Toll-like receptor 2 expression on c-kit + cells tracks the emergence of embryonic definitive hematopoietic progenitors

International audienceHematopoiesis in mammalian embryos proceeds through three successive waves of hema-topoietic progenitors. Since their emergence spatially and temporally overlap and phenotypic markers are often shared, the specifics regarding their origin, development, lineage restriction and mutual relationships have not been fully determined. The identification of wave-specific markers would aid to resolve these uncertainties. Here, we show that toll-like receptors (TLRs) are expressed during early mouse embryogenesis. We provide phenotypic and functional evidence that the expression of TLR2 on E7.5 c-kit + cells marks the emergence of precursors of erythro-myeloid progenitors (EMPs) and provides resolution for separate tracking of EMPs from primitive progenitors. Using in vivo fate mapping, we show that at E8.5 the Tlr2 locus is already active in emerging EMPs and in progenitors of adult hematopoietic stem cells (HSC). Together, this data demonstrates that the activation of the Tlr2 locus tracks the earliest events in the process of EMP and HSC specification

Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms

A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of q-expansions, and a transformation law under \Gamma_0(4) for the second complete elliptic integral is derived. More generally, it is shown how Picard-Fuchs equations of triangle subgroups of PSL(2,R) which are hypergeometric equations, yield systems of nonlinear equations for weight-1 forms, and generalized Chazy equations. Each triangle group commensurable with \Gamma(1) is treated.Comment: 40 pages, final version, accepted by Manuscripta Mathematic

K moduli of log del Pezzo pairs

We establish the full explicit wall-crossing for K-moduli space $\overline{P}^K_c$ of degree $8$ del Pezzo pairs $(X,cC)$ where generically X \cong \bbF_1 and $C \sim -2K_X$. We also show K-moduli spaces $\overline{P}^K_c$ coincide with Hassett-Keel-Looijenga(HKL) models \cF(s) of a $18$-dimensional locally symmetric spaces associated to the lattice $E_8\oplus U^2\oplus E_7\oplus A_1$.Comment: 42 pages, comments welcome