Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps

We prove a global implicit function theorem. In particular we show that any Lipschitz map ${f : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}}$ (with n-dim. image) can be precomposed with a bi-Lipschitz map ${\bar{g} : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n} \times \mathbb{R}^{m}}$ such that ${f \circ \bar{g}}$ will satisfy, when we restrict to a large portion of the domain ${E \subset \mathbb{R}^{n} \times \mathbb{R}^{m}}$ , that ${f \circ \bar{g}}$ is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the map ${\bar{g}}$ distorts ${\mathbb{R}^{n+m}}$ in a controlled manner so that the fibers of f are straightened out. Furthermore, our results stay valid when the target space is replaced by any metric space. A main point is that our results are quantitative: the size of the set E on which behavior is good is a significant part of the discussion. Our estimates are motivated by examples such as Kaufman’s 1979 construction of a C ¹ map from [0, 1]³ onto [0, 1]² with rank ≤ 1 everywhere. On route we prove an extension theorem which is of independent interest. We show that for any D ≥ n, any Lipschitz function ${f : [0,1]^{n} \rightarrow \mathbb{R}^{D}}$ gives rise to a large (in an appropriate sense) subset ${E \subset [0,1]^{n}}$ such that ${f|_E}$ is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on all of ${\mathbb{R}^{n}}$ . This extends results of Jones and David, from 1988. As a simple corollary, we show that n-dimensional Ahlfors–David regular spaces lying in ${\mathbb{R}^{D}}$ having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in ${\mathbb{R}^{D}}$ . This was previously known only for D ≥ 2n?+?1 by a result of David and Semmes.     

Charge-transfer and spin dynamics in DNA hairpin conjugates with perylenediimide as a base-pair surrogate

A perylenediimide chromophore (P) was incorporated into DNA hairpins as a base-pair surrogate to prevent the self-aggregation of P that is typical when it is used as the hairpin linker. The photoinduced charge-transfer and spin dynamics of these hairpins were studied using femtosecond transient absorption spectroscopy and time-resolved EPR spectroscopy (TREPR). P is a photooxidant that is sufficiently powerful to quantitatively inject holes into adjacent adenine (A) and guanine (G) nucleobases. The charge-transfer dynamics observed following hole injection from P into the A-tract of the DNA hairpins is consistent with formation of a polaron involving an estimated 3-4 A bases. Trapping of the (A 3-4) (+*) polaron by a G base at the opposite end of the A-tract from P is competitive with charge recombination of the polaron and P (-*) only at short P-G distances. In a hairpin having 3 A-T base pairs between P and G ( 4G), the radical ion pair that results from trapping of the hole by G is spin-correlated and displays TREPR spectra at 295 and 85 K that are consistent with its formation from (1*)P by the radical-pair intersystem crossing mechanism. Charge recombination is spin-selective and produces (3*)P, which at 85 K exhibits a spin-polarized TREPR spectrum that is diagnostic of its origin from the spin-correlated radical ion pair. Interestingly, in a hairpin having no G bases ( 0G), TREPR spectra at 85 K revealed a spin-correlated radical pair with a dipolar interaction identical to that of 4G, implying that the A-base in the fourth A-T base pair away from the P chromophore serves as a hole trap. Our data suggest that hole injection and transport in these hairpins is completely dominated by polaron generation and movement to a trap site rather than by superexchange. On the other hand, the barrier for charge injection from G (+*) back onto the A-T base pairs is strongly activated, so charge recombination from G (or even A trap sites at 85 K) most likely proceeds by a superexchange mechanism.     

Proton positions in the Mn(2+) binding site of concanavalin A as determined by single-crystal high-field ENDOR spectroscopy

High-field (95 GHz) pulsed EPR and electron-nuclear double resonance (ENDOR) techniques have been used for the first time to determine coordinates of ligand protons of a high-spin metal center in a protein single crystal. The protein concanavalin A contains a Mn(2+) ion which is coordinated to two water molecules, a histidine residue, and three carboxylates. Single crystals of concanavalin A were grown in H(2)O and in D(2)O to distinguish the exchangeable water protons from the nonexchangeable protons of the imidazole group. Distinct EPR transitions were selected by performing the ENDOR measurements at different magnetic fields within the EPR spectrum. This selection, combined with the large thermal polarization achieved at 4.5 K and a magnetic field of approximately 3.4 T allowed us to assign the ENDOR signals to their respective M(S) manifolds, thus providing the signs of the hyperfine couplings. Rotation patterns were acquired in the ac and ab crystallographic planes. Two distinct crystallographic sites were identified in each plane, and the hyperfine tensors of two of the imidazole protons and the four water protons were determined by simulations of the rotation patterns. All protons have axially symmetric hyperfine tensors and, by applying the point-dipole approximation, the positions of the various protons relative to the Mn(2+) ion were determined. Likewise, the water protons involved in H-bonding to neighboring residues were identified using the published, ultrahigh-resolution X-ray crystallographic coordinates of the protein (Deacon et al. J. Chem. Soc., Faraday Trans. 1997, 93(24), 4305-4312).     

Anti-Herpes Simplex 1 Activity of Simmondsia chinensis (Jojoba) Wax

Jojoba (Simmondsia chinensis (Link) Schneider) wax is used for various dermatological and pharmaceutical applications. Several reports have previously shown beneficial properties of Jojoba wax and extracts, including antimicrobial activity. The current research aimed to elucidate the impact of Jojoba wax on skin residential bacterial (Staphylococcus aureus and Staphylococcus epidermidis), fungal (Malassezia furfur), and virus infection (herpes simplex 1; HSV-1). First, the capacity of four commercial wax preparations to attenuate their growth was evaluated. The results suggest that the growth of Staphylococcus aureus, Staphylococcus epidermidis, and Malassezia furfur was unaffected by Jojoba in pharmacologically relevant concentrations. However, the wax significantly attenuated HSV-1 plaque formation. Next, a complete dose–response analysis of four different Jojoba varieties (Benzioni, Shiloah, Hatzerim, and Sheva) revealed a similar anti-viral effect with high potency (EC₅₀ of 0.96 ± 0.4 µg/mL) that blocked HSV-1 plaque formation. The antiviral activity of the wax was also confirmed by real-time PCR, as well as viral protein expression by immunohistochemical staining. Chemical characterization of the fatty acid and fatty alcohol composition was performed, showing high similarity between the wax of the investigated varieties. Lastly, our results demonstrate that the observed effects are independent of simmondsin, repeatedly associated with the medicinal impact of Jojoba wax, and that Jojoba wax presence is required to gain protection against HSV-1 infection. Collectively, our results support the use of Jojoba wax against HSV-1 skin infections.     

The catalytic Mn2+ sites in the enolase-inhibitor complex: crystallography, single-crystal EPR, and DFT calculations

Crystals of Zn2+/Mn2+ yeast enolase with the inhibitor PhAH (phosphonoacetohydroxamate) were grown under conditions with a slight preference for binding of Zn2+ at the higher affinity site, site I. The structure of the Zn2+/Mn2+-PhAH complex was solved at a resolution of 1.54 A, and the two catalytic metal binding sites, I and II, show only subtle displacement compared to that of the corresponding complex with the native Mg2+ ions. Low-temperature echo-detected high-field (W-band, 95 GHz) EPR (electron paramagnetic resonance) and 1H ENDOR (electron-nuclear double resonance) were carried out on a single crystal, and rotation patterns were acquired in two perpendicular planes. Analysis of the rotation patterns resolved a total of six Mn2+ sites, four symmetry-related sites of one type and two out of the four of the other type. The observation of two chemically inequivalent Mn2+ sites shows that Mn2+ ions populate both sites I and II and the zero-field splitting (ZFS) tensors of the Mn2+ in the two sites were determined. The Mn2+ site with the larger D value was assigned to site I based on the 1H ENDOR spectra, which identified the relevant water ligands. This assignment is consistent with the seemingly larger deviation of site I from octahedral symmetry, compared to that of site II. The ENDOR results gave the coordinates of the protons of two water ligands, and adding them to the crystal structure revealed their involvement in a network of H bonds stabilizing the binding of the metal ions and PhAH. Although specific hyperfine interactions with the inhibitor were not determined, the spectroscopic properties of the Mn2+ in the two sites were consistent with the crystal structure. Density functional theory (DFT) calculations carried out on a cluster representing the catalytic site, with Mn2+ in site I and Zn2+ in site II, and vice versa, gave overestimated D values on the order of the experimental ones, although the larger D value was found for Mn2+ in site II rather than in site I. This discrepancy was attributed to the high sensitivity of the ZFS parameters to the Mn-O bond lengths and orientations, such that small, but significant, differences between the optimized and crystal structures alter the ZFS considerably, well above the difference between the two sites.     

Single crystal 55Mn ENDOR of concanavalin a: detection of two Mn2+ sites with different 55Mn quadrupole tensors

Concanavalin A is a member of the plant hemeagglutinin (or plant lectin) family that contains two metal binding sites; one, called S1, is occupied by Mn2+ and the other, S2, by Ca2+. 55Mn electron-nuclear double resonance (ENDOR) measurements were performed on a single crystal of concanavalin A at W-band (95 GHz, ~3.5 T) to determine the 55Mn nuclear quadrupole interaction in a protein binding site and its relation to structural parameters. Such measurements are easier at a high field because of the high sensitivity for size-limited samples and the reduction of second-order effects on the spectrum which simplifies spectral analysis. The analysis of the 55Mn ENDOR rotation patterns showed that two chemically inequivalent Mn2+ types are present at low temperatures, although the high-resolution X-ray structure reported only one site. Their quadrupole coupling constants, e2Qq/h, are significantly different; 10.7 +/- 0.6 MHz for Mand only -2.7 +/-0.6 MHz for M. The ENDOR data also refined the hyperfine coupling determined earlier by single-crystal EPR measurements, yielding a small but significant difference between the two: -262.5 MHz for M and -263.5 MHz for M. The principal z-axis for M is not aligned with any of the Mn-ligand directions, but is 25 off the Mn-asp10 direction, and its orientation is different than that of the zero-field splitting (ZFS) interaction. Because of the small quadrupole interaction of M the orientation dependence was very mild, leading to larger uncertainties in the asymmetry parameter. Nonetheless, there too z is not along the Mn-ligand bonds and is rotated 90 with respect to MnA. These results show, that similar to the ZFS, the quadrupolar interaction is highly sensitive to small differences in the coordination sphere of the Mn2+, and the resolution of the two types is in agreement with the earlier observation of a two-site conformational dynamic detected through the ZFS interaction, which is frozen out at low temperatures and averaged at room temperature. To account for the structural origin of the different e2Qq/h values, the electric field gradient tensor was calculated using the point-charge model. The calculations showed that a relatively small displacement of the oxygen ligand of asp10 can lead to differences on the order observed experimentally.     

Global stability of equilibrium manifolds,and “peaking” behavior in quadratic differential systems related to viscoelastic models

We study a model inspired by the Oldroyd-B equations for viscoelastic fluids. The objective is to better understand the nonlinear coupling between the stress and velocity fields in viscoelastic flows, and thus gain insight into the reasons that cause the loss of accuracy of numerical computations at high Weissenberg number. We derive a model system by discarding the stress-advection and stress-relaxation terms in the Oldroyd-B model. The reduced (unphysical) model, which bears some resemblance to a viscoelastic solid, only retains the stretching of the stress due to velocity gradients and the induction of velocity by the stress field. Our conjecture is that such a system always evolves toward an equilibrium in which the stress builds up such to cancel the external forces. This conjecture is supported by numerous simulations. We then turn our attention to a finite dimensional model (i.e., a set of ordinary differential equations) that has the same algebraic structure as our model system. Numerical simulations indicate that the finite-dimensional analog has a globally attracting equilibrium manifold. In particular, it is found that subsets of the equilibrium manifold may be unstable, leading to a “peaking” behavior, where trajectories are repelled from the equilibrium manifold at one point, and are eventually attracted to a stable equilibrium point on the same manifold. Generalizations and implications to solutions of the Oldroyd-B model are discussed.