Polynomial growth of volume of balls for zero-entropy geodesic systems

The aim of this paper is to state and prove polynomial analogues of the classical Manning inequality relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim we use two numerical conjugacy invariants, the {\em strong polynomial entropy $h_{pol}$} and the {\em weak polynomial entropy $h_{pol}^*$}. Both are infinite when the topological entropy is positive and they satisfy $h_{pol}^*\leq h_{pol}$. We first prove that the growth rate of the volume of balls is bounded above by means of the strong polynomial entropy and we show that for the flat torus this inequality becomes an equality. We then study the explicit example of the torus of revolution for which we can give an exact asymptotic equivalent of the growth rate of volume of balls, which we relate to the weak polynomial entropy.Comment: 22 page

Phenotypic covariance of longevity, immunity and stress resistance in the Caenorhabditis nematodes

Background \ud Ageing, immunity and stresstolerance are inherent characteristics of all organisms. In animals, these traits are regulated, at least in part, by forkhead transcription factors in response to upstream signals from the Insulin/Insulin– like growth factor signalling (IIS) pathway. In the nematode Caenorhabditis elegans, these phenotypes are molecularly linked such that activation of the forkhead transcription factor DAF-16 both extends lifespan and simultaneously increases immunity and stress resistance. It is known that lifespan varies significantly among the Caenorhabditis species but, although DAF-16 signalling is highly conserved, it is unclear whether this phenotypic linkage occurs in other species. Here we investigate this phenotypic covariance by comparing longevity, stress resistance and immunity in four \ud Caenorhabditis species. \ud \ud Methodology/Principal Findings \ud We show using phenotypic analysis of DAF-16 influenced phenotypes that among four closely related Caenorhabditis nematodes, the gonochoristic species (Caenorhabditis remanei and Caenorhabditis brenneri) have diverged \ud significantly with a longer lifespan, improved stress resistance and higher immunity than the hermaphroditic species (C. elegans and Caenorhabditis briggsae). Interestingly, we also observe significant differences in expression levels between the daf-16 homologues in these species using Real-Time PCR, which positively correlate with the observed phenotypes. Finally, we provide additional evidence in support of a role for DAF-16 in regulating phenotypic coupling by using a combination of wildtype isolates, constitutively active daf-16 mutants and bioinformatic analysis. \ud \ud Conclusions \ud The gonochoristic species display a significantly longer lifespan (p < 0.0001)and more robust immune and stress response (p<0.0001, thermal stress; p<0.01, heavy metal stress; p<0.0001, pathogenic stress) than the hermaphroditic species. Our data suggests that divergence in DAF-16 mediated phenotypes may underlie many of the differences observed between these four species of Caenorhabditis nematodes. These findings are further supported by the correlative higher daf-16 expression levels among the gonochoristic species and significantly higher lifespan, immunity and stress tolerance in the constitutively active daf-16 hermaphroditic mutants

Phenotypic covariance of Longevity, Immunity and Stress Resistance in the Caenorhabditis Nematodes

Background: Ageing, immunity and stresstolerance are inherent characteristics of all organisms. In animals, these traits are regulated, at least in part, by forkhead transcription factors in response to upstream signals from the Insulin/Insulin–like growth factor signalling (IIS) pathway. In the nematode Caenorhabditis elegans, these phenotypes are molecularly linked such that activation of the forkhead transcription factor DAF-16 both extends lifespan and simultaneously increases immunity and stress resistance. It is known that lifespan varies significantly among the Caenorhabditis species but, although DAF-16 signalling is highly conserved, it is unclear whether this phenotypic linkage occurs in other species. Here we investigate this phenotypic covariance by comparing longevity, stress resistance and immunity in four Caenorhabditis species. \ud \ud Methodology/Principal Findings: We show using phenotypic analysis of DAF-16 influenced phenotypes that among four closely related Caenorhabditis nematodes, the gonochoristic species (Caenorhabditis remanei and Caenorhabditis brenneri) have diverged significantly with a longer lifespan, improved stress resistance and higher immunity than the hermaphroditic species (C. elegans and Caenorhabditis briggsae). Interestingly, we also observe significant differences in expression levels between the daf-16 homologues in these species using Real-Time PCR, which positively correlate with the observed phenotypes. Finally, we provide additional evidence in support of a role for DAF-16 in regulating phenotypic coupling by using a combination of wildtype isolates, constitutively active daf-16 mutants and bioinformatic analysis. \ud \ud Conclusions: The gonochoristic species display a significantly longer lifespan (p<0.0001) and more robust immune and stress response (p<0.0001, thermal stress; p<0.01, heavy metal stress; p<0.0001, pathogenic stress) than the hermaphroditic species. Our data suggests that divergence in DAF-16 mediated phenotypes may underlie many of the differences observed between these four species of Caenorhabditis nematodes. These findings are further supported by the correlative higher daf-16 expression levels among the gonochoristic species and significantly higher lifespan, immunity and stress tolerance in the constitutively active daf-16 hermaphroditic mutants