Stability Analysis of Turing Patterns Generated by the Schnakenberg Model

We consider the following Schnakenberg model on the interval (−1, 1): ut = D1u − u + vu2 in (−1, 1), vt = D2v + B − vu2 in (−1, 1), u (−1) = u (1) = v (−1) = v (1) = 0, where D1 > 0, D2 > 0, B>0. We rigorously show that the stability of symmetric N−peaked steady-states can be reduced to computing two matrices in terms of the diffusion coefficients D1,D2 and the number N of peaks. These matrices and their spectra are calculated explicitly and sharp conditions for linear stability are derived. The results are verified by some numerical simulations

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

The Artis Problem

The Artis aquarium has had difficulty maintaining a reasonable temperature in the recently install mammoth sea water tanks during the peak of summer. At this time the approximately 400 000 liters of water may be as much as 3 degrees Celsius too hot. This represents a considerable amount of energy to dissipate. Any solution to this problem must take into account the limited budget of the zoo, the heritage status of the building and the health of the fish in the tank. In this report, we analyse the major sources of energy entering and leaving the system. From this analysis, we find that the most effective method of reducing the water temperature is to increase the amount of evaporation from the system

Flow-distributed spikes for Schnakenberg kinetics

This is the post-print version of the final published paper. The final publication is available at link.springer.com by following the link below. Copyright @ 2011 Springer-Verlag.We study a system of reaction–diffusion–convection equations which combine a reaction–diffusion system with Schnakenberg kinetics and the convective flow equations. It serves as a simple model for flow-distributed pattern formation. We show how the choice of boundary conditions and the size of the flow influence the positions of the emerging spiky patterns and give conditions when they are shifted to the right or to the left. Further, we analyze the shape and prove the stability of the spikes. This paper is the first providing a rigorous analysis of spiky patterns for reaction-diffusion systems coupled with convective flow. The importance of these results for biological applications, in particular the formation of left–right asymmetry in the mouse, is indicated.RGC of Hong Kon

Overcoming artificial broadening in Gd³⁺–Gd³⁺ distance distributions arising from dipolar pseudo-secular terms in DEER experiments

By providing accurate distance measurements between spin labels site-specifically attached to bio-macromolecules, double electron–electron resonance (DEER) spectroscopy provides a unique tool to probe the structural and conformational changes in these molecules. Gd3+-tags present an important family of spin-labels for such purposes, as they feature high chemical stability and high sensitivity in high-field DEER measurements. The high sensitivity of the Gd3+ ion is associated with its high spin (S = 7/2) and small zero field splitting (ZFS), resulting in a narrow spectral width of its central transition at high fields. However, under the conditions of short distances and exceptionally small ZFS, the weak coupling approximation, which is essential for straightforward DEER data analysis, becomes invalid and the pseudo-secular terms of the dipolar Hamiltonian can no longer be ignored. This work further explores the effects of pseudo-secular terms on Gd3+–Gd3+ DEER measurements using a specifically designed ruler molecule; a rigid bis-Gd3+-DOTA model compound with an expected Gd3+–Gd3+ distance of 2.35 nm and a very narrow central transition at the W-band (95 GHz). We show that the DEER dipolar modulations are damped under the standard W-band DEER measurement conditions with a frequency separation, Δν, of 100 MHz between the pump and observe pulses. Consequently, the DEER spectrum deviates considerably from the expected Pake pattern. We show that the Pake pattern and the associated dipolar modulations can be restored with the aid of a dual mode cavity by increasing Δν from 100 MHz to 1.09 GHz, allowing for a straightforward measurement of a Gd3+–Gd3+ distance of 2.35 nm. The increase in Δν increases the contribution of the |−5/2〉 → |−3/2〉 and |−7/2〉 → |−5/2〉 transitions to the signal at the expense of the |−3/2 〉 → |−1/2〉 transition, thus minimizing the effect of dipolar pseudo-secular terms and restoring the validity of the weak coupling approximation. We apply this approach to the A93C/N140C mutant of T4 lysozyme labeled with two different Gd3+ tags that have narrow central transitions and show that even for a distance of 4 nm there is still a significant (about two-fold) broadening that is removed by increasing Δν to 636 MHz and 898 MHz.This research was supported by the Israeli Science Foundation (grant 334/14) and made possible in part by the historic generosity of the Harold Perlman Family. D. G. holds the Erich Klieger professorial chair in Chemical Physic

Stability of cluster solutions in a cooperative consumer chain model

This is the author's accepted manuscript. The final published article is available from the link below. Copyright @ Springer-Verlag Berlin Heidelberg 2012.We study a cooperative consumer chain model which consists of one producer and two consumers. It is an extension of the Schnakenberg model suggested in Gierer and Meinhardt [Kybernetik (Berlin), 12:30-39, 1972] and Schnakenberg (J Theor Biol, 81:389-400, 1979) for which there is only one producer and one consumer. In this consumer chain model there is a middle component which plays a hybrid role: it acts both as consumer and as producer. It is assumed that the producer diffuses much faster than the first consumer and the first consumer much faster than the second consumer. The system also serves as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir. In the small diffusion limit we construct cluster solutions in an interval which have the following properties: The spatial profile of the third component is a spike. The profile for the middle component is that of two partial spikes connected by a thin transition layer. The first component in leading order is given by a Green's function. In this profile multiple scales are involved: The spikes for the middle component are on the small scale, the spike for the third on the very small scale, the width of the transition layer for the middle component is between the small and the very small scale. The first component acts on the large scale. To the best of our knowledge, this type of spiky pattern has never before been studied rigorously. It is shown that, if the feedrates are small enough, there exist two such patterns which differ by their amplitudes.We also study the stability properties of these cluster solutions. We use a rigorous analysis to investigate the linearized operator around cluster solutions which is based on nonlocal eigenvalue problems and rigorous asymptotic analysis. The following result is established: If the time-relaxation constants are small enough, one cluster solution is stable and the other one is unstable. The instability arises through large eigenvalues of order O(1). Further, there are small eigenvalues of order o(1) which do not cause any instabilities. Our approach requires some new ideas: (i) The analysis of the large eigenvalues of order O(1) leads to a novel system of nonlocal eigenvalue problems with inhomogeneous Robin boundary conditions whose stability properties have been investigated rigorously. (ii) The analysis of the small eigenvalues of order o(1) needs a careful study of the interaction of two small length scales and is based on a suitable inner/outer expansion with rigorous error analysis. It is found that the order of these small eigenvalues is given by the smallest diffusion constant ε22.RGC of Hong Kon

Existence and Stability of a Spike in the Central Component for a Consumer Chain Model

We study a three-component consumer chain model which is based on Schnakenberg type kinetics. In this model there is one consumer feeding on the producer and a second consumer feeding on the first consumer. This means that the first consumer (central component) plays a hybrid role: it acts both as consumer and producer. The model is an extension of the Schnakenberg model suggested in \cite{gm,schn1} for which there is only one producer and one consumer. It is assumed that both the producer and second consumer diffuse much faster than the central component. We construct single spike solutions on an interval for which the profile of the first consumer is that of a spike. The profiles of the producer and the second consumer only vary on a much larger spatial scale due to faster diffusion of these components. It is shown that there exist two different single spike solutions if the feed rates are small enough: a large-amplitude and a small-amplitude spike. We study the stability properties of these solutions in terms of the system parameters. We use a rigorous analysis for the linearized operator around single spike solutions based on nonlocal eigenvalue problems. The following result is established: If the time-relaxation constants for both producer and second consumer vanish, the large-amplitude spike solution is stable and the small-amplitude spike solution is unstable. We also derive results on the stability of solutions when these two time-relaxation constants are small. We show a new effect: if the time-relaxation constant of the second consumer is very small, the large-amplitude spike solution becomes unstable. To the best of our knowledge this phenomenon has not been observed before for the stability of spike patterns. It seems that this behavior is not possible for two-component reaction-diffusion systems but that at least three components are required. Our main motivation to study this system is mathematical since the novel interaction of a spike in the central component with two other components results in new types of conditions for the existence and stability of a spike. This model is realistic if several assumptions are made: (i) cooperation of consumers is prevalent in the system, (ii) the producer and the second consumer diffuse much faster than the first consumer, and (iii) there is practically an unlimited pool of producer. The first assumption has been proven to be correct in many types of consumer groups or populations, the second assumption occurs if the central component has a much smaller mobility than the other two, the third assumption is realistic if the consumers do not feel the impact of the limited amount of producer due to its large quantity. This chain model plays a role in population biology, where consumer and producer are often called predator and prey. This system can also be used as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir

Identifying factors likely to influence compliance with diagnostic imaging guideline recommendations for spine disorders among chiropractors in North America: a focus group study using the Theoretical Domains Framework

Background: The Theoretical Domains Framework (TDF) was developed to investigate determinants of specific clinical behaviors and inform the design of interventions to change professional behavior. This framework was used to explore the beliefs of chiropractors in an American Provider Network and two Canadian provinces about their adherence to evidence-based recommendations for spine radiography for uncomplicated back pain. The primary objective of the study was to identify chiropractors’ beliefs about managing uncomplicated back pain without xrays and to explore barriers and facilitators to implementing evidence-based recommendations on lumbar spine xrays. A secondary objective was to compare chiropractors in the United States and Canada on their beliefs regarding the use of spine x-rays. Methods: Six focus groups exploring beliefs about managing back pain without x-rays were conducted with a purposive sample. The interview guide was based upon the TDF. Focus groups were digitally recorded, transcribed verbatim, and analyzed by two independent assessors using thematic content analysis based on the TDF. Results: Five domains were identified as likely relevant. Key beliefs within these domains included the following: conflicting comments about the potential consequences of not ordering x-rays (risk of missing a pathology, avoiding adverse treatment effects, risks of litigation, determining the treatment plan, and using x-ray-driven techniques contrasted with perceived benefits of minimizing patient radiation exposure and reducing costs; beliefs about consequences); beliefs regarding professional autonomy, professional credibility, lack of standardization, and agreement with guidelines widely varied (social/professional role & identity); the influence of formal training, colleagues, and patients also appeared to be important factors (social influences); conflicting comments regarding levels of confidence and comfort in managing patients without x-rays (belief about capabilities); and guideline awareness and agreements (knowledge). Conclusions: Chiropractors’ use of diagnostic imaging appears to be influenced by a number of factors. Five key domains may be important considering the presence of conflicting beliefs, evidence of strong beliefs likely to impact the behavior of interest, and high frequency of beliefs. The results will inform the development of a theorybased survey to help identify potential targets for behavioral-change strategies

Factors associated with initiation and completion of the quadrivalent human papillomavirus vaccine series in an ontario cohort of grade 8 girls

Abstract Background Although over a hundred million dollars have been invested in offering free quadrivalent human papillomavirus (HPV) vaccination to young girls in Ontario, there continues to be very little information about its usage. In order to successfully guide future HPV vaccine programming, it is important to monitor HPV vaccine use and determine factors associated with use in this population. Methods Linking administrative health and immunization databases, we conducted a population-based, retrospective cohort study of girls eligible for Ontario's Grade 8 HPV vaccination program in Kingston, Frontenac, Lennox, and Addington. We determined the proportion of girls who initiated (at least one dose) and completed (all three doses) the vaccination series overall and according to socio-demographics, vaccination history, health services utilization, medical history, and program year. Multivariable logistic regression was used to estimate the strength of association between individual factors and initiation and completion, adjusted for all other factors. Results We identified a cohort of 2519 girls, 56.6% of whom received at least one dose of the HPV vaccine. Among vaccinated girls, 85.3% received all three doses. Vaccination history was the strongest predictor of initiation in that girls who received the measles-mumps-rubella, meningococcal C, and hepatitis B vaccines were considerably more likely to also receive the HPV vaccine (odds ratio 4.89; 95% confidence interval 4.04-5.92). Nevertheless, HPV vaccine uptake was more than 20% lower than that of these other vaccines. In addition, while series initiation was not influenced by income, series completion was. In particular, girls of low income were the least likely to receive all three indicated doses of the HPV vaccine (odds ratio 0.45; 95% confidence interval 0.28-0.72). Conclusions The current low level of HPV vaccine acceptance in Kingston, Frontenac, Lennox, and Addington will likely have important implications in terms of the health benefits and cost-effectiveness of its publicly funded program. We identified important factors associated with series initiation and completion that should be considered in efforts to improve HPV vaccine use in this population