Existence uniqueness and ratio decomposition for Gibbs states via duality

We give an elementary proof of existence and uniqueness of Gibbs states for Hölder weight systems on subshifts of finite type. This uses a notion of duality for such subshifts. The approach of Paterson [2] is used to construct a measure with a prescribed Jacobian and the duality is used to produce an invariant measure from this

Rigidity of hyperbolic sets on surfaces

Given a hyperbolic invariant set of a diffeomorphism on a surface, it is proved that, if the holonomies are sufficiently smooth, then the diffeomorphism on the hyperbolic invariant set is rigid in the sense that it is C1+ conjugate to a hyperbolic affine model

Smoothness of holonomies for codimension 1 hyperbolic dynamics

Hyperbolic invariant sets {Lambda} of C1+{gamma} diffeomorphisms where either the stable or unstable leaves are 1-dimensional are considered in this paper. Under the assumption that the {Lambda} has local product structure, the authors prove that the holonomies between the 1-dimensional leaves are C1+{alpha} for some 0 < {alpha} < 1

Teichmüller spaces and HR structures for hyperbolic surface dynamics

We construct a Teichmüller space for the C^{1+}-conjugacy classes of hyperbolic dynamical systems on surfaces. After introducing the notion of an HR structure which associates an affine structure with each of the stable and unstable laminations, we show that there is a one-to-one correspondence between these HR structures and the C^{1+}-conjugacy classes. As part of the proof we construct a canonical representative dynamical system for each HR structure. This has the smoothest holonomies of any representative of the corresponding C^{1+}-conjugacy class. Finally, we introduce solenoid functions and show that they provide a good Teichmüller space

A temporal switch model for estimating transcriptional activity in gene expression

Motivation: The analysis and mechanistic modelling of time series gene expression data provided by techniques such as microarrays, NanoString, reverse transcription–polymerase chain reaction and advanced sequencing are invaluable for developing an understanding of the variation in key biological processes. We address this by proposing the estimation of a flexible dynamic model, which decouples temporal synthesis and degradation of mRNA and, hence, allows for transcriptional activity to switch between different states. Results: The model is flexible enough to capture a variety of observed transcriptional dynamics, including oscillatory behaviour, in a way that is compatible with the demands imposed by the quality, time-resolution and quantity of the data. We show that the timing and number of switch events in transcriptional activity can be estimated alongside individual gene mRNA stability with the help of a Bayesian reversible jump Markov chain Monte Carlo algorithm. To demonstrate the methodology, we focus on modelling the wild-type behaviour of a selection of 200 circadian genes of the model plant Arabidopsis thaliana. The results support the idea that using a mechanistic model to identify transcriptional switch points is likely to strongly contribute to efforts in elucidating and understanding key biological processes, such as transcription and degradation

Robustness from flexibility in the fungal circadian clock

Background Robustness is a central property of living systems, enabling function to be maintained against environmental perturbations. A key challenge is to identify the structures in biological circuits that confer system-level properties such as robustness. Circadian clocks allow organisms to adapt to the predictable changes of the 24-hour day/night cycle by generating endogenous rhythms that can be entrained to the external cycle. In all organisms, the clock circuits typically comprise multiple interlocked feedback loops controlling the rhythmic expression of key genes. Previously, we showed that such architectures increase the flexibility of the clock's rhythmic behaviour. We now test the relationship between flexibility and robustness, using a mathematical model of the circuit controlling conidiation in the fungus Neurospora crassa. Results The circuit modelled in this work consists of a central negative feedback loop, in which the frequency (frq) gene inhibits its transcriptional activator white collar-1 (wc-1), interlocked with a positive feedback loop in which FRQ protein upregulates WC-1 production. Importantly, our model reproduces the observed entrainment of this circuit under light/dark cycles with varying photoperiod and cycle duration. Our simulations show that whilst the level of frq mRNA is driven directly by the light input, the falling phase of FRQ protein, a molecular correlate of conidiation, maintains a constant phase that is uncoupled from the times of dawn and dusk. The model predicts the behaviour of mutants that uncouple WC-1 production from FRQ's positive feedback, and shows that the positive loop enhances the buffering of conidiation phase against seasonal photoperiod changes. This property is quantified using Kitano's measure for the overall robustness of a regulated system output. Further analysis demonstrates that this functional robustness is a consequence of the greater evolutionary flexibility conferred on the circuit by the interlocking loop structure. Conclusions Our model shows that the behaviour of the fungal clock in light-dark cycles can be accounted for by a transcription-translation feedback model of the central FRQ-WC oscillator. More generally, we provide an example of a biological circuit in which greater flexibility yields improved robustness, while also introducing novel sensitivity analysis techniques applicable to a broader range of cellular oscillators

The Evolution of Antisocial Punishment in Optional Public Goods Games

Cooperation, where one individual incurs a cost to help another, is a fundamental building block of the natural world and human society. It has been suggested that costly punishment can promote the evolution of cooperation, with the threat of punishment deterring free-riders. Recent experiments, however, have revealed the existence of 'antisocial' punishment, where non-cooperators punish cooperators. While various theoretical models find that punishment can promote the evolution of cooperation, these models a priori exclude the possibility of antisocial punishment. Here we extend the standard theory of optional public goods games to include the full set of punishment strategies. We find that punishment no longer increases cooperation, and that selection favours substantial levels of antisocial punishment for a wide range of parameters. Furthermore, we conduct behavioural experiments, showing results consistent with our model predictions. As opposed to an altruistic act that promotes cooperation, punishment is mostly a self-interested tool for protecting oneself against potential competitors.MathematicsPsycholog

Direct measurement of transcription rates reveals multiple mechanisms for configuration of the Arabidopsis ambient temperature response

Background Sensing and responding to ambient temperature is important for controlling growth and development of many organisms, in part by regulating mRNA levels. mRNA abundance can change with temperature, but it is unclear whether this results from changes in transcription or decay rates, and whether passive or active temperature regulation is involved. Results Using a base analog labelling method, we directly measured the temperature coefficient, Q10, of mRNA synthesis and degradation rates of the Arabidopsis transcriptome. We show that for most genes, transcript levels are buffered against passive increases in transcription rates by balancing passive increases in the rate of decay. Strikingly, for temperature-responsive transcripts, increasing temperature raises transcript abundance primarily by promoting faster transcription relative to decay and not vice versa, suggesting a global transcriptional process exists that controls mRNA abundance by temperature. This is partly accounted for by gene body H2A.Z which is associated with low transcription rate Q10, but is also influenced by other marks and transcription factor activities. Conclusions Our data show that less frequent chromatin states can produce temperature responses simply by virtue of their rarity and the difference between their thermal properties and those of the most common states, and underline the advantages of directly measuring transcription rate changes in dynamic systems, rather than inferring rates from changes in mRNA abundance. Background The mechanism for ambient temperature sensing in plants is unclear. Control of transcript levels is believed to be important in responses to temperature [1-4] but affects of ambient temperature on transcription and mRNA decay rates have not been measured. According to the work of Arrhenius [5] the temperature coefficient (Q10) of biochemical reactions is expected to be 2 to 3 at biological temperatures: yet less than 2% of Arabidopsis thaliana genes have a two-fold or greater difference in expression level between 17°C and 27°C [6]. The remaining genes either have rates buffered against changing temperatures, or passive increases in transcription rate must be offset by a balanced increase in decay rate, leading to higher turnover but static steady state levels. Despite this fundamental uncertainty, steady state transcriptomic responses to ambient temperature have been used to infer a role for chromatin modifications in temperature signaling [2,7]. 4-Thiouracil (4SU) is a non-toxic base analogue that has been shown to be incorporated into mammalian and yeast mRNA during transcription [8-12]. Biotinylation and column separation allow 4SU-labeled RNA to be separated from unlabeled RNA, and transcriptomic analysis using the separated samples can be used to simultaneously calculate mRNA synthesis and decay rates [8]. Here we use 4SU labeling to measure transcription rates and determine the Q10 genome-wide of mRNA synthesis and decay rates in Arabidopsis thaliana. We show that ambient temperature has large passive effects on both mRNA synthesis and decay rates, and that where temperature controls transcript abundance it does so by regulating transcription relative to decay and not vice versa. Our analysis suggests that transcription factor binding sites and epigenetic state combine to create a complex network of temperature responses in plants. Results Cells incorporate 4SU into RNA and this has been exploited in mammalian cells [8,11,12] and in yeast [13] to measure mRNA synthesis and decay rates. In order to determine whether plants can take up 4SU we floated intact seedlings in MS medium and monitored 4SU incorporation into RNA by biotinylation and dot blot (Figure S1a in Additional file 1). This clearly showed that plants incorporate 4SU from the environment into RNA and that concentrations as low as 1 mM lead to a signal detectable above background within 1 hour (Figure 1B). The resulting RNA could be separated from unlabeled RNA by biotinylation and passage through a streptavidin column as described previously. At 1.5 mM the flow-through can be depleted of detectable 4SU-labeled RNA, whilst labeled plant RNA is highly concentrated in the fraction recovered from the column [8,13] (Figure S1c in Additional file 1). To maximize recovery we chose a low concentration of 4SU at 1.5 mM [8] as high labeling frequencies are known to lead to binding of fewer more frequently labeled transcripts to the columns and reduce recovery. At this concentration Arabidopsis plants treated with 4SU showed the same growth and survival as control plants (Figure S2a in Additional file 1), suggesting 4SU has low toxicity in plants, as in other organisms. Therefore, 4SU dynamics in Arabidopsis seedlings resemble those described for other experimental systems. Preliminary experiments showed that RNA turnover was faster at 27°C compared to 12°C (Figure S2b in Additional file 1), suggesting that temperature generally affected transcription rates

Renormalization of circle diffeomorphism sequences and markov sequences

We show a one-to-one correspondence between circle diffeomorphism sequences that are C^{ 1+n}-periodic points of renormalization and smooth Markov sequences.We thank the financial support of LIAAD–INESC TEC through program PEst, USP-UP project, Faculty of Sciences, University of Porto, Calouste Gulbenkian Foundation, FEDER and COMPETE Programmes, PTDC/MAT/121107/2010 and Fundação para a Ciência e a Tecnologia (FCT). J. P.Almeida acknowledges the FCT support given through Grant SFRH/PROTEC/49754/2009