Linear Potentials in Nonlinear Potential Theory

Pointwise gradient bounds via Riesz potentials, such as those available for the linear Poisson equation, actually hold for general quasilinear degenerate equations of p-Laplacean type. The regularity theory of such equations completely reduces to that of the classical Poisson equation up to the C ¹-level.     

Riesz Potentials and Nonlinear Parabolic Equations

The spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric Riesz potential of the right hand side datum, exactly as in the case of the heat equation. Heat kernels type estimates persist in the nonlinear case.     

Molecular dynamics simulations for human CAR inverse agonists

Constitutive androstane receptor (CAR), along with pregnane x receptor (PXR), is an important metabolic sensor in the hepatocytes. Like all other nuclear receptors (NRs), CAR works in concert with coregulator proteins, coactivators, and corepressors which bind to the NRs. The main basis for the receptor to distinguish between coactivators and corepressors is the position of the C-terminal helix 12 (H12), which is determined by the bound NR ligand. CAR, having constitutive activity, can be repressed or further activated by its ligands. Crystal structure of human CAR bound to an agonist and a coactivator peptide is available, but no structural information on an inverse agonist-bound human CAR and a corepressor exists. In our previous molecular dynamics (MD) studies, no corepressor peptide was included. Therefore, probably due to the strong interactions which keep the relatively short H12 of CAR in the active position, the structural changes elicited by inverse agonists were very subtle, and H12 of CAR seemed to more or less retain its active conformation. Here, we have run a series of MD simulations to study the movement of H12 in the presence of both activating and repressing ligands as well as a corepressor peptide. The presence of the corepressor on the coregulator surface of CAR induced a clear shift of H12 of the inverse agonists-bound CAR. In general, H12 moved toward H10 and not away from the ligand binding domain, as seen in some other NRs. However, H12 of CAR is short enough that this movement seems to be adequate to accommodate the binding of the corepressor.     

H?lder continuity for Trudinger’s equation in measure spaces

We complete the study of the regularity for Trudinger’s equation by proving that weak solutions are H?lder continuous also in the singular case. The setting is that of a measure space with a doubling non-trivial Borel measure supporting a Poincaré inequality. The proof uses the Harnack inequality and intrinsic scaling.     

Large-Scale Analyticity and Unique Continuation for Periodic Elliptic Equations

We prove that a solution of an elliptic operator with periodic coefficients behaves on large scales like an analytic function in the sense of approximation by polynomials with periodic corrections. Equivalently, the constants in the large-scale C^{k, 1} estimate scale exponentially in k , just as for the classical estimate for harmonic functions, and the minimal scale grows at most linearly in k . As a consequence, we characterize entire solutions of periodic, uniformly elliptic equations that exhibit growth like O(exp(δ| x| )) for small δ > 0 . The large-scale analyticity also implies quantitative unique continuation results, namely a three-ball theorem with an optimal error term as well as a proof of the nonexistence of L² eigenfunctions at the bottom of the spectrum. © 2020 Wiley Periodicals LLC.     

Production of milligram quantities of affinity tagged-proteins using automated multistep chromatographic purification

A new chromatography system, AKTAxpress (GE Healthcare, Amersham Biosciences, Uppsala, Sweden) has been designed to meet the demand for high-throughput purification of proteins in structural genomics and drug discovery. The system offers a number of automated multistep purification protocols for affinity-tagged proteins. All protocols start with affinity chromatography followed by combinations of desalting, ion exchange chromatography and gel filtration. As an option, tag removal can be included in the purification protocols. Up to 16 proteins can be purified per day and the yield can be as high as 50 mg of each protein at > 90% purity. To highlight the versatility of the system, this paper presents several case studies; purifications of hexahistidine- and glutathione S-transferase-tagged proteins using different protocols, automated on-column tag cleavage and optimization studies for a hexahistidine-tagged kinase.