Geometry of Dirac Operators

Let $M$ be a compact, oriented, even dimensional Riemannian manifold and let $S$ be a Clifford bundle over $M$ with Dirac operator $D$. Then \[ \textsc{Atiyah Singer: } \quad \text{Ind } \mathsf{D}= \int_M \hat{\mathcal{A}}(TM)\wedge \text{ch}(\mathcal{V}) \] where $\mathcal{V} =\text{Hom}_{\mathbb{C}l(TM)}(\slashed{\mathsf{S}},S)$. We prove the above statement with the means of the heat kernel of the heat semigroup $e^{-tD^2}$. The first outstanding result is the McKean-Singer theorem that describes the index in terms of the supertrace of the heat kernel. The trace of heat kernel is obtained from local geometric information. Moreover, if we use the asymptotic expansion of the kernel we will see that in the computation of the index only one term matters. The Berezin formula tells us that the supertrace is nothing but the coefficient of the Clifford top part, and at the end, Getzler calculus enables us to find the integral of these top parts in terms of characteristic classes.

IMPACT OF RNA VIRUSES ON THE REGULATION OF IL-23 IN MOUSE AND HUMAN MODELS OF INFECTION.

Interluekin-23 (IL-23) is a pro-inflammatory cytokine critical to the regulation of innate and adaptive immune responses. The main role for this cytokine is in the proliferation and differentiation of the IL-17 producing CD4 T helper cell, Th17. Virus infection deregulates IL-23 expression and function, but little is known about the mechanism behind this phenomena. Here, I demonstrate a reduction of Toll like receptor (TLR) ligand-induced IL-23 expression in lymphocytic choriomeningitis virus (LCMV)-infected bone marrow-derived dendritic cells (BMDCs), indicating that a function of these cells is disrupted during virus infection. I propose a mechanism of TLR ligand-induced IL-23 expression inhibition upon LCMV infection via the deactivation of p38, AP-1, and NF-κB. Further analysis revealed a direct relationship between LCMV infection with the IL-10 and SOCS3 expression. To understand IL-23 function, I characterized IL-23-induced JAK/STAT signalling pathway and IL-23 receptor expression on human CD4 T cells. My results demonstrate that IL-23 induces activation of p-JAK2, p-Tyk2, p-STAT1, p-STAT3, and p-STAT4 in CD4 T cells. For the first time I show that IL-23 alone induces the expression of its own receptor components, IL-12Rβ1 and IL-23Rα, in CD4 T cells. Blocking JAK2, STAT1, and STAT3 activation with specific inhibitors detrimentally effected expression of IL-23 receptor demonstrating that activation of JAK/STAT signalling is important for IL-23 receptor expression. I also addressed the effect of viral infection on IL-23 function and receptor expression in CD4 T cells using cells isolated from HIV positive individuals. These studies were based on earlier reports that the expression of IL-23 and the IL-23 receptor are impaired during HIV infection. I demonstrate that the phosphorylation of JAK2, STAT1, and STAT3 induced by IL-23, as well as IL-23 receptor expression are deregulated in CD4 T cells isolated from HIV positive individuals. This study has furthered the understanding of how the expression and function of IL-23 is regulated during viral infections.

THE IMPACT OF VIRUS INFECTION IN DEREGULATION OF CYTOKINE PRODUCTION UPON SECONDARY BACTERIAL INFECTION

Dendritic cells (DCs) secrete cytokines such as interleukin-23 (IL-23) when stimulated with certain Toll-like receptor (TLR) agonists and infected with pathogens such as P. aeruginosa. IL- 23 is a proinflammatory cytokine that plays a critical role in the proliferation and differentiation of the IL-17 producing Th17- CD4 T helper cells. The lack of efficient cytokine production from antigen-presenting cells, such as DCs, can impact CD4 differentiation and thus impair the immune responses against pathogens. Clearance of some bacterial infections, such as Klebsiella pneumonia and Listeria monocytogenes has been shown to be dependent on the induction of IL-23 and therefore, deregulation of these cytokines as a direct result of virus infection may impede immune responses to secondary infections. Here, an inhibition of TLR ligand or P. aeruginosa-induced IL- 23 expression in Lymphocytic Choriomeningitis Virus (LCMV)-infected bone marrow-derived dendritic cells (BMDCs) has been demonstrated, indicating that an important function of these cells is disrupted during virus/bacterial coinfection. While production of TNF-α was unaffected in LPS stimulated cells, TNF-α was significantly inhibited in bacterium infected cells by LCMV. Type I IFN in LPS or LCMV infected cell was not detected and therefore, ruling out the possibility of cytokine suppression by Type I IFN. The production of IL-10 was high in BMDCs infected with LCMV and stimulated with LPS or bacteria. Analysis of multiple cytokines produced in this coinfection model demonstrated that LCMV infection impacts specific cytokine production upon LPS or bacterium infection, which may be important for bacterial clearance. This data is important for future immunotherapy use in viral/bacterial coinfection scenarios.