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.     

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.     

A Potent Leukocyte Transmigration Blocker: GT-73 Showed a Protective Effect against LPS-Induced ARDS in Mice

We recently developed a molecule (GT-73) that blocked leukocyte transendothelial migration from blood to the peripheral tissues, supposedly by affecting the platelet endothelial cell adhesion molecule (PECAM-1) function. GT-73 was tested in an LPS-induced acute respiratory distress syndrome (ARDS) mouse model. The rationale for this is based on the finding that the mortality of COVID-19 patients is partly caused by ARDS induced by a massive migration of leukocytes to the lungs. In addition, the role of tert-butyl and methyl ester moieties in the biological effect of GT-73 was investigated. A human leukocyte, transendothelial migration assay was applied to validate the blocking effect of GT-73 derivatives. Finally, a mouse model of LPS-induced ARDS was used to evaluate the histological and biochemical effects of GT-73. The obtained results showed that GT-73 has a unique structure that is responsible for its biological activity; two of its chemical moieties (tert-butyl and a methyl ester) are critical for this effect. GT-73 is a prodrug, and its lipophilic tail covalently binds to PECAM-1 via Lys536. GT-73 significantly decreased the number of infiltrating leukocytes in the lungs and reduced the inflammation level. Finally, GT-73 reduced the levels of IL-1β, IL-6, and MCP-1 in bronchoalveolar lavage fluid (BALF). In summary, we concluded that GT-73, a blocker of white blood cell transendothelial migration, has a favorable profile as a drug candidate for the treatment of ARDS in COVID-19 patients.     

Structure and dynamics of photogenerated triplet radical ion pairs in DNA hairpin conjugates with anthraquinone end caps

A series of DNA hairpins (AqGn) possessing a tethered anthraquinone (Aq) end-capping group were synthesized in which the distance between the Aq and a guanine-cytosine (G-C) base pair was systematically varied by changing the number (n - 1) of adenine-thymine (A-T) base pairs between them. The photophysics and photochemistry of these hairpins were investigated using nanosecond transient absorption and time-resolved electron paramagnetic resonance (TREPR) spectroscopy. Upon photoexcitation, (1*)Aq undergoes rapid intersystem crossing to yield (3*)Aq, which is capable of oxidizing purine nucleobases resulting in the formation of (3)(Aq(-?)Gn(+?)). All (3)(Aq(-?)Gn(+?)) radical ion pairs exhibit asymmetric TREPR spectra with an electron spin polarization phase pattern of absorption and enhanced emission (A/E) due to their different triplet spin sublevel populations, which are derived from the corresponding non-Boltzmann spin sublevel populations of the (3*)Aq precursor. The TREPR spectra of the (3)(Aq(-?)Gn(+?)) radical ion pairs depend strongly on their spin-spin dipolar interaction and weakly on their spin-spin exchange coupling. The anisotropy of (3)(Aq(-?)Gn(+?)) makes it possible to determine that the π systems of Aq(-?) and G(+?) within the radical ion pair are parallel to one another. Charge recombination of the long-lived (3)(Aq(-?)Gn(+?)) radical ion pair displays an unusual bimodal distance dependence that results from a change in the rate-determining step for charge recombination from radical pair intersystem crossing for n < 4 to coherent superexchange for n > 4.     

Fast photodriven electron spin coherence transfer: a quantum gate based on a spin exchange j-jump

Photoexcitation of the electron donor (D) within a linear, covalent donor-acceptor-acceptor molecule (D-A(1)-A(2)) in which A(1) = A(2) results in sub-nanosecond formation of a spin-coherent singlet radical ion pair state, (1)(D(+?)-A(1)(-?)-A(2)), for which the spin-spin exchange interaction is large: 2J = 79 ± 1 mT. Subsequent laser excitation of A(1)(-?) during the lifetime of (1)(D(+?)-A(1)(-?)-A(2)) rapidly produces (1)(D(+?)-A(1)-A(2)(-?)), which abruptly decreases 2J 3600-fold. Subsequent coherent spin evolution mixes (1)(D(+?)-A(1)-A(2)(-?)) with (3)(D(+?)-A(1)-A(2)(-?)), resulting in mixed states which display transient spin-polarized EPR transitions characteristic of a spin-correlated radical ion pair. These photodriven J-jump experiments show that it is possible to use fast laser pulses to transfer electron spin coherence between organic radical ion pairs and observe the results using an essentially background-free time-resolved EPR experiment.     

Mechanically stabilized tetrathiafulvalene radical dimers

Two donor-acceptor [3]catenanes-composed of a tetracationic molecular square, cyclobis(paraquat-4,4'-biphenylene), as the π-electron deficient ring and either two tetrathiafulvalene (TTF) and 1,5-dioxynaphthalene (DNP) containing macrocycles or two TTF-butadiyne-containing macrocycles as the π-electron rich components-have been investigated in order to study their ability to form TTF radical dimers. It has been proven that the mechanically interlocked nature of the [3]catenanes facilitates the formation of the TTF radical dimers under redox control, allowing an investigation to be performed on these intermolecular interactions in a so-called "molecular flask" under ambient conditions in considerable detail. In addition, it has also been shown that the stability of the TTF radical-cation dimers can be tuned by varying the secondary binding motifs in the [3]catenanes. By replacing the DNP station with a butadiyne group, the distribution of the TTF radical-cation dimer can be changed from 60% to 100%. These findings have been established by several techniques including cyclic voltammetry, spectroelectrochemistry and UV-vis-NIR and EPR spectroscopies, as well as with X-ray diffraction analysis which has provided a range of solid-state crystal structures. The experimental data are also supported by high-level DFT calculations. The results contribute significantly to our fundamental understanding of the interactions within the TTF radical dimers.