Asymptotic analysis of a semi-linear elliptic system in perforated domains: well-posedness and correctors for the homogenization limit

In this study, we prove results on the weak solvability and homogenization of a microscopic semi-linear elliptic system posed in perforated media. The model presented here explores the interplay between stationary diffusion and both surface and volume chemical reactions in porous media. Our interest lies in deriving homogenization limits (upscaling) for alike systems and particularly in justifying rigorously the obtained averaged descriptions. Essentially, we prove the well-posedness of the microscopic problem ensuring also the positivity and boundedness of the involved concentrations and then use the structure of the two scale expansions to derive corrector estimates delimitating this way the convergence rate of the asymptotic approximates to the macroscopic limit concentrations. Our techniques include Moser-like iteration techniques, a variational formulation, two-scale asymptotic expansions as well as energy-like estimates.Comment: 22 pages, 1 figur

Molecular characterization of porcine genes encoding complement components of the terminal lytic pathway and their association with hemolytic complement activity

Activation of the complement system from three different pathways (classical, alternative and lectin pathway) results in the generation of the C3-convertase enzyme, which plays a key role in formation of the membrane attack complex (C5b-C9) causing the death of target cells. The porcine C3 and C5 complement components were characterized and studied for association with hemolytic complement activity (Kumar et al. 2004, Wimmers et al. 2003). In order to gain understanding for the membrane attack complex action in the innate immune mechanism, in this study it was focussed on the terminal complement components C6, C7, C8, and C9 to characterize their molecular structure, to detect single nucleotide polymorphisms (SNPs), to establish their location on chromosome, and to associate their genetic variation with hemolytic complement activity in both classical and alternative pathway in the pig. The entire length of cDNA sequence of the candidate genes C6, C7, C8A, C8B, C8G and C9 were identified with 3306, 3561, 2146, 2461, 840 and 2536 bp encoding 935, 843, 589, 611, 202 and 543 amino acids, respectively. The porcine deduced protein sequence of the candidate genes showed 67-83% identities with human analogue. Respectively, screening the coding region revealed five, six, seven, nine, and two SNPs in the porcine C6, C7, C8A, C8B, and C9 but non in C8G by polymerase chain reactionrestriction fragment length polymorphism (PCR-RFLP). Most of the SNPs belong to the functional protein domains such as TSP1, LDLa, MACPF, CCP and FIMAC. Genotyping for several SNP sites in three porcine breeds German Landrace (LR), Pietrain (PIE) and Muong Khuong (MK) showed that European breeds (LR and PIE) had higher allelic variation than the Asian breed (MK). All genotypic frequencies fit to Hardy-Weinberg equilibrium rule. Using the INRA-Minnesota porcine Radiation Hybrid mapping panel, the porcine C6, C7, and C9 were assigned to the q-arm of chromosome 16 (q1.4) whereas the porcine C8A, and C8B were mapped to chromosome 6 (q3.1-q3.5). Particularly the porcine C8G was located on chromosome 1 (q2.13). Genetic association with hemolytic complement activity in both classical (CH50) and alternative pathway (AH50) was carried out in 417 animals of a F2 DUMI resource population derived from cross between Duroc and Berlin Miniature Pig. Therefore, the F2 DUMI animals were immunized with Mycoplasma hyopneumoniae (Mh), Aujeszky (ADV) and porcine reproductive and respiratory syndrome (PRRSV) vaccine. Sera were isolated from blood samples taken prior and post vaccinations and measurement for CH50 and AH50 was conducted thereafter. For each gene except the porcine C8G, the SNP site with amino acid substitution 862A→G for C6, 881A→G for C7, 1544C→T for C8A, 222C→T for C8B, and 407C→G for C9, segregating in the DUMI, were used for genotyping the F2 animals using PCR-RFLP with the restriction enzymes TaqI, MboII, Hin6I, FnuDII, and HpyCH4III, respectively. The association results illustrated that significant difference in hemolysis among genotypes was found in CH50 for C7 (p=0.0080), and C9 (p=0.0488). However, this was close to significance for C6 (p=0.0853) and C8A (p=0.0650) in CH50. Therefore between homozygous genotypes CC and TT for C8A hemolytic activity showed significant difference (p=0.0522). There was no association of any of the candidate gene with hemolytic complement activity in the alternative pathway. Analyzing the interaction between genotypes and eight different immunization time points in AH50 revealed significant differences for C8A (p=0.0027), C8B (p=0.0231), and C9 (p=0.0340) whereas in CH50 this interaction was found significant for C8B (p=0.0048). Hemolytic complement activity showed the highest values at the fourth day after immunization with ADV vaccine for CH50 whereas linear increment during the experiment was performed for AH50. Along the vaccination program after each of complement stimulation by different vaccines, a short termed increment of complement activity was found, especially with ADV vaccine. Also male animals always performed higher hemolysis than females in both pathways. These results show that hemolytic complement activity depends on the genetic variation, sex, age, kind of vaccine, and interaction of complement components. In summary, the obtained results provide the means for further understanding the role of C6, C7, C8, and C9 in natural immune response of the host against pathogens. It also promotes the porcine C6, C7, C8, and C9 as candidate genes in efforts to genetically improve general animal health, a goal of breeding programmes for food animals

A high-order corrector estimate for a semi-linear elliptic system in perforated domains

We derive in this note a high-order corrector estimate for the homogenization of a microscopic semi-linear elliptic system posed in perforated domains. The major challenges are the presence of nonlinear volume and surface reaction rates. This type of correctors justifies mathematically the convergence rate of formal asymptotic expansions for the two-scale homogenization settings. As main tool, we follow the standard approach by the energy-like method to investigate the error estimate between the micro and macro concentrations and micro and macro concentration gradients. This work aims at generalizing the results reported in [2, 7].Comment: 6 pages, 1 figur