The growth rate and dimension theory of beta-expansions

In a recent paper of Feng and Sidorov they show that for β∈(1,(1+5√)/2) the set of β-expansions grows exponentially for every x∈(0,1/(β−1)). In this paper we study this growth rate further. We also consider the set of β-expansions from a dimension theory perspective.

Differential crosstalk between estrogen receptor (ER) alpha and ER beta and the thyroid hormone receptor isoforms results in flexible regulation of the consensus ERE

Crosstalk between nuclear receptors is important for conversion of external and internal stimuli to a physiologically meaningful response by cells. Previous studies from this laboratory have demonstrated crosstalk between the estrogen (ER) and thyroid hormone receptors (TR) on two estrogen responsive physiological promoters, the preproenkephalin and oxytocin receptor gene promoter. Since ERa and ERb are isoforms possessing overlapping and distinct transactivation properties, we hypothesized that the interaction of ERa and b with the various TR isoforms would not be equivalent. To explore this hypothesis, the consensus estrogen response element (ERE)derived from the Xenopus vitellogenin gene is used to investigate the differences in interaction between ERa and b isoforms and the different TR isoforms in fibroblast cells. Both the ER isoforms transactivate from the consensus ERE, though ERa transactivates to a greater extent than ERb. Although neither of the TRb isoforms have an effect on ERa transactivation from the consensus ERE, the liganded TRa1 inhibits the ERa transactivation from the consensus ERE. In contrast, the liganded TRa1 facilitates ERb-mediated transactivation. The crosstalk between the TRb isoforms with the ERa isoform, on the consensus ERE, is different from that with the ERb isoform. The use of a TRa1 mutant, which is unable to bind DNA, abolishes the ability of the TRa1 isoform to interact with either of the ER isoforms. These differences in nuclear receptor crosstalk reveal an important functional difference between isoforms, which provides a novel mechanism for neuroendocrine integration.