On the Rademacher maximal function

Tools known as maximal functions are frequently used in harmonic analysis when studying local behaviour of functions. Typically they measure the suprema of local averages of non-negative functions. It is essential that the size (more precisely, the L^p-norm) of the maximal function is comparable to the size of the original function. When dealing with families of operators between Banach spaces we are often forced to replace the uniform bound with the larger R-bound. Hence such a replacement is also needed in the maximal function for functions taking values in spaces of operators. More specifically, the suprema of norms of local averages (i.e. their uniform bound in the operator norm) has to be replaced by their R-bound. This procedure gives us the Rademacher maximal function, which was introduced by Hytönen, McIntosh and Portal in order to prove a certain vector-valued Carleson's embedding theorem. They noticed that the sizes of an operator-valued function and its Rademacher maximal function are comparable for many common range spaces, but not for all. Certain requirements on the type and cotype of the spaces involved are necessary for this comparability, henceforth referred to as the “RMF-property”. It was shown, that other objects and parameters appearing in the definition, such as the domain of functions and the exponent p of the norm, make no difference to this. After a short introduction to randomized norms and geometry in Banach spaces we study the Rademacher maximal function on Euclidean spaces. The requirements on the type and cotype are considered, providing examples of spaces without RMF. L^p-spaces are shown to have RMF not only for p greater or equal to 2 (when it is trivial) but also for 1 < p < 2. A dyadic version of Carleson's embedding theorem is proven for scalar- and operator-valued functions. As the analysis with dyadic cubes can be generalized to filtrations on sigma-finite measure spaces, we consider the Rademacher maximal function in this case as well. It turns out that the RMF-property is independent of the filtration and the underlying measure space and that it is enough to consider very simple ones known as Haar filtrations. Scalar- and operator-valued analogues of Carleson's embedding theorem are also provided. With the RMF-property proven independent of the underlying measure space, we can use probabilistic notions and formulate it for martingales. Following a similar result for UMD-spaces, a weak type inequality is shown to be (necessary and) sufficient for the RMF-property. The RMF-property is also studied using concave functions giving yet another proof of its independence from various parameters.

DNA Packaging and Host Cell Lysis: Late Events in Bacteriophage PRD1 Infection

The object of this study is a tailless internal membrane-containing bacteriophage PRD1. It has a dsDNA genome with covalently bound terminal proteins required for replication. The uniqueness of the structure makes this phage a desirable object of research. PRD1 has been studied for some 30 years during which time a lot of information has accumulated on its structure and life-cycle. The two least characterised steps of the PRD1 life-cycle, the genome packaging and virus release are investigated here. PRD1 shares the main principles of virion assembly (DNA packaging in particular) and host cell lysis with other dsDNA bacteriophages. However, this phage has some fascinating individual peculiarities, such as DNA packaging into a membrane vesicle inside the capsid, absence of apparent portal protein, holin inhibitor and procapsid expansion. In the course of this study we have identified the components of the DNA packaging vertex of the capsid, and determined the function of protein P6 in packaging. We managed to purify the procapsids for an in vitro packaging system, optimise the reaction and significantly increase its efficiency. We developed a new method to determine DNA translocation and were able to quantify the efficiency and the rate of packaging. A model for PRD1 DNA packaging was also proposed. Another part of this study covers the lysis of the host cell. As other dsDNA bacteriophages PRD1 has been proposed to utilise a two-component lysis system. The existence of this lysis system in PRD1 has been proven by experiments using recombinant proteins and the multi-step nature of the lysis process has been established.

The genome packaging machinery of dsDNA bacteriophage PRD1

Viral genomes are encapsidated within protective protein shells. This encapsidation can be achieved either by a co-condensation reaction of the nucleic acid and coat proteins, or by first forming empty viral particles which are subsequently packaged with nucleic acid, the latter mechanism being typical for many dsDNA bacteriophages. Bacteriophage PRD1 is an icosahedral, non-tailed dsDNA virus that has an internal lipid membrane, the hallmark of the Tectiviridae family. Although PRD1 has been known to assemble empty particles into which the genome is subsequently packaged, the mechanism for this has been unknown, and there has been no evidence for a separate packaging vertex, similar to the portal structures used for packaging in the tailed bacteriophages and herpesviruses. In this study, a unique DNA packaging vertex was identified for PRD1, containing the packaging ATPase P9, packaging factor P6 and two small membrane proteins, P20 and P22, extending the packaging vertex to the internal membrane. Lack of small membrane protein P20 was shown to totally abolish packaging, making it an essential part of the PRD1 packaging mechanism. The minor capsid proteins P6 was shown to be an important packaging factor, its absence leading to greatly reduced packaging efficiency. An in vitro DNA packaging mechanism consisting of recombinant packaging ATPase P9, empty procapsids and mutant PRD1 DNA with a LacZ-insert was developed for the analysis of PRD1 packaging, the first such system ever for a virus containing an internal membrane. A new tectiviral sequence, a linear plasmid called pBClin15, was identified in Bacillus cereus, providing material for sequence analysis of the tectiviruses. Analysis of PRD1 P9 and other putative tectiviral ATPase sequences revealed several conserved sequence motifs, among them a new tectiviral packaging ATPase motif. Mutagenesis studies on PRD1 P9 were used to confirm the significance of the motifs. P9-type putative ATPase sequences carrying a similar sequence motif were identified in several other membrane containing dsDNA viruses of bacterial, archaeal and eukaryotic hosts, suggesting that these viruses may have similar packaging mechanisms. Interestingly, almost the same set of viruses that were found to have similar putative packaging ATPases had earlier been found to share similar coat protein folds and capsid structures, and a common origin for these viruses had been suggested. The finding in this study of similar packaging proteins further supports the idea that these viruses are descendants of a common ancestor.