The ergodic hierarchy, randomness and Hamiltonian chaos

Various processes are often classified as both deterministic and random or chaotic. The main difficulty in analysing the randomness of such processes is the apparent tension between the notions of randomness and determinism: what type of randomness could exist in a deterministic process? Ergodic theory seems to offer a particularly promising theoretical tool for tackling this problem by positing a hierarchy, the so-called ‘ergodic hierarchy’ (EH), which is commonly assumed to provide a hierarchy of increasing degrees of randomness. However, that notion of randomness requires clarification. The mathematical definition of EH does not make explicit appeal to randomness; nor does the usual way of presenting EH involve a specification of the notion of randomness that is supposed to underlie the hierarchy. In this paper we argue that EH is best understood as a hierarchy of random behaviour if randomness is explicated in terms of unpredictability. We then show that, contrary to common wisdom, EH is useful in characterising the behaviour of Hamiltonian dynamical systems.     

Population density drives the local evolution of a threshold dimorphism

Evolution can favour more than one reproductive tactic among conspecifics of the same sex. Under the conditional evolutionarily stable strategy, individuals adopt the tactic that generates the highest fitness return for their status: large males guard females, whereas small males sneak copulations. Tactics change at the status at which fitness benefits switch from favouring one tactic to favouring the alternative. This 'switchpoint' is expressed in many species as a threshold between divergent morphologies. Environmental and demographic parameters that influence the relative fitness of male tactics are predicted to determine a population's switchpoint and consequently whether the population is monomorphic or dimorphic. Here we show threshold evolution in the forceps dimorphism of the European earwig Forficula auricularia and document the transition from completely monomorphic to classical male-dimorphic populations over a distance of only 40 km. Because the superior fighting ability of the dominant morph will be more frequently rewarded at high encounter rates, population density is likely to be a key determinant of the relative fitness of the alternative tactics, and consequently the threshold. We show that, as predicted, population density correlates strongly with the shift in threshold, and that this factor drives the local evolution of the male dimorphism in these island populations. Our data provide evidence for the origin of phenotypic diversity within populations, through the evolution of a switchpoint in a conditional strategy that has responded to local population density.     

Insights into assembly from structural analysis of bacteriophage PRD1

The structure of the membrane-containing bacteriophage PRD1 has been determined by X-ray crystallography at about 4 A resolution. Here we describe the structure and location of proteins P3, P16, P30 and P31. Different structural proteins seem to have specialist roles in controlling virus assembly. The linearly extended P30 appears to nucleate the formation of the icosahedral facets (composed of trimers of the major capsid protein, P3) and acts as a molecular tape-measure, defining the size of the virus and cementing the facets together. Pentamers of P31 form the vertex base, interlocking with subunits of P3 and interacting with the membrane protein P16. The architectural similarities with adenovirus and one of the largest known virus particles PBCV-1 support the notion that the mechanism of assembly of PRD1 is scaleable and applies across the major viral lineage formed by these viruses.     

Membrane structure and interactions with protein and DNA in bacteriophage PRD1

Membranes are essential for selectively controlling the passage of molecules in and out of cells and mediating the response of cells to their environment. Biological membranes and their associated proteins present considerable difficulties for structural analysis. Although enveloped viruses have been imaged at about 9 A resolution by cryo-electron microscopy and image reconstruction, no detailed crystallographic structure of a membrane system has been described. The structure of the bacteriophage PRD1 particle, determined by X-ray crystallography at about 4 A resolution, allows the first detailed analysis of a membrane-containing virus. The architecture of the viral capsid and its implications for virus assembly are presented in the accompanying paper. Here we show that the electron density also reveals the icosahedral lipid bilayer, beneath the protein capsid, enveloping the viral DNA. The viral membrane contains about 26,000 lipid molecules asymmetrically distributed between the membrane leaflets. The inner leaflet is composed predominantly of zwitterionic phosphatidylethanolamine molecules, facilitating a very close interaction with the viral DNA, which we estimate to be packaged to a pressure of about 45 atm, factors that are likely to be important during membrane-mediated DNA translocation into the host cell. In contrast, the outer leaflet is enriched in phosphatidylglycerol and cardiolipin, which show a marked lateral segregation within the icosahedral asymmetric unit. In addition, the lipid headgroups show a surprising degree of order.     

A critical role for the protein tyrosine phosphatase receptor type Z in functional recovery from demyelinating lesions

Several lines of evidence suggest that tyrosine phosphorylation is a key element in myelin formation, differentiation of oligodendrocytes and Schwann cells, and recovery from demyelinating lesions. Multiple sclerosis is a demyelinating disease of the human central nervous system, and studies of experimental demyelination indicate that remyelination in vivo requires the local generation, migration or maturation of new oligodendrocytes, or some combination of these. Failure of remyelination in multiple sclerosis could result from the failure of any of these processes or from the death of oligodendrocytes. Ptprz encodes protein tyrosine phosphatase receptor type Z (Ptpz, also designated Rptpbeta), which is expressed primarily in the nervous system but also in oligodendrocytes, astrocytes and neurons. Here we examine the susceptibility of mice deficient in Ptprz to experimental autoimmune encephalomyelitis (EAE), a model of multiple sclerosis. We observe that mice deficient in Ptprz show impaired recovery from EAE induced by myelin oligodendrocyte glycoprotein (MOG) peptide. This sustained paralysis is associated with increased apoptosis of mature oligodendrocytes in the spinal cords of mutant mice at the peak of inflammation. We further demonstrate that expression of PTPRZ1, the human homolog of Ptprz, is induced in multiple sclerosis lesions and that the gene is specifically expressed in remyelinating oligodendrocytes in these lesions. These results support a role for Ptprz in oligodendrocyte survival and in recovery from demyelinating disease.     

Biochemical networking contributes more to genetic buffering in human and mouse metabolic pathways than does gene duplication

During evolution different genes evolve at unequal rates, reflecting the varying functional constraints on phenotype. An important contributor to this variation is genetic buffering, which reduces the potential detrimental effects of mutations. We studied whether gene duplication and redundant metabolic networks affect genetic buffering by comparing the evolutionary rate of 242 human and mouse orthologous genes involved in metabolic pathways. A gene with a redundant network is defined as one for which the structural layout of metabolic pathways provides an alternative metabolic route that can, in principle, compensate for the loss of a protein function encoded by the gene. We found that genes with redundant networks evolve at similar rates as did genes without redundant networks, [corrected] but no significant difference was detected between single-copy genes and gene families. This implies that redundancy in metabolic networks provides significantly more genetic buffering than do gene families. We also found that genes encoding proteins involved in glycolysis and gluconeogenesis showed as a group a distinct pattern of variation, in contrast to genes involved in other pathways. These results suggest that redundant networks provide genetic buffering and contribute to the functional diversification of metabolic pathways.     

A gravitationally lensed quasar with quadruple images separated by 14.62 arcseconds

Gravitational lensing is a powerful tool for the study of the distribution of dark matter in the Universe. The cold-dark-matter model of the formation of large-scale structures (that is, clusters of galaxies and even larger assemblies) predicts the existence of quasars gravitationally lensed by concentrations of dark matter so massive that the quasar images would be split by over 7 arcsec. Numerous searches for large-separation lensed quasars have, however, been unsuccessful. All of the roughly 70 lensed quasars known, including the first lensed quasar discovered, have smaller separations that can be explained in terms of galaxy-scale concentrations of baryonic matter. Although gravitationally lensed galaxies with large separations are known, quasars are more useful cosmological probes because of the simplicity of the resulting lens systems. Here we report the discovery of a lensed quasar, SDSS J1004 + 4112, which has a maximum separation between the components of 14.62 arcsec. Such a large separation means that the lensing object must be dominated by dark matter. Our results are fully consistent with theoretical expectations based on the cold-dark-matter model.