Sharp eigenvalue estimates for rank one perturbations of nonnegative operators in Krein spaces

Let A and B be selfadjoint operators in a Krein space and assume that the resolvent difference of A and B is of rank one. In the case that A is nonnegative and I is an open interval such that σ(A)∩I consists of isolated eigenvalues we prove sharp estimates on the number and multiplicities of eigenvalues of B in I. The general result is illustrated with eigenvalue estimates for singular indefinite Sturm–Liouville problems.Fil: Behrndt, Jussi. Technische Universität Graz; AustriaFil: Leben, Leslie. Technische Universität Ilmenau; AlemaniaFil: Martinez Peria, Francisco Dardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderon; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matematicas; ArgentinaFil: Möws, Roland. Technische Universität Ilmenau; AlemaniaFil: Trunk, Carsten. Technische Universität Ilmenau; Alemani

Alloreactive T Cell Receptor Diversity against Structurally Similar or Dissimilar HLA-DP Antigens Assessed by Deep Sequencing

T cell alloreactivity is mediated by a self-human leukocyte antigen (HLA)-restricted T cell receptor (TCR) repertoire able to recognize both structurally similar and dissimilar allogeneic HLA molecules (i.e., differing by a single or several amino acids in their peptide-binding groove). We hypothesized that thymic selection on self-HLA molecules could have an indirect impact on the size and diversity of the alloreactive response. To test this possibility, we used TCR Vβ immunophenotyping and immunosequencing technology in a model of alloreactivity between self-HLA selected T cells and allogeneic HLA-DPB1 (DPB1) differing from self-DPB1*04:02 by a single (DPB1*02:01) or several (DPB1*09:01) amino acids in the peptide-binding groove. CD4+ T cells from three different self-DPB1*04:01,*04:02 individuals were stimulated with HeLa cells stably transduced with the relevant peptide processing machinery, co-stimulatory molecules, and HLA-DP. Flow cytometric quantification of the DPB1-specific T cell response measured as upregulation of the activation marker CD137 revealed significantly lower levels of alloreactivity against DPB1*02:01 compared with DPB1*09:01 (mean CD4+CD137+ frequency 35.2 ± 9.9 vs. 61.5 ± 7.7%, respectively, p < 0.0001). These quantitative differences were, however, not reflected by differences in the breadth of the alloreactive response at the Vβ level, with both alloantigens eliciting specific responses from all TCR-Vβ specificities tested by flow cytometry, albeit with higher levels of reactivity from most Vβ specificities against DPB1*09:01. In line with these observations, TCRB-CDR3 immunosequencing showed no significant differences in mean clonality of sorted CD137+CD4+ cells alloreactive against DPB1*02:01 or DPB1*09:01 [0.39 (0.36–0.45) and 0.39 (0.30–0.46), respectively], or in the cumulative frequencies of the 10 most frequent responding clones (55–67 and 58–62%, respectively). Most of the clones alloreactive against DPB1*02:01 (68.3%) or DPB1*09:01 (75.3%) were characterized by low-abundance (i.e., they were not appreciable among the pre-culture T cells). Interestingly, however, their cumulative frequency was lower against DPB1*02:01 compared with DPB1*09:01 (mean cumulative frequency 35.3 vs. 50.6%, respectively). Our data show that, despite lower levels of alloreactivity, a similar clonal diversity can be elicited by structurally similar compared with structurally dissimilar HLA-DPB1 alloantigens and demonstrate the power of TCRB immunosequencing in unraveling subtle qualitative changes not appreciable by conventional methods