Auxiliary kinetics transformation

In the first case, the limit (for t- co) distribution for the auxiliary kinetics is the well-studied stationary distribution of the cycle A A , +2, described in Section 2 (ID-QS), (15). The set A j+], A . c+2, , n is the only ergodic component for the whole network too, and the limit distribution for that system is nonzero on vertices only. The stationary distribution for the cycle A i+] A t+2. ., A A t+i approximates the stationary distribution for the whole system. To approximate the relaxation process, let us delete the limiting step A A j+] from this cycle. By this deletion we produce an acyclic system with one fixed point, A , and auxiliary kinetic equation (33) transforms into... 

Again we should analyze, whether this new cycle is a sink in the new reaction network, etc. Finally, after a chain of transformations, we should come to an auxiliary discrete dynamical system with one attractor, a cycle, that is the sink of the transformed whole reaction network. After that, we can find stationary distribution by restoring of glued cycles in auxiliary kinetic system and applying formulas (11)-(13) and (15) from Section 2. First, we find the stationary state of the cycle constructed on the last iteration, after that for each vertex Ay that is a glued cycle we know its concentration (the sum of all concentrations) and can find the stationary distribution, then if there remain some vertices that are glued cycles we find distribution of concentrations in these cycles, etc. At the end of this process we find all stationary concentrations with high accuracy, with probability close to one. 

The fast stage of relaxation of a complex reaction network could be described as mass transfer from nodes to correspondent attractors of auxiliary dynamical system and mass distribution in the attractors. After that, a slower process of mass redistribution between attractors should play a more important role. To study the next stage of relaxation, we should glue cycles of the first auxiliary system (each cycle transforms into a point), define constants of the first derivative network on this new set of nodes, construct for this new network an (first) auxiliary discrete dynamical system, etc. The process terminates when we get a discrete dynamical system with one attractor. Then the inverse process of cycle restoration and cutting starts. As a result, we create an explicit description of the relaxation process in the reaction network, find estimates of eigenvalues and eigenvectors for the kinetic equation, and provide full analysis of steady states for systems with well-separated constants. 

The expression KR emphasizes that the racemic mixture undergoes a separation under a chiral influence in a kinetically controlled process. In principle, the word resolution refers to the isolation of one of the enantiomers of racemic mixture after a partial transformation of the initial mixture. If the reaction product is chiral, as in the esterification of a racemic alcohol, then the KR will afford a product with some enantiomeric excess. The full transformation of a racemic mixture by coupling with a chiral auxiliary will give a 1 1 mixture of diastereomers and is not considered as a KR process, unless the reaction is stopped at an intermediate stage, leaving some enantioenriched starting material. 

In racemic resolution processes a racemic mixture of the desired product is produced first. There are several techniques by which this mixture can be separated into its two enantiomers. A favorable option is to react the racemic mixture with another chiral compound to form diastereomers. The latter have different physicochemical properties and thus they can be separated, for example, by chromatographic or crystallization processes. After separation of the diasteromers the chiral auxiliary compound is split-off and separated to re-obtain the desired compound as pure enantiomer. In an alternative concept, called kinetic racemic resolution, the initial racemic mixture is reacted with a chiral reactant or in the presence of a chiral catalyst (e.g., an enzyme) and only one of the two enantiomers of the desired product is transformed into a new compound. The reacted and non-reacted enantiomers are usually easily separated. All processes of racemic resolution have the common disadvantage that both enantiomers, the desired and the undesired one, have to be synthesized initially. Consequently, half of the initial racemic mixture is the undesired enantiomer, which usually has no or very little commercial value. This problem is partialy solved by applying racemization processes in which after separation the pure wrong enantiomer is re-converted into the racemic mixture. The latter is then applied in another round of racemic resolution again to increase the final yield of the desired enantiomer. 

The enzyme-catalyzed kinetic asymmetric transformation (KAT) of a diastereomeric 1 1 syn anti mixture is limited to a maximum theoretical yield of 25% of one enantiomer. This important drawback has been overcome by the combination of the actions of a ruthenium complex and a lipase in a dynamic kinetic asymmetric transformation (DYKAT), the desymmetrization of racemic or diastereomeric mixtures involving interconverting diastereomeric intermediates, implying different equilibration rates of the stereoisomers. Thus, this strategy allows the preparation of optically active diols, widely employed in organic and medicinal chemistry, as they are an important source of chiral auxiliaries and ligands and they can be easily employed as precursors of much other functionality. 

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