Congruence conditions, parcels, and Tutte polynomials of graphs and matroids

Let $G$ be a matrix and $M(G)$ be the matroid defined by linear dependence on the set $E$ of column vectors of $G.$ Roughly speaking, a parcel is a subset of pairs $(f,g)$ of functions defined on $E$ to an Abelian group $A$ satisfying a coboundary condition (that $f-g$ is a flow over $A$ relative to $G$) and a congruence condition (that the size of the supports of $f$ and $g$ satisfy some congruence condition modulo an integer). We prove several theorems of the form: a linear combination of sizes of parcels, with coefficients roots of unity, equals an evaluation of the Tutte polynomial of $M(G)$ at a point $(\lambda-1,x-1)$ on the complex hyperbola $(\lambda - 1)(x-1) = |A|.

Maximum size binary matroids with no AG(3,2)-minor are graphic

We prove that the maximum size of a simple binary matroid of rank $r \geq 5$ with no AG(3,2)-minor is $\binom{r+1}{2}$ and characterise those matroids achieving this bound. When $r \geq 6$, the graphic matroid $M(K_{r+1})$ is the unique matroid meeting the bound, but there are a handful of smaller examples. In addition, we determine the size function for non-regular simple binary matroids with no AG(3,2)-minor and characterise the matroids of maximum size for each rank

Microenvironmental Influence on Pre-Clinical Activity of Polo-Like Kinase Inhibition in Multiple Myeloma: Implications for Clinical Translation

Polo-like kinases (PLKs) play an important role in cell cycle progression, checkpoint control and mitosis. The high mitotic index and chromosomal instability of advanced cancers suggest that PLK inhibitors may be an attractive therapeutic option for presently incurable advanced neoplasias with systemic involvement, such as multiple myeloma (MM). We studied the PLK 1, 2, 3 inhibitor BI 2536 and observed potent (IC50<40 nM) and rapid (commitment to cell death <24 hrs) in vitro activity against MM cells in isolation, as well as in vivo activity against a traditional subcutaneous xenograft mouse model. Tumor cells in MM patients, however, don't exist in isolation, but reside in and interact with the bone microenvironment. Therefore conventional in vitro and in vivo preclinical assays don't take into account how interactions between MM cells and the bone microenvironment can potentially confer drug resistance. To probe this question, we performed tumor cell compartment-specific bioluminescence imaging assays to compare the preclinical anti-MM activity of BI 2536 in vitro in the presence vs. absence of stromal cells or osteoclasts. We observed that the presence of these bone marrow non-malignant cells led to decreased anti-MM activity of BI 2536. We further validated these results in an orthotopic in vivo mouse model of diffuse MM bone lesions where tumor cells interact with non-malignant cells of the bone microenvironment. We again observed that BI 2536 had decreased activity in this in vivo model of tumor-bone microenvironment interactions highlighting that, despite BI 2536's promising activity in conventional assays, its lack of activity in microenvironmental models raises concerns for its clinical development for MM. More broadly, preclinical drug testing in the absence of relevant tumor microenvironment interactions may overestimate potential clinical activity, thus explaining at least in part the gap between preclinical vs. clinical efficacy in MM and other cancers