We construct a planar cubic system and demonstrate that it has at least 13 limit cycles. The construction is essentially based on counting the number of zeros of some Abelian integrals.

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The innate immune-dependent bactericidal effects are critical for preventing microbial colonization in the urinary system. However, the mechanisms involved in establishing innate immune responses in kidney are not completely understood. Here we describe the role of a novel member of the LRR (leucine-rich repeat) class of transmembrane proteins, LRRC19 (LRR-containing 19) in eliminating uropathogenic bacteria. LRRC19 is predominantly expressed in human and mouse kidney tubular epithelial cells and LRRC19-deficient mice are more susceptible to uropathogenic Escherichia coli (UPEC) infection than wild-type or TLR4 knockout mice. Recognition of UPEC by LRRC19 induces the production of cytokines, chemokines and antimicrobial substances through TRAF2-and TRAF6-mediated NF-kB and MAPK signalling pathways. Thus, LRRC19 may be a critical pathogen-recognition receptor in kidney mediating the elimination of UPEC infection.