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Congenital abnormalities of the kidney and urinary tract collectively form the most common type of prenatally diagnosed malformations. Whilst many of the crucial genes that direct the kidney developmental program are known, the mechanisms by which kidney organogenesis is achieved is still largely unclear. In this paper, we propose a mathematical model for the localisation of the ureteric bud, the precursor to the ureter and collecting duct system of the kidney. The mathematical model presented fundamentally implicates Schnakenberg-like ligand-receptor Turing patterning as the mechanism by which the ureteric bud is localised on the Wolfian duct as proposed by Menshykaul and Iber (2013) . This model explores the specific roles of regulatory proteins GREM1 and BMP as well as the domain properties of GDNF production. Our model demonstrates that this proposed pattern formation mechanism is capable of naturally predicting the phenotypical outcomes of many genetic experiments from the literature. Furthermore, we conclude that whilst BMP inhibits GDNF away from the budding site and GREM1 permits GDNF to signal, GREM1 also stabilises the effect of BMP on GDNF signalling from fluctuations in BMP sensitivity but not signal strength. [ABSTRACT FROM AUTHOR]

In this paper, three multiscale methods for coupling of mesoscopic (compartment-based) and microscopic (molecular-based) stochastic reaction–diffusion simulations are investigated. Two of the three methods that will be discussed in detail have been previously reported in the literature; the two-regime method (TRM) and the compartment–placement method (CPM). The third method that is introduced and analysed in this paper is called the ghost cell method (GCM), since it works by constructing a “ghost cell” in which molecules can disappear and jump into the compartment-based simulation. Presented is a comparison of sources of error. The convergent properties of this error are studied as the time step Δ t (for updating the molecular-based part of the model) approaches zero. It is found that the error behaviour depends on another fundamental computational parameter h , the compartment size in the mesoscopic part of the model. Two important limiting cases, which appear in applications, are considered: (i) Δ t → 0 and h is fixed; (ii) Δ t → 0 and h → 0 such that Δ t / h is fixed. The error for previously developed approaches (the TRM and CPM) converges to zero only in the limiting case (ii), but not in case (i). It is shown that the error of the GCM converges in the limiting case (i). Thus the GCM is superior to previous coupling techniques if the mesoscopic description is much coarser than the microscopic part of the model. [ABSTRACT FROM AUTHOR]

Abstract: Nonhealing wounds are a major burden for health care systems worldwide. In addition, a patient who suffers from this type of wound usually has a reduced quality of life. While the wound healing process is undoubtedly complex, in this paper we develop a deterministic mathematical model, formulated as a system of partial differential equations, that focusses on an important aspect of successful healing: oxygen supply to the wound bed by a combination of diffusion from the surrounding unwounded tissue and delivery from newly formed blood vessels. While the model equations can be solved numerically, the emphasis here is on the use of asymptotic methods to establish conditions under which new blood vessel growth can be initiated and wound-bed angiogenesis can progress. These conditions are given in terms of key model parameters including the rate of oxygen supply and its rate of consumption in the wound. We use our model to discuss the clinical use of treatments such as hyperbaric oxygen therapy, wound bed debridement, and revascularisation therapy that have the potential to initiate healing in chronic, stalled wounds. [Copyright &y& Elsevier]

• Comparable simple Wnt-signalling model dynamical and delayed systems analysed. • General conditions required for Wnt-driven oscillations discussed. The Wnt signalling pathway plays an important role in development, disease, and normal tissue function. Mathematical models for Wnt signalling have predominantly focused on quantitatively predicting changes in steady-state β -catenin concentrations (the main downstream protein regulated by canonical Wnt signalling). One of the genes targeted for expression by Wnt/ β -catenin signalling is the negative Wnt regulator Axin2. Recently, a number of authors have indicated a potential theoretical role of Axin2 feedback to induce oscillatory behaviour in the pathway and this has been observed in a number of detailed mathematical models. Due to the complexity of these models, the investigations to date have been limited to numerical experiments and parameter sensitivity analyses. In this manuscript, we study the fundamental structure of the dynamical system underlying the Wnt signalling mechanism with Axin2 feedback to gain some insight into why and when oscillations occur in models with this structure. We semi-rigorously analyse three simple models and, for these models, gain deep understanding of the characteristic set of conditions that are necessary and sufficient for oscillations to be induced. We discuss the possible biological consequences of these findings for Wnt signalling pathway oscillations. They include; to promote oscillations (1) Keeping all other parameters constant, the Wnt signal strength should neither be too high or too low but within a single finite window of values, (2) Wnt receptor complexes should fully deactivate Axin rather than temporarily deconstruct it from other scaffold proteins, (3) In the absence of stochastic effects or more complicated mechanisms, a critical delay in Axin2 feedback in the system is necessary, (4) Deactivation of Axin by the Wnt receptor complex needs to be critically efficient relative to β -catenin removal by Axin, and (5) conditions necessary are less strict if Axin2 feedback occurs after a fixed time rather than a Poisson-distributed time with the same average. [ABSTRACT FROM AUTHOR]

Highlights • A reaction-advection-diffusion model for the effectiveness of using oxygen to treat surgical wound infection is developed and its behaviour investigated. • Numerical solutions of the full model and analysis of a reduced model allowed us to determine that two model behaviours are possible: bacterial elimination or bacterial persistence. • The model predicts that the outcome of treating a surgical wound with oxygen therapy depends on the system parameters, but is especially sensitive to the initial bacterial load and the rate at which the immune system can deal with the bacteria. Abstract Infections are a common complication of any surgery, often requiring a recovery period in hospital. Supplemental oxygen therapy administered during and immediately after surgery is thought to enhance the immune response to bacterial contamination. However, aerobic bacteria thrive in oxygen-rich environments, and so it is unclear whether oxygen has a net positive effect on recovery. Here, we develop a mathematical model of post-surgery infection to investigate the efficacy of supplemental oxygen therapy on surgical-site infections. A 4-species, coupled, set of non-linear partial differential equations that describes the space-time dependence of neutrophils, bacteria, chemoattractant and oxygen is developed and analysed to determine its underlying properties. Through numerical solutions, we quantify the efficacy of different supplemental oxygen regimes on the treatment of surgical site infections in wounds of different initial bacterial load. A sensitivity analysis is performed to investigate the robustness of the predictions to changes in the model parameters. The numerical results are in good agreement with analyses of the associated well-mixed model. Our model findings provide insight into how the nature of the contaminant and its initial density influence bacterial infection dynamics in the surgical wound. [ABSTRACT FROM AUTHOR]

The immune system mounts a response to an infection by activating T cells. T cell activation occurs when dendritic cells, which have already interacted with the pathogen, scan a T cell that is cognate for (responsive to) the pathogen. This often occurs inside lymph nodes. The time it takes for this scanning event to occur, indeed the probability that it will occur at all, depends on many factors, including the rate that T cells and dendritic cells enter and leave the lymph node as well as the geometry of the lymph node and of course other cellular and molecular parameters. In this paper, we develop a hybrid stochastic-deterministic mathematical model at the tissue scale of the lymph node and simulate dendritic cells and cognate T cells to investigate the most important physiological factors leading to a successful and timely immune response after a vaccination. We use an agent-based model to describe the small population of cognate naive T cells and a partial differential equation description for the concentration of mature dendritic cells. We estimate the model parameters based on the known literature and measurements previously taken in our lab. We perform a parameter sensitivity analysis to quantify the sensitivity of the model results to the parameters. The results show that increasing T cell inflow through high endothelial venules, restricting cellular egress via the efferent lymph and increasing the total dendritic cell count by improving vaccinations are the among the most important physiological factors leading to an improved immune response. We also find that increasing the physical size of lymph nodes improves the overall likelihood that an immune response will take place but has a fairly weak effect on the response rate. The nature of dendritic cell trafficking through the LN (either passive or active transport) seems to have little effect on the overall immune response except if a change in overall egress time is observed. [ABSTRACT FROM AUTHOR]