Effectiveness of Hindman's theorem for bounded sums

We consider the strength and effective content of restricted versions of Hindman's Theorem in which the number of colors is specified and the length of the sums has a specified finite bound. Let $\mathsf{HT}^{\leq n}_k$ denote the assertion that for each $k$-coloring $c$ of $\mathbb{N}$ there is an infinite set $X \subseteq \mathbb{N}$ such that all sums $\sum_{x \in F} x$ for $F \subseteq X$ and $0 < |F| \leq n$ have the same color. We prove that there is a computable $2$-coloring $c$ of $\mathbb{N}$ such that there is no infinite computable set $X$ such that all nonempty sums of at most $2$ elements of $X$ have the same color. It follows that $\mathsf{HT}^{\leq 2}_2$ is not provable in $\mathsf{RCA}_0$ and in fact we show that it implies $\mathsf{SRT}^2_2$ in $\mathsf{RCA}_0$. We also show that there is a computable instance of $\mathsf{HT}^{\leq 3}_3$ with all solutions computing $0'$. The proof of this result shows that $\mathsf{HT}^{\leq 3}_3$ implies $\mathsf{ACA}_0$ in $\mathsf{RCA}_0$

Hypernatural Numbers as Ultrafilters

In this paper we present a use of nonstandard methods in the theory of ultrafilters and in related applications to combinatorics of numbers

Sortase A Substrate Specificity in GBS Pilus 2a Cell Wall Anchoring

Streptococcus agalactiae, also referred to as Group B Streptococcus (GBS), is one of the most common causes of life-threatening bacterial infections in infants. In recent years cell surface pili have been identified in several Gram-positive bacteria, including GBS, as important virulence factors and promising vaccine candidates. In GBS, three structurally distinct types of pili have been discovered (pilus 1, 2a and 2b), whose structural subunits are assembled in high-molecular weight polymers by specific class C sortases. In addition, the highly conserved housekeeping sortase A (SrtA), whose main role is to link surface proteins to bacterial cell wall peptidoglycan by a transpeptidation reaction, is also involved in pili cell wall anchoring in many bacteria. Through in vivo mutagenesis, we demonstrate that the LPXTG sorting signal of the minor ancillary protein (AP2) is essential for pilus 2a anchoring. We successfully produced a highly purified recombinant SrtA (SrtAΔN40) able to specifically hydrolyze the sorting signal of pilus 2a minor ancillary protein (AP2-2a) and catalyze in vitro the transpeptidation reaction between peptidoglycan analogues and the LPXTG motif, using both synthetic fluorescent peptides and recombinant proteins. By contrast, SrtAΔN40 does not catalyze the transpeptidation reaction with substrate-peptides mimicking sorting signals of the other pilus 2a subunits (the backbone protein and the major ancillary protein). Thus, our results add further insight into the proposed model of GBS pilus 2a assembly, in which SrtA is required for pili cell wall covalent attachment, acting exclusively on the minor accessory pilin, representing the terminal subunit located at the base of the pilus

Structural Differences between the Streptococcus agalactiae Housekeeping and Pilus-Specific Sortases: SrtA and SrtC1

The assembly of pili on the cell wall of Gram-positive bacteria requires transpeptidase enzymes called sortases. In Streptococcus agalactiae, the PI-1 pilus island of strain 2603V/R encodes two pilus-specific sortases (SrtC1 and SrtC2) and three pilins (GBS80, GBS52 and GBS104). Although either pilus-specific sortase is sufficient for the polymerization of the major pilin, GBS80, incorporation of the minor pilins GBS52 and GBS104 into the pilus structure requires SrtC1 and SrtC2, respectively. The S. agalactiae housekeeping sortase, SrtA, whose gene is present at a different location and does not catalyze pilus polymerization, was shown to be involved in cell wall anchoring of pilus polymers. To understand the structural basis of sortases involved in such diverse functions, we determined the crystal structures of S. agalactiae SrtC1 and SrtA. Both enzymes are made of an eight-stranded beta-barrel core with variations in their active site architecture. SrtA exhibits a catalytic triad arrangement similar to that in Streptococcus pyogenes SrtA but different from that in Staphylococcus aureus SrtA. In contrast, the SrtC1 enzyme contains an N-terminal helical domain and a ‘lid’ in its putative active site, which is similar to that seen in Streptococcus pneumoniae pilus-specific sortases, although with subtle differences in positioning and composition. To understand the effect of such differences on substrate recognition, we have also determined the crystal structure of a SrtC1 mutant, in which the conserved DP(W/F/Y) motif was replaced with the sorting signal motif of GBS80, IPNTG. By comparing the structures of WT wild type SrtA and SrtC1 and the ‘lid’ mutant of SrtC1, we propose that structural elements within the active site and the lid may be important for defining the role of specific sortase in pili biogenesis

Exopolysaccharide-associated protein sorting in environmental organisms: the PEP-CTERM/EpsH system. Application of a novel phylogenetic profiling heuristic

BACKGROUND: Protein translocation to the proper cellular destination may be guided by various classes of sorting signals recognizable in the primary sequence. Detection in some genomes, but not others, may reveal sorting system components by comparison of the phylogenetic profile of the class of sorting signal to that of various protein families. RESULTS: We describe a short C-terminal homology domain, sporadically distributed in bacteria, with several key characteristics of protein sorting signals. The domain includes a near-invariant motif Pro-Glu-Pro (PEP). This possible recognition or processing site is followed by a predicted transmembrane helix and a cluster rich in basic amino acids. We designate this domain PEP-CTERM. It tends to occur multiple times in a genome if it occurs at all, with a median count of eight instances; Verrucomicrobium spinosum has sixty-five. PEP-CTERM-containing proteins generally contain an N-terminal signal peptide and exhibit high diversity and little homology to known proteins. All bacteria with PEP-CTERM have both an outer membrane and exopolysaccharide (EPS) production genes. By a simple heuristic for screening phylogenetic profiles in the absence of pre-formed protein families, we discovered that a homolog of the membrane protein EpsH (exopolysaccharide locus protein H) occurs in a species when PEP-CTERM domains are found. The EpsH family contains invariant residues consistent with a transpeptidase function. Most PEP-CTERM proteins are encoded by single-gene operons preceded by large intergenic regions. In the Proteobacteria, most of these upstream regions share a DNA sequence, a probable cis-regulatory site that contains a sigma-54 binding motif. The phylogenetic profile for this DNA sequence exactly matches that of three proteins: a sigma-54-interacting response regulator (PrsR), a transmembrane histidine kinase (PrsK), and a TPR protein (PrsT). CONCLUSION: These findings are consistent with the hypothesis that PEP-CTERM and EpsH form a protein export sorting system, analogous to the LPXTG/sortase system of Gram-positive bacteria, and correlated to EPS expression. It occurs preferentially in bacteria from sediments, soils, and biofilms. The novel method that led to these findings, partial phylogenetic profiling, requires neither global sequence clustering nor arbitrary similarity cutoffs and appears to be a rapid, effective alternative to other profiling methods