Self-Healing Multiblock Copolypeptide Hydrogels via Polyion Complexation

Diblock, triblock, and pentablock copolypeptides were designed and prepared for formation of polyion complex hydrogels in aqueous media. Increasing the number of block segments was found to allow formation of hydrogels with substantially enhanced stiffness at equivalent concentrations. Use of similar length ionic segments also allowed mixing of different block architectures to fine-tune hydrogel properties. The pentablock hydrogels possess a promising combination of high stiffness, rapid self-healing properties, and cell compatible surface chemistry that makes them promising candidates for applications requiring injectable or printable hydrogel scaffolds

Genus Distributions of cubic series-parallel graphs

We derive a quadratic-time algorithm for the genus distribution of any 3-regular, biconnected series-parallel graph, which we extend to any biconnected series-parallel graph of maximum degree at most 3. Since the biconnected components of every graph of treewidth 2 are series-parallel graphs, this yields, by use of bar-amalgamation, a quadratic-time algorithm for every graph of treewidth at most 2 and maximum degree at most 3.Comment: 21 page

Self-Sorting Microscale Compartmentalized Block Copolypeptide Hydrogels

Multicomponent interpenetrating network hydrogels possessing enhanced mechanical stiffness compared to their individual components were prepared via physical mixing of diblock copolypeptides that assemble by either hydrophobic association or polyion complexation in aqueous media. Optical microscopy analysis of fluorescent-probe-labeled multicomponent hydrogels revealed that the diblock copolypeptide components rapidly and spontaneously self-sort to form distinct hydrogel networks that interpenetrate at micron length scales. These materials represent a class of microscale compartmentalized hydrogels composed of degradable, cell-compatible components, which possess rapid self-healing properties and independently tunable domains for downstream applications in biology and additive manufacturing

Testing Convexity and Acyclicity, and New Constructions for Dense Graph Embeddings

Property testing, especially that of geometric and graph properties, is an ongoing area of research. In this thesis, we present a result from each of the two areas. For the problem of convexity testing in high dimensions, we give nearly matching upper and lower bounds for the sample complexity of algorithms have one-sided and two-sided error, where algorithms only have access to labeled samples independently drawn from the standard multivariate Gaussian. In the realm of graph property testing, we give an improved lower bound for testing acyclicity in directed graphs of bounded degree. Central to the area of topological graph theory is the genus parameter, but the complexity of determining the genus of a graph is poorly understood when graphs become nearly complete. We summarize recent progress in understanding the space of minimum genus embeddings of such dense graphs. In particular, we classify all possible face distributions realizable by minimum genus embeddings of complete graphs, present new constructions for genus embeddings of the complete graphs, and find unified constructions for minimum triangulations of surfaces

Jungerman ladders and index 2 constructions for genus embeddings of dense regular graphs

We construct several families of genus embeddings of near-complete graphs using index 2 current graphs. In particular, we give the first known minimum genus embeddings of certain families of octahedral graphs, solving a longstanding conjecture of Jungerman and Ringel, and Hamiltonian cycle complements, making partial progress on a problem of White. Index 2 current graphs are also applied to various cases of the Map Color Theorem, in some cases yielding significantly simpler solutions, e.g., the nonorientable genus of $K_{12s+8}-K_2$. We give a complete description of the method, originally due to Jungerman, from which all these results were obtained.Comment: 23 pages, 21 figures; fixed 2 figures from previous versio