Central limit theorems of partial sums for large segmental values

AbstractLet (Xi,Ui) be i.i.d., Xi real valued and Ui vector valued, bounded random variables or governed by a finite state Markov chain. Assuming that E[X]<0 and P(X> 0) > 0, central limit theorems are derived for ΣiUi on segments conditioned that ΣiXi is increasingly high, going to +∞. While these segments are exponentially rare, they are of importance in many models of stochastic analysis including queueing systems and molecular sequence comparisons. Particular applications give central limit theorems for the empirical frequencies over such segments and for their length

Some inequalities for generalized concave functions

Digitalitzat per Artypla

Protein length in eukaryotic and prokaryotic proteomes

We analyzed length differences of eukaryotic, bacterial and archaeal proteins in relation to function, conservation and environmental factors. Comparing Eukaryotes and Prokaryotes, we found that the greater length of eukaryotic proteins is pervasive over all functional categories and involves the vast majority of protein families. The magnitude of these differences suggests that the evolution of eukaryotic proteins was influenced by processes of fusion of single-function proteins into extended multi-functional and multi-domain proteins. Comparing Bacteria and Archaea, we determined that the small but significant length difference observed between their proteins results from a combination of three factors: (i) bacterial proteomes include a greater proportion than archaeal proteomes of longer proteins involved in metabolism or cellular processes, (ii) within most functional classes, protein families unique to Bacteria are generally longer than protein families unique to Archaea and (iii) within the same protein family, homologs from Bacteria tend to be longer than the corresponding homologs from Archaea. These differences are interpreted with respect to evolutionary trends and prevailing environmental conditions within the two prokaryotic groups

Quantile distributions of amino acid usage in protein classes

A comparative study of the compositional properties of various protein sets from both cellular and viral organisms is presented. Invariants and contrasts of amino acid usages have been discerned for different protein function classes and for different species using robust statistical methods based on quantile distributions and stochastic ordering relationships. In addition, a quantitative criterion to assess amino acid compositional extremes relative to a reference protein set is proposed and applied. Invariants of amino acid usage relate mainly to the central range of quantile distributions, whereas contrasts occur mainly in the tails of the distributions, especially contrasts between eukaryote and prokaryote species. Influences from genomic constraint are evident, for example, in the arginine:lysine ratios and the usage frequencies of residues encoded by G + C-rich versus A + T-rich codon types. The structurally similar amino acids, glutamate versus aspartate and phenylalanine versus tyrosine, show stochastic dominance relationships for most species protein sets favoring glutamate and phenylalanine respectively. The quantile distribution of hydrophobic amino acid usages in prokaryote data dominates the corresponding quantile distribution in human data. In contrast, glutamate, cysteine, proline and serine usages in human proteins dominate the corresponding quantile distributions in Escherichia coli. E.coli dominates human in the use of basic residues, but no dominance ordering applies to acidic residues. The discussion centers on commonalities and anomalies of the amino acid compositional spectrum in relation to species, function, cellular localization, biochemical and steric attributes, complexity of the amino acid biosynthetic pathway, amino acid relative abundances and founder effect

Discs around giant protoplanets

This thesis contains a study of circumplanetary discs (CPDs) around giant protoplanets. They are modelled with three-dimensional, hydrodynamical multifluid simulations, with the aim of understanding their dust grain size distributions and thus opacities and dust masses. By comparing 2-fluid (gas + 1 dust grain size) and multifluid (gas + multiple grain size) simulations for a 1 M_Jup protoplanet at 10 AU, it is ascertained that every dust grain size accretes onto the CPD with the same efficiency as if it and the gas were a 2-fluid system. Dust grains of size 1um-100um accrete with comparable efficiency. 1mm dust grains are blocked at the outer gap edge or taken into the horseshoe region, where they reach high concentration. They are extremely inefficient at accreting onto the CPD. This leads to low CPD dust-to-gas ratio ~ 8 × 10^-4. By comparing 9 multifluid simulations of 10 M_Earth, 100 M_Earth and 1000 M_Earth protoplanets at 5 AU, 15 AU and 30 AU, it is demonstrated that the thermal criterion R_Hill > H accurately predicts which can form gaps and CPDs and which only envelopes. The crucial governing parameter is shown to be a_dec the grain size at which accretion efficiency decreases. Small a_dec means low CPD dust mass, because most dust mass is in large grains. A parametrisation with a_dec is an excellent fit to the grain size distribution. Knowing a_dec gives that distribution, thus giving opacity, to translate observed fluxes into masses. a_dec falls as semimajor axis rises. High protoplanet mass also makes a_dec smaller because of a deeper gap. Therefore CPD dust mass sometimes falls as protoplanet mass rises. The results suggest that massive giant planets at >~ 30 AU will have extremely low dust-to-gas mass ratios (~ 2 × 10^-4). They will be unable to form rocky satellites and will be very poor in Fe and silicates and rich in H/He

Determination of the Critical Concentration of Neutrophils Required to Block Bacterial Growth in Tissues

We showed previously that the competition between bacterial killing by neutrophils and bacterial growth in stirred serum-containing suspensions could be modeled as the competition between a first-order reaction (bacterial growth) and a second-order reaction (bacterial killing by neutrophils). The model provided a useful parameter, the critical neutrophil concentration (CNC), below which bacterial concentration increased and above which it decreased, independent of the initial bacterial concentration. We report here that this model applies to neutrophil killing of bacteria in three-dimensional fibrin matrices and in rabbit dermis. We measured killing of 103–108 colony forming units/ml Staphylococcus epidermidis by 105–108 human neutrophils/ml in fibrin gels. The CNC was ∼4 × 106 neutrophils/ml gel in the presence of normal serum and ∼1.6 × 107 neutrophils/ml gel in the presence of C5-deficient serum. Application of our model to published data of others on killing of ∼5 × 107 to 2 × 108 E. coli/ml rabbit dermis yielded CNCs from ∼4 × 106 to ∼8 × 106 neutrophils/ml dermis. Thus, in disparate tissues and tissuelike environments, our model fits the kinetics of bacterial killing and gives similar lower limits (CNCs) to the neutrophil concentration required to control bacterial growth