Enzymatic Conversion of Flavonoids using Bacterial Chalcone Isomerase and Enoate Reductase

Flavonoids are a large group of plant secondary metabolites with a variety of biological properties and are therefore of interest to many scientists, as they can lead to industrially interesting intermediates. The anaerobic gut bacterium Eubacterium ramulus can catabolize flavonoids, but until now, the pathway has not been experimentally confirmed. In the present work, a chalcone isomerase (CHI) and an enoate reductase (ERED) could be identified through whole genome sequencing and gene motif search. These two enzymes were successfully cloned and expressed in Escherichia coli in their active form, even under aerobic conditions. The catabolic pathway of E. ramulus was confirmed by biotransformations of flavanones into dihydrochalcones. The engineered E. coli strain that expresses both enzymes was used for the conversion of several flavanones, underlining the applicability of this biocatalytic cascade reaction.     

On the Convergence Behaviour of Variable Stepsize Multistep Methods for Singularly Perturbed Problems

In this note, we investigate the convergence behaviour of linear multistep discretizations for singularly perturbed systems, emphasising the features of variable stepsizes. We derive a convergence result for A()-stable linear multistep methods and specify a refined error estimate for backward differentiation formulas. Important ingredients in our convergence analysis are stability bounds for non-autonomous linear problems that are obtained by perturbation techniques.     

Convergence analysis of high-order time-splitting pseudo-spectral methods for rotational Gross–Pitaevskii equations

A convergence analysis of time-splitting pseudo-spectral methods adapted for time-dependent Gross–Pitaevskii equations with additional rotation term is given. For the time integration high-order exponential operator splitting methods are studied, and the space discretization relies on the generalized-Laguerre–Fourier spectral method with respect to the $(x,y)$ -variables as well as the Hermite spectral method in the $z$ -direction. Essential ingredients in the stability and error analysis are a general functional analytic framework of abstract nonlinear evolution equations, fractional power spaces defined by the principal linear part, a Sobolev-type inequality in a curved rectangle, and results on the asymptotical distribution of the nodes and weights associated with Gauß–Laguerre quadrature. The obtained global error estimate ensures that the nonstiff convergence order of the time integrator and the spectral accuracy of the spatial discretization are retained, provided that the problem data satisfy suitable regularity requirements. A numerical example confirms the theoretical convergence estimate.     

Full Discretisations for Nonlinear Evolutionary Inequalities Based on Stiffly Accurate Runge–Kutta and hp-Finite Element Methods

The convergence of full discretisations by implicit Runge–Kutta and nonconforming Galerkin methods applied to nonlinear evolutionary inequalities is studied. The scope of applications includes differential inclusions governed by a nonlinear operator that is monotone and fulfills a certain growth condition. A basic assumption on the considered class of stiffly accurate Runge–Kutta time discretisations is a stability criterion which is in particular satisfied by the Radau IIA and Lobatto IIIC methods. In order to allow nonconforming hp-finite element approximations of unilateral constraints, set convergence of convex subsets in the sense of Glowinski–Mosco–Stummel is utilised. An appropriate formulation of the fully discrete variational inequality is deduced on the basis of a characteristic example of use, a Signorini-type initial-boundary value problem. Under hypotheses close to the existence theory of nonlinear first-order evolutionary equations and inequalities involving a monotone main part, a convergence result for the piecewise constant in time interpolant is established.     

Backward Euler discretization of fully nonlinear parabolic problems

This paper is concerned with the time discretization of nonlinear evolution equations. We work in an abstract Banach space setting of analytic semigroups that covers fully nonlinear parabolic initial-boundary value problems with smooth coefficients. We prove convergence of variable stepsize backward Euler discretizations under various smoothness assumptions on the exact solution. We further show that the geometric properties near a hyperbolic equilibrium are well captured by the discretization. A numerical example is given.