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Tissues form from collections of cells that interact together mechanically via cell-to-cell adhesion, mediated by transmembrane cell adhesion molecules. Under a sufficiently large amount of induced stress, these tissues can undergo elastic deformation in the direction of tension, where they then elongate without any topological changes, and experience plastic deformation within the tissue. In this work, we present a novel mathematical model describing the deformation of cells, where tissues are elongated in a controlled manner. In doing so, the cells are able to undergo remodelling through elastic and then plastic deformation, in accordance with experimental observation. Our model describes bistable sizes of a cell that actively deform under stress to elongate the cell. In the absence of remodelling, the model reduces to the standard linear interaction model. In the presence of instant remodelling, we provide a bifurcation analysis to describe the existence of the bistable cell sizes. In the case of general remodelling, we show numerically that cells within a tissue may populate both the initial and elongated cell sizes, following a sufficiently large degree of stress. [ABSTRACT FROM AUTHOR]

The intracellular release of calcium from the endoplasmic reticulum is controlled by ion channels. The resulting calcium signals exhibit a rich spatio-temporal signature, which originates at least partly from microscopic fluctuations. While stochasticity in the gating transition of ion channels has been incorporated into many models, the distribution of calcium is usually described by deterministic reaction-diffusion equations. Here we test the validity of the latter modeling approach by using two different models to calculate the frequency of localized calcium signals (calcium puffs) from clustered IP3 receptor channels. The complexity of the full calcium system is here limited to the basic opening mechanism of the ion channels and, in the mathematical reduction simplifies to the calculation of a first passage time. Two models are then studied: (i) a hybrid model, where channel gating is treated stochastically, while calcium concentration is deterministic and (ii) a fully stochastic model with noisy channel gating and Brownian calcium ion motion. The second model utilises the recently developed two-regime method [M. B. Flegg, S. J. Chapman, and R. Erban, 'The two-regime method for optimizing stochastic reaction-diffusion simulations,' J. R. Soc., Interface 9, 859-868 (2012)] in order to simulate a large domain with precision required only near the Ca2+ absorbing channels. The expected time for a first channel opening that results in a calcium puff event is calculated. It is found that for a large diffusion constant, predictions of the interpuff time are significantly overestimated using the model (i) with a deterministic non-spatial calcium variable. It is thus demonstrated that the presence of diffusive noise in local concentrations of intracellular Ca2+ ions can substantially influence the occurrence of calcium signals. The presented approach and results may also be relevant for other cell-physiological first-passage time problems with small ligand concentration and high cooperativity. [ABSTRACT FROM AUTHOR]

In this paper we present a novel method for finding unknown parameters for an unknown morphogen. We postulate the existence of an unknown morphogen in a given three-dimensional domain due to the spontaneous arrangement of a downstream species on the domain boundary for which data is known. Assuming a modified Helmholtz model for the morphogen and that it is produced from a single source in the domain, our method accurately estimates the source location and other model parameters. Notably, our method does not require the forward solution of the model to be computed which can often be a challenge for three-dimensional PDE model parameter fitting. Instead, an extension is made from the problem domain to an infinite domain and the analytic nature of the fundamental solution is exploited. We explore in this manuscript strategies for best conditioning the problem and rigorously explore the accuracy of the method on two test problems. Our tests focus on the effect of source location on accuracy but also the robustness of the algorithm to experimental noise. [ABSTRACT FROM AUTHOR]

We develop a computationally efficient method for the theoretical analysis of thermophoresis of nanoparticles and adatoms on crystalline surfaces (thermal tweezers) for efficient parallel nanofabrication. The analysis of surface diffusion of particles or adatoms in the presence of strong temperature gradients is conducted through the direct determination of probability distributions for diffusing particles, using the numerical solution of the Smoluchowski diffusion equation with varying (temperature-dependent) diffusion constant. The local values of the diffusion constant are determined from the Fokker–Planck equation for the considered crystalline potential of the substrate and local temperature. Steady-state and nonsteady-state particle distributions on the surface are obtained and analyzed in the presence of optically-induced strong temperature gradients. Detailed comparison of this approach with the previously obtained results from the Monte Carlo simulations of the Langevin equation is conducted, demonstrating high computational efficiency, and accuracy of the new method in the high-friction regime. Applicability conditions for the developed method are also determined and discussed. [ABSTRACT FROM AUTHOR]

In recent years, modeling of biochemical systems has attracted much attention. Traditionally, these systems have been modeled with differential equations. However, specific biological processes tend to produce a lot of intrinsic noise, which has led to the development of a number of stochastic models for such processes. One such framework is that of Smoluchowski, which is used to model bimolecular reactions. In this model, particles diffuse with Brownian motion until they come within a critical separation distance and undergo a chemical reaction. This theory has undergone a number of developments, notably the extension of the theory to deal with reversible reactions, which has made available various software packages for stochastic reaction-diffusion simulations. Recent work has extended Smoluchowski theory to irreversible reactions of order greater than two. In this paper, we build on that theory and develop a model that includes reversible reactions of any order. This type of chemical kinetics is critical to many catalytic systems in biology. We use our new mod- eling framework to simulate the response of the core destruction regulation cycle to sharp changes in Wnt signaling, generating behavior not seen in classical ODE models. [ABSTRACT FROM AUTHOR]

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]

The two-regime method (TRM) has been recently developed for optimizing stochastic reaction-diffusion simulations [M. Flegg, J. Chapman, and R. Erban, J. Roy. Soc. Interface, 9 (2012), pp. 859--868]. It is a multiscale (hybrid) algorithm which uses stochastic reaction-diffusion models with different levels of detail in different parts of the computational domain. The coupling condition on the interface between different modeling regimes of the TRM was previously derived for one-dimensional models. In this paper, the TRM is generalized to higher dimensional reaction-diffusion systems. Coupling Brownian dynamics models with compartment-based models on regular (square) two-dimensional lattices is studied in detail. In this case, the interface between different modeling regimes contains either flat parts or right-angle corners. Both cases are studied in the paper. For flat interfaces, it is shown that the one-dimensional theory can be used along the line perpendicular to the TRM interface. In the direction tangential to the interface, two choices of the TRM parameters are presented. Their applicability depends on the compartment size and the time step used in the molecular-based regime. The two-dimensional generalization of the TRM is also discussed in the case of corners. [ABSTRACT FROM AUTHOR]

The acceptance of broadband ultrasound attenuation for the assessment of osteoporosis suffers from a limited understanding of ultrasound wave propagation through cancellous bone. It has recently been proposed that the ultrasound wave propagation can be described by a concept of parallel sonic rays. This concept approximates the detected transmission signal to be the superposition of all sonic rays that travel directly from transmitting to receiving transducer. The transit time of each ray is defined by the proportion of bone and marrow propagated. An ultrasound transit time spectrum describes the proportion of sonic rays having a particular transit time, effectively describing lateral inhomogeneity of transit times over the surface of the receiving ultrasound transducer. The aim of this study was to provide a proof of concept that a transit time spectrum may be derived from digital deconvolution of input and output ultrasound signals. We have applied the active-set method deconvolution algorithm to determine the ultrasound transit time spectra in the three orthogonal directions of four cancellous bone replica samples and have compared experimental data with the prediction from the computer simulation. The agreement between experimental and predicted ultrasound transit time spectrum analyses derived from Bland–Altman analysis ranged from 92% to 99%, thereby supporting the concept of parallel sonic rays for ultrasound propagation in cancellous bone. In addition to further validation of the parallel sonic ray concept, this technique offers the opportunity to consider quantitative characterisation of the material and structural properties of cancellous bone, not previously available utilising ultrasound. [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]

Abstract: In this paper we report the results of the detailed monitoring and analysis of combustion emissions from fast moving diesel trains. A new highly efficient monitoring methodology is proposed based on the measurements of the total number concentration (TNC) of combustion aerosols at a fixed point (on a bridge overpassing the railway) inside the violently mixing zone created by a fast moving train. Applicability conditions for the proposed methodology are presented, discussed and linked to the formation of the stable and uniform mixing zone. In particular, it is demonstrated that if such a mixing zone is formed, the monitoring results are highly consistent, repeatable (with typically negligible statistical errors and dispersion), stable with respect to the external atmospheric turbulence and result in an unusual pattern of the aerosol evolution with two or three distinct TNC maximums. It is also shown that the stability and uniformity of the created mixing zone (as well as the repeatability of the monitoring results) increase with increasing length of the train (with an estimated critical train length of ~10 carriages, at the speed of ~150km/h). The analysis of the obtained evolutionary dependencies of aerosol TNC suggests that the major possible mechanisms responsible for the formation of the distinct concentration maximums are condensation (the second maximum) and thermal fragmentation of solid nanoparticle aggregates (third maximum). The obtained results and the new methodology will be important for monitoring and analysis of combustion emissions from fast moving trains, and for the determination of the impact of rail networks on the atmospheric environment and human exposure to combustion emissions. [ABSTRACT FROM AUTHOR]

This work investigates multi-resolution methodologies for simulating dimer models. The solvent particles which make up the heat bath interact with the monomers of the dimer either through direct collisions (short-range) or through harmonic springs (long-range). Two types of multi-resolution methodologies are considered in detail: (a) describing parts of the solvent far away from the dimer by a coarser approach; (b) describing each monomer of the dimer by using a model with different level of resolution. These methodologies are then used to investigate the effect of a shared heat bath versus two uncoupled heat baths, one for each monomer. Furthermore, the validity of the multi-resolution methods is discussed by comparison to dynamics of macroscopic Langevin equations., Competing Interests: We declare we have no competing interests.

Spatial reaction-diffusion models have been employed to describe many emergent phenomena in biological systems. The modelling technique most commonly adopted in the literature implements systems of partial differential equations (PDEs), which assumes there are sufficient densities of particles that a continuum approximation is valid. However, owing to recent advances in computational power, the simulation and therefore postulation, of computationally intensive individual-based models has become a popular way to investigate the effects of noise in reaction-diffusion systems in which regions of low copy numbers exist. The specific stochastic models with which we shall be concerned in this manuscript are referred to as 'compartment-based' or 'on-lattice'. These models are characterized by a discretization of the computational domain into a grid/lattice of 'compartments'. Within each compartment, particles are assumed to be well mixed and are permitted to react with other particles within their compartment or to transfer between neighbouring compartments. Stochastic models provide accuracy, but at the cost of significant computational resources. For models that have regions of both low and high concentrations, it is often desirable, for reasons of efficiency, to employ coupled multi-scale modelling paradigms. In this work, we develop two hybrid algorithms in which a PDE in one region of the domain is coupled to a compartment-based model in the other. Rather than attempting to balance average fluxes, our algorithms answer a more fundamental question: 'how are individual particles transported between the vastly different model descriptions?' First, we present an algorithm derived by carefully redefining the continuous PDE concentration as a probability distribution. While this first algorithm shows very strong convergence to analytical solutions of test problems, it can be cumbersome to simulate. Our second algorithm is a simplified and more efficient implementation of the first, it is derived in the continuum limit over the PDE region alone. We test our hybrid methods for functionality and accuracy in a variety of different scenarios by comparing the averaged simulations with analytical solutions of PDEs for mean concentrations., (© 2015 The Author(s) Published by the Royal Society. All rights reserved.)

Spatial organization and noise play an important role in molecular systems biology. In recent years, a number of software packages have been developed for stochastic spatio-temporal simulation, ranging from detailed molecular-based approaches to less detailed compartment-based simulations. Compartment-based approaches yield quick and accurate mesoscopic results, but lack the level of detail that is characteristic of the computationally intensive molecular-based models. Often microscopic detail is only required in a small region (e.g. close to the cell membrane). Currently, the best way to achieve microscopic detail is to use a resource-intensive simulation over the whole domain. We develop the two-regime method (TRM) in which a molecular-based algorithm is used where desired and a compartment-based approach is used elsewhere. We present easy-to-implement coupling conditions which ensure that the TRM results have the same accuracy as a detailed molecular-based model in the whole simulation domain. Therefore, the TRM combines strengths of previously developed stochastic reaction-diffusion software to efficiently explore the behaviour of biological models. Illustrative examples and the mathematical justification of the TRM are also presented.

This paper presents a statistical theory of stochastic evaporation and degradation processes in complex polymerlike ring and chain aggregates with multiple degrading bonds between the primary particles (monomers). The exact kinetic solution fully describing fragmentation processes is obtained for such aggregates with arbitrary number of primary particles (monomers) and bonds between them. The effects of additional interaction of multiple bonds with each other is shown to have a drastic impact on the predicted kinetic processes and time-dependent particle size distributions during aggregate degradation. Structural effects associated with different distributions of multiple bonds and bonding configurations in the aggregates are also investigated and shown to have a significant impact on typical fragmentation time and accumulation of fragmenting aggregates in intermediate modes. The developed theory and its results will be important for degradation of multistranded polymers, polymer networks, self-assembling structures, surface nanoclusters and nanotechnology, and formation and evolution of aerosol aggregates resulting from transport and industry emissions.