Projective toric varieties of codimension 2 with maximal Castelnuovo--Mumford regularity

The Eisenbud--Goto conjecture states that $\operatorname{reg} X\le\operatorname{deg} X -\operatorname{codim} X+1$ for a nondegenerate irreducible projective variety $X$ over an algebraically closed field. While this conjecture is known to be false in general, it has been proven in several special cases, including when $X$ is a projective toric variety of codimension $2$. We classify the projective toric varieties of codimension $2$ having maximal regularity, that is, for which equality holds in the Eisenbud--Goto bound. We also give combinatorial characterizations of the arithmetically Cohen--Macaulay toric varieties of maximal regularity in characteristic $0$.Comment: 26 page

Biconed graphs, edge-rooted forests, and h-vectors of matroid complexes

A well-known conjecture of Richard Stanley posits that the $h$-vector of the independence complex of a matroid is a pure ${\mathcal O}$-sequence. The conjecture has been established for various classes but is open for graphic matroids. A biconed graph is a graph with two specified `coning vertices', such that every vertex of the graph is connected to at least one coning vertex. The class of biconed graphs includes coned graphs, Ferrers graphs, and complete multipartite graphs. We study the $h$-vectors of graphic matroids arising from biconed graphs, providing a combinatorial interpretation of their entries in terms of `edge-rooted forests' of the underlying graph. This generalizes constructions of Kook and Lee who studied the M\"obius coinvariant (the last nonzero entry of the $h$-vector) of graphic matroids of complete bipartite graphs. We show that allowing for partially edge-rooted forests gives rise to a pure multicomplex whose face count recovers the $h$-vector, establishing Stanley's conjecture for this class of matroids.Comment: 15 pages, 3 figures; V2: added omitted author to metadat

An Asymptotic Model of Electroporation-Mediated Molecular Delivery in Skeletal Muscle Tissue

<p>Electroporation is a biological cell's natural reaction to strong electric fields, where transient pores are created in the cell membrane. While electroporation holds promise of being a safe and effective tool for enhancing molecular delivery in numerous medical applications, it remains largely confined to preclinical research and clinical trials due to an incomplete understanding of the exact mechanisms involved. Muscle fibers are an important delivery target, but traditional theoretical studies of electroporation ignore the individual fiber geometry, making it impossible to study the unique transverse and longitudinal effects from the pulse stimulus. In these long, thin muscle fibers, the total reaction of the fiber to the electric field is due to fundamentally different effects from the constituent longitudinal and transverse components of the electric field generated by the pulse stimulus. While effects from the transverse component have been studied to some degree, the effects from the longitudinal component have not been considered. </p><p>This study develops a model of electroporation and delivery of small molecules in muscle tissue that includes effects from both the transverse and longitudinal components of the electric field. First, an asymptotic model of electric potential in an individual muscle fiber is derived that separates the full 3D boundary value problem into transverse and a longitudinal problems. The transverse and longitudinal problems each have their own respective source functions: the new "transverse activating function" and the well known longitudinal activating function (AF). This separation enhances analysis of the different effects from these two AFs and drastically reduces computational intensity. Electroporation is added to the asymptotic fiber model, and simplified two-compartment mass transport equations are derived from the full 3D conservation of mass equations to allow simulation of molecular uptake due to diffusion and the electric field. Special emphasis is placed on choosing model geometry, electrical, and pulsing parameters that are in accordance with experiments that study electroporation-mediated delivery of small molecules in the skeletal muscle of small mammals.</p><p>Simulations reveal that for fibers close to the electrodes the transverse AF dominates, but for fibers far from the electrodes the longitudinal AF enhances uptake by as much as 2000%. However, on the macroscopic tissue level, the increase in uptake from the longitudinal AF is no more than 10%, given that fibers far from the electrodes contribute so little to the total uptake in the tissue. The mechanism underlying the smaller effect from the longitudinal AF is found to be unique to the process of electroporation itself. Electroporation occurs on the short time scale of polarization via the transverse AF, drastically increases membrane conductance, and effectively precludes further creation of pores from charging of the membrane via the longitudinal AF. The exact value of enhancement in uptake from the longitudinal AF is shown to depend on pulsing, membrane, and tissue parameters. Finally, simulation results reproduce qualitative, and in some cases quantitative, behavior of uptake observed in experiments.</p><p>Overall, percent increase in total tissue uptake from the longitudinal AF is on the order of experimental variability, and this study corroborates previous theoretical models that neglect the effects from the longitudinal AF. However, previous models neglect the longitudinal AF without explanation, while the asymptotic fiber model is able to detail the mechanisms involved. Mechanisms revealed by the model offer insight into interpreting experimental results and increasing efficiency of delivery protocols. The model also rigorously derives a new transverse AF based on individual fiber geometry, which affects the spatial distribution of uptake in tissue differently than predicting uptake based on the magnitude of the electric field, as used in many published models. Results of this study are strictly valid for transport of small molecules through small non-growing pores. For gene therapy applications the model must be extended to transport of large DNA molecules through large pores, which may alter the importance of the longitudinal AF. In broader terms, the asymptotic model also provides a new, computationally efficient tool that may be used in studying the effect of transverse and longitudinal components of the field for other types of membrane dynamics in muscle and nerves.</p>Dissertatio