Existence theorems for a crystal surface model involving the p-Laplace operator

The manufacturing of crystal films lies at the heart of modern nanotechnology. How to accurately predict the motion of a crystal surface is of fundamental importance. Many continuum models have been developed for this purpose, including a number of PDE models, which are often obtained as the continuum limit of a family of kinetic Monte Carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete Gaussian models. In this paper we offer an analytical perspective into some of these models. To be specific, we study the existence of a weak solution to the boundary value problem for the equation - \Delta e^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right)}+au=f, where $p>1, a>0$ are given numbers and $f$ is a given function. This problem is derived from a crystal surface model proposed by J.L.~Marzuola and J.~Weare (2013 Physical Review, E 88, 032403). The mathematical challenge is due to the fact that the principal term in our equation is an exponential function of a p-Laplacian. Existence of a suitably-defined weak solution is established under the assumptions that $p\in(1,2], \ N\leq 4$, and $f\in W^{1,p}$. Our investigations reveal that the key to our existence assertion is how to control the set where -\mbox{div}\left(|\nabla u|^{p-2}\nabla u\right) is $\pm\infty$

Immunization with a streptococcal multiple-epitope recombinant protein protects mice against invasive group A streptococcal infection

<div><p><i>Streptococcus pyogenes</i> (group A <i>Streptococcus</i>; GAS) causes clinical diseases, including pharyngitis, scarlet fever, impetigo, necrotizing fasciitis and streptococcal toxic shock syndrome. A number of group A streptococcus vaccine candidates have been developed, but only one 26-valent recombinant M protein vaccine has entered clinical trials. Differing from the design of a 26-valent recombinant M protein vaccine, we provide here a vaccination using the polyvalence epitope recombinant FSBM protein (rFSBM), which contains four different epitopes, including the fibronectin-binding repeats domain of streptococcal fibronectin binding protein Sfb1, the C-terminal immunogenic segment of streptolysin S, the C3-binding motif of streptococcal pyrogenic exotoxin B, and the C-terminal conserved segment of M protein. Vaccination with the rFSBM protein successfully prevented mortality and skin lesions caused by several <i>emm</i> strains of GAS infection. Anti-FSBM antibodies collected from the rFSBM-immunized mice were able to opsonize at least six <i>emm</i> strains and can neutralize the hemolytic activity of streptolysin S. Furthermore, the internalization of GAS into nonphagocytic cells is also reduced by anti-FSBM serum. These findings suggest that rFSBM can be applied as a vaccine candidate to prevent different <i>emm</i> strains of GAS infection.</p></div

Construction and purification of the recombinant FSBM protein.

<p>(A) The map of the pET24a-<i>fsbm</i> plasmid and the sequence of the synthetic <i>fsbm</i> gene. The plasmid was transformed, and the expression of rFSBM was induced. The recombinant protein was purified as described in Materials and Methods. Purified recombinant rFSBM protein was separated by SDS-PAGE (B) and the antigenic epitope of C-terminal domain of SPE B within rFSBM protein was analyzed by Western blotting (C) and ELISA (D) using the anti-SPE B antibody.</p

Immunization of recombinant FSBM protein protected mice against invasive <i>emm1</i> strain GAS-induced death.

<p>(A) The average titers of rFSBM-specific serum IgG collected from BALB/c mice that were immunized with recombinant FSBM proteins alone or rFSBM proteins combined with Freund's Adjuvant or AddaVax. (B) The survival rates of mice immunized with the rFSBM protein combined with different adjuvants were significantly higher than the adjuvant controls, including the mice immunized with CFA adjuvant only or immunized with AddaVax only controls, post subcutaneously challenged with 2 x 10⁸ CFU of <i>emm1</i> strain GAS. The survival curves were compared for significance using the log-rank test for the immunized mice compared to the adjuvant-only control groups (<i>P</i> < 0.01).</p

<i>In vitro</i> opsonization and macrophage-associated bacteria were enhanced by the rFSBM antiserum.

<p>(A) The M protein-specific reactivity of rFSBM-immunized sera was determined by flow cytometry with <i>emm6</i> wild type GAS and the <i>emm6</i>-deletion mutant, as described in Materials and Methods. (B) Sera from rFSBM-immunized mice or CFA adjuvant-only control mice were incubated with the wild type <i>emm6</i>-type GAS or the <i>emm6</i>-deletion mutant for 1 h, and then serum-treated GAS was taken to infect RAW 264.7 cells at a MOI of 25 for 1 h. The unbound bacteria were removed by washing with sterile PBS and the total number of bacteria bound with the cells and intracellular bacteria were quantified by plating on THY agar plates as described in Materials and Methods. The macrophage-associated bacterial number of anti-rFSBM serum-treated wild type GAS was increased compared to that of the CFA adjuvant-only control serum-treated GAS. The results of three experiments are shown and expressed as the mean ± SD. *<i>P</i>< 0.05, compared with the values determined for the control serum-treated <i>emm6</i> wild type GAS.**<i>P</i>< 0.01, compared with the values determined for the PBS-treated <i>emm6</i> wild type GAS or the anti-rFSBM serum-treated <i>emm6</i>-deletion mutant.</p

Evaluation of TAA-induced liver injury in rat models using Ishak and Metavir scores.

<p>Group 1 (TAA only); Group 2 (TAA+0.25 g/kg SST); Group 3 (TAA+1 g/kg SST).</p><p>TAA: Thioacetamide; STT: Sho-saiko-to; Fibrosis score: Ishak: 0–6; Metavir: F0 = 0, F1 = 1, F2 = 2, F3 = 3. Data represent as median.</p><p>Evaluation of TAA-induced liver injury in rat models using Ishak and Metavir scores.</p

Immunization with recombinant FSBM protein protected mice against different <i>emm</i> type GAS-induced tissue damage and mortality.

<p>The severity of the skin lesions (A) and the mortality (B) in the rFSBM-immunized mice were significantly decreased compared to the control mice after air pouch inoculation with 2 x 10⁸ CFU of different <i>emm</i> types of GAS (A) or intraperitoneal infection with 1 x 10⁸ CFU of different <i>emm</i> types of GAS (B). The survival curves were compared for significance using the log-rank test for the immunized mice versus the control group (<i>P</i>< 0.01).</p

Examination of thioacetamide (TAA)-induced liver fibrosis in rats by sonoelastography.

<p>(A) Sonoelastographic images before and after SST administration (weeks 0 and 6) in Groups 1 (TAA only, upper panel), 2 (TAA +0.25 g/kg SST, middle panel) and 3 (TAA +1 g/kg SST, lower panel). In the elastography frame, blue color stands for hard issues and red stands for soft issues. (B) Quantification of difference in liver stiffness of each group at weeks 0 and 6 by sonoelastography. Significant decrease in liver stiffness was observed in Groups 1 and 3. Bar, SE; *p<0.1.</p

Comparison of biochemical markers in TAA-induced liver fibrosis in rats.

<p>Group 1 (TAA only); Group 2 (TAA+0.25 g/kg SST); Group 3 (TAA+1 g/kg SST).</p><p>TAA: Thioacetamide; STT: Sho-saiko-to; AST: Alanine aminotranferease; ALT: Aspartate aminotransferase; GGT: Gamma-glutamyl transpeptidase; ALP:Alkaline phosphatase; LDH: Lactic acid dehydrogenase. Data represent as mean±SE.</p><p>Comparison of biochemical markers in TAA-induced liver fibrosis in rats.</p

Detection of Mrp2, Oatp1 and α-Sma expression in liver fibrosis induced by thioacetamide (TAA) in rats by immunohistochemistry and Western blots.

<p>(A) Immunohistochemistry of Mrp2 (upper panel), Oatp1 (middle panel) and α-Sma (lower panel) in rat liver sections of normal controls, Group 1 (TAA only), 2 (TAA+0.25 g/kg SST) and 3 (TAA+1 g/kg SST) was shown under light-field microscope with 200×magnifications. (B) Graph showed quantification of percentage of positive Mrp2 in Groups 1, 2 and 3. Bar, SE; **p<0.01; ***p<0.001. (C) Western blots of liver tissues from Mrp2, Oatp1 and α-Sma in rat liver tissues of normal controls, Groups 1, 2 and 3. α-tubulin serves as internal control. (D) Graph showed quantification of percentage of positive Oatp1 in Groups 1, 2 and 3. Bar, SE. (E) Graph showed quantification of percentage of positive α-Sma in Groups 1, 2 and 3. Bar, SE.</p