Abstract. We introduce Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type. The definition is relative to a given pair of pants decomposition of the surface. We start by establishing conditions under which any pair of pants decomposition on a hyperbolic surface of infinite type can be turned into a geometric decomposition, that is, a decomposition into hyperbolic pairs of pants. This is expressed in terms of a condition we introduce and which we call Nielsen-convexity. This condition is related to Nielsen cores of Fuchsian groups. We use this to define the Fenchel-Nielsen Teichmüller space relative to a geometric pair of pants decomposition. We study a metric, called the Fenchel-Nielsen metric, on such a Teichmüller space, and we compare it to the (quasiconformal) Teichmüller metric. We study conditions under which there is an equality between the Fenchel-Nielsen Teichmüller space and the familiar Teichmüller space defined using quasiconformal mappings, and we study topological and metric properties of the identity map between these two spaces when this map exists.

G-quadruplex is a special secondary structure of nucleic acids in guanine-rich sequences of genome. G-quadruplexes have been proved to be involved in the regulation of replication, DNA damage repair, and transcription and translation of oncogenes or other cancer-related genes. Therefore, targeting G-quadruplexes has become a novel promising anti-tumor strategy. Different kinds of small molecules targeting the G-quadruplexes have been designed, synthesized, and identified as potential anti-tumor agents, including molecules directly bind to the G-quadruplex and molecules interfering with the binding between the G-quadruplex structures and related binding proteins. This review will explore the feasibility of G-quadruplex ligands acting as anti-tumor drugs, from basis to application. Meanwhile, since helicase is the most well-defined G-quadruplex-related protein, the most extensive research on the relationship between helicase and G-quadruplexes, and its meaning in drug design, is emphasized.

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