Comparing a discrete and continuum model of the intestinal crypt

The integration of processes at different scales is a key problem in the modelling of cell populations. Owing to increased computational resources and the accumulation of data at the cellular and subcellular scales, the use of discrete, cell-level models, which are typically solved using numerical simulations, has become prominent. One of the merits of this approach is that important biological factors, such as cell heterogeneity and noise, can be easily incorporated. However, it can be difficult to efficiently draw generalizations from the simulation results, as, often, many simulation runs are required to investigate model behaviour in typically large parameter spaces. In some cases, discrete cell-level models can be coarse-grained, yielding continuum models whose analysis can lead to the development of insight into the underlying simulations. In this paper we apply such an approach to the case of a discrete model of cell dynamics in the intestinal crypt. An analysis of the resulting continuum model demonstrates that there is a limited region of parameter space within which steady-state (and hence biologically realistic) solutions exist. Continuum model predictions show good agreement with corresponding results from the underlying simulations and experimental data taken from murine intestinal crypts.

Discrete time representation of continuous time ARMA processes

This paper derives exact discrete time representations for data generated by a continuous time autoregressive moving average (ARMA) system with mixed stock and flow data. The representations for systems comprised entirely of stocks or of flows are also given. In each case the discrete time representations are shown to be of ARMA form, the orders depending on those of the continuous time system. Three examples and applications are also provided, two of which concern the stationary ARMA(2, 1) model with stock variables (with applications to sunspot data and a short-term interest rate) and one concerning the nonstationary ARMA(2, 1) model with a flow variable (with an application to U.S. nondurable consumers’ expenditure). In all three examples the presence of an MA(1) component in the continuous time system has a dramatic impact on eradicating unaccounted-for serial correlation that is present in the discrete time version of the ARMA(2, 0) specification, even though the form of the discrete time model is ARMA(2, 1) for both models.

Riluzole elevates GLT-1 activity and levels in striatal astrocytes

Drugs which upregulate astrocyte glutamate transport may be useful neuroprotective compounds by preventing excitotoxicity. We set up a new system to identify potential neuroprotective drugs which act through GLT-1. Primary mouse striatal astrocytes grown in the presence of the growth-factor supplement G5 express high levels of the functional glutamate transporter, GLT-1 (also known as EAAT2) as assessed by Western blotting and (3)H-glutamate uptake assay, and levels decline following growth factor withdrawal. The GLT-1 transcriptional enhancer dexamethasone (0.1 or 1muM) was able to prevent loss of GLT-1 levels and activity following growth factor withdrawal. In contrast, ceftriaxone, a compound previously reported to enhance GLT-1 expression, failed to regulate GLT-1 in this system. The neuroprotective compound riluzole (100muM) upregulated GLT-1 levels and activity, through a mechanism that was not dependent on blockade of voltage-sensitive ion channels, since zonasimide (1mM) did not regulate GLT-1. Finally, CDP-choline (10muM-1mM), a compound which promotes association of GLT-1/EAAT2 with lipid rafts was unable to prevent GLT-1 loss under these conditions. This observation extends the known pharmacological actions of riluzole, and suggests that this compound may exert its neuroprotective effects through an astrocyte-dependent mechanism.

The first international workshop on the role and impact of mathematics in medicine: a collective account

The First International Workshop on The Role and Impact of Mathematics in Medicine (RIMM) convened in Paris in June 2010. A broad range of researchers discussed the difficulties, challenges and opportunities faced by those wishing to see mathematical methods contribute to improved medical outcomes. Finding mechanisms for inter- disciplinary meetings, developing a common language, staying focused on the medical problem at hand, deriving realistic mathematical solutions, obtaining

Modelling acidosis and the cell cycle in multicellular tumour spheroids

A partial differential equation model is developed to understand the effect that nutrient and acidosis have on the distribution of proliferating and quiescent cells and dead cell material (necrotic and apopotic) within a multicellular tumour spheroid. The rates of cell quiescence and necrosis depend upon the local nutrient and acid concentrations and quiescent cells are assumed to consume less nutrient and produce less acid than proliferating cells. Analysis of the differences in nutrient consumption and acid production by quiescent and proliferating cells shows low nutrient levels do not necessarily lead to increased acid concentration via anaerobic metabolism. Rather, it is the balance between proliferating and quiescent cells within the tumour which is important; decreased nutrient levels lead to more quiescent cells, which produce less acid than proliferating cells. We examine this effect via a sensitivity analysis which also includes a quantification of the effect that nutrient and acid concentrations have on the rates of cell quiescence and necrosis.

Modulation of neuroinflammation by dietary flavonoids

There is increasing evidence to suggest neuroinflammatory processes contribute to the cascade of events that lead to the progressive neuronal damage observed in neurodegenerative disorders such as Parkinson’s disease and Alzheimer’s disease. The molecular mechanisms underlying such neurodegenerative processes are rather complex and involve modulation of the mitogen-activated protein kinase (MAPK) and NF-κB pathways leading to the generation of nitric oxide (NO). Such a small molecule may diffuse to the neighbouring neurons and trigger neuronal death through the inhibition of mitochondrial respiration and increases in the reactive oxygen and nitrogen species. Recently, attention has focused on the neuroprotective effects of flavonoids which have been effective in protecting against both age-related cognitive and motor decline in vivo. Although, the precise mechanisms by which flavonoids may exert their neuroprotective effects remain unclear, accumulating evidence suggest that they may exert their neuroprotective effects through the modulation of the MAP Kinase and PI3 Kinase signaling pathways. The aim of the present chapter is to highlight the potential neuroprotective role of dietary flavonoids in terms of their ability to modulate neuroinflammation in the central nervous system. We will provide an outline of the role glial cells play in neuroinflammation and describe the involvement of inflammatory mediators, produced by glia, in the cascade of events leading to neuronal degeneration. We will then present the evidence that flavonoids may modulate neuroinflammation by inhibiting the production of these inflammatory agents and summarise their potential mechanisms of action.

Classifying general nonlinear force laws in cell-based models via the continuum limit

though discrete cell-based frameworks are now commonly used to simulate a whole range of biological phenomena, it is typically not obvious how the numerous different types of model are related to one another, nor which one is most appropriate in a given context. Here we demonstrate how individual cell movement on the discrete scale modeled using nonlinear force laws can be described by nonlinear diffusion coefficients on the continuum scale. A general relationship between nonlinear force laws and their respective diffusion coefficients is derived in one spatial dimension and, subsequently, a range of particular examples is considered. For each case excellent agreement is observed between numerical solutions of the discrete and corresponding continuum models. Three case studies are considered in which we demonstrate how the derived nonlinear diffusion coefficients can be used to (a) relate different discrete models of cell behavior; (b) derive discrete, intercell force laws from previously posed diffusion coefficients, and (c) describe aggregative behavior in discrete simulations.

Regulation of NF-κB activity in astrocytes: effects of flavonoids at dietary-relevant concentrations

Neuroinflammation plays an important role in the progression of neurodegenerative disorders such as Alzheimer’s disease and Parkinson’s disease. Sustained activation of nuclear transcription factor κB (NF-κB) is thought to play an important role in the pathogenesis of neurodegenerative disorders. Flavonoids have been shown to possess antioxidant and anti-inflammatory properties and we investigated whether flavonoids, at submicromolar concentrations relevant to their bioavailability from the diet, were able to modulate NF-κB signalling in astrocytes. Using luciferase reporter assays, we found that tumour necrosis factor (TNFα, 150 ng/ml) increased NF-κB-mediated transcription in primary cultures of mouse cortical astrocytes, which was abolished on co-transfection of a dominant-negative IκBα construct. In addition, TNFα increased nuclear localisation of p65 as shown by immunocytochemistry. To investigate potential flavonoid modulation of NF-κB activity, astrocytes were treated with flavonoids from different classes; flavan-3-ols ((−)-epicatechin and (+)-catechin), flavones (luteolin and chrysin), a flavonol (kaempferol) or the flavanones (naringenin and hesperetin) at dietary-relevant concentrations (0.1–1 μM) for 18 h. None of the flavonoids modulated constitutive or TNFα-induced NF-κB activity. Therefore, we conclude that NF-κB signalling in astrocytes is not a major target for flavonoids.

New insights on regulation of LMTK2, a membrane kinase integrating pathways central to neurodegeneration

In a short communication in this issue (Manser et al. 2012), Christopher Miller’s group at the Institute of Psychiatry, King’s College London present an elegant and convincing set of experiments using molecular techniques to show that a brain-enriched membrane-associated protein kinase, lemur tyrosine kinase-2 (LMTK2), is directly phosphorylated by the cyclin-dependent kinase-5/p35 and this event is sufficient for LMTK2 to phosphorylate an abundant protein phosphatase, PP1C. LMTK2 has been little studied to date and, despite its name, is a kinase which phosphorylates serine or threonine residues of protein substrates. The paper adds to the evidence that this enzyme is a potentially important mediator positioned to integrate a number of intracellular signalling pathways relevant to neurodegeneration.

Everything and nothing project

An EPRSC ‘Partnerships for Public Engagement’ scheme 2010. FEC 122,545.56/UoR 10K everything and nothing is a performance and workshop which engages the public creatively with mathematical concepts: the Poincare conjecture, the shape of the universe, topology, and the nature of infinity are explored through an original, thought provoking piece of music theatre. Jorge Luis Borges' short story 'The Library of Babel' and the aviator Amelia Earhart’s attempt to circumnavigate the globe combine to communicate to audience key mathematical concepts of Poincare’s conjecture. The project builds on a 2008 EPSRC early development project (EP/G001650/1) and is led by an interdisciplinary team the19thstep consisting of composer Dorothy Ker, sculptor Kate Allen and mathematician Marcus du Sautoy. everything and nothing has been devised by Dorothy Ker and Kate Allen, is performed by percussionist Chris Brannick, mezzo soprano Lucy Stevens and sound designer Kelcey Swain. The UK tour targets arts-going audiences, from the Green Man Festival to the British Science Festival. Each performance is accompanied with a workshop led by Topologist Katie Steckles. Alongside the performances and workshops is a website, http://www.everythingandnothingproject.com/ The Public engagement evaluation and monitoring for the project are carried out by evaluator Bea Jefferson. The project is significant in its timely relation to contemporary mathematics and arts-science themes delivering an extensive programme of public engagement.