SUB-OPTIMAL CONVERGENCE OF DISCONTINUOUS GALERKIN METHODS WITH CENTRAL FLUXES FOR LINEAR HYPERBOLIC EQUATIONS WITH EVEN DEGREE POLYNOMIAL APPROXIMATIONS

In this paper,we theoretically and numerically verify that the discontinuous Galerkin(DG)methods with central fluxes for linear hyperbolic equations on non-uniform meshes have sub-optimal convergence properties when measured in the L^(2)-norm for even degree polynomial approximations.On uniform meshes,the optimal error estimates are provided for arbitrary number of cells in one and multi-dimensions,improving previous results.The theoretical findings are found to be sharp and consistent with numerical results.

Unstructured kinetic models to simulate an arabinose switch that decouples cell growth from metabolite production

Modeling synthetic gene circuits to implement dynamic flux balancing is crucial in teaching and exploring metabolic engineering strategies to repartition metabolic precursors and construct efficient microbial cell factories.Microbial fitness and production rates are often complex phenotypes that are governed by highly nonlinear,multivariable functions which are intrinsically linked through carbon metabolism.The solution of such dynamic system can be difficult for synthetic biologists to visualize or conceptualize.Recently,researchers(Santala et al.,Metab.Eng.Comm.,2018)have implemented an arabinose based genetic switch to dynamically partition the central carbon flux between cell growth and product formation.The autonomous switch allowed dynamic shift from arabinose-associated cell growth to acetate-associated product(wax ester)formation.This system clearly demonstrates the effectiveness of using a genetic switch to decouple cell growth from product formation in a one-pot bioreactor to minimize operational cost.Coupled with Michaelis-Menten kinetics,and Luedeking-Piret equations,we were able to reconstruct and analyze this metabolic switch in silica and achieved graphical solutions that qualitatively match with the experimental data.By assessing physiologically-accessible parameter space,we observed a wide range of dynamic behavior and examined the different limiting cases.Graphical solutions for this dynamic system can be viewed simultaneously and resolved in real time via buttons on the graphical user interface(GUI).Metabolic bottlenecks in the system can be accurately predicted by varying the respective rate constants.The GUI serves as a diagnosis toolkit to troubleshoot genetic circuits design constraints and as an interactive workflow of using this arabinose based genetic switch to dynamically control carbon flux,which may provide a valuable computational toolbox for metabolic engineers and synthetic biologists to simulate and understand complex genetic-metabolic system.