Weighted Homology of Bi-Structures over Certain Discrete Valuation Rings

An RNA bi-structure is a pair of RNA secondary structures that are considered as arc-diagrams. We present a novel weighted homology theory for RNA bi-structures, which was obtained through the intersections of loops. The weighted homology of the intersection complex X features a new boundary operator and is formulated over a discrete valuation ring, R. We establish basic properties of the weighted complex and show how to deform it in order to eliminate any 3-simplices. We connect the simplicial homology, Hi(X), and weighted homology, Hi,R(X), in two ways: first, via chain maps, and second, via the relative homology. We compute H0,R(X) by means of a recursive contraction procedure on a weighted spanning tree and H1,R(X) via an inflation map, by which the simplicial homology of the 1-skeleton allows us to determine the weighted homology H1,R(X). The homology module H2,R(X) is naturally obtained from H2(X) via chain maps. Furthermore, we show that all weighted homology modules Hi,R(X) are trivial for i>2. The invariant factors of our structure theorems, as well as the weighted Whitehead moves facilitating the removal of filled tetrahedra, are given a combinatorial interpretation. The weighted homology of bi-structures augments the simplicial counterpart by introducing novel torsion submodules and preserving the free submodules that appear in the simplicial homology.

Incidence coefficients in the Novikov Complex for Morse forms: rationality and exponential growth properties

Let f : M → S 1 be a Morse map, v a transverse f -gradient. Theconstruction of the Novikov complex associates to these data a free chain complexC ∗ (f, v) over the ring Z[t]][t −1 ], generated by the critical points of f and computingthe completed homology module of the corresponding infinite cyclic covering of M .Novikov’s Exponential Growth Conjecture says that the boundary operators in thiscomplex are power series of non-zero convergence raduis.In [12] the author announced the proof of the Novikov conjecture for the case ofC 0 -generic gradients together with several generalizations. The proofs of the firstpart of this work were published in [13]. The present article contains the proofs ofthe second part.There is a refined version of the Novikov complex, defined over a suitable com-pletion of the group ring of the fundamental group. We prove that for a C 0 -genericf -gradient the corresponding incidence coefficients belong to the image in the Novikovring of a (non commutative) localization of the fundamental group ring.The Novikov construction generalizes also to the case of Morse 1-forms. In thiscase the corresponding incidence coefiicients belong to a certain completion of thering of integral Laurent polynomials of several variables. We prove that for a givenMorse form ω and a C 0 -generic ω-gradient these incidence coefficients are rationalfunctions.The incidence coefficients in the Novikov complex are obtained by counting thealgebraic number of the trajectories of the gradient, joining the zeros of the Morseform. There is V.I.Arnold’s version of the exponential growth conjecture, whichconcerns the total number of trajectories. We confirm this stronger form of theconjecture for any given Morse form and a C 0 -dense set of its gradients.We give an example of explicit computation of the Novikov complex.

Amenable covers and ℓ1-invisibility

Let [Formula: see text] be a topological space admitting an amenable cover of multiplicity [Formula: see text]. We show that, for every [Formula: see text] and every [Formula: see text], the image of [Formula: see text] in the [Formula: see text]-homology module [Formula: see text] vanishes. This strengthens previous results by Gromov and Ivanov, who proved, under the same assumptions, that the [Formula: see text]-seminorm of [Formula: see text] vanishes.

Novikov homology and non-commutative Alexander polynomials

In the early 2000's Cochran and Harvey introduced non-commutative Alexander polynomials for 3-manifolds. Their degrees give strong lower bounds on the Thurston norm. In this paper, we make the case that the vanishing of a certain Novikov–Sikorav homology module is the correct notion of a monic non-commutative Alexander polynomial. Furthermore we will use the opportunity to give new proofs of several statements about Novikov–Sikorav homology in the three-dimensional context.

Comparison of Family 9 Cellulases from Mesophilic and Thermophilic Bacteria

ABSTRACTCellulases containing a family 9 catalytic domain and a family 3c cellulose binding module (CBM3c) are important components of bacterial cellulolytic systems. We measured the temperature dependence of the activities of three homologs:Clostridium cellulolyticumCel9G,Thermobifida fuscaCel9A, andC. thermocellumCel9I. To directly compare their catalytic activities, we constructed six new versions of the enzymes in which the three GH9-CBM3c domains were fused to a dockerin both with and without aT. fuscafibronectin type 3 homology module (Fn3). We studied the activities of these enzymes on crystalline cellulose alone and in complex with a miniscaffoldin containing a cohesin and a CBM3a. The presence of Fn3 had no measurable effect on thermostability or cellulase activity. The GH9-CBM3c domains of Cel9A and Cel9I, however, were more active than the wild type when fused to a dockerin complexed to scaffoldin. The three cellulases in complex have similar activities on crystalline cellulose up to 60°C, butC. thermocellumCel9I, the most thermostable of the three, remains highly active up to 80°C, where its activity is 1.9 times higher than at 60°C. We also compared the temperature-dependent activities of different versions of Cel9I (wild type or in complex with a miniscaffoldin) and found that the thermostable CBM is necessary for activity on crystalline cellulose at high temperatures. These results illustrate the significant benefits of working with thermostable enzymes at high temperatures, as well as the importance of retaining the stability of all modules involved in cellulose degradation.

Deletion of the Extracellular Membrane-Distal Cytokine Receptor Homology Module of Mpl Results in Constitutive Cell Growth and Loss of Thrombopoietin Binding

The thrombopoietin receptor, Mpl, is a member of the cytokine receptor superfamily. The extracellular domain of Mpl contains two copies of the cytokine receptor homology module (CRM). Mpl is encoded by c-mpl, the cellular homologue of the oncogene v-mpl.The oncogenic potential of v-mpl may arise from deletion of all but the 43 most membrane-proximal amino acids of the extracellular domain of the wild-type receptor. To test the hypothesis that the extracellular domain of Mpl plays a role in controlling receptor activity, we created mutants of murine Mpl in which the membrane-distal CRM was either deleted or replaced by the membrane-proximal CRM. Introduction of these mutant receptors into factor-dependent BaF3 cells led to constitutive cell growth in the absence of growth factor. Both mutant receptors failed to bind 125I-Tpo. These results suggest that the membrane-distal CRM of Mpl acts as a brake on cell proliferation and that this region is required for ligand binding.