A graded subring of an inverse limit of polynomial rings

We study the power series ring R= K[[x1,x2,x3,...]]on countably infinitely many variables, over a field K, and two particular K-subalgebras of it: the ring S, which is isomorphic to an inverse limit of the polynomial rings in finitely many variables over K, and the ring R', which is the largest graded subalgebra of R. Of particular interest are the homogeneous, finitely generated ideals in R', among them the generic ideals. The definition of S as an inverse limit yields a set of truncation homomorphisms from S to K[x1,...,xn] which restrict to R'. We have that the truncation of a generic I in R' is a generic ideal in K[x1,...,xn]. It is shown in Initial ideals of Truncated Homogeneous Ideals that the initial ideal of such an ideal converge to the initial ideal of the corresponding ideal in R'. This initial ideal need no longer be finitely generated, but it is always locally finitely generated: this is proved in Gröbner Bases in R'. We show in Reverse lexicographic initial ideals of generic ideals are finitely generated that the initial ideal of a generic ideal in R' is finitely generated. This contrast to the lexicographic term order. If I in R' is a homogeneous, locally finitely generated ideal, and if we write the Hilbert series of the truncated algebras K[x1,...,xn] module the truncation of I as qn(t)/(1-t)n, then we show in Generalized Hilbert Numerators that the qn's converge to a power series in t which we call the generalized Hilbert numerator of the algebra R'/I. In Gröbner bases for non-homogeneous ideals in R' we show that the calculations of Gröbner bases and initial ideals in R' can be done also for some non-homogeneous ideals, namely those which have an associated homogeneous ideal which is locally finitely generated. The fact that S is an inverse limit of polynomial rings, which are naturally endowed with the discrete topology, provides S with a topology which makes it into a complete Hausdorff topological ring. The ring R', with the subspace topology, is dense in R, and the latter ring is the Cauchy completion of the former. In Topological properties of R' we show that with respect to this topology, locally finitely generated ideals in R'are closed.

Theoretical studies of Membrane Proteins : Properties, Prediction Methods and Genome-wide analysis

Membrane proteins are a large and important class of proteins. They are responsible for several of the key functions in a living cell, e.g. transport of nutrients and ions, cell-cell signaling, and cell-cell adhesion. Despite their importance it has not been possible to study their structure and organization in much detail because of the difficulty to obtain 3D structures. In this thesis theoretical studies of membrane protein sequences and structures have been carried out by analyzing existing experimental data. The data comes from several sources including sequence databases, genome sequencing projects, and 3D structures. Prediction of the membrane spanning regions by hydrophobicity analysis is a key technique used in several of the studies. A novel method for this is also presented and compared to other methods. The primary questions addressed in the thesis are: What properties are common to all membrane proteins? What is the overall architecture of a membrane protein? What properties govern the integration into the membrane? How many membrane proteins are there and how are they distributed in different organisms? Several of the findings have now been backed up by experiments. An analysis of the large family of G-protein coupled receptors pinpoints differences in length and amino acid composition of loops between proteins with and without a signal peptide and also differences between extra- and intracellular loops. Known 3D structures of membrane proteins have been studied in terms of hydrophobicity, distribution of secondary structure and amino acid types, position specific residue variability, and differences between loops and membrane spanning regions. An analysis of several fully and partially sequenced genomes from eukaryotes, prokaryotes, and archaea has been carried out. Several differences in the membrane protein content between organisms were found, the most important being the total number of membrane proteins and the distribution of membrane proteins with a given number of transmembrane segments. Of the properties that were found to be similar in all organisms, the most obvious is the bias in the distribution of positive charges between the extra- and intracellular loops. Finally, an analysis of homologues to membrane proteins with known topology uncovered two related, multi-spanning proteins with opposite predicted orientations. The predicted topologies were verified experimentally, providing a first example of "divergent topology evolution".