Nathan Kaplan

ON DELTA SETS OF NUMERICAL MONOIDS

Let S be a numerical monoid (i.e. an additive submonoid of ℕ0) with minimal generating set 〈n1,…,nt〉. For m ∈ S, if , then is called a factorization length of m. We denote by (where mi < mi+1 for each 1 ≤ i < k) the set of all possible factorization lengths of m. The Delta set of m is defined by Δ(m) = {mi+1 - mi|1 ≤ i < k} and the Delta set of S by Δ(S) = ∪m∈SΔ(m). In this paper, we address some basic questions concerning the structure of the set Δ(S). In Sec. 2, we find upper and lower bounds on Δ(S) by finding such bounds on the Delta set of any monoid S where the associated reduced monoid Sred is finitely generated. We prove in Sec. 3 that if S = 〈n, n + k, n + 2k,…,n + bk〉, then Δ(S) = {k}. In Sec. 4 we offer some specific constructions which yield for any k and v in ℕ a numerical monoid S with Δ(S) = {k, 2k,…,vk}. Moreover, we show that Delta sets of numerical monoids may contain natural gaps by arguing that Δ(〈n, n + 1, n2 - n - 1〉) = {1,2,…,n - 2, 2n - 5}.

High-throughput Phenotyping of Lung Cancer Somatic Mutations

Recent genome sequencing efforts have identified millions of somatic mutations in cancer. However, the functional impact of most variants is poorly understood. Here we characterize 194 somatic mutations identified in primary lung adenocarcinomas. We present an expression-based variant-impact phenotyping (eVIP) method that uses gene expression changes to distinguish impactful from neutral somatic mutations. eVIP identified 69% of mutations analyzed as impactful and 31% as functionally neutral. A subset of the impactful mutations induces xenograft tumor formation in mice and/or confers resistance to cellular EGFR inhibition. Among these impactful variants are rare somatic, clinically actionable variants including EGFR S645C, ARAF S214C and S214F, ERBB2 S418T, and multiple BRAF variants, demonstrating that rare mutations can be functionally important in cancer.

Oncogenic RIT1 mutations in lung adenocarcinoma.

Lung adenocarcinoma is comprised of distinct mutational subtypes characterized by mutually exclusive oncogenic mutations in RTK/RAS pathway members KRAS, EGFR, BRAF and ERBB2, and translocations involving ALK, RET and ROS1. Identification of these oncogenic events has transformed the treatment of lung adenocarcinoma via application of therapies targeted toward specific genetic lesions in stratified patient populations. However, such mutations have been reported in only ∼55% of lung adenocarcinoma cases in the United States, suggesting other mechanisms of malignancy are involved in the remaining cases. Here we report somatic mutations in the small GTPase gene RIT1 in ∼2% of lung adenocarcinoma cases that cluster in a hotspot near the switch II domain of the protein. RIT1 switch II domain mutations are mutually exclusive with all other known lung adenocarcinoma driver mutations. Ectopic expression of mutated RIT1 induces cellular transformation in vitro and in vivo, which can be reversed by combined PI3K and MEK inhibition. These data identify RIT1 as a driver oncogene in a specific subset of lung adenocarcinomas and suggest PI3K and MEK inhibition as a potential therapeutic strategy in RIT1-mutated tumors.

Dynamic Epigenetic Regulation by Menin During Pancreatic Islet Tumor Formation

The tumor suppressor gene MEN1 is frequently mutated in sporadic pancreatic neuroendocrine tumors (PanNET) and is responsible for the familial multiple endocrine neoplasia type 1 (MEN-1) cancer syndrome. Menin, the protein product of MEN1, associates with the histone methyltransferases (HMT) MLL1 (KMT2A) and MLL4 (KMT2B) to form menin–HMT complexes in both human and mouse model systems. To elucidate the role of methylation of histone H3 at lysine 4 (H3K4) mediated by menin–HMT complexes during PanNET formation, genome-wide histone H3 lysine 4 trimethylation (H3K4me3) signals were mapped in pancreatic islets using unbiased chromatin immunoprecipitation coupled with next-generation sequencing (ChIP-seq). Integrative analysis of gene expression profiles and histone H3K4me3 levels identified a number of transcripts and target genes dependent on menin. In the absence of Men1, histone H3K27me3 levels are enriched, with a concomitant decrease in H3K4me3 within the promoters of these target genes. In particular, expression of the insulin-like growth factor 2 mRNA binding protein 2 (IGF2BP2) gene is subject to dynamic epigenetic regulation by Men1-dependent histone modification in a time-dependent manner. Decreased expression of IGF2BP2 in Men1-deficient hyperplastic pancreatic islets is partially reversed by ablation of RBP2 (KDM5A), a histone H3K4-specific demethylase of the jumonji, AT-rich interactive domain 1 (JARID1) family. Taken together, these data demonstrate that loss of Men1 in pancreatic islet cells alters the epigenetic landscape of its target genes. Implications: Epigenetic profiling and gene expression analysis in Men1-deficient pancreatic islet cells reveals vital insight into the molecular events that occur during the progression of pancreatic islet tumorigenesis. Mol Cancer Res; 13(4); 689–98. ©2014 AACR.

Shifts of generators and delta sets of numerical monoids

Let S be a numerical monoid with minimal generating set 〈n1, …, nt〉. For m ∈ S, if

Delta Sets of Numerical Monoids Using Nonminimal Sets of Generators

m = sum_{i = 1}^{t} x_{i}n_{i}

Identification of an “Exceptional Responder” Cell Line to MEK1 Inhibition: Clinical Implications for MEK-Targeted Therapy

, then

Flat Cyclotomic Polynomials of Order Four and Higher

sum_{i = 1}^{t} x_{i}

Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks

is called a factorization length of m. We denote by ℒ(m) = {m1, …, mk} (where mi N then |Δ(Mn)| = 1. If t = 2 and r1 and r2 are relatively prime, then we determine a value for N which is sharp.

The proportion of Weierstrass semigroups

Several recent articles have studied the structure of the delta set of a numerical monoid. We continue this work with the assumption that the generating set S chosen for the numerical monoid M is not necessarily minimal. We show that for certain choices of S, the resulting delta set can be made (in terms of cardinality) arbitrarily large or small. We close with a close analysis of the case where M =⟨n 1, n 2, in 1 + jn 2⟩for non-negative i and j.