Regular homomorphisms of minimal sets

The classification of minimal sets is a central theme in abstract topological dynamics. Recently this work has been strengthened and extended by consideration of homomorphisms. Background material is presented in Chapter I. Given a flow on a compact Hausdorff space, the action extends naturally to the space of closed subsets, taken with the Hausdorff topology. These hyperspaces are discussed and used to give a new characterization of almost periodic homomorphisms. Regular minimal sets may be described as minimal subsets of enveloping semigroups. Regular homomorphisms are defined in Chapter II by extending this notion to homomorphisms with minimal range. Several characterizations are obtained. In Chapter III, some additional results on homomorphisms are obtained by relativizing enveloping semigroup notions. In Veech's paper on point distal flows, hyperspaces are used to associate an almost one-to-one homomorphism with a given homomorphism of metric minimal sets. In Chapter IV, a non-metric generalization of this construction is studied in detail using the new notion of a highly proximal homomorphism. An abstract characterization is obtained, involving only the abstract properties of homomorphisms. A strengthened version of the Veech Structure Theorem for point distal flows is proved. In Chapter V, the work in the earlier chapters is applied to the study of homomorphisms for which the almost periodic elements of the associated hyperspace are all finite. In the metric case, this is equivalent to having at least one fiber finite. Strong results are obtained by first assuming regularity, and then assuming that the relative proximal relation is closed as well.

Exploring the Function and Regulation of Arabidopsis EIN2 in Ethylene Signaling

Ethylene is an essential plant hormone involved in nearly all stages of plant growth and development. EIN2 (ETHYLENE INSENSITIVE2) is a master positive regulator in the ethylene signaling pathway, consisting of an N-terminal domain and a C-terminal domain. The EIN2 N-terminal domain localizes to the endoplasmic reticulum (ER) membrane and shows sequence similarity to Nramp metal ion transporters. The cytosolic C-terminal domain is unique to plants and signals downstream. There have been several major gaps in our knowledge of EIN2 function. It was unknown how the ethylene signal gets relayed from the known upstream component CTR1 (CONSTITUTIVE RESPONSE1) a Ser/Thr kinase at the ER, to EIN2. How the ethylene signal was transduced from EIN2 to the next downstream component transcription factor EIN3 (ETHYLENE INSENSITIVE3) in the nucleus was also unknown. The N-terminal domain of EIN2 shows homology to Nramp metal ion transporters and whether EIN2 can also function as a metal transporter has been a question plaguing the ethylene field for almost two decades. Here, EIN2 was found to interact with the CTR1 protein kinase, leading to the discovery that CTR1 phosphorylates the C-terminal domain of EIN2 in Arabidopsis thaliana. Using tags at the termini of EIN2, it was deduced that in the presence of ethylene, the EIN2 C-terminal domain is cleaved and translocates into the nucleus, where it could somehow activate downstream ethylene responses. The EIN2 C-terminal domain interacts with nuclear proteins, RTE3 and EER5, which are components of the TREX-2 mRNA export complex, although the role of these interactions remains unclear. The EIN2 N-terminal domain was found to be capable of divalent metal transport when expressed in E. coli and S. cerevisiae leading to the hypothesis that metal transport plays a role in ethylene signaling. This hypothesis was tested using a novel missense allele, ein2 G36E, substituting a highly conserved residue that is required for metal transport in Nramp proteins. This G36E substitution did not disrupt metal ion transport of EIN2, but the ethylene insensitive phenotype of this mutant indicates that the EIN2 N-terminal domain is important for positively regulating the C-terminal domain. The defect of the ein2 G36E mutant does not prevent proper expression or subcellular localization, but might affect protein modifications. The ein2 G36E allele is partially dominant, mostly likely displaying haploinsufficiency. Overexpression of the EIN2 N-terminal domain in the ein2 G36E mutant did not rescue ethylene insensitivity, suggesting the N-terminal domain functions in cis to regulate the C-terminal domain. These findings advance our knowledge of EIN2, which is critical to understanding ethylene signaling.