Tamari Lattices and the symmetric Thompson monoid

We investigate the connection between Tamari lattices and the Thompson group F, summarized in the fact that F is a group of fractions for a certain monoid F+sym whose Cayley graph includes all Tamari lattices. Under this correspondence, the Tamari lattice operations are the counterparts of the least common multiple and greatest common divisor operations in F+sym. As an application, we show that, for every n, there exists a length l chain in the nth Tamari lattice whose endpoints are at distance at most 12l/n.Comment: 35page

On classification of groups generated by 3-state automata over a 2-letter alphabet

We show that the class of groups generated by 3-state automata over a 2-letter alphabet has no more than 122 members. For each group in the class we provide some basic information, such as short relators, a few initial values of the growth function, a few initial values of the sizes of the quotients by level stabilizers (congruence quotients), and hystogram of the spectrum of the adjacency operator of the Schreier graph of the action on level 9. In most cases we provide more information, such as whether the group is contracting, self-replicating, or (weakly) branch group, and exhibit elements of infinite order (we show that no group in the class is an infinite torsion group). A GAP package, written by Muntyan and Savchuk, was used to perform some necessary calculations. For some of the examples, we establish that they are (virtually) iterated monodromy groups of post-critically finite rational functions, in which cases we describe the functions and the limit spaces. There are exactly 6 finite groups in the class (of order no greater than 16), two free abelian groups (of rank 1 and 2), and only one free nonabelian group (of rank 3). The other examples in the class range from familiar (some virtually abelian groups, lamplighter group, Baumslag-Solitar groups BS(1±3), and a free product C2 ∗ C2 ∗ C2) to enticing (Basilica group and a few other iterated monodromy groups)

Insulin-like growth factor binding proteins in bovine articular and ovine growth-plate chondrocyte cultures: Their regulation by IGF's and modulation of poteoglycan synthesis

Cultured chondrocytes respond to insulin-like growth factors (IGFs) by increasing the production of proteoglycans and insulin-like growth factor binding proteins (IGF-BPs). To investigate the biological effects of IGFs and IGF-BPs, isolated bovine articular and ovine growth-plate chondrocytes were cultured at high density in the presence of IGF-1, and its truncated form, des (1-3) IGF-I. Both growth factors stimulated the production of IGF-BPs in articular and growth-plate chondrocyte monolayers. Western ligand blots showed that bovine articular chondrocytes released two forms of IGF-BPs into conditioned medium with molecular weights of 29 and 31 kDa. Ovine growth-plate chondrocytes released four different forms of IGF-BPs of approx. 22, 24; 29-30 and 34 kDa. IGF-I and des (1-3) IGF-I stimulated total proteoglycan synthesis by articular chondrocytes up to 1.5-fold. The truncated analogue was more potent at lower concentrations, particularly in stimulating incorporation of newly synthesized proteoglycans into the cell-layer. The maximal stimulation of proteoglycan synthesis in ovine growth-plate chondrocyte culture was 3-fold with des (1-3) IGF-I, while IGF-I enhanced proteoglycan production by only 2-fold over the concentrations used. Our results suggest that endogenous IGF-BPs in chondrocyte cultures act as a part of a feed-back mechanism which diminishes the bioactivity of IGF-I.Sunic, Damir; Belford, David A.; McNeil, Julian D.; Wiebkin, Ole W

Regulation of insulin-like growth factor-binding protein-5 by insulin-like growth factor I and interleukin-1alpha in ovine articular chondrocytes

Insulin-like growth factors (IGFs) contribute to the maintenance of the cartilage matrix by stimulating proteoglycan synthesis. In contrast, interleukin-1 (IL-1), an inflammatory cytokine, suppresses the synthesis of proteoglycans. In pathological conditions the chondrocytes’ responsiveness to IGF-I is decreased, and elevated levels of IGF-binding proteins (IGFBPs) have been implicated as a possible cause. The aim of this study was to investigate the effects of IGF-I and IL-1 on IGFBP production by ovine articular chondrocytes (OAC) and the roles of these IGFBPs in the regulation of proteoglycan synthesis. As revealed by Western ligand and immunoblotting, OACs secreted IGFBP-2 and a 24-kDa IGFBP in culture medium under basal conditions. Exposure of the cells to IGF-I for 48 h resulted in the appearance of IGFBP-5 in the medium. Des(1–3)IGF-I, an IGF-I analog with reduced affinity for IGFBPs, also increased the level of IGFBP-5, but to a lesser extent than IGF-I, whereas LR3IGF-I, which has virtually no affinity for IGFBPs, had no effect on IGFBP-5. Furthermore, IGFBP-5 underwent a time-dependent limited proteolysis when incubated with OAC-conditioned medium, degrading into 22- and 16-kDa fragments. The degradation of IGFBP-5 was significantly inhibited by IGF-I, but not by des(1–3)IGF-I or LR3IGF-I. Basic fibroblast growth factor, transforming growth factor-β, and platelet-derived growth factor had no effect on OAC IGFBPs. However, IL-1α increased the IGFBP-5 level in a dose-dependent manner, showing maximum activity at 200 U/ml. Furthermore, IL-1α, but not IGF-I, induced IGFBP-5 messenger RNA expression, as assessed by Northern blot analysis. Coincubation of IGF-I with IL-1α resulted in a substantially increased IGFBP-5 protein level, suggesting a synergism between the mechanisms of action of these two factors. Des(1–3)IGF-I and LR3IGF-I were 10 times more potent than IGF-I in stimulating proteoglycan synthesis, indicating inhibition of IGF-I activity by endogenous IGFBPs. IL-1α reduced the IGF-I bioactivity, but had no effect on the activities of the IGF-I analogs, thus implying that locally produced IGFBPs, particularly IGFBP-5, which was substantially increased when IGF-I and IL-1α were coincubated, mediated the reduction of the IGF-I activity. Our results demonstrate that IGF-I and IL-1α synergistically increase the level of IGFBP-5 in OAC by inhibiting the proteolysis and stimulating the expression of IGFBP-5, respectively. Furthermore, the attenuation of IGF-I-stimulated proteoglycan synthesis by IL-1α in OAC appears to be mediated by chondrocyte IGFBPs. We conclude that locally produced IGFBPs, in particular IGFBP-5, may play a critical role in the regulation of cartilage matrix degradation in inflammatory and degenerative arthritides.Damir Sunic, Julian D. McNeil, Timothy E. Rayner, Dennis L. Andress, and David A. Belfor