination of its eigenvalues and eigenvectors. The Euler angles of the eigenvectors are computed. A small computer algebra program is used to compute some of the identities, and a C++ program for testing the formulas has been uploaded to arXiv.Comment: Corrected formula 4.1

Properties of epoxide hydrolase from the yeast Rhodotorula glutinis

Epoxide hydrolases are ubiquitous enzymes that can be found in nearly all living organisms. Some of the enzymes play an important role in detoxifying xenobiotic and metabolic compounds. Others are important in the growth of organisms like the juvenile hormone in some insects. The role of these enzymes in some organisms is still not fully understood.Epoxides are highly reactive valuable intermediates used by the pharmaceutical industry. Enantiopure epoxides are of high value in the production of pharmaceuticals like pain-killers or protease-inhibitors. There are a number of ways to produce enantiopure epoxides, but nowadays an environmentally friendly manner has a high preference. One such environmentally friendly method is the use of the epoxide hydrolases. These enzymes are able to enantioselectively hydrolyze one epoxide-enantiomer to its vicinal diol. By this so-called kinetic resolution, it is possible to obtain both the epoxide and diol enantiopure. Enantiopure diols are also of high value in the fine and pharmaceutical chemistry.The first goal of this project is achieved: the epoxide hydrolase from Rhodotorula glutinis has been isolated and purified. The second goal (optimization of the reaction conditions) has been performed but it is still favorable to further optimize them. Initial experiments of enzyme stability towards temperature and pH has been performed with crude enzyme extracts, not with the purified enzyme. With respect to the third goal (a suitable method for the isolation and separation of epoxide and diol), the use of a recently described membrane reactor is recommended. The performance of this reactor, however, has not been verified for the EH studied.The enzyme is partially characterized. The EH was found to be a membrane associated enzyme. Whether or not it is actually a membrane bound enzyme (and how many times it passes the membrane) is still unknown. The enzyme consists of two (most probably) identical subunits with a molecular mass of 45 kDa. Amino acid analysis revealed that the enzyme belongs to thea/b-hydrolase fold family, because the characteristic catalytic histidine motive (G H F) can be found in the amino acid sequence. The N-terminal sequence, however, could not be detected. The amount of purified enzyme was too low to establish its the three-dimensional structure.With partially purified enzyme sample, the specific activity and enantioselectivity could be enhanced when detergents were added. Non-ionic detergents had the largest positive effects, e.g. the specific activity for 1,2-epoxyhexane and styrene oxide was enhanced three and eight times, respectively. In the same way, the enantioselectivity for 1,2-epoxyhexane and styrene oxide could be enhanced over 10 and nearly 5 times, respectively. In addition, non-ionic detergents had an enzyme stabilizing effect. Anionic detergents had a very clear negative effect: enzyme activities were reduced to 20%.Another method investigated to influence the stability, the activity and the enantioselectivity consists of polymerizing the epoxide hydrolase in a network. The enantioselective conversion of (±)-1,2-epoxyoctane was reversed from a preference for ( R )-1,2-epoxyoctane to ( S )-1,2-epoxyoctane when the enzyme had been imprinted with ( S )-1,2-epoxyoctane prior to co-polymerization. This is the first time that the above mentioned method was successfully performed with a membrane-associated enzyme of thea/b-hydrolase fold family to which EH belongs. The half-life of the immobilized and imprinted biocatalyst was enhanced at least 7-fold. Most remarkable was that washing the immobilized EH with HCl, followed by washing it with buffer, resulted in about 50% of the residual activity, while native EH completely lost its activityThe effect of increasing epoxide amounts (up to 10 mmol per 10 mL of water, leading to phase separations) on both the activity and enantioselectivity has been studied, including the effect of detergents on such two-phase enzymatic conversions. It appeared that cell-free extracts without detergents gave the highest activity at 10 mmol epoxide per 10 mL of water added, without loss of enantioselectivity as compared to 1 mmol epoxide per 10 mL of water emulsions.It is recommended either to use whole cells (overexpressing EH or not) in a bioreactor to produce enantiopure epoxides and diols in large quantities or to use an immobilized-imprinted enzyme polymer for the conversion of smaller quantities of epoxides in an enantioselectivity of your choice. Further investigations to improve both methods (whole cells, the enzyme properties and the immobilization and imprinting procedures) are required to optimize this type of conversions for practical applications