Generalised golden ratios over integer alphabets

It is a well known result that for β ∈ (1,1+√52) and x ∈ (0,1β−1) there exists uncountably many (ǫi)∞i=1 ∈ {0,1}N such that x = P∞i=1ǫiβ−i. When β ∈ (1+√52,2] there exists x ∈ (0,1β−1) for which there exists a unique (ǫi)∞i=1 ∈ {0,1}N such that x=P∞i=1ǫiβ−i. In this paper we consider the more general case when our sequences are elements of {0, . . . , m}N. We show that an analogue of the golden ratio exists and give an explicit formula for it.

Regulation of early steps of GPVI signal transduction by phosphatases: a systems biology approach

We present a data-driven mathematical model of a key initiating step in platelet activation, a central process in the prevention of bleeding following Injury. In vascular disease, this process is activated inappropriately and causes thrombosis, heart attacks and stroke. The collagen receptor GPVI is the primary trigger for platelet activation at sites of injury. Understanding the complex molecular mechanisms initiated by this receptor is important for development of more effective antithrombotic medicines. In this work we developed a series of nonlinear ordinary differential equation models that are direct representations of biological hypotheses surrounding the initial steps in GPVI-stimulated signal transduction. At each stage model simulations were compared to our own quantitative, high-temporal experimental data that guides further experimental design, data collection and model refinement. Much is known about the linear forward reactions within platelet signalling pathways but knowledge of the roles of putative reverse reactions are poorly understood. An initial model, that includes a simple constitutively active phosphatase, was unable to explain experimental data. Model revisions, incorporating a complex pathway of interactions (and specifically the phosphatase TULA-2), provided a good description of the experimental data both based on observations of phosphorylation in samples from one donor and in those of a wider population. Our model was used to investigate the levels of proteins involved in regulating the pathway and the effect of low GPVI levels that have been associated with disease. Results indicate a clear separation in healthy and GPVI deficient states in respect of the signalling cascade dynamics associated with Syk tyrosine phosphorylation and activation. Our approach reveals the central importance of this negative feedback pathway that results in the temporal regulation of a specific class of protein tyrosine phosphatases in controlling the rate, and therefore extent, of GPVI-stimulated platelet activation.

Characterisation of convective regimes over the British Isles

Convection-permitting modelling has led to a step change in forecasting convective events. However, convection occurs within different regimes which exhibit different forecast behaviour. A convective adjustment timescale can be used to distinguish between these regimes and examine their associated predictability. The convective adjustment timescale is calculated from radiosonde ascents and found to be consistent with that derived from convection-permitting model forecasts. The model-derived convective adjustment timescale is then examined for three summers in the British Isles to determine characteristics of the convective regimes for this maritime region. Convection in the British Isles is predominantly in convective quasi-equilibrium with 85%of convection having a timescale less than or equal to three hours. This percentage varies spatially with more non-equilibriumevents occurring in the south and southwest. The convective adjustment timescale exhibits a diurnal cycle over land. The nonequilibrium regime occurs more frequently at mid-range wind speeds and with winds from southerly to westerly sectors. Most non-equilibrium convective events in the British Isles are initiated near large coastal orographic gradients or on the European continent. Thus, the convective adjustment timescale is greatest when the location being examined is immediately downstream of large orographic gradients and decreases with distance from the convective initiation region. The dominance of convective quasiequilibrium conditions over the British Isles argues for the use of large-member ensembles in probabilistic forecasts for this region.

Using problem-based learning in a chemistry practical class for pharmacy students and engaging them with feedback

Objective: To introduce a new approach to problem based learning (PBL) used in the context of medicinal chemistry practical class teaching pharmacy students. Design: The described chemistry practical is based on independent studies by small groups of undergraduate students (4-5), who design their own practical work taking relevant professional standards into account. Students are carefully guided by feedback and acquire a set of skills important to their future profession as healthcare professionals. This model has been tailored to the application of PBL in a chemistry practical class setting for a large student cohort (150 students). Assessment: The achievement of learning outcomes is based on the submission of relevant documentation including a certificate of analysis, in addition to peer assessment. Some of the learning outcomes are also assessed in the final written examination at the end of the academic year. Conclusion: The described design of a novel PBL chemistry laboratory course for pharmacy students has been found to be successful. Self-reflective learning and engagement with feedback were encouraged, and students enjoyed the challenging learning experience. Skills that are highly essential for the students’ future careers as healthcare professionals are promoted.