Mode transitions in a model reaction-diffusion system driven by domain growth and noise

Pattern formation in many biological systems takes place during growth of the underlying domain. We study a specific example of a reaction–diffusion (Turing) model in which peak splitting, driven by domain growth, generates a sequence of patterns. We have previously shown that the pattern sequences which are presented when the domain growth rate is sufficiently rapid exhibit a mode-doubling phenomenon. Such pattern sequences afford reliable selection of certain final patterns, thus addressing the robustness problem inherent of the Turing mechanism. At slower domain growth rates this regular mode doubling breaks down in the presence of small perturbations to the dynamics. In this paper we examine the breaking down of the mode doubling sequence and consider the implications of this behaviour in increasing the range of reliably selectable final patterns

A mechanism for morphogen-controlled domain growth

Many developmental systems are organised via the action of graded distributions of morphogens. In the Drosophila wing disc, for example, recent experimental evidence has shown that graded expression of the morphogen Dpp controls cell proliferation and hence disc growth. Our goal is to explore a simple model for regulation of wing growth via the Dpp gradient: we use a system of reaction-diffusion equations to model the dynamics of Dpp and its receptor Tkv, with advection arising as a result of the flow generated by cell proliferation. We analyse the model both numerically and analytically, showing that uniform domain growth across the disc produces an exponentially growing wing disc

Gene expression time delays & Turing pattern formation systems

The incorporation of time delays can greatly affect the behaviour of partial differential equations and dynamical systems. In addition, there is evidence that time delays in gene expression due to transcription and translation play an important role in the dynamics of cellular systems. In this paper, we investigate the effects of incorporating gene expression time delays into a one-dimensional putative reaction diffusion pattern formation mechanism on both stationary domains and domains with spatially uniform exponential growth. While oscillatory behaviour is rare, we find that the time taken to initiate and stabilise patterns increases dramatically as the time delay is increased. In addition, we observe that on rapidly growing domains the time delay can induce a failure of the Turing instability which cannot be predicted by a naive linear analysis of the underlying equations about the homogeneous steady state. The dramatic lag in the induction of patterning, or even its complete absence on occasions, highlights the importance of considering explicit gene expression time delays in models for cellular reaction diffusion patterning

Aberrant behaviours of reaction diffusion self-organisation models on growing domains in the presence of gene expression time delays

Turing’s pattern formation mechanism exhibits sensitivity to the details of the initial conditions suggesting that, in isolation, it cannot robustly generate pattern within noisy biological environments. Nonetheless, secondary aspects of developmental self-organisation, such as a growing domain, have been shown to ameliorate this aberrant model behaviour. Furthermore, while in-situ hybridisation reveals the presence of gene expression in developmental processes, the influence of such dynamics on Turing’s model has received limited attention. Here, we novelly focus on the Gierer–Meinhardt reaction diffusion system considering delays due the time taken for gene expression, while incorporating a number of different domain growth profiles to further explore the influence and interplay of domain growth and gene expression on Turing’s mechanism. We find extensive pathological model behaviour, exhibiting one or more of the following: temporal oscillations with no spatial structure, a failure of the Turing instability and an extreme sensitivity to the initial conditions, the growth profile and the duration of gene expression. This deviant behaviour is even more severe than observed in previous studies of Schnakenberg kinetics on exponentially growing domains in the presence of gene expression (Gaffney and Monk in Bull. Math. Biol. 68:99–130, 2006). Our results emphasise that gene expression dynamics induce unrealistic behaviour in Turing’s model for multiple choices of kinetics and thus such aberrant modelling predictions are likely to be generic. They also highlight that domain growth can no longer ameliorate the excessive sensitivity of Turing’s mechanism in the presence of gene expression time delays. The above, extensive, pathologies suggest that, in the presence of gene expression, Turing’s mechanism would generally require a novel and extensive secondary mechanism to control reaction diffusion patterning

The influence of gene expression time delays on Gierer-Meinhardt pattern formation systems

There are numerous examples of morphogen gradients controlling long range signalling in developmental and cellular systems. The prospect of two such interacting morphogens instigating long range self-organisation in biological systems via a Turing bifurcation has been explored, postulated, or implicated in the context of numerous developmental processes. However, modelling investigations of cellular systems typically neglect the influence of gene expression on such dynamics, even though transcription and translation are observed to be important in morphogenetic systems. In particular, the influence of gene expression on a large class of Turing bifurcation models, namely those with pure kinetics such as the Gierer–Meinhardt system, is unexplored. Our investigations demonstrate that the behaviour of the Gierer–Meinhardt model profoundly changes on the inclusion of gene expression dynamics and is sensitive to the sub-cellular details of gene expression. Features such as concentration blow up, morphogen oscillations and radical sensitivities to the duration of gene expression are observed and, at best, severely restrict the possible parameter spaces for feasible biological behaviour. These results also indicate that the behaviour of Turing pattern formation systems on the inclusion of gene expression time delays may provide a means of distinguishing between possible forms of interaction kinetics. Finally, this study also emphasises that sub-cellular and gene expression dynamics should not be simply neglected in models of long range biological pattern formation via morphogens

Rising rural body-mass index is the main driver of the global obesity epidemic in adults

Body-mass index (BMI) has increased steadily in most countries in parallel with a rise in the proportion of the population who live in cities 1,2 . This has led to a widely reported view that urbanization is one of the most important drivers of the global rise in obesity 3�6 . Here we use 2,009 population-based studies, with measurements of height and weight in more than 112 million adults, to report national, regional and global trends in mean BMI segregated by place of residence (a rural or urban area) from 1985 to 2017. We show that, contrary to the dominant paradigm, more than 55 of the global rise in mean BMI from 1985 to 2017�and more than 80 in some low- and middle-income regions�was due to increases in BMI in rural areas. This large contribution stems from the fact that, with the exception of women in sub-Saharan Africa, BMI is increasing at the same rate or faster in rural areas than in cities in low- and middle-income regions. These trends have in turn resulted in a closing�and in some countries reversal�of the gap in BMI between urban and rural areas in low- and middle-income countries, especially for women. In high-income and industrialized countries, we noted a persistently higher rural BMI, especially for women. There is an urgent need for an integrated approach to rural nutrition that enhances financial and physical access to healthy foods, to avoid replacing the rural undernutrition disadvantage in poor countries with a more general malnutrition disadvantage that entails excessive consumption of low-quality calories. © 2019, The Author(s)

Reaction-diffusion patterns on growing domains

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Development of models of active ion transport for whole−cell modelling: cardiac sodium−potassium pump as a case study

This study presents a method for the reduction of biophysically-based kinetic models for the active transport of ions. A lumping scheme is presented which exploits the differences in timescales associated with fast and slow transitions between model states, while maintaining the thermodynamic properties of the model. The goal of this approach is to contribute to modelling of the effects of disturbances to metabolism, associated with ischaemic heart disease, on cardiac cell function. The approach is illustrated for the sodium-potassium pump in the myocyte. The lumping scheme is applied to produce a 4-state representation from the detailed 15-state model of Lauger and Apell, Eur. Biophys. J. 13 (1986) 309, for which the principles of free energy transduction are used to link the free energy released from ATP hydrolysis (deltaGATP) to the transition rates between states of the model. An iterative minimisation algorithm is implemented to determine the transition rate parameters based on the model fit to experimental data. Finally, the relationship between deltaGATP and pump cycling direction is investigated and compared with recent experimental findings