Techniques for analysing pattern formation in populations of stem cells and their progeny

Background

To investigate how patterns of cell differentiation are related to underlying intra- and inter-cellular signalling pathways, we use a stochastic individual-based model to simulate pattern formation when stem cells and their progeny are cultured as a monolayer. We assume that the fate of an individual cell is regulated by the signals it receives from neighbouring cells via either diffusive or juxtacrine signalling. We analyse simulated patterns using two different spatial statistical measures that are suited to planar multicellular systems: pair correlation functions (PCFs) and quadrat histograms (QHs).

Results

With a diffusive signalling mechanism, pattern size (revealed by PCFs) is determined by both morphogen decay rate and a sensitivity parameter that determines the degree to which morphogen biases differentiation; high sensitivity and slow decay give rise to large-scale patterns. In contrast, with juxtacrine signalling, high sensitivity produces well-defined patterns over shorter lengthscales. QHs are simpler to compute than PCFs and allow us to distinguish between random differentiation at low sensitivities and patterned states generated at higher sensitivities.

Conclusions

PCFs and QHs together provide an effective means of characterising emergent patterns of differentiation in planar multicellular aggregates.

Theoretical models for coronary vascular biomechanics: progress & challenges

A key aim of the cardiac Physiome Project is to develop theoretical models to simulate the functional behaviour of the heart under physiological and pathophysiological conditions. Heart function is critically dependent on the delivery of an adequate blood supply to the myocardium via the coronary vasculature. Key to this critical function of the coronary vasculature is system dynamics that emerge via the interactions of the numerous constituent components at a range of spatial and temporal scales. Here, we focus on several components for which theoretical approaches can be applied, including vascular structure and mechanics, blood flow and mass transport, flow regulation, angiogenesis and vascular remodelling, and vascular cellular mechanics. For each component, we summarise the current state of the art in model development, and discuss areas requiring further research. We highlight the major challenges associated with integrating the component models to develop a computational tool that can ultimately be used to simulate the responses of the coronary vascular system to changing demands and to diseases and therapies.

A hybrid approach to multi-scale modelling of cancer

In this paper, we review multi-scale models of solid tumour growth and discuss a middle-out framework that tracks individual cells. By focusing on the cellular dynamics of a healthy colorectal crypt and its invasion by mutant, cancerous cells, we compare a cell-centre, a cell-vertex and a continuum model of cell proliferation and movement. All models reproduce the basic features of a healthy crypt: cells proliferate near the crypt base, they migrate upwards and are sloughed off near the top. The models are used to establish conditions under which mutant cells are able to colonize the crypt either by top-down or by bottom-up invasion. While the continuum model is quicker and easier to implement, it can be difficult to relate system parameters to measurable biophysical quantities. Conversely, the greater detail inherent in the multi-scale models means that experimentally derived parameters can be incorporated and, therefore, these models offer greater scope for understanding normal and diseased crypts, for testing and identifying new therapeutic targets and for predicting their impacts.

Predicting the onset of high-frequency self-excited oscillations in elastic-walled tubes

We present a theoretical description of flow-induced self-excited oscillations in the Starling resistor—a pre-stretched thin-walled elastic tube that is mounted on two rigid tubes and enclosed in a pressure chamber. Assuming that the flow through the elastic tube is driven by imposing the flow rate at the downstream end, we study the development of small-amplitude long-wavelength high-frequency oscillations, combining the results of two previous studies in which we analysed the fluid and solid mechanics of the problem in isolation. We derive a one-dimensional eigenvalue problem for the frequencies and mode shapes of the oscillations, and determine the slow growth or decay of the normal modes by considering the system’s energy budget. We compare the theoretical predictions for the mode shapes, frequencies and growth rates with the results of direct numerical simulations, based on the solution of the three-dimensional Navier–Stokes equations, coupled to the equations of shell theory, and find good agreement between the results. Our results provide the first asymptotic predictions for the onset of self-excited oscillations in three-dimensional collapsible tube flows.

A biomechanical model of agonist-initiated contraction in the asthmatic airway

This paper presents a modelling framework in which the local stress environment of airway smooth muscle (ASM) cells may be predicted and cellular responses to local stress may be investigated. We consider an elastic axisymmetric model of a layer of connective tissue and circumferential ASM fibres embedded in parenchymal tissue and model the active contractile force generated by ASM via a stress acting along the fibres. A constitutive law is proposed that accounts for active and passive material properties as well as the proportion of muscle to connective tissue. The model predicts significantly different contractile responses depending on the proportion of muscle to connective tissue in the remodelled airway. We find that radial and hoop-stress distributions in remodelled muscle layers are highly heterogenous with distinct regions of compression and tension. Such patterns of stress are likely to have important implications, from a mechano-transduction perspective, on contractility, short-term cytoskeletal adaptation and long-term airway remodelling in asthma.

A model of carbon dioxide dissolution and mineral carbonation kinetics

The kinetics of the dissolution of carbon dioxide in water and subsequent chemical reactions through to the formation of calcium carbonate, a system of reactions integral to carbon sequestration and anthropogenic ocean acidification, is mathematically modelled using the mass action law. This group of reactions is expressed as a system of five coupled nonlinear ordinary differential equations, with 14 independent parameters. The evolution of this system to equilibrium at 25 ° C and 1 atm, following an instantaneous injection of gaseous carbon dioxide, is simulated. An asymptotic analysis captures the leading-order behaviour of the system over six disparate time scales, yielding expressions for all species in each time scale. These approximations show excellent agreement with simulations of the full system, and give remarkably simple formulae for the equilibrium concentrations.

A mathematical model of intervillous blood flow in the human placentone

We present a mathematical model for maternal blood flow in a placental circulatory unit (a placentone), describing flow of maternal blood via Darcy's law and steady advective transport of a dissolved nutrient. The method of images and computational integration along streamlines are employed to find flow and solute concentration distributions, which are illustrated for a range of governing system parameters. The model shows how the calibre of the basal vessels can be a dominant determinant of the maternal blood flow rate through the placentone, given a driving pressure difference between the spiral arteries and decidual veins. The model supports the hypothesis that basal veins are located on the periphery of the placentone in order to optimise delivery of nutrients and suggests the existence of an optimal volume fraction of villous tissue.

An integrative computational model for intestinal tissue renewal

OBJECTIVES

The luminal surface of the gut is lined with a monolayer of epithelial cells that acts as a nutrient absorptive engine and protective barrier. To maintain its integrity and functionality, the epithelium is renewed every few days. Theoretical models are powerful tools that can be used to test hypotheses concerning the regulation of this renewal process, to investigate how its dysfunction can lead to loss of homeostasis and neoplasia, and to identify potential therapeutic interventions. Here we propose a new multiscale model for crypt dynamics that links phenomena occurring at the subcellular, cellular and tissue levels of organisation.

METHODS

At the subcellular level, deterministic models characterise molecular networks, such as cell-cycle control and Wnt signalling. The output of these models determines the behaviour of each epithelial cell in response to intra-, inter- and extracellular cues. The modular nature of the model enables us to easily modify individual assumptions and analyse their effects on the system as a whole.

RESULTS

We perform virtual microdissection and labelling-index experiments, evaluate the impact of various model extensions, obtain new insight into clonal expansion in the crypt, and compare our predictions with recent mitochondrial DNA mutation data.

CONCLUSIONS

We demonstrate that relaxing the assumption that stem-cell positions are fixed enables clonal expansion and niche succession to occur. We also predict that the presence of extracellular factors near the base of the crypt alone suffices to explain the observed spatial variation in nuclear beta-catenin levels along the crypt axis.

Intracellular flow in optic nerve axons: a mechanism for cell death in glaucoma

PURPOSE

In glaucoma, elevated intraocular pressure causes a progressive loss of retinal ganglion cells and results in optic neuropathy. The authors propose a potential mechanism for cell death, whereby elevated intraocular pressure causes fluid to permeate axonal membranes, creating a passive intracellular fluid flow within the axons. It is hypothesized that this intracellular flow locally depletes the adenosine triphosphate (ATP) concentration, disrupting axonal transport and leading to cell death.

METHODS

A mathematical model was developed that takes into account the biomechanical principles underpinning the proposed hypothesis, and was solved to determine the implications of the mechanism.

RESULTS

The model suggests that the raised intraocular pressures present in glaucoma are adequate to produce significant intracellular fluid flow. In the periphery of the optic nerve head, this flow may be sufficient to disrupt the diffusion of ATP and hence interrupt active axonal transport.

CONCLUSIONS

The mathematical model demonstrates that it is physically plausible that a passive intracellular fluid flow could significantly contribute to the pathophysiology of the retinal ganglion cell axon in glaucoma.

Three-dimensional elastohydrodynamics of a thin plate oscillating above a wall

We consider deflections of a thin rectangular elastic plate that is submerged within a Newtonian fluid. The plate is clamped along one edge and supported horizontally over a plane horizontal wall. We consider both external driving, where the clamped edge is vibrated vertically at high frequencies, and thermal driving, where the plate fluctuates under Brownian motion. In both cases, the amplitude of oscillation is assumed sufficiently small that the resulting flow has little convective inertia, although the oscillation frequency is sufficiently high to generate substantial unsteady inertia in the flow, a common scenario in many nano- and microdevices. We exploit the plate's thinness to develop an integral-equation representation for the three-dimensional flow (a so-called thin-plate theory) which offers considerable computational savings over a full boundary-integral formulation. Limiting cases of high oscillation frequencies and small wall-plate separation distances are studied separately, leading to further simplified descriptions for the hydrodynamics. We validate these reduced integral representations against full boundary-integral computations, and identify the parameter ranges over which these simplified formulations are valid. Addressing the full flow-structure interaction, we also examine the limits of simpler two-dimensional hydrodynamic models. We compare the responses of a narrow plate under two- and three-dimensional hydrodynamic loading, and report differences in the frequency response curves that occur when the plate operates in water, in contrast to the excellent agreement observed in air.

Shock formation and non-linear dispersion in a microvascular capillary network

Temporal and spatial fluctuations are a common feature of blood flow in microvascular networks. Among many possible causes, previous authors have suggested that the non-linear rheological properties of capillary blood flow (notably the Fåhraeus effect, the Fåhraeus-Lindqvist effect and the phase-separation effect at bifurcations) may be sufficient to generate temporal fluctuations even in very simple networks. We have simulated blood flow driven by a fixed pressure drop through a simple arcade network using coupled hyperbolic partial differential equations (PDEs) that incorporate well-established empirical descriptions of these rheological effects, accounting in particular for spatially varying haematocrit distributions; we solved the PDE system using a characteristic-based method. Our computations indicate that, under physiologically realistic conditions, there is a unique steady flow in an arcade network which is linearly stable and that plasma skimming suppresses the oscillatory decay of perturbations. In addition, we find that non-linear perturbations to haematocrit distributions can develop shocks via the Fåhraeus effect, providing a novel mechanism for non-linear dispersion in microvascular networks.

Elucidating the interactions between the adhesive and transcriptional functions of -catenin in normal and cancerous cells

Wnt signalling is involved in a wide range of physiological and pathological processes. The presence of an extracellular Wnt stimulus induces cytoplasmic stabilisation and nuclear translocation of beta-catenin, a protein that also plays an essential role in cadherin-mediated adhesion. Two main hypotheses have been proposed concerning the balance between beta-catenin's adhesive and transcriptional functions: either beta-catenin's fate is determined by competition between its binding partners, or Wnt induces folding of beta-catenin into a conformation allocated preferentially to transcription. The experimental data supporting each hypotheses remain inconclusive. In this paper we present a new mathematical model of the Wnt pathway that incorporates beta-catenin's dual function. We use this model to carry out a series of in silico experiments and compare the behaviour of systems governed by each hypothesis. Our analytical results and model simulations provide further insight into the current understanding of Wnt signalling and, in particular, reveal differences in the response of the two modes of interaction between adhesion and signalling in certain in silico settings. We also exploit our model to investigate the impact of the mutations most commonly observed in human colorectal cancer. Simulations show that the amount of functional APC required to maintain a normal phenotype increases with increasing strength of the Wnt signal, a result which illustrates that the environment can substantially influence both tumour initiation and phenotype.

T-cell motility in the early stages of the immune response modeled as a random walk amongst targets

The transport process by which a T cell makes high-frequency encounters with antigen-presenting cells following infection is an important element of adaptive immunity. Recent experimental work has allowed in vivo cell motility to be characterized in detail. On the basis of experimental data we develop a quantitative model for encounters between T cells and antigen-presenting cells. We model this as a transport-limited chemical reaction with the dynamics dependent on physical contact between randomly moving reactants. We use asymptotic methods to calculate a time distribution which characterizes the delay before a T cell is activated and use Monte Carlo simulations to verify the analysis. We find that the density of antigen-primed dendritic cells within the lymph node paracortex must be greater than 35 cells/mm3 for a T cell to have a more than 50% chance of encountering a dendritic cell within 24 h. This density is much larger than existing estimates based on calculations which neglect the transport process. We also use simulations to compare a T cell which re-orients isotropically with a T cell which turns according to an experimentally observed distribution and find that the effects of anisotropy on the solution are small.

Crypt dynamics and colorectal cancer: advances in mathematical modelling

Mathematical modelling forms a key component of systems biology, offering insights that complement and stimulate experimental studies. In this review, we illustrate the role of theoretical models in elucidating the mechanisms involved in normal intestinal crypt dynamics and colorectal cancer. We discuss a range of modelling approaches, including models that describe cell proliferation, migration, differentiation, crypt fission, genetic instability, APC inactivation and tumour heterogeneity. We focus on the model assumptions, limitations and applications, rather than on the technical details. We also present a new stochastic model for stem-cell dynamics, which predicts that, on average, APC inactivation occurs more quickly in the stem-cell pool in the absence of symmetric cell division. This suggests that natural niche succession may protect stem cells against malignant transformation in the gut. Finally, we explain how we aim to gain further understanding of the crypt system and of colorectal carcinogenesis with the aid of multiscale models that cover all levels of organization from the molecular to the whole organ.

Stochastic elastohydrodynamics of a microcantilever oscillating near a wall

We consider the thermally driven motion of a microcantilever in a fluid environment near a wall, a configuration characteristic of the atomic force microscope. A theoretical model is presented which accounts for hydrodynamic interactions between the cantilever and wall over a wide range of frequencies and which exploits the fluctuation-dissipation theorem to capture the Brownian dynamics of the coupled fluid-cantilever system. Model predictions are tested against experimental thermal spectra for a cantilever in air and water. The model shows how, in a liquid environment, the effects of non-delta-correlated Brownian forcing appear in the power spectrum, particularly at low frequencies. The model also predicts accurately changes in the spectrum in liquid arising through hydrodynamic wall effects, which we show are strongly mediated by the angle at which the cantilever is tilted relative to the wall.

Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory

This paper outlines the framework of a porous flow mixture theory for the mathematical modelling of in vitro tissue growth, and gives an application of this theory to an aspect of tissue engineering. The problem is formulated as a set of partial differential equations governing the space and time dependence of the amounts of each component of the tissue (phase), together with the physical stresses in each component. The theory requires constitutive relations to specify the material properties of each phase, and also requires relations to specify the stresses developed due to mechanical interactions, both within each phase and between different phases. An application of the theory is given to the study of the mobility and aggregation of a population of cells seeded into an artificial polymeric scaffold. Stability analysis techniques show that the interplay of the forces between the tissue constituents results in two different regimes: either the cells form aggregates or disperse through the scaffold.

Epithelial cell deformation during surfactant-mediated airway reopening: a theoretical model

A theoretical model is presented describing the reopening by an advancing air bubble of an initially liquid-filled collapsed airway lined with deformable epithelial cells. The model integrates descriptions of flow-structure interaction (accounting for nonlinear deformation of the airway wall and viscous resistance of the airway liquid flow), surfactant transport around the bubble tip (incorporating physicochemical parameters appropriate for Infasurf), and cell deformation (due to stretching of the airway wall and airway liquid flows). It is shown how the pressure required to drive a bubble into a flooded airway, peeling apart the wet airway walls, can be reduced substantially by surfactant, although the effectiveness of Infasurf is limited by slow adsorption at high concentrations. The model demonstrates how the addition of surfactant can lead to the spontaneous reopening of a collapsed airway, depending on the degree of initial airway collapse. The effective elastic modulus of the epithelial layer is shown to be a key determinant of the relative magnitude of strains generated by flow-induced shear stresses and by airway wall stretch. The model also shows how epithelial-layer compressibility can mediate strains arising from flow-induced normal stresses and stress gradients.

An Asymptotic Model of Unsteady Airway Reopening

We consider a simple physical model for the reopening of a collapsed lung airway involving the unsteady propagation of a long bubble of air, driven at a prescribed flow-rate, into a liquid-filled channel formed by two flexible membranes that are held under large longitudinal tension and are confined between two parallel rigid plates. This system is described theoretically using an asymptotic approximation, valid for uniformly small membrane slopes, which reduces to a fourth-order nonlinear evolution equation for the channel width ahead of the bubble tip, from which the time-evolution of the bubble pressure pb* and bubble speed may be determined. The model shows that there can be a substantial delay between the time at which the bubble starts to grow in volume and the time at which its tip starts to move. Under certain conditions, the start of the bubble’s motion is accompanied by a transient overshoot in pb*, as seen previously in experiment; the model predicts that the overshoot is greatest in narrow channels when the bubble is driven with a large volume flux. It is also shown how the threshold pressure for steady bubble propagation in wide channels has distinct contributions from the capillary pressure drop across the bubble tip and viscous dissipation in the channel ahead of the bubble.

Capillary-elastic instabilities of liquid-lined lung airways

To model the competition between capillary and elastic forces in controlling the shape of a small lung airway and its interior liquid lining, we compute the equilibrium configurations of a liquid-lined, externally pressurized, buckled elastic tube. We impose axial uniformity and assume that the liquid wets the tube wall with zero contact angle. Non-zero surface tension has a profound effect on the tube's quasi-steady inflation-deflation characteristics. At low liquid volumes, hysteresis arises through two distinct mechanisms, depending on the buckling wavenumber. Sufficient compression always leads to abrupt and irreversible collapse and flooding of the tube; flooding is promoted by increasing liquid volumes or surface tension. The model captures mechanisms whereby capillary-elastic instabilities can lead to airway closure.

Surfactant transport over airway liquid lining of nonuniform depth

Numerous effects (e.g., airway wall buckling, gravity, airway curvature, capillary instabilities) give rise to nonuniformities in the depth of the liquid lining of peripheral lung airways. The effects of such thickness variations on the unsteady spreading of a surfactant monolayer along an airway are explored theoretically here. Flow-induced film deformations are shown to have only a modest influence on spreading rates, motivating the use of a simplified model in which the liquid-lining depth is prescribed and the monolayer concentration satisfies a spatially inhomogeneous nonlinear diffusion equation. Two generic situations are considered: spreading along a continuous annular liquid lining of nonuniform depth, and spreading along a rivulet that wets the airway wall with zero contact angle. In both cases, transverse averaging at large times yields a one-dimensional approximation of axial spreading that is valid for the majority of the monolayer. However, a localized monolayer remains persistently two dimensional in a region at its leading edge having axial length scales comparable to the length scale of transverse depth variation. It is also shown how the transverse spreading of a monolayer may be arrested as it approaches a static contact line at the edge of a rivulet. Implications for Surfactant Replacement Therapy are discussed.

Draining Collars and Lenses in Liquid-Lined Vertical Tubes

The speed at which an annular liquid collar drains under gravity g in a vertical tube of radius a, when the tube has an otherwise thin viscous liquid lining on its interior, is determined by a balance between the collar's weight and viscous shear stresses confined to narrow regions in the neighborhood of the collar's effective contact lines. Whether a collar grows or shrinks in volume as it drains depends on the modified Bond number B=rho g a(2)/(sigmaepsilon), where rho is the fluid density, sigma is its surface tension, and epsilona is the thickness of the thin film immediately ahead of the collar. Asymptotic methods are used here to determine the following nonlinear stability criteria for an individual collar, valid in the limit of small epsilon. For 0<B<0.5960, all draining collars grow in volume and, in sufficiently long tubes, ultimately "snap off" to form stable lenses. For 0.5960<B<1.769, small collars may shrink but in long tubes sufficiently large collars will snap off. For 1.769<B<11.235, both stable collars and lenses may arise, although most collars will shrink. If B<11.235, all collars and lenses shrink in volume as they drain, so that any lens ultimately ruptures, unless stabilizing intermolecular forces allow the formation of a lamella supported by a macroscopic Plateau border. If surfactant immobilizes the liquid's free surface, these critical values of B are reduced by a factor of 2 but the distance a collar must travel before it snaps off is unchanged. Gravitationally driven snap off is therefore most likely to occur in long tubes with radii substantially less than the capillary lengthscale sigma/rhog)(1/2). Copyright 2000 Academic Press.

An asymptotic model of viscous flow limitation in a highly collapsed channel

A viscous flow through a long two-dimensional channel, one wall of which is formed by a finite-length membrane, experiences flow limitation when the channel is highly collapsed over a narrow region under high external pressure. Simple approximate relations between flow rate and pressure drop are obtained for this configuration by the use of matched asymptotic expansions. Weak inertial effects are also considered.