## Graduate Theses, Dissertations, and Problem Reports

#### Title

From Graph Coloring to Receptor Clustering

Spring

2014

Dissertation

PhD

#### College

Eberly College of Arts and Sciences

Mathematics

Hong-Jian Lai

John Goldwasser

Guodong Guo

#### Committee Member

Jerzy Wojciechowski

Cun-Quan Zhang

#### Abstract

1. Hued colorings for planar graphs, graphs of higher genus and K4-minor free graphs.;For integers k, r > 0, a (k,r) -coloring of a graph G is a proper coloring of the vertices of G with k colors such that every vertex v of degree d(v) is adjacent to vertices with at least min{lcub}d(v) ,r{rcub} different colors. The r-hued chromatic number, denoted by Xr (G), is the smallest integer k for which a graph G has a ( k,r)-coloring. A list assignment L of G is a function that assigns to every vertex v of G a set L(v) of positive integers. For a given list assignment L of G, an ( L,r)-coloring of G is a proper coloring c of the vertices such that every vertex v of degree d(v) is adjacent to vertices with at least min{lcub} d(v),r{rcub} different colors and c(v) epsilon L(v). The r-hued choice number of G, XL,r(G), is the least integer k such that every list assignment L with | L(v)| = k, ∀ v epsilon V(G), permits an (L,r)-coloring. It is known that for any graph G, Xr(G) ≤ XL,r( G). Using Euler distributions, we proved the following results, where (ii) and (iii) are best possible. (i) If G is planar, then XL,2(G) ≤ 6. Moreover, XL,2( (G) ≤ 5 when Delta (G) ≤ 4. (ii) If G is planar, then X2( G) ≤ 5. (iii) If G is a graph with genus g(G) ≥ 1, then XL,2 (G) ≤ ½ 7+1+48gG .;Let K(r) = r + 3 if 2 ≤ r ≤ 3, and K(r) = 3r/2+1 if r≥ 4. We proved that if G is a K4-minor free graph, then (i) Xr(G) ≤ K(r), and the bound can be attained; (ii) XL,r(G) ≤ K( r)+1. This extends a previous result in [Discrete Math. 269 (2003) 303--309].;2. Quantitative description and impact of VEGF receptor clustering .;Cell membrane-bound receptors control signal initiation in many important cellular signaling pathways. Microscopic imaging and modern labeling techniques reveal that certain receptor types tend to co-localize in clusters, ranging from a few to hundreds of members. Here, we further develop a method of defining receptor clusters in the membrane based on their mutual distance, and apply it to a set of transmission microscopy (TEM) images of vascular endothelial growth factor (VEGF) receptors. We clarify the difference between the observed distributions and random placement. Moreover, we outline a model of clustering based on the hypothesis of pre-existing domains that have a high affinity for receptors. The observed results are consistent with the combination of two distributions, one corresponding to the placement of clusters, and the other to that of random placement of individual receptors within the clusters. Further, we use the preexisting domain model to calculate the probability distribution of cluster sizes. By comparing to the experimental result, we estimate the likely area and attractiveness of the clustering domains.;Furthermore, as VEGF signaling is involved in the process of blood vessel development and maintenance, it is of our interest to investigate the impact of VEGF receptors (VEGFR) clustering. VEGF signaling is initiated by binding of the bivalent VEGF ligand to the membrane-bound receptors (VEGFR), which in turn stimulates receptor dimerization. To address these questions, we have formulated the simplest possible model. We have postulated the existence of a single high affinity region in the cell membrane, which acts as a transient trap for receptors. We have defined an ODE model by introducing high- and low-density receptor variables and introduce the corresponding reactions from a realistic model of VEGF signal initiation. Finally, we use the model to investigate the relation between the degree of VEGFR concentration, ligand availability, and signaling. In conclusion, our simulation results provide a deeper understanding of the role of receptor clustering in cell signaling.

COinS