A mathematical model on the propagation of tau pathology in neurodegenerative diseases.

A system of partial differential equations is developed to study the spreading of tau pathology in the brain for Alzheimer's and other neurodegenerative diseases. Two cases are considered with one assuming intracellular diffusion through synaptic activities or the nanotubes that connect the adjacent cells. The other, in addition to intracellular spreading, takes into account of the secretion of the tau species which are able to diffuse, move with the interstitial fluid flow and subsequently taken up by the surrounding cells providing an alternative pathway for disease spreading. Cross membrane transport of the tau species are considered enabling us to examine the role of extracellular clearance of tau protein on the disease status. Bifurcation analysis is carried out for the steady states of the spatially homogeneous system yielding the results that fast cross-membrane transport combined with effective extracellular clearance is key to maintain the brain's healthy status. Numerical simulations of the first case exhibit solutions of travelling wave form describing the gradual outward spreading of the pathology; whereas the second case shows faster spreading with the buildup of neurofibrillary tangles quickly elevated throughout. Our investigation thus indicates that the gradual progression of the intracellular spreading case is more consistent with the clinical observations of the development of Alzheimer's disease.

A mathematical model demonstrating the role of interstitial fluid flow on the clearance and accumulation of amyloid ß in the brain.

A system of partial differential equations is developed to describe the formation and clearance of amyloid ß (Aß) and the subsequent buildup of Aß plaques in the brain, which are associated with Alzheimer's disease. The Aß related proteins are divided into five distinct categories depending on their size. In addition to enzymatic degradation, the clearance via diffusion and the outflow of interstitial fluid (ISF) into the surrounding cerebral spinal fluid (CSF) are considered. Treating the brain tissue as a porous medium, a simplified two-dimensional circular geometry is assumed for the transverse section of the brain leading to a nonlinear, coupled system of PDEs. Asymptotic analysis is carried out for the steady states of the spatially homogeneous system in the vanishingly small limit of Aß clearance rate. The PDE model is studied numerically for two cases, a spherically symmetric case and a more realistic 2D asymmetric case, allowing for non-uniform boundary conditions. Our investigations demonstrate that ISF advection is a key component in reproducing the clinically observed accumulation of plaques on the outer boundaries. Furthermore, ISF circulation serves to enhance Aß clearance over diffusion alone and that non-uniformities in ISF drainage into the CSF can lead to local clustering of plaques. Analysis of the model also demonstrates that plaque formation does not directly correspond to the high presence of toxic oligomers.

Mathematical modelling of contact dermatitis from nickel and chromium.

Dermal exposure to metal allergens can lead to irritant and allergic contact dermatitis (ACD). In this paper we present a mathematical model of the absorption of metal ions, hexavalent chromium and nickel, into the viable epidermis and compare the localised irritant and T-lymphocyte (T-cell) mediated immune responses. The model accounts for the spatial-temporal variation of skin health, extra and intracellular allergen concentrations, innate immune cells, T-cells, cytokine signalling and lymph node activity up to about 6 days after contact with these metals; repair processes associated with withdrawal of exposure to both metals is not considered in the current model, being assumed secondary during the initial phases of exposure. Simulations of the resulting system of PDEs are studied in one-dimension, i.e. across skin depth, and three-dimensional scenarios with the aim of comparing the responses to the two ions in the cases of first contact (no T-cells initially present) and second contact (T-cells initially present). The results show that on continuous contact, chromium ions elicit stronger skin inflammation, but for nickel, subsequent re-exposure stimulates stronger responses due to an accumulation of cytotoxic T-cell mediated responses which characterise ACD. Furthermore, the surface area of contact to these metals has little effect on the speed of response, whilst sensitivity is predicted to increase with the thickness of skin. The modelling approach is generic and should be applicable to describe contact dermatitis from a wide range of allergens.

Modelling the outbreak of infectious disease following mutation from a non-transmissible strain.

In-host mutation of a cross-species infectious disease to a form that is transmissible between humans has resulted with devastating global pandemics in the past. We use simple mathematical models to describe this process with the aim to better understand the emergence of an epidemic resulting from such a mutation and the extent of measures that are needed to control it. The feared outbreak of a human-human transmissible form of avian influenza leading to a global epidemic is the paradigm for this study. We extend the SIR approach to derive a deterministic and a stochastic formulation to describe the evolution of two classes of susceptible and infected states and a removed state, leading to a system of ordinary differential equations and a stochastic equivalent based on a Markov process. For the deterministic model, the contrasting timescale of the mutation process and disease infectiousness is exploited in two limits using asymptotic analysis in order to determine, in terms of the model parameters, necessary conditions for an epidemic to take place and timescales for the onset of the epidemic, the size and duration of the epidemic and the maximum level of the infected individuals at one time. Furthermore, the basic reproduction number R0 is determined from asymptotic analysis of a distinguished limit. Comparisons between the deterministic and stochastic model demonstrate that stochasticity has little effect on most aspects of an epidemic, but does have significant impact on its onset particularly for smaller populations and lower mutation rates for representatively large populations. The deterministic model is extended to investigate a range of quarantine and vaccination programmes, whereby in the two asymptotic limits analysed, quantitative estimates on the outcomes and effectiveness of these control measures are established.

Author Correction: Checkpoint blockade immunotherapy reshapes the high-dimensional phenotypic heterogeneity of murine intratumoural neoantigen-specific CD8+ T cells.

The original version of this Article omitted a declaration from the competing interests statement, which should have included the following: 'R.D.S. is a cofounder, stock holder, and scientific advisory board member of Jounce Therapeutics and Neon Therapeutics, and a member of the scientific advisory boards of BioLegend, Constellation, Lytix, and NGM. He also received research funding from Janssen and Agios.'. This has now been corrected in both the PDF and HTML versions of the Article.

Phenotype and Functional Features of Human Telomerase Reverse Transcriptase Immortalized Human Airway Smooth Muscle Cells from Asthmatic and Non-Asthmatic Donors.

Asthma is an obstructive respiratory disease characterised by chronic inflammation with airway hyperresponsiveness. In asthmatic airways, there is an increase in airway smooth muscle (ASM) cell bulk, which differs from non-asthmatic ASM in characteristics. This study aimed to assess the usefulness of hTERT immortalisation of human ASM cells as a research tool. Specifically we compared proliferative capacity, inflammatory mediator release and extracellular matrix (ECM) production in hTERT immortalised and parent primary ASM cells from asthmatic and non-asthmatic donors. Our studies revealed no significant differences in proliferation, IL-6 and eotaxin-1 production, or CTGF synthesis between donor-matched parent and hTERT immortalised ASM cell lines. However, deposition of ECM proteins fibronectin and fibulin-1 was significantly lower in immortalised ASM cells compared to corresponding primary cells. Notably, previously reported differences in proliferation and inflammatory mediator release between asthmatic and non-asthmatic ASM cells were retained, but excessive ECM protein deposition in asthmatic ASM cells was lost in hTERT ASM cells. This study shows that hTERT immortalised ASM cells mirror primary ASM cells in proliferation and inflammatory profile characteristics. Moreover, we demonstrate both strengths and weaknesses of this immortalised cell model as a representation of primary ASM cells for future asthma pathophysiological research.

Checkpoint blockade immunotherapy reshapes the high-dimensional phenotypic heterogeneity of murine intratumoural neoantigen-specific CD8+ T cells.

The analysis of neoantigen-specific CD8+ T cells in tumour-bearing individuals is challenging due to the small pool of tumour antigen-specific T cells. Here we show that mass cytometry with multiplex combinatorial tetramer staining can identify and characterize neoantigen-specific CD8+ T cells in mice bearing T3 methylcholanthrene-induced sarcomas that are susceptible to checkpoint blockade immunotherapy. Among 81 candidate antigens tested, we identify T cells restricted to two known neoantigens simultaneously in tumours, spleens and lymph nodes in tumour-bearing mice. High-dimensional phenotypic profiling reveals that antigen-specific, tumour-infiltrating T cells are highly heterogeneous. We further show that neoantigen-specific T cells display a different phenotypic profile in mice treated with anti-CTLA-4 or anti-PD-1 immunotherapy, whereas their peripheral counterparts are not affected by the treatments. Our results provide insights into the nature of neoantigen-specific T cells and the effects of checkpoint blockade immunotherapy.Immune checkpoint blockade (ICB) therapies can unleash anti-tumour T-cell responses. Here the authors show, by integrating MHC tetramer multiplexing, mass cytometry and high-dimensional analyses, that neoantigen-specific, tumour-infiltrating T cells are highly heterogeneous and are subjected to ICB modulations.

Latrophilin receptors: novel bronchodilator targets in asthma.

BACKGROUND: Asthma affects 300 million people worldwide. In asthma, the major cause of morbidity and mortality is acute airway narrowing, due to airway smooth muscle (ASM) hypercontraction, associated with airway remodelling. However, little is known about the transcriptional differences between healthy and asthmatic ASM cells. OBJECTIVES: To investigate the transcriptional differences between asthmatic and healthy airway smooth muscle cells (ASMC) in culture and investigate the identified targets using in vitro and ex vivo techniques. METHODS: Human asthmatic and healthy ASMC grown in culture were run on Affymetrix_Hugene_1.0_ST microarrays. Identified candidates were confirmed by PCR, and immunohistochemistry. Functional analysis was conducted using in vitro ASMC proliferation, attachment and contraction assays and ex vivo contraction of mouse airways. RESULTS: We suggest a novel role for latrophilin (LPHN) receptors, finding increased expression on ASMC from asthmatics, compared with non-asthmatics in vivo and in vitro, suggesting a role in mediating airway function. A single nucleotide polymorphism in LPHN1 was associated with asthma and with increased LPHN1 expression in lung tissue. When activated, LPHNs regulated ASMC adhesion and proliferation in vitro, and promoted contraction of mouse airways and ASMC. CONCLUSIONS: Given the need for novel inhibitors of airway remodelling and bronchodilators in asthma, the LPHN family may represent promising novel targets for future dual therapeutic intervention.

A mathematical model of the growth of uterine myomas.

Uterine myomas or fibroids are common, benign smooth muscle tumours that can grow to 10  cm or more in diameter and are routinely removed surgically. They are typically slow- growing, well-vascularised, spherical tumours that, on a macro-scale, are a structurally uniform, hard elastic material. We present a multi-phase mathematical model of a fully vascularised myoma growing within a surrounding elastic tissue. Adopting a continuum approach, the model assumes the conservation of mass and momentum of four phases, namely cells/collagen, extracellular fluid, arterial and venous phases. The cell/collagen phase is treated as a poro-elastic material, based on a linear stress-strain relationship, and Darcy's law is applied to describe flow in the extracellular fluid and the two vascular phases. The supply of extracellular fluid is dependent on the capillary flow rate and mean capillary pressure expressed in terms of the arterial and venous pressures. Cell growth and division is limited to the myoma domain and dependent on the local stress in the material. The resulting model consists of a system of nonlinear partial differential equations with two moving boundaries. Numerical solutions of the model successfully reproduce qualitatively the clinically observed three-phase "fast-slow-fast" growth profile that is typical for myomas. The results suggest that this growth profile requires stress-induced resistance to growth by the surrounding tissue and a switch-like cell growth response to stress. Analysis of large-time solutions reveal that while there is a functioning vasculature throughout the myoma, exponential growth results, otherwise power-law growth is predicted. An extensive survey of the effect of parameters on model solutions is also presented, and in particular, the enhanced growth caused by factors such as oestrogen is predicted by the model.

Sphingosine-1-phosphate induces pro-remodelling response in airway smooth muscle cells.

BACKGROUND: Increased proliferation of airway smooth muscle (ASM) cells leading to hyperplasia and increased ASM mass is one of the most characteristic features of airway remodelling in asthma. A bioactive lipid, sphingosine-1-phosphate (S1P), has been suggested to affect airway remodelling by stimulation of human ASM cell proliferation. OBJECTIVE: To investigate the effect of S1P on signalling and regulation of gene expression in ASM cells from healthy and asthmatic individuals. METHODS: Airway smooth muscle cells grown from bronchial biopsies of healthy and asthmatic individuals were exposed to S1P. Gene expression was analysed using microarray, real-time PCR and Western blotting. Receptor signalling and function were determined by mRNA knockdown and intracellular calcium mobilization experiments. RESULTS: S1P potently regulated the expression of more than 80 genes in human ASM cells, including several genes known to be involved in the regulation of cell proliferation and airway remodelling (HBEGF, TGFB3, TXNIP, PLAUR, SERPINE1, RGS4). S1P acting through S1P2 and S1P3 receptors activated intracellular calcium mobilization and extracellular signal-regulated and Rho-associated kinases to regulate gene expression. S1P-induced responses were not inhibited by corticosteroids and did not differ significantly between ASM cells from healthy and asthmatic individuals. CONCLUSION: S1P induces a steroid-resistant, pro-remodelling pathway in ASM cells. Targeting S1P or its receptors could be a novel treatment strategy for inhibiting airway remodelling in asthma.

Ca2+ handling and sensitivity in airway smooth muscle: emerging concepts for mechanistic understanding and therapeutic targeting.

Free calcium ions within the cytosol serve as a key secondary messenger system for a diverse range of cellular processes. Dysregulation of cytosolic Ca(2+) handling in airway smooth muscle (ASM) has been implicated in asthma, and it has been hypothesised that this leads, at least in part, to associated changes in both the architecture and function of the lung. Significant research is therefore directed towards furthering our understanding of the mechanisms which control ASM cytosolic calcium, in addition to those regulating the sensitivity of its downstream effector targets to calcium. Key aspects of the recent developments in this field were discussed at the 8th Young Investigators' Symposium on Smooth Muscle (2013, Groningen, The Netherlands), and are outlined in this review.

A mathematical model for the human menstrual cycle.

A simple mathematical model framework is developed to describe the hormonal interactions of the human menstrual cycle along the hypothalamus-pituitary-ovaries axis. The framework is designed so that it can be readily extended to model processes that disrupt the normal functioning cycle. The model in its most basic formulation exhibits multiple periodic solutions, one of which shows the key characteristics of a menstrual cycle, while the others indicate possible abnormalities sometimes observed in women of reproductive age. The basic model is extended to encompass receptor down-regulation as a mechanism to describe the desensitization of the pituitary to continuous stimulation of hypothalamic hormone, a hormonal therapy that is commonly prescribed prior to the surgical procedure for the removal of uterine myomas. Though the mechanisms for desensitization are likely to be more complex, the model results are in good qualitative agreement with physiological observations.

Regulation of GPCR-mediated smooth muscle contraction: implications for asthma and pulmonary hypertension.

Contractile G-protein-coupled receptors (GPCRs) have emerged as key regulators of smooth muscle contraction, both under healthy and diseased conditions. This brief review will discuss some key topics and novel insights regarding GPCR-mediated airway and vascular smooth muscle contraction as discussed at the 7th International Young Investigators' Symposium on Smooth Muscle (2011, Winnipeg, Manitoba, Canada) and will in particular focus on processes driving Ca(2+)-mobilization and -sensitization.

A mathematical model of the in vitro keratinocyte response to chromium and nickel exposure.

A mathematical model describing the main mechanistic processes involved in keratinocyte response to chromium and nickel has been developed and compared to experimental in vitro data. Accounting for the interactions between the metal ions and the keratinocytes, the law of mass action was used to generate ordinary differential equations which predict the time evolution and ion concentration dependency of keratinocyte viability, the amount of metal associated with the keratinocytes and the release of cytokines by the keratinocytes. Good agreement between model predictions and existing experimental data of these endpoints was observed, supporting the use of this model to explore physiochemical parameters that influence the toxicological response of keratinocytes to these two metals.

A multi-phase mathematical model of quorum sensing in a maturing Pseudomonas aeruginosa biofilm.

It is well known that sessile bacteria have a strong tendency to exist in a biofilm phenotype, whereby bacterial cells aggregate and produce a gel-like extracellular matrix, which, in an infection scenario, offers a significant barrier to attack by conventional antibiotics and the immune system. In this paper we develop a multi-phase model of a maturing Pseudomonas aeruginosa biofilm, allowing for the production and secretion of exopolysaccharide (EPS). The primary quorum-sensing system of P. aeruginosa (namely the lasR system) is believed to be required for full biofilm development, and we thus take the synthesis of EPS to be regulated by the cognate signal molecule, 3-oxo-C12-HSL. We also take EPS and signal production, along with bacterial growth, to be limited by oxygen availability, thus factoring in the nutrient poor conditions deep inside the biofilm. We use simulations to examine the role played by quorum sensing in the biofilm maturation process, and to investigate the effect of anti-quorum sensing and antibiotic treatments on EPS concentration, signal level, bacterial numbers and biofilm growth rate. In addition, we undertake analysis of the associated travelling-wave behaviour.

Modelling antibiotic- and anti-quorum sensing treatment of a spatially-structured Pseudomonas aeruginosa population.

The bacterial cell to cell signalling system known as quorum sensing (QS) is essential for the regulation of virulence in many pathogens and offers a specific biochemical target for novel antibacterial therapies. Expanding on earlier work, in which consideration was given to the primary QS system (lasR system) in a homogeneous population of the common human pathogen Pseudomonas aeruginosa, we build a simple spatial model of an early-stage P. aeruginosa biofilm subject to treatment with topically applied anti-QS drugs (of two specific kinds) and conventional antibiotics. In the case of a slowly growing biofilm we show that both kinds of anti-quorum sensing drug are effective in reducing the level of the relevant signal molecule (3-oxo-C12-homoserine lactone; henceforth AHL), in each case obtaining an explicit bound on the steady-state AHL profile in terms of a prescribed surface drug concentration. Using numerical methods, we are also able to reproduce the hysteretic phenomena exhibited by the homogeneous model, in particular showing that for each kind of anti-QS drug there is a parameter regime in which a catastrophic collapse occurs in the steady-state AHL concentration as the surface drug concentration passes some critical value; an alternative way of interpreting this result is to say that, for a prescribed surface drug concentration, there is a critical biofilm depth such that treatment is successful until this depth is reached, but fails thereafter. In the thick-biofilm limit we show that the critical concentration of each drug increases exponentially with the biofilm thickness (or, conversely, that the critical depth increases logarithmically with surface drug concentration); this is dramatically different to the behaviour observed in the corresponding homogeneous model, where the critical concentrations grow linearly with bacterial carrying capacity, and thus highlights the relative difficulty of treating a large, spatially-structured population with diffusing antibacterials.

Mathematical modelling of therapies targeted at bacterial quorum sensing.

Bacteria commonly use diffusible signal molecules to synchronise their behaviour by facilitating population dependent co-ordination. This cell-to-cell signalling mechanism is known as quorum sensing (QS) and provides a way of ensuring that certain genes are 'switched on' only when a certain signal concentration (typically corresponding to a large population density) has been reached. In this paper we focus on the QS system of the human pathogen Pseudomonas aeruginosa, which employs a complex hierarchy of QS signalling systems, which regulate the formation of multiple exoproducts, swarming and biofilm differentiation. In P. aeruginosa, the signal molecules are N-acylated homoserine lactones (AHLs; e.g., N-(3-oxododecanoyl)-homoserine lactone [3-oxo-C12-HSL]), which bind to transcriptional regulator proteins (LasR in the case of 3-oxo-C12-HSL) to activate the expression of target genes including lasI, which codes for the 3-oxo-C12-HSL synthase. Since the virulence of P. aeruginosa is controlled by QS, agents (QSBs) designed to block this cell-to-cell communication have potential as novel antibacterials. By drawing on existing models for the reaction kinetics of this system, we model a growing population subject to treatment with two kinds of QSB, together with a conventional antibiotic. The first kind of QSB is assumed to act by diffusing through the cell membrane and then destabilising/sequestering LasR, while the second kind remains outside the cell and degrades the AHL signal molecule itself. Numerical and mathematical analysis of the resulting systems of ordinary differential equations reveals in particular that, while a sufficiently high dose of QSB is, in all cases considered, able to reduce the AHL concentration (and hence virulence) to a negligible level, the qualitative response to treatment is sensitive to parameter values.