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.

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

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

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

Design of a Mach-3 Nozzle for TBCC Testing in the NASA LaRC 8-ft High Temperature Tunnel

A new nozzle is being constructed for the NASA Langley Research Center 8-Foot High Temperature Tunnel. The axisymmetric nozzle was designed with a Mach-3 exit flow for testing Turbine-Based Combined-Cycle engines at a Mach number in the vicinity of the transition from turbojet to ramjet operation. The nozzle contour was designed using the NASA Langley IMOCND computer program which solves the potential equation using the classical method of characteristics. To include viscous effects, the design procedure iterated the MOC contour generation with CFD Navier-Stokes calculations, adjusting MOC input parameters until target nozzle-exit conditions were achieved in the Navier-Stokes calculations. The design process was complicated by a requirement to use the final 29.5 inches of an existing 54.5-inch exit-diameter Mach-5 nozzle contour. This was accomplished by generating a Mach-3 contour that matched the radius of the Mach-5 contour at the match point and using a 3rd order polynomial to create a smooth transition between the two contours. During the final evaluation of the design it was realized that the throat diameter is more than half that of the upstream mixing chamber. This led to the concern that large vortical structures generated in the mixer would persist downstream, affecting nozzle-exit flow. This concern was addressed by analyzing the results of three-dimensional, viscous, numerical simulations of the entire flowfield, from the exit of the facility combustor to the nozzle exit. An analysis of the solution indicated that large scale structures do not pass through the throat and that both the total temperature and species (CO2) are well mixed in the mixer, providing uniform flow to the nozzle and subsequently the test cabin

Unveiling the prehistoric landscape at Stonehenge through multi-receiver EMI

YesArchaeological research at Stonehenge (UK) is increasingly aimed at understanding the dynamic of the wider archaeological landscape. Through the application of state-of-the-art geophysical techniques, unprecedented insight is being gathered into the buried archaeological features of the area. However, applied survey techniques have rarely targeted natural soil variation, and the detailed knowledge of the palaeotopography is consequently less complete. In addition, metallic topsoil debris, scattered over different parts of the Stonehenge landscape, often impacts the interpretation of geophysical datasets. The research presented here demonstrates how a single multi-receiver electromagnetic induction (EMI) survey, conducted over a 22 ha area within the Stonehenge landscape, offers detailed insight into natural and anthropogenic soil variation at Stonehenge. The soil variations that were detected through recording the electrical and magnetic soil variability, shed light on the genesis of the landscape, and allow for a better definition of potential palaeoenvironmental and archaeological sampling locations. Based on the multi-layered dataset, a procedure was developed to remove the influence of topsoil metal from the survey data, which enabled a more straightforward identification of the detected archaeology. The results provide a robust basis for further geoarchaeological research, while potential to differentiate between modern soil disturbances and the underlying sub-surface variations can help in solving conservation and management issues. Through expanding this approach over the wider area, we aim at a fuller understanding of the human–landscape interactions that have shaped the Stonehenge landscape

Splenic Injury and ERCP: A Possible Risk for Patients with Advanced Chronic Pancreatitis

Splenic injury is a rare but potentially life-threatening complication of endoscopy, with very few cases of endoscopic retrograde cholangiopancreatography (ERCP)-induced injury reported in the literature. Here we report a patient with chronic alcoholic pancreatitis who was diagnosed with a sub-capsular splenic laceration nearly 6 days after an ERCP. Clinicians should be alerted to the potential post-procedure complications associated with ERCP, particularly as this procedure is being utilized more frequently for the management of patients with complex hepatobiliary and pancreatic conditions