This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first, concentrating on stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. Much effort has gone into making these subjects as accessible as possible by providing many concrete examples that illustrate techniques of calculation, and by treating all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appeared for the first time in this book. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

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Summary SARS-CoV-2 Spike protein is critical for virus infection via engagement of ACE2 1 , and is a major antibody target. Here we report chronic SARS-CoV-2 with reduced sensitivity to neutralising antibodies in an immune suppressed individual treated with convalescent plasma, generating whole genome ultradeep sequences over 23 time points spanning 101 days. Little change was observed in the overall viral population structure following two courses of remdesivir over the first 57 days. However, following convalescent plasma therapy we observed large, dynamic virus population shifts, with the emergence of a dominant viral strain bearing D796H in S2 and ΔH69/ΔV70 in the S1 N-terminal domain NTD of the Spike protein. As passively transferred serum antibodies diminished, viruses with the escape genotype diminished in frequency, before returning during a final, unsuccessful course of convalescent plasma. In vitro , the Spike escape double mutant bearing ΔH69/ΔV70 and D796H conferred modestly decreased sensitivity to convalescent plasma, whilst maintaining infectivity similar to wild type. D796H appeared to be the main contributor to decreased susceptibility but incurred an infectivity defect. The ΔH69/ΔV70 single mutant had two-fold higher infectivity compared to wild type, possibly compensating for the reduced infectivity of D796H. These data reveal strong selection on SARS-CoV-2 during convalescent plasma therapy associated with emergence of viral variants with evidence of reduced susceptibility to neutralising antibodies.

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Although phospholipid bilayers are ubiquitous in modern cells, their impermeability, lack of dynamic properties, and synthetic complexity are difficult to reconcile with plausible pathways of proto-metabolism, growth and division. Here, we present an alternative membrane-free model, which demonstrates that low-molecular-weight mononucleotides and simple cationic peptides spontaneously accumulate in water into microdroplets that are stable to changes in temperature and salt concentration, undergo pH-induced cycles of growth and decay, and promote α-helical peptide secondary structure. Moreover, the microdroplets selectively sequester porphyrins, inorganic nanoparticles and enzymes to generate supramolecular stacked arrays of light-harvesting molecules, nanoparticle-mediated oxidase activity, and enhanced rates of glucose phosphorylation, respectively. Taken together, our results suggest that peptide-nucleotide microdroplets can be considered as a new type of protocell model that could be used to develop novel bioreactors, primitive artificial cells and plausible pathways to prebiotic organization before the emergence of lipid-based compartmentalization on the early Earth.

The mechanism of spontaneous assembly of microscale compartments is a central question for the origin of life, and has technological repercussions in diverse areas such as materials science, catalysis, biotechnology and biomedicine. Such compartments need to be semipermeable, structurally robust and capable of housing assemblages of functional components for internalized chemical transformations. In principle, proteins should be ideal building blocks for the construction of membrane-bound compartments but protein vesicles with cell-like properties are extremely rare. Here we present an approach to the interfacial assembly of protein-based micro-compartments (proteinosomes) that are delineated by a semi-permeable, stimulus-responsive, enzymatically active, elastic membrane consisting of a closely packed monolayer of conjugated protein-polymer building blocks. The proteinosomes can be dispersed in oil or water, thermally cycled to temperatures of 70°C, and partially dried and re-inflated without loss of structural integrity. As a consequence, they exhibit protocellular properties such as guest molecule encapsulation, selective permeability, gene-directed protein synthesis and membrane-gated internalized enzyme catalysis.

Complex coacervate microdroplets are finding increased utility in synthetic cell applications due to their cytomimetic properties. However, their intrinsic membrane-free nature results in instability that limits their application in protocell research. Herein, we present the development of a new protocell model through the spontaneous interfacial self-assembly of copolymer molecules on biopolymer coacervate microdroplets. This hierarchical protocell model not only incorporates the favorable properties of coacervates (such as spontaneous assembly and macromolecular condensation) but also assimilates the essential features of a semipermeable copolymeric membrane (such as discretization and stabilization). This was accomplished by engineering an asymmetric, biodegradable triblock copolymer molecule comprising hydrophilic, hydrophobic, and polyanionic components capable of direct coacervate membranization via electrostatic surface anchoring and chain self-association. The resulting hierarchical protocell demonstrated striking integrity as a result of membrane formation, successfully stabilizing enzymatic cargo against coalescence and fusion in discrete protocellular populations. The semipermeable nature of the copolymeric membrane enabled the incorporation of a simple enzymatic cascade, demonstrating chemical communication between discrete populations of neighboring protocells. In this way, we pave the way for the development of new synthetic cell constructs.

Now available in paperback, this celebrated book has been prepared with readers' needs in mind, remaining a systematic guide to a large part of the modern theory of Probability, whilst retaining its vitality. The authors' aim is to present the subject of Brownian motion not as a dry part of mathematical analysis, but to convey its real meaning and fascination. The opening, heuristic chapter does just this, and it is followed by a comprehensive and self-contained account of the foundations of theory of stochastic processes. Chapter 3 is a lively and readable account of the theory of Markov processes. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.