Turing patterns, characterised by spatial self-organisation in reaction-diffusion systems, exhibit sensitivity to initial conditions. This sensitivity, known as the robustness problem, results in different final patterns emerging even from small initial perturbations. In this paper, we introduce a mechanism of pattern mode isolation, where we investigate parameter regimes that promote the isolation of bifurcation branches, thereby delineating the conditions under which distinct pattern modes emerge and evolve independently. Pattern mode isolation can provide a means of enhancing the predictability of Turing pattern mode transitions and enhance the robustness and reproducibility of the patterning outputs.

Abstract Circuit theory has been successfully applied to ecological connectivity modelling, notably via the Circuitscape [1] software, which is typically run locally on a laptop or via a server. For downstream geospatial web applications relying on connectivity analysis, backend infrastructure is required, which can be costly and require advanced data governance. Recent developments in WebAssembly (Wasm) now allow fast C ++ or Rust code to be run directly in a sandboxed browser environment for edge computing. We present a WebAssembly/Rust toolset with a geospatial data pipeline and efficient implementation of connectivity analysis. This approach may be useful for geospatial modelling software where rasters and memory footprint are small enough for the browser context. Our approach has the potential to reduce backend operational complexity, resource requirements, and overall cost.

Spatial and temporal pattern formation in reaction-diffusion systems is typically studied with two or more equations, as scalar reaction-diffusion equations confined to convex domains do not admit stable inhomogeneous states in time or space on long timescales. Here, we show that a single morphogen diffusing across layered two-dimensional media, with nonlinear coupling between layers, is able to generate stable patterns in time and space. This N-layer model is analysed via a thin-domain limit, which reduces to an N-component reaction-diffusion system on a homogeneous one-dimensional domain. This reduced model can be analysed via linear stability techniques, showing that non-diffusive, or reactive, coupling between regions is necessary for pattern-forming instabilities, at least in the reduced model. This reduced system can exhibit Turing, Hopf, and Turing-wave instabilities, with emergent structures that are numerically shown to persist even away from the thin-domain regime of the full 2D single-morphogen system. These results suggest that heterogeneous stratification and nonlinear coupling can broaden the class of systems which exhibit complex spatiotemporal behaviours, which may be relevant in scenarios where only a single morphogen is known to act.

RESEARCH QUESTION:What are the perspectives, challenges and current practices of clinical reproductive practitioners worldwide regarding environmental sustainability in the IVF laboratory? DESIGN:A cross-sectional online survey was distributed exclusively among clinical reproductive scientists. The survey was accessible for 10 weeks from July 2024 to September 2024. It was designed to assess different aspects of sustainability in the IVF laboratory, and explore the role of demographic variables. RESULTS:In total, 893 recipients responded, of which 754 were eligible to participate and spanned 83 countries. Awareness of sustainability was high, with >90% being familiar with the term, and over half expressing strong concern about the environmental impact of the medical field. Three-quarters (74.2%, 559/753) considered sustainability in assisted reproductive technology (ART) to be critical, and many believed that adopting greener practices would appear more attractive to patients, although this opinion was not shared equally among different age groups. The survey explored the frequency of daily laboratory activities. 'Switching off microscope lights when not in use' was performed most often, while 'feedback to suppliers regarding excessive packaging' was seen in <10% of responses. Respondents understood the need for a collaborative approach; however, ART suppliers were identified as most responsible for driving a greener culture. CONCLUSIONS:While most practitioners who participated in the study are concerned about the environmental impact of ART and are keen to adopt more sustainable practices, concerns about regulatory and economic consequences, as well as negative impact on treatment outcomes, hinder meaningful action. This study highlights areas of concern and responsibilities for ART stakeholders to address going forwards in fostering a greener culture in IVF.

Turing patterns have been extensively studied on simple geometries such as lines, squares, rectangles, and circles. Consequently, many biological and physical applications of Turing’s theory approximate their domains to have simple geometries. In particular, thin domains are often approximated as one-dimensional lines or rectangles, whereas the actual geometry may be curved and closer to a stretched ellipse. Thus, we investigate Turing patterns on ellipses and show that they exhibit two distinct limiting behaviours: (i) they tend to those on the circular domain as the ellipse’s aspect ratio approaches unity; (ii) they do not converge to the behaviour of a one-dimensional line as the ellipse becomes thin. This contrasts with rectangular domains, where the bifurcation structure smoothly tends to that of a one-dimensional line as the rectangle’s height is reduced. Using a combination of analytical methods involving Mathieu equations and numerical bifurcation tracking, we demonstrate that the bifurcation modes in an elliptical domain are intrinsically coupled in both radial and angular directions, preventing simple interpolation between circular and linear limits. The results provide insights into the role of domain geometry in governing Turing instabilities and pattern selection, highlighting the distinctive behaviour of ellipses compared to other commonly studied geometries.

Epidemiological models can inform policymaking on disease control strategies, and these models often rely on sampled contact networks. The Random Walk (RW) sampling algorithm, commonly used for network sampling, produces size-biased samples that over-represent highly connected individuals, leading to biased estimates of disease spread. The Metropolis-Hastings Random Walk (MHRW) addresses this by providing samples representative of the underlying network's connectivity distribution. We compare MHRW and RW in reducing size bias across four network types: Erdös-Rényi (ER), Small-world (SW), Negative-binomial (NB), and Scale-free (SF). We simulate disease spread using a stochastic Susceptible-Infected-Recovered (SIR) framework. RW tends to overestimate infections (by 25 % in ER, SW, NB) and secondary infections (by 25 % in ER, SW and 80 % in NB), and underestimate time-to-infection in NB networks. MHRW reduces the size bias, except on SF networks, where both algorithms provide non-representative samples and highly variable estimates. We find that RW is appropriate for fast-spreading, high-mortality epidemics in homogeneous or moderately random networks (ER, SW). In contrast, MHRW is better suited for slower and low-severity epidemics and can be effective in both homogeneous and heterogeneous networks (ER, SW, NB). However, MHRW is computationally expensive and less accurate when duplicate nodes are removed. We also analyse real-world data from cattle movement and human contact networks; MHRW generates disease spread estimates closer to the underlying network than RW. Our findings guide the selection of sampling algorithms based on network structure and epidemic characteristics, enhancing the reliability of disease modelling for policymaking.

Epidemiological models can aid policymakers in reducing disease spread by predicting outcomes based on disease dynamics and contact network characteristics. Calibrating these models requires representative network samples. In this connection, we investigate two sampling algorithms, Random Walk (RW), and Metropolis-Hastings Random Walk (MHRW), across three network types: Erd\H{o}s-R\'enyi (ER), Small-world (SW), and Scale-free (SF). Disease transmission is simulated using a susceptible-infected-recovered (SIR) framework. Our findings show that RW overestimates infected individuals and secondary infections by $25\%$ for ER and SW networks due to size bias, favouring highly connected nodes. MHRW, which corrects for size bias, provides estimates that are more consistent with the underlying network. Also, both methods yield estimates significantly closer to the underlying network for time-to-infection. However, sampling SF networks exhibits significant variability, for both algorithms. Removing duplicate sampled nodes reduces MHRW's accuracy across all network types. We apply both algorithms to a cattle movement network of $46,512$ farms, exhibiting ER, SW, and SF network features. RW overestimates infected farms by approximately $100\%$ and secondary infections by $>900\%$, reflecting size bias whereas MHRW estimates align closely with the cattle network dynamics. Time-to-infection estimates reveal that RW underestimates by approximately $40\%$, while MHRW slightly overestimates by $10\%$. Estimates differ greatly when duplicate nodes are removed. These findings underscore choosing algorithms based on network structure and disease severity. RW's conservative estimates suit high-mortality, fast-spreading diseases, while MHRW provides precise interventions suitable for less severe outbreaks. These insights can guide policymakers in optimizing resource allocation and disease control strategies.

This paper introduces an integrated method that combines computer-aided modelling of indoor environments, multi-agent movement simulation and airborne viral transmission modelling to analyse how spatial design and occupant behaviour affect disease spread. Using TopologicPy, interior spaces are represented as connected networks that support navigation-graph generation and agent movement based on schedules, walking speeds and activities. Agents move incrementally along shortest paths, while the system calculates precise inter-agent distances and respects architectural constraints such as walls and doorways. Viral aerosol concentrations are modelled via a reaction-diffusion equation, and infection risk is estimated using an extended Wells-Riley model. By capturing detailed spatio-temporal and topological interactions, the framework offers realistic infection-risk assessments. The resulting tool serves as a rapid decision-support system for policymakers, facility managers and designers, enabling evaluation of mitigation strategies and informing future building design. A comparative study of cellular and open-plan offices demonstrates the method's capabilities.

Replacing cells lost during the progression of neurodegenerative disorders holds potential as a therapeutic strategy. Unfortunately, the majority of cells die post-transplantation, which creates logistical and biological challenges for cell therapy approaches. The cause of cell death is likely to be multifactorial in nature but has previously been correlated with hypoxia in the graft core. Here we use mathematical modelling to highlight that grafted cells experiencing hypoxia will also face a rapid decline in glucose availability. Interestingly, three neuron progenitor types derived from stem cell sources, and primary human fetal ventral mesencephalic (VM) cells all remained highly viable in severe hypoxia (0.1 ​% oxygen), countering the idea of rapid hypoxia-induced death in grafts. However, we demonstrate that glucose deprivation, not a paucity of oxygen, was a driver of rapid cell death, which was compounded in ischemic conditions of both oxygen and glucose deprivation. Supplementation of glucose to rat embryonic VM cells transplanted to the adult rat brain failed to improve survival at the dose administered and highlighted the problems of using osmotic minipumps in assisting neural grafting. The data shows that maintaining sufficient glucose in grafts is likely to be of critical importance for cell survival, but better means of achieving sustained glucose delivery is required.

RESEARCH QUESTION:What is the awareness, adoption and comprehension of artificial intelligence (AI) among assisted reproductive technology (ART) laboratory professionals? DESIGN:A cross-sectional survey consisting of 32 questions was conducted among clinical embryologists worldwide using an online questionnaire between 17 July and 31 August 2023. The survey assessed familiarity with AI technology; current knowledge within laboratories; understanding of AI principles and limitations; and views on ethical concerns, job impacts and scientist-patient relationships. RESULTS:In total, there were 702 survey respondents. The results revealed a high degree of awareness of AI concepts. The participants recognized the potential benefits of AI in embryology, but acknowledged known limitations. While open to the adoption of AI, they expressed reservations surrounding ethics, effects on jobs, and maintaining positive patient relationships. The study uncovered differences in embryologists' opinions based on their years of experience. Most embryologists, independent of age, were positive regarding AI, but workplace concerns diminished with age. CONCLUSIONS:ART professionals are broadly receptive to AI, but ethical and practical uncertainties were raised. Further engagement between developers and end-users can align AI innovation with the values and needs of human practitioners.

Isolated patterning systems have been repeatedly investigated. However, biological systems rarely work on their own. This paper presents a theoretical and quantitative analysis of a two-domain interconnected geometry, or bilayer, coupling two two-species reaction-diffusion systems mimicking interlayer communication, such as in mammary organoids. Each layer has identical kinetics and parameters, but differing diffusion coefficients. Critically, we show that despite a linear coupling between the layers, the model demonstrates nonlinear behavior; the coupling can lead to pattern suppression or pattern enhancement. Using the Routh-Hurwitz stability criterion multiple times, we investigate the pattern-forming capabilities of the uncoupled system, the weakly coupled system, and the strongly coupled system, using numerical simulations to back up the analysis. We show that although the dispersion relation of the entire system is a nontrivial octic polynomial, the patterning wave modes in the strongly coupled case can be approximated by a quartic polynomial, whose features are easier to understand.

Turing patterns offer a mechanism for understanding self-organization in biological systems. However, due to their flexibility, it is a mechanism that can often be abused. Here, we construct a minimal Turing system defined by just four parameters controlling the: diffusion rate, steady state, linear dynamics and nonlinear dynamics. Using just these four parameters, we can construct a set of kinetics with a number of desirable properties. Firstly, we can turn any homogeneous steady state into a Turing unstable steady state. Secondly, we can ensure that the Turing instability appears within any chosen parameter region. Thirdly, this formulation provides an unbounded patterning parameter space with guaranteed positive solutions. Finally, using weakly nonlinear analysis, we demonstrate that if we have freedom in any two of the parameters, then we can define any required pattern transition (i.e. spots-to-stripes, or stripes-to-spots) under any given changes of one of the parameters. Thus, if a Turing system is going to be applied to understand a specific biological system and, moreover, if it is going to be used to extrapolate predictions for experimental perturbations, then our findings underscore the necessity of heavily restricting the modelling components and parameter values, since any freedom could be exploited to generate potentially contradictory predictions.

BACKGROUND AND OBJECTIVE:Conventional (100 ms) pan-retinal photocoagulation (PRP) laser burns are larger than short-pulse (10 ms to 20 ms) PRP burns. This study investigates the effect of PRP burns of different sizes on retinal oxygenation. METHOD:A mathematical model using COMSOL Multiphysics 6 was used to create a three-dimensional abstraction of the coupled biology of the choroid, photoreceptor, and retinal tissues. Laser burn sizes were varied in the model, specifically considering burn diameters of 500 μm, 250 μm, and 125 μm, while keeping the total burn area constant. RESULTS:Total increase in retinal oxygenation was the same for different burn sizes, but the oxygen distribution differed. Smaller burns resulted in a more even lateral oxygen distribution but with reduced penetration into the inner retina. CONCLUSIONS:Conventional and short-pulse PRP may affect retinal oxygenation differently, even when total burn area is the same. Further investigation into optimum burn size and pattern is required. [Ophthalmic Surg Lasers Imaging Retina 2024;55:40-45.].

We investigate the formation of Turing patterns on regular polygonal domains, as the number of edges grow, leading to the limiting case of the circle. Using linear and weakly nonlinear analysis, and evidence by simulations, we demonstrate how the domain shape can fundamentally change the expected bifurcation structure. Specifically, on the square domain we are able to derive pitchfork bifurcations for stripe and spot solutions, as well as show that both branches cannot bifurcate to produce stable patterns. This compares with the case of the equilateral triangle domain that causes the Turing bifurcation to be generically transcritical and, in some cases, none of the bifurcating branches are stable. Moreover, we find a monotonically increasing, but nonlinear relationship, between the minimal bifurcation area and the number of edges. Thus, patterns can occur on triangles with much smaller areas than circles. Overall, this work raises questions for researchers who are simulating applications on domains with simple shapes. Specifically, even small changes to domain geometry can have large impacts on the produced patterns; thus, domain perturbations should be considered in any sensitivity analyses.

We present models of bat motion derived from radio-tracking data collected over 14 nights. The data presents an initial dispersal period and a return to roost period. Although a simple diffusion model fits the initial dispersal motion we show that simple convection cannot provide a description of the bats returning to their roost. By extending our model to include non-autonomous parameters, or a leap frogging form of motion, where bats on the exterior move back first, we find we are able to accurately capture the bat’s motion. We discuss ways of distinguishing between the two movement descriptions and, finally, consider how the different motion descriptions would impact a bat’s hunting strategy.

Periodic patterning requires coordinated cell-cell interactions at the tissue level. Turing showed, using mathematical modeling, how spatial patterns could arise from the reactions of a diffusive activator-inhibitor pair in an initially homogenous two-dimensional field. Most activators and inhibitors studied in biological systems are proteins, and the roles of cell-cell interaction, ions, bioelectricity, etc. are only now being identified. Gap junctions (GJs) mediate direct exchanges of ions or small molecules between cells, enabling rapid long-distance communications in a cell collective. They are therefore good candidates for propagating non-protein-based patterning signals that may act according to the Turing principles. Here, we explore the possible roles of GJs in Turing-type patterning using feather pattern formation as a model. We found seven of the twelve investigated GJ isoforms are highly dynamically expressed in the developing chicken skin. In ovo functional perturbations of the GJ isoform, connexin 30, by siRNA and the dominant-negative mutant applied before placode development led to disrupted primary feather bud formation, including patches of smooth skin and buds of irregular sizes. Later, after the primary feather arrays were laid out, inhibition of gap junctional intercellular communication in the ex vivo skin explant culture allowed the emergence of new feather buds in temporal waves at specific spatial locations relative to the existing primary buds. The results suggest that gap junctional communication may facilitate the propagation of long-distance inhibitory signals. Thus, the removal of GJ activity would enable the emergence of new feather buds if the local environment is competent and the threshold to form buds is reached. We propose Turing-based computational simulations that can predict the appearance of these ectopic bud waves. Our models demonstrate how a Turing activator-inhibitor system can continue to generate patterns in the competent morphogenetic field when the level of intercellular communication at the tissue scale is modulated.

A Viral Infection Risk Indoor Simulator (VIRIS) has been developed to quickly assess and compare mitigations for airborne disease spread. This agent-based simulator combines people movement in an indoor space, viral transmission modelling and detailed architectural design, and it is powered by topologicpy, an open-source Python library. VIRIS generates very fast predictions of the viral concentration and the spatiotemporal infection risk for individuals as they move through a given space. The simulator is validated with data from a courtroom superspreader event. A sensitivity study for unknown parameter values is also performed. We compare several non-pharmaceutical interventions (NPIs) issued in UK government guidance, for two indoor settings: a care home and a supermarket. Additionally, we have developed the user-friendly VIRIS web app that allows quick exploration of diverse scenarios of interest and visualisation, allowing policymakers, architects and space managers to easily design or assess infection risk in an indoor space.

A Viral Infection Risk Indoor Simulator (VIRIS) has been developed to quickly assess and compare mitigations for airborne disease spread. This agent-based simulator combines people movement in an indoor space, viral transmission modelling and detailed architectural design, and it is powered by topologicpy, an open-source Python library. VIRIS generates very fast predictions of the viral concentration and the spatiotemporal infection risk for individuals as they move through a given space. The simulator is validated with data from a courtroom superspreader event. A sensitivity study for unknown parameter values is also performed. We compare several non-pharmaceutical interventions (NPIs) issued in UK government guidance, for two indoor settings: a care home and a supermarket. Additionally, we have developed the user-friendly VIRIS web app that allows quick exploration of diverse scenarios of interest and visualisation, allowing policymakers, architects and space managers to easily design or assess infection risk in an indoor space.