Stochastic reaction & diffusion on growing domains: understanding the breakdown of robust pattern formation

Many biological patterns, from population densities to animal coat markings, can be thought of as heterogeneous spatiotemporal distributions of mobile agents. Many mathematical models have been proposed to account for the emergence of this complexity, but, in general, they have consisted of deterministic systems of differential equations, which do not take into account the stochastic nature of population interactions. One particular, pertinent criticism of these deterministic systems is that the exhibited patterns can often be highly sensitive to changes in initial conditions, domain geometry, parameter values, etc. Due to this sensitivity, we seek to understand the effects of stochasticity and growth on paradigm biological patterning models. In this paper, we extend spatial Fourier analysis and growing domain mapping techniques to encompass stochastic Turing systems. Through this we find that the stochastic systems are able to realize much richer dynamics than their deterministic counterparts, in that patterns are able to exist outside the standard Turing parameter range. Further, it is seen that the inherent stochasticity in the reactions appears to be more important than the noise generated by growth, when considering which wave modes are excited. Finally, although growth is able to generate robust pattern sequences in the deterministic case, we see that stochastic effects destroy this mechanism for conferring robustness. However, through Fourier analysis we are able to suggest a reason behind this lack of robustness and identify possible mechanisms by which to reclaim it

Power spectra methods for a stochastic description of diffusion on deterministically growing domains

A central challenge in developmental biology is understanding the creation of robust spatiotemporal heterogeneity. Generally, the mathematical treatments of biological systems have used continuum, mean-field hypotheses for their constituent parts, which ignores any sources of intrinsic stochastic effects. In this paper we consider a stochastic space-jump process as a description of diffusion, i.e., particles are able to undergo a random walk on a discretized domain. By developing analytical Fourier methods we are able to probe this probabilistic framework, which gives us insight into the patterning potential of diffusive systems. Further, an alternative description of domain growth is introduced, with which we are able to rigorously link the mean-field and stochastic descriptions. Finally, through combining these ideas, it is shown that such stochastic descriptions of diffusion on a deterministically growing domain are able to support the nucleation of states that are far removed from the deterministic mean-field steady state

Influence of stochastic domain growth on pattern nucleation for diffusive systems with internal noise

Numerous mathematical models exploring the emergence of complexity within developmental biology incorporate diffusion as the dominant mechanism of transport. However, self-organizing paradigms can exhibit the biologically undesirable property of extensive sensitivity, as illustrated by the behavior of the French-flag model in response to intrinsic noise and Turing’s model when subjected to fluctuations in initial conditions. Domain growth is known to be a stabilizing factor for the latter, though the interaction of intrinsic noise and domain growth is underexplored, even in the simplest of biophysical settings. Previously, we developed analytical Fourier methods and a description of domain growth that allowed us to characterize the effects of deterministic domain growth on stochastically diffusing systems. In this paper we extend our analysis to encompass stochastically growing domains. This form of growth can be used only to link the meso- and macroscopic domains as the “box-splitting” form of growth on the microscopic scale has an ill-defined thermodynamic limit. The extension is achieved by allowing the simulated particles to undergo random walks on a discretized domain, while stochastically controlling the length of each discretized compartment. Due to the dependence of diffusion on the domain discretization, we find that the description of diffusion cannot be uniquely derived. We apply these analytical methods to two justified descriptions, where it is shown that, under certain conditions, diffusion is able to support a consistent inhomogeneous state that is far removed from the deterministic equilibrium, without additional kinetics. Finally, a logistically growing domain is considered. Not only does this show that we can deal with nonmonotonic descriptions of stochastic growth, but it is also seen that diffusion on a stationary domain produces different effects to diffusion on a domain that is stationary “on average.

Effects of intrinsic stochasticity on delayed reaction-diffusion patterning systems

Cellular gene expression is a complex process involving many steps, including the transcription of DNA and translation of mRNA; hence the synthesis of proteins requires a considerable amount of time, from ten minutes to several hours. Since diffusion-driven instability has been observed to be sensitive to perturbations in kinetic delays, the application of Turing patterning mechanisms to the problem of producing spatially heterogeneous differential gene expression has been questioned. In deterministic systems a small delay in the reactions can cause a large increase in the time it takes a system to pattern. Recently, it has been observed that in undelayed systems intrinsic stochasticity can cause pattern initiation to occur earlier than in the analogous deterministic simulations. Here we are interested in adding both stochasticity and delays to Turing systems in order to assess whether stochasticity can reduce the patterning time scale in delayed Turing systems. As analytical insights to this problem are difficult to attain and often limited in their use, we focus on stochastically simulating delayed systems. We consider four different Turing systems and two different forms of delay. Our results are mixed and lead to the conclusion that, although the sensitivity to delays in the Turing mechanism is not completely removed by the addition of intrinsic noise, the effects of the delays are clearly ameliorated in certain specific cases

Non-linear effects on Turing patterns: time oscillations and chaos.

We show that a model reaction-diffusion system with two species in a monostable regime and over a large region of parameter space, produces Turing patterns coexisting with a limit cycle which cannot be discerned from the linear analysis. As a consequence, Turing patterns oscillate in time, a phenomenon which is expected to occur only in a three morphogen system. When varying a single parameter, a series of bifurcations lead to period doubling, quasi-periodic and chaotic oscillations without modifying the underlying Turing pattern. A Ruelle-Takens-Newhouse route to chaos is identified. We also examined the Turing conditions for obtaining a diffusion driven instability and discovered that the patterns obtained are not necessarily stationary for certain values of the diffusion coefficients. All this results demonstrates the limitations of the linear analysis for reaction-diffusion systems

Modelling biological invasions: individual to population scales at interfaces

Extracting the population level behaviour of biological systems from that of the individual is critical in understanding dynamics across multiple scales and thus has been the subject of numerous investigations. Here, the influence of spatial heterogeneity in such contexts is explored for interfaces with a separation of the length scales characterising the individual and the interface, a situation that can arise in applications involving cellular modelling. As an illustrative example, we consider cell movement between white and grey matter in the brain which may be relevant in considering the invasive dynamics of glioma. We show that while one can safely neglect intrinsic noise, at least when considering glioma cell invasion, profound differences in population behaviours emerge in the presence of interfaces with only subtle alterations in the dynamics at the individual level. Transport driven by local cell sensing generates predictions of cell accumulations along interfaces where cell motility changes. This behaviour is not predicted with the commonly used Fickian diffusion transport model, but can be extracted from preliminary observations of specific cell lines in recent, novel, cryo-imaging. Consequently, these findings suggest a need to consider the impact of individual behaviour, spatial heterogeneity and especially interfaces in experimental and modelling frameworks of cellular dynamics, for instance in the characterisation of glioma cell motility

Nexus researching church toddler groups

The Headlines This unique research project revealed fascinating insights into the spiritual flourishing of very young children and how this is nourished within church toddler groups in England. The findings have uncovered evidence of very young children exploring self-understanding and the nurturing of relationships with others. We found that church toddler groups contribute to children’s spiritual flourishing through: • Providing an environment that fosters a non-threatening connection with spirituality and faith. • Nurturing and sustaining relationships with families. • Offering a safe place for young families to come together, play together, and feel supported in their spiritual well-being. • Enabling a sense of belonging to the church community through engagement in the church toddler group. However, we also found that: • There is a need for adults to have a deeper awareness of the significant role they play in enabling the spiritual flourishing of very young children. • A common misunderstanding of spiritual nurture in terms of faith development hampers focused attention on the spiritual nurture of very young children

Clinical competence in performing and recognising a mediolateral episiotomy of protective angle and length: a systematic review

Objective: It is assumed that all doctors and midwives understand and apply evidence‐based principles in performing episiotomies in their everyday practice. However, remarkable discrepancies between even the most reputable literature sources in defining and describing the technique of performing mediolateral episiotomy (MLE) suggest that there is much ambiguity and confusion for both researchers and clinicians alike. Design: The systematic review protocol was written prior to starting the review and registered in the international prospective register of systematic reviews (PROSPERO/ID CRD42017070523) last updated on December 15, 2017. The review is reported using the Preferred Reporting Items for Systematic Reviews and Meta‐Analysis (PRISMA) guidelines. Methods: A database search was performed using: Medline, CINAHL, Scopus, Informit, the Cochrane Library and PubMed from database inception to 17 September 2017, with a final search on 10 February 2017. Studies were included if they examined clinicians' competency in performing an 'ideal' or 'correct' mediolateral episiotomy, as well as those studies that compared the performance of different professional roles. Studies usually defined an 'ideal' incision as one that met the criteria of an acceptable angle of incision from the midline, starting incision point distance from the midline and in terms of the length of the incision created. Results: While many of the studies included in this review were not of high quality (author self‐assessment) and had their own study criteria for a MLE, the literature suggests clinicians are generally unable to perform or simulate episiotomies within such standards. Overall, most of the literature reported doctors were performing more 'ideal', lateral and longer incisions compared to midwives; however, there were studies that found the opposite, showing statistically significant results in favour of midwives performing more protective episiotomies. There was no association between clinicians' participation in formal training courses and their ability to perform the 'ideal' incision, though one study did find an increased number of episiotomies performed under supervision improved clinicians competency. Conclusion: The obvious lack of understanding around defining and performing MLE for clinicians of various professional roles suggests the need to produce a uniform set of guidelines, and to develop a universal, low‐cost approach for teaching and performing the MLE technique in any clinical environment around the world

A near-infrared spectroscopic screening of the red giant populations in omega Centauri

Near-infrared spectra of 21 giants in omega Centauri, spanning the whole range of metallicities observed in this cluster, are presented. This work is part of a coordinated photometric and spectroscopic campaign in the optical and in the infrared, aimed at studying the complex stellar population of omega Centauri and understanding its formation and chemical evolution. By analyzing the several CO and OH molecular bands and atomic lines in the spectra of the selected giants, metal abundances and abundance ratios have been obtained. The existence of three major metallicity regimes at [Fe/H]=-1.6, -1.2 and [Fe/H]<-0.5 has been confirmed. The most metal-rich stars in our sample show a lower (if any) alpha-enhancement when compared to the more metal-poor components, suggesting that they should have formed in a medium significantly polluted by type Ia supernova ejecta. Isotopic carbon abundances have been also inferred, providing an average 13C/12C=4, which clearly indicates that extra-mixing processes occurred in the stellar interiors during the ascent on the Red Giant Branch.Comment: 22 pages, 7 .ps figures. aastex. Accepted for pubilcation in the Astrophysical Journa