Modeling ion channel dynamics through reflected stochastic differential equations

Ion channels are membrane proteins that open and close at random and play a vital role in the electrical dynamics of excitable cells. The stochastic nature of the conformational changes these proteins undergo can be significant, however current stochastic modeling methodologies limit the ability to study such systems. Discrete-state Markov chain models are seen as the "gold standard," but are computationally intensive, restricting investigation of stochastic effects to the single-cell level. Continuous stochastic methods that use stochastic differential equations (SDEs) to model the system are more efficient but can lead to simulations that have no biological meaning. In this paper we show that modeling the behavior of ion channel dynamics by a reflected SDE ensures biologically realistic simulations, and we argue that this model follows from the continuous approximation of the discrete-state Markov chain model. Open channel and action potential statistics from simulations of ion channel dynamics using the reflected SDE are compared with those of a discrete-state Markov chain method. Results show that the reflected SDE simulations are in good agreement with the discrete-state approach. The reflected SDE model therefore provides a computationally efficient method to simulate ion channel dynamics while preserving the distributional properties of the discrete-state Markov chain model and also ensuring biologically realistic solutions. This framework could easily be extended to other biochemical reaction networks. © 2012 American Physical Society.

Mixed convection flow of micropolar fluid in an open ended arc-shape cavity

Numerical simulations for mixed convection of micropolar fluid in an open ended arc-shape cavity have been carried out in this study. Computation is performed using the Alternate Direct Implicit (ADI) method together with the Successive Over Relaxation (SOR) technique for the solution of governing partial differential equations. The flow phenomenon is examined for a range of values of Rayleigh number, 102 ≤ Ra ≤ 106, Prandtl number, 7 ≤ Pr ≤ 50, and Reynolds number, 10 ≤ Re ≤ 100. The study is mainly focused on how the micropolar fluid parameters affect the fluid properties in the flow domain. It was found that despite the reduction of flow in the core region, the heat transfer rate increases, whereas the skin friction and microrotation decrease with the increase in the vortex viscosity parameter, Δ.

Anomalous subdiffusion equations by the meshless collocation method

This paper aims to develop an implicit meshless collocation technique based on the moving least squares approximation for numerical simulation of the anomalous subdiffusion equation(ASDE). The discrete system of equations is obtained by using the MLS meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach related to the time discretization are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling of ASDEs.

Analytical solutions of the multi-term modified power law wave equations in a finite domain

Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development. In this paper, multi-term modified power law wave equations in a finite domain are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, n) (n > 2), respectively. Analytical solutions of the multi-term modified power law wave equations are derived. These new techniques are based on Luchko’s Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. Then these general methods are applied to the special cases of the Szabo wave equation and the power law wave equation. These methods and techniques can also be extended to other kinds of the multi term time-space fractional models including fractional Laplacian.

Numerical methods for solving the multi-term time-fractional wave-diffusion equations

In this paper, the multi-term time-fractional wave diffusion equations are considered. The multiterm time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

Analytical solutions for the two- and three-dimensional time-fractional telegraph equations by the method of separating variables

In this paper, a method of separating variables is effectively implemented for solving a time-fractional telegraph equation (TFTE) in two and three dimensions. We discuss and derive the analytical solution of the TFTE in two and three dimensions with nonhomogeneous Dirichlet boundary condition. This method can be extended to other kinds of the boundary conditions.

Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time-space Caputo-Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0, 1], [1, 2] and [0, 2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchko's Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations. © 2012.

Relative entropy rate based model selection for linear hybrid system filters of uncertain nonlinear systems

Hybrid system representations have been exploited in a number of challenging modelling situations, including situations where the original nonlinear dynamics are too complex (or too imprecisely known) to be directly filtered. Unfortunately, the question of how to best design suitable hybrid system models has not yet been fully addressed, particularly in the situations involving model uncertainty. This paper proposes a novel joint state-measurement relative entropy rate based approach for design of hybrid system filters in the presence of (parameterised) model uncertainty. We also present a design approach suitable for suboptimal hybrid system filters. The benefits of our proposed approaches are illustrated through design examples and simulation studies.

Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain

Multi-term time-fractional differential equations have been used for describing important physical phenomena. However, studies of the multi-term time-fractional partial differential equations with three kinds of nonhomogeneous boundary conditions are still limited. In this paper, a method of separating variables is used to solve the multi-term time-fractional diffusion-wave equation and the multi-term time-fractional diffusion equation in a finite domain. In the two equations, the time-fractional derivative is defined in the Caputo sense. We discuss and derive the analytical solutions of the two equations with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin conditions, respectively.

Surface plasmon hybridization and exciton coupling

We derive a semianalytical model to describe the interaction of a single photon emitter and a collection of arbitrarily shaped metal nanoparticles. The theory treats the metal nanoparticles classically within the electrostatic eigenmode method, wherein the surface plasmon resonances of collections of nanoparticles are represented by the hybridization of the plasmon modes of the noninteracting particles. The single photon emitter is represented by a quantum mechanical two-level system that exhibits line broadening due to a finite spontaneous decay rate. Plasmon-emitter coupling is described by solving the resulting Bloch equations. We illustrate the theory by studying model systems consisting of a single emitter coupled to one, two, and three nanoparticles, and we also compare the predictions of our model to published experimental data. ©2012 American Physical Society.

Mixed-dimensional consistent coupling by multi-point constraint equations for efficient multi-scale modeling

Finding an appropriate linking method to connect different dimensional element types in a single finite element model is a key issue in the multi-scale modeling. This paper presents a mixed dimensional coupling method using multi-point constraint equations derived by equating the work done on either side of interface connecting beam elements and shell elements for constructing a finite element multiscale model. A typical steel truss frame structure is selected as case example and the reduced scale specimen of this truss section is then studied in the laboratory to measure its dynamic and static behavior in global truss and local welded details while the different analytical models are developed for numerical simulation. Comparison of dynamic and static response of the calculated results among different numerical models as well as the good agreement with those from experimental results indicates that the proposed multi-scale model is efficient and accurate.

Application of an appearance-based intervention to improve sun protection outcomes of outdoor workers in Queensland, Australia

Outdoor workers are exposed to high levels of ultraviolet radiation (UVR) and may thus be at greater risk to experience UVR-related health effects such as skin cancer, sun burn, and cataracts. A number of intervention trials (n=14) have aimed to improve outdoor workers’ work-related sun protection cognitions and behaviours. Only one study however has reported the use of UV-photography as part of a multi-component intervention. This study was performed in the USA and showed long-term (12 months) improvements in work-related sun protection behaviours. Intervention effects of the other studies have varied greatly, depending on the population studied, intervention applied, and measurement of effect. Previous studies have not assessed whether: - Interventions are similarly effective for workers in stringent and less stringent policy organisations; - Policy effect is translated into workers’ leisure time protection; - Implemented interventions are effective in the long-term; - The facial UV-photograph technique is effective in Australian male outdoor workers without a large additional intervention package, and; - Such interventions will also affect workers’ leisure time sun-related cognitions and behaviours. Therefore, the present Protection of Outdoor Workers from Environmental Radiation [POWER]-study aimed to fill these gaps and had the objectives of: a) assessing outdoor workers’ sun-related cognitions and behaviours at work and during leisure time in stringent and less stringent sun protection policy environments; b) assessing the effect of an appearance-based intervention on workers’ risk perceptions, intentions and behaviours over time; c) assessing whether the intervention was equally effective within the two policy settings; and d) assessing the immediate post-intervention effect. Effectiveness was described in terms of changes in sun-related risk perceptions and intentions (as these factors were shown to be main precursors of behaviour change in many health promotion theories) and behaviour. The study purposefully selected and recruited two organisations with a large outdoor worker contingent in Queensland, Australia within a 40 kilometre radius of Brisbane. The two organisations differed in the stringency of implementation and reinforcement of their organisational sun protection policy. Data were collected from 154 male predominantly Australian born outdoor workers with an average age of 37 years and predominantly medium to fair skin (83%). Sun-related cognitions and behaviours of workers were assessed using self-report questionnaires at baseline and six to twelve months later. Variation in follow-up time was due to a time difference in the recruitment of the two organisations. Participants within each organisation were assigned to an intervention or control group. The intervention group participants received a one-off personalised Skin Cancer Risk Assessment Tool [SCRAT]-letter and a facial UV-photograph with detailed verbal information. This was followed by an immediate post-intervention questionnaire within three months of the start of the study. The control group only received the baseline and follow-up questionnaire. Data were analysed using a variety of techniques including: descriptive analyses, parametric and non-parametric tests, and generalised estimating equations. A 15% proportional difference observed was deemed of clinical significance, with the addition of reported statistical significance (p<0.05) where applicable. Objective 1: Assess and compare the current sun-related risk perceptions, intentions, behaviours, and policy awareness of outdoor workers in stringent and less stringent sun protection policy settings. Workers within the two organisations (stringent n=89 and less stringent n=65) were similar in their knowledge about skin cancer, self efficacy, attitudes, and social norms regarding sun protection at work and during leisure time. Participants were predominantly in favour of sun protection. Results highlighted that compared to workers in a less stringent policy organisation working for an organisation with stringent sun protection policies and practices resulted in more desirable sun protection intentions (less willing to tan p=0.03) ; actual behaviours at work (sufficient use of upper and lower body protection, headgear, and sunglasses (p<0.001 for all comparisons), and greater policy awareness (awareness of repercussions if Personal Protective Equipment (PPE) was not used, p<0.001)). However the effect of the work-related sun protection policy was found not to extend to leisure time sun protection. Objective 2: Compare changes in sun-related risk perceptions, intentions, and behaviours between the intervention and control group. The effect of the intervention was minimal and mainly resulted in a clinically significant reduction in work-related self-perceived risk of developing skin cancer in the intervention compared to the control group (16% and 32% for intervention and control group, respectively estimated their risk higher compared to other outdoor workers: , p=0.11). No other clinical significant effects were observed at 12 months follow-up. Objective 3: Assess whether the intervention was equally effective in the stringent sun protection policy organisation and the less stringent sun protection policy organisation. The appearance-based intervention resulted in a clinically significant improvement in the stringent policy intervention group participants’ intention to protect from the sun at work (workplace*time interaction, p=0.01). In addition to a reduction in their willingness to tan both at work (will tan at baseline: 17% and 61%, p=0.06, at follow-up: 54% and 33%, p=0.07, stringent and less stringent policy intervention group respectively. The workplace*time interaction was significant p<0.001) and during leisure time (will tan at baseline: 42% and 78%, p=0.01, at follow-up: 50% and 63%, p=0.43, stringent and less stringent policy intervention group respectively. The workplace*time interaction was significant p=0.01) over the course of the study compared to the less stringent policy intervention group. However, no changes in actual sun protection behaviours were found. Objective 4: Examine the effect of the intervention on level of alarm and concern regarding the health of the skin as well as sun protection behaviours in both organisations. The immediate post-intervention results showed that the stringent policy organisation participants indicated to be less alarmed (p=0.04) and concerned (p<0.01) about the health of their skin and less likely to show the facial UV-photograph to others (family p=0.03) compared to the less stringent policy participants. A clinically significantly larger proportion of participants from the stringent policy organisation reported they worried more about skin cancer (65%) and skin freckling (43%) compared to those in the less stringent policy organisation (46%,and 23% respectively , after seeing the UV-photograph). In summary the results of this study suggest that the having a stringent work-related sun protection policy was significantly related to for work-time sun protection practices, but did not extend to leisure time sun protection. This could reflect the insufficient level of sun protection found in the general Australian population during leisure time. Alternatively, reactance caused by being restricted in personal decisions through work-time policy could have contributed to lower leisure time sun protection. Finally, other factors could have also contributed to the less than optimal leisure time sun protection behaviours reported, such as unmeasured personal or cultural barriers. All these factors combined may have lead to reduced willingness to take proper preventive action during leisure time exposure. The intervention did not result in any measurable difference between the intervention and control groups in sun protection behaviours in this population, potentially due to the long lag time between the implementation of the intervention and assessment at 12-months follow-up. In addition, high levels of sun protection behaviours were found at baseline (ceiling effect) which left little room for improvement. Further, this study did not assess sunscreen use, which was the predominant behaviour assessed in previous effective appearance-based interventions trials. Additionally, previous trials were mainly conducted in female populations, whilst the POWER-study’s population was all male. The observed immediate post-intervention result could be due to more emphasis being placed on sun protection and risks related to sun exposure in the stringent policy organisation. Therefore participants from the stringent policy organisation could have been more aware of harmful effects of UVR and hence, by knowing that they usually protect adequately, not be as alarmed or concerned as the participants from the less stringent policy organisation. In conclusion, a facial UV-photograph and SCRAT-letter information alone may not achieve large changes in sun-related cognitions and behaviour, especially of assessed 6-12 months after the intervention was implemented and in workers who are already quite well protected. Differences found between workers in the present study appear to be more attributable to organisational policy. However, against a background of organisational policy, this intervention may be a useful addition to sun-related workplace health and safety programs. The study findings have been interpreted while respecting a number of limitations. These have included non-random allocation of participants due to pre-organised allocation of participants to study group in one organisation and difficulty in separating participants from either study group. Due to the transient nature of the outdoor worker population, only 105 of 154 workers available at baseline could be reached for follow-up. (attrition rate=32%). In addition the discrepancy in the time to follow-up assessment between the two organisations was a limitation of the current study. Given the caveats of this research, the following recommendations were made for future research: - Consensus should be reached to define "outdoor worker" in terms of time spent outside at work as well as in the way sun protection behaviours are measured and reported. - Future studies should implement and assess the value of the facial UV-photographs in a wide range of outdoor worker organisations and countries. - More timely and frequent follow-up assessments should be implemented in intervention studies to determine the intervention effect and to identify the best timing of booster sessions to optimise results. - Future research should continue to aim to target outdoor workers’ leisure time cognitions and behaviours and improve these if possible. Overall, policy appears to be an important factor in workers’ compliance with work-time use of sun protection. Given the evidence generated by this research, organisations employing outdoor workers should consider stringent implementation and reinforcement of a sun protection policy. Finally, more research is needed to improve ways to generate desirable behaviour in this population during leisure time.

A least squares based finite volume method for the Cahn-Hilliard and Cahn-Hilliard-reaction equations

A vertex-centred finite volume method (FVM) for the Cahn-Hilliard (CH) and recently proposed Cahn-Hilliard-reaction (CHR) equations is presented. Information at control volume faces is computed using a high-order least-squares approach based on Taylor series approximations. This least-squares problem explicitly includes the variational boundary condition (VBC) that ensures that the discrete equations satisfy all of the boundary conditions. We use this approach to solve the CH and CHR equations in one and two dimensions and show that our scheme satisfies the VBC to at least second order. For the CH equation we show evidence of conservative, gradient stable solutions, however for the CHR equation, strict gradient-stability is more challenging to achieve.

Solution methods for advection-diffusion-reaction equations on growing domains and subdomains, with application to modelling skin substitutes

Problems involving the solution of advection-diffusion-reaction equations on domains and subdomains whose growth affects and is affected by these equations, commonly arise in developmental biology. Here, a mathematical framework for these situations, together with methods for obtaining spatio-temporal solutions and steady states of models built from this framework, is presented. The framework and methods are applied to a recently published model of epidermal skin substitutes. Despite the use of Eulerian schemes, excellent agreement is obtained between the numerical spatio-temporal, numerical steady state, and analytical solutions of the model.