Strong shift equivalence, algebraic K-theory, and isolating zero-dimensional dynamics on manifolds

We study the relations of shift equivalence and strong shift equivalence for matrices over a ring $\mathcal{R}$, and establish a connection between these relations and algebraic K-theory. We utilize this connection to obtain results in two areas where the shift and strong shift equivalence relations play an important role: the study of finite group extensions of shifts of finite type, and the Generalized Spectral Conjectures of Boyle and Handelman for nonnegative matrices over subrings of the real numbers. We show the refinement of the shift equivalence class of a matrix $A$ over a ring $\mathcal{R}$ by strong shift equivalence classes over the ring is classified by a quotient $NK_{1}(\mathcal{R}) / E(A,\mathcal{R})$ of the algebraic K-group $NK_{1}(\calR)$. We use the K-theory of non-commutative localizations to show that in certain cases the subgroup $E(A,\mathcal{R})$ must vanish, including the case $A$ is invertible over $\mathcal{R}$. We use the K-theory connection to clarify the structure of algebraic invariants for finite group extensions of shifts of finite type. In particular, we give a strong negative answer to a question of Parry, who asked whether the dynamical zeta function determines up to finitely many topological conjugacy classes the extensions by $G$ of a fixed mixing shift of finite type. We apply the K-theory connection to prove the equivalence of a strong and weak form of the Generalized Spectral Conjecture of Boyle and Handelman for primitive matrices over subrings of $\mathbb{R}$. We construct explicit matrices whose class in the algebraic K-group $NK_{1}(\mathcal{R})$ is non-zero for certain rings $\mathcal{R}$ motivated by applications. We study the possible dynamics of the restriction of a homeomorphism of a compact manifold to an isolated zero-dimensional set. We prove that for $n \ge 3$ every compact zero-dimensional system can arise as an isolated invariant set for a homeomorphism of a compact $n$-manifold. In dimension two, we provide obstructions and examples.

ROLE OF ICAM-1-MEDIATED ENDOCYTOSIS IN ENDOTHELIAL FUNCTION AND IMPLICATIONS FOR CARRIER-ASSISTED DRUG DELIVERY

Intercellular adhesion molecule 1 (ICAM-1) is a transmembrane protein found on the surface of vascular endothelial cells (ECs). Its expression is upregulated at inflammatory sites, allowing for targeted delivery of therapeutics using ICAM-1-binding drug carriers. Engagement of multiple copies of ICAM-1 by these drug carriers induces cell adhesion molecule (CAM)-mediated endocytosis, which results in trafficking of carriers to lysosomes and across ECs. Knowledge about the regulation behind CAM-mediated endocytosis can help improve drug delivery, but questions remain about these regulatory mechanisms. Furthermore, little is known about the natural function of this endocytic pathway. To address these gaps in knowledge, we focused on two natural binding partners of ICAM-1 that potentially elicit CAM-mediated endocytosis: leukocytes (which bind ICAM-1 via β₂ integrins) and fibrin polymers (a main component of blood clots which binds ICAM-1 via the γ3 sequence). First, inspired by properties of these natural binding partners, we varied the size and targeting moiety of model drug carriers to determine how these parameters affect CAM-mediated endocytosis. Increasing ICAM-1-targeted carrier size slowed carrier uptake kinetics, reduced carrier trafficking to lysosomes, and increased carrier transport across ECs. Changing targeting moieties from antibodies to peptides decreased particle binding and uptake, lowered trafficking to lysosomes, and increased transport across ECs. Second, using cell culture models of leukocyte/EC interactions, inhibiting regulatory elements of the CAM-mediated pathway disrupted leukocyte sampling, a process crucial to leukocyte crossing of endothelial layers (transmigration). This inhibition also decreased leukocyte transmigration across ECs, specifically through the transcellular route, which occurs through a single EC without disassembly of cell-cell junctions. Third, fibrin meshes, which mimic blood clot fragments/remnants, bound to ECs at ICAM-1-enriched sites and were internalized by the endothelium. Inhibiting the CAM-mediated pathway disrupted this uptake. Following endocytosis, fibrin meshes trafficked to lysosomes where they were degraded. In mouse models, CAM-mediated endocytosis of fibrin meshes appeared to remove fibrin remnants at the endothelial surface, preventing re-initiation of the coagulation cascade. Overall, these results support a link between CAM-mediated endocytosis and leukocyte transmigration as well as uptake of fibrin materials by ECs. Furthermore, these results will guide the future design of ICAM-1-targeted carrier-assisted therapies.