Sandpiles, spanning trees, and plane duality

Let G be a connected, loopless multigraph. The sandpile group of G is a finite abelian group associated to G whose order is equal to the number of spanning trees in G. Holroyd et al. used a dynamical process on graphs called rotor-routing to define a simply transitive action of the sandpile group of G on its set of spanning trees. Their definition depends on two pieces of auxiliary data: a choice of a ribbon graph structure on G, and a choice of a root vertex. Chan, Church, and Grochow showed that if G is a planar ribbon graph, it has a canonical rotor-routing action associated to it, i.e., the rotor-routing action is actually independent of the choice of root vertex. It is well-known that the spanning trees of a planar graph G are in canonical bijection with those of its planar dual G*, and furthermore that the sandpile groups of G and G* are isomorphic. Thus, one can ask: are the two rotor-routing actions, of the sandpile group of G on its spanning trees, and of the sandpile group of G* on its spanning trees, compatible under plane duality? In this paper, we give an affirmative answer to this question, which had been conjectured by Baker.Comment: 13 pages, 9 figure

LPA signaling is regulated through the primary cilium: a novel target in glioblastoma

The primary cilium is a ubiquitous organelle presented on most human cells. It is a crucial signaling hub for multiple pathways including growth factor and G-protein coupled receptors. Loss of primary cilia, observed in various cancers, has been shown to affect cell proliferation. Primary cilia formation is drastically decreased in glioblastoma (GBM), however, the role of cilia in normal astrocyte or glioblastoma proliferation has not been explored. Here we report that loss of primary cilia in human astrocytes stimulates growth rate in a lysophosphatidic acid (LPA)-dependent manner. We show that lysophosphatidic acid receptor 1 (LPAR1) is accumulated in primary cilia. LPAR1 signaling through Gα12/Gαq was previously reported to be responsible for cancer cell proliferation. We found that in ciliated cells, Gα12 and Gαq are excluded from the cilium, creating a barrier against unlimited proliferation, one of the hallmarks of cancer. Upon loss of primary cilia, LPAR1 redistributes to the plasma membrane with a concomitant increase in LPAR1 association with Gα12 and Gαq. Inhibition of LPA signaling with the small molecule compound Ki16425 in deciliated highly proliferative astrocytes or glioblastoma patient-derived cells/xenografts drastically suppresses their growth both in vitro and in vivo. Moreover, Ki16425 brain delivery via PEG-PLGA nanoparticles inhibited tumor progression in an intracranial glioblastoma PDX model. Overall, our findings establish a novel mechanism by which primary cilium restricts proliferation and indicate that loss of primary cilia is sufficient to increase mitogenic signaling, and is important for the maintenance of a highly proliferative phenotype. Clinical application of LPA inhibitors may prove beneficial to restrict glioblastoma growth and ensure local control of disease