Exact solutions of coupled multispecies linear reaction–diffusion equations on a uniformly growing domain

Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction–diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction–diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction–diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The first example calculation corresponds to a situation where the initially–confined population diffuses sufficiently slowly that it is unable to reach the moving boundary at x = L(t). In contrast, the second example calculation corresponds to a situation where the initially–confined population is able to overcome the domain growth and reach the moving boundary at x = L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit.

Survival probability for a diffusive process on a growing domain

We consider the motion of a diffusive population on a growing domain, 0 < x < L(t ), which is motivated by various applications in developmental biology. Individuals in the diffusing population, which could represent molecules or cells in a developmental scenario, undergo two different kinds of motion: (i) undirected movement, characterized by a diffusion coefficient, D, and (ii) directed movement, associated with the underlying domain growth. For a general class of problems with a reflecting boundary at x = 0, and an absorbing boundary at x = L(t ), we provide an exact solution to the partial differential equation describing the evolution of the population density function, C(x,t ). Using this solution, we derive an exact expression for the survival probability, S(t ), and an accurate approximation for the long-time limit, S = limt→∞ S(t ). Unlike traditional analyses on a nongrowing domain, where S ≡ 0, we show that domain growth leads to a very different situation where S can be positive. The theoretical tools developed and validated in this study allow us to distinguish between situations where the diffusive population reaches the moving boundary at x = L(t ) from other situations where the diffusive population never reaches the moving boundary at x = L(t ). Making this distinction is relevant to certain applications in developmental biology, such as the development of the enteric nervous system (ENS). All theoretical predictions are verified by implementing a discrete stochastic model.

Exact solutions of linear reaction-diffusion on a uniformly growing domain: criteria for successful colonization

Many processes during embryonic development involve transport and reaction of molecules, or transport and proliferation of cells, within growing tissues. Mathematical models of such processes usually take the form of a reaction-diffusion partial differential equation (PDE) on a growing domain. Previous analyses of such models have mainly involved solving the PDEs numerically. Here, we present a framework for calculating the exact solution of a linear reaction-diffusion PDE on a growing domain. We derive an exact solution for a general class of one-dimensional linear reaction—diffusion process on 0domain. Comparing our exact solutions with numerical approximations confirms the veracity of the method. Furthermore, our examples illustrate a delicate interplay between: (i) the rate at which the domain elongates, (ii) the diffusivity associated with the spreading density profile, (iii) the reaction rate, and (iv) the initial condition. Altering the balance between these four features leads to different outcomes in terms of whether an initial profile, located near x = 0, eventually overcomes the domain growth and colonizes the entire length of the domain by reaching the boundary where x = L(t).

Higher-resolution structure of the human insulin receptor ectodomain: Multi-modal inclusion of the insert domain

Insulin receptor (IR) signaling is critical to controlling nutrient uptake and metabolism. However, only a low-resolution (3.8 Å) structure currently exists for the IR ectodomain, with some segments ill-defined or unmodeled due to disorder. Here, we revise this structure using new diffraction data to 3.3 Å resolution that allow improved modeling of the N-linked glycans, the first and third fibronectin type III domains, and the insert domain. A novel haptic interactive molecular dynamics strategy was used to aid fitting to low-resolution electron density maps. The resulting model provides a foundation for investigation of structural transitions in IR upon ligand binding.

A cyclostationary multi-domain analysis of fluid instability in Kaplan turbines

Hydraulic instabilities represent a critical problem for Francis and Kaplan turbines, reducing their useful life due to increase of fatigue on the components and cavitation phenomena. Whereas an exhaustive list of publications on computational fluid-dynamic models of hydraulic instability is available, the possibility of applying diagnostic techniques based on vibration measurements has not been investigated sufficiently, also because the appropriate sensors seldom equip hydro turbine units. The aim of this study is to fill this knowledge gap and to exploit fully, for this purpose, the potentiality of combining cyclostationary analysis tools, able to describe complex dynamics such as those of fluid-structure interactions, with order tracking procedures, allowing domain transformations and consequently the separation of synchronous and non-synchronous components. This paper will focus on experimental data obtained on a full-scale Kaplan turbine unit, operating in a real power plant, tackling the issues of adapting such diagnostic tools for the analysis of hydraulic instabilities and proposing techniques and methodologies for a highly automated condition monitoring system. © 2015 Elsevier Ltd.

Computational analyses of the catalytic and heparin-binding sites and their interactions with glycosaminoglycans in glycoside hydrolase family 79 endo-d-glucuronidase (heparanase)

Mammalian heparanase is an endo-β-glucuronidase associated with cell invasion in cancer metastasis, angiogenesis and inflammation. Heparanase cleaves heparan sulfate proteoglycans in the extracellular matrix and basement membrane, releasing heparin/heparan sulfate oligosaccharides of appreciable size. This in turn causes the release of growth factors, which accelerate tumor growth and metastasis. Heparanase has two glycosaminoglycan-binding domains; however, no three-dimensional structure information is available for human heparanase that can provide insights into how the two domains interact to degrade heparin fragments. We have constructed a new homology model of heparanase that takes into account the most recent structural and bioinformatics data available. Heparin analogs and glycosaminoglycan mimetics were computationally docked into the active site with energetically stable ring conformations and their interaction energies were compared. The resulting docked structures were used to propose a model for substrates and conformer selectivity based on the dimensions of the active site. The docking of substrates and inhibitors indicates the existence of a large binding site extending at least two saccharide units beyond the cleavage site (toward the nonreducing end) and at least three saccharides toward the reducing end (toward heparin-binding site 2). The docking of substrates suggests that heparanase recognizes the N-sulfated and O-sulfated glucosamines at subsite +1 and glucuronic acid at the cleavage site, whereas in the absence of 6-O-sulfation in glucosamine, glucuronic acid is docked at subsite +2. These findings will help us to focus on the rational design of heparanase-inhibiting molecules for anticancer drug development by targeting the two heparin/heparan sulfate recognition domains.

Electronic coupling and catalytic effect on H2 evolution of MoS2/graphene nanocatalyst

Inorganic nano-graphene hybrid materials that are strongly coupled via chemical bonding usually present superior electrochemical performance. However, how the chemical bond forms and the synergistic catalytic mechanism remain fundamental questions. In this study, the chemical bonding of the MoS2 nanolayer supported on vacancy mediated graphene and the hydrogen evolution reaction of this nanocatalyst system were investigated. An obvious reduction of the metallic state of the MoS2 nanolayer is noticed as electrons are transferred to form a strong contact with the reduced graphene support. The missing metallic state associated with the unsaturated atoms at the peripheral sites in turn modifies the hydrogen evolution activity. The easiest evolution path is from the Mo edge sites, with the presence of the graphene resulting in a decrease in the energy barrier from 0.17 to 0.11 eV. Evolution of H2 from the S edge becomes more difficult due to an increase in the energy barrier from 0.43 to 0.84 eV. The clarification of the chemical bonding and catalytic mechanisms for hydrogen evolution using this strongly coupled MoS2/graphene nanocatalyst provide a valuable source of reference and motivation for further investigation for improved hydrogen evolution using chemically active nanocoupled systems.

Mutational analysis of the insulin‑like growth factor 1 receptor tyrosine kinase domain in non-small cell lung cancer patients

The insulin‑like growth factor 1 receptor (IGF1R) pathway plays an important role in the pathogenesis of non‑small cell lung cancer (NSCLC) and also provides a mechanism of resistance to targeted therapies. IGF1R is therefore an ideal therapeutic target and several inhibitors have entered clinical trials. However, thus far the response to these inhibitors has been poor, highlighting the importance of predictive biomarkers to identify patient cohorts who will benefit from these targeted agents. It is well‑documented that mutations and/or deletions in the epidermal growth factor receptor (EGFR) tyrosine kinase (TK) domain predict sensitivity of NSCLC patients to EGFR TK inhibitors. Single‑nucleotide polymorphisms (SNPs) in the IGF pathway have been associated with disease, including breast and prostate cancer. The aim of the present study was to elucidate whether the IGF1R TK domain harbours SNPs, somatic mutations or deletions in NSCLC patients and correlates the mutation status to patient clinicopathological data and prognosis. Initially 100 NSCLC patients were screened for mutations/deletions in the IGF1R TK domain (exons 16‑21) by sequencing analysis. Following the identification of SNP rs2229765, a further 98 NSCLC patients and 866 healthy disease‑free control patients were genotyped using an SNP assay. The synonymous SNP (rs2229765) was the only aberrant base change identified in the IGF1R TK domain of 100 NSCLC patients initially analysed. SNP rs2229765 was detected in exon 16 and was found to have no significant association between IGF1R expression and survival. The GA genotype was identified in 53.5 and 49.4% of NSCLC patients and control individuals, respectively. No significant difference was found in the genotype (P=0.5487) or allele (P=0.9082) frequencies between the case and control group. The present findings indicate that in contrast to the EGFR TK domain, the IGF1R TK domain is not frequently mutated in NSCLC patients. The synonymous SNP (rs2229765) had no significant association between IGF1R expression and survival in the cohort of NSCLC patients.