David L. Powers

The distance spectrum of the path Pn and The First Distance Eigenvector of Connected Graphs

Forumulas are given for all of the eigenvalues and eigenvectors of the distance matrix of the path Pn on n vertices. It is shown that Pn has the maximum distance spectral radius among all connected graphs of order n, and an ordering property of the entries of the Perron-Frobenius eigenvector is presented.

Graph Partitioning by Eigenvectors

Let A be the adjacency matrix of a connected graph 9. If z is a column vector, we say that a vertex of ?? is positive, nonnegative, null, etc. if the corresponding entry of z has that property. For z such that AZ > OLZ, we bound the number of components in the subgraph induced by positive vertices. For eigenvectors z having a null element, we bound the number of components in the graph induced by nonnull vertices. Finally, bounds are established for the number of null elements in an eigenvector, for the multiplicity of an eigenvalue and for the magnitudes of the second and last eigenvalues of a general or a bipartite graph.

Glycosylation of the enhanced aromatic sequon is similarly stabilizing in three distinct reverse turn contexts

Cotranslational N-glycosylation can accelerate protein folding, slow protein unfolding, and increase protein stability, but the molecular basis for these energetic effects is incompletely understood. N-glycosylation of proteins at naïve sites could be a useful strategy for stabilizing proteins in therapeutic and research applications, but without engineering guidelines, often results in unpredictable changes to protein energetics. We recently introduced the enhanced aromatic sequon as a family of portable structural motifs that are stabilized upon glycosylation in specific reverse turn contexts: a five-residue type I β-turn harboring a G1 β-bulge (using a Phe–Yyy–Asn–Xxx–Thr sequon) and a type II β-turn within a six-residue loop (using a Phe–Yyy–Zzz–Asn–Xxx–Thr sequon) [Culyba EK, et al. (2011) Science 331:571–575]. Here we show that glycosylating a new enhanced aromatic sequon, Phe–Asn–Xxx–Thr, in a type I′ β-turn stabilizes the Pin 1 WW domain. Comparing the energetic effects of glycosylating these three enhanced aromatic sequons in the same host WW domain revealed that the glycosylation-mediated stabilization is greatest for the enhanced aromatic sequon complementary to the type I β-turn with a G1 β-bulge. However, the portion of the stabilization from the tripartite interaction between Phe, Asn(GlcNAc), and Thr is similar for each enhanced aromatic sequon in its respective reverse turn context. Adding the Phe–Asn–Xxx–Thr motif (in a type I′ β-turn) to the enhanced aromatic sequon family doubles the number of proteins that can be stabilized by glycosylation without having to alter the native reverse turn type.

Structural and energetic basis of carbohydrate-aromatic packing interactions in proteins.

Carbohydrate-aromatic interactions mediate many biological processes. However, the structure-energy relationships underpinning direct carbohydrate-aromatic packing interactions in aqueous solution have been difficult to assess experimentally and remain elusive. Here, we determine the structures and folding energetics of chemically synthesized glycoproteins to quantify the contributions of the hydrophobic effect and CH-π interactions to carbohydrate-aromatic packing interactions in proteins. We find that the hydrophobic effect contributes significantly to protein-carbohydrate interactions. Interactions between carbohydrates and aromatic amino acid side chains, however, are supplemented by CH-π interactions. The strengths of experimentally determined carbohydrate CH-π interactions do not correlate with the electrostatic properties of the involved aromatic residues, suggesting that the electrostatic component of CH-π interactions in aqueous solution is small. Thus, tight binding of carbohydrates and aromatic residues is driven by the hydrophobic effect and CH-π interactions featuring a dominating dispersive component.

A Perspective on Mechanisms of Protein Tetramer Formation

Homotetrameric proteins can assemble by several different pathways, but have only been observed to use one, in which two monomers associate to form a homodimer, and then two homodimers associate to form a homotetramer. To determine why this pathway should be so uniformly dominant, we have modeled the kinetics of tetramerization for the possible pathways as a function of the rate constants for each step. We have found that competition with the other pathways, in which homotetramers can be formed either by the association of two different types of homodimers or by the successive addition of monomers to homodimers and homotrimers, can cause substantial amounts of protein to be trapped as intermediates of the assembly pathway. We propose that this could lead to undesirable consequences for an organism, and that selective pressure may have caused homotetrameric proteins to evolve to assemble by a single pathway.

Bounds on graph eigenvalues

Abstract Upper and lower estimates are found for the maximum of the kth eigenvalue of a graph as a function of the number of vertices or edges.

Some Hamiltonian Cayley Graphs

No negative example or positive proof is known for the conjecture that every Cayley graph is hamiltonian. Trivalent Cayley graphs are especially interesting, being at the same time the simplest nontrivial Cayley graphs and those most likely to be nonhamiltonian, because of the small number of edges. In this note, we use the eulerian or hamiltonian structure of one graph to find a hamiltonian cycle in another. This technique is then used to expand certain trivalent Cayley graphs into hamiltonian Cayley graphs at the expense of higher valency.

Heat conduction on graphs

Abstract A mathematical model for heat conduction in a graph-like object provides a setting for interpreting an algebraic eigenvalue problem associated with a graph. Applications include bounds for eigenvalues and a way to find eigenvalues and eigenvectors for a subdivision graph.