Oliver E. Jensen

Insoluble surfactant spreading on a thin viscous film: shock evolution and film rupture

Lubrication theory and similarity methods are used to determine the spreading rate of a localized monolayer of insoluble surfactant on the surface of a thin viscous film, in the limit of weak capillarity and weak surface diffusion. If the total mass of surfactant increases as t(alpha), then at early times, when spreading is driven predominantly by Marangoni forces, a planar (axisymmetric) region of surfactant is shown to spread as t(1 + alpha)/3 (t(1 + alpha)/4) . A shock exists at the leading edge of the monolayer; asymptotic methods are used to show that a wavetrain due to capillary forces exists ahead of the shock at small times, but that after a finite time it is swamped by diffusive effects. For alpha < 1/2 (alpha < 1), the diffusive lengthscale at the shock grows faster than the length of the monolayer, ultimately destroying the shock; subsequently, spreading is driven by diffusion, and proceeds as t1/2. The asymptotic results are shown to be good approximations of numerical solutions of the governing partial differential equations in the appropriate limits. Additional forces are also considered: weak vertical gravity can also destroy the shock in finite time, while effects usually neglected from lubrication theory are important only early in spreading. Experiments have shown that the severe thinning of the film behind the shock can cause it to rupture: the dryout process is modelled by introducing van der Waals forces.

Auxin regulates aquaporin function to facilitate lateral root emergence.

Aquaporins are membrane channels that facilitate water movement across cell membranes. In plants, aquaporins contribute to water relations. Here, we establish a new link between aquaporin-dependent tissue hydraulics and auxin-regulated root development in Arabidopsis thaliana. We report that most aquaporin genes are repressed during lateral root formation and by exogenous auxin treatment. Auxin reduces root hydraulic conductivity both at the cell and whole-organ levels. The highly expressed aquaporin PIP2;1 is progressively excluded from the site of the auxin response maximum in lateral root primordia (LRP) whilst being maintained at their base and underlying vascular tissues. Modelling predicts that the positive and negative perturbations of PIP2;1 expression alter water flow into LRP, thereby slowing lateral root emergence (LRE). Consistent with this mechanism, pip2;1 mutants and PIP2;1-overexpressing lines exhibit delayed LRE. We conclude that auxin promotes LRE by regulating the spatial and temporal distribution of aquaporin-dependent root tissue water transport.

The spreading of heat or soluble surfactant along a thin liquid film

The spreading of a localized distribution of surfactant on a thin viscous film is considered, in the situation in which the surfactant is soluble in the bulk layer and the boundary beneath the fluid is impermeable to surfactant. The surfactant distribution is controlled by advection and diffusion both at the surface of the film, where the surfactant forms a monolayer, and in the bulk. The bulk and surface surfactant concentrations are related by linearized sorption kinetics. The surfactant diffuses rapidly across the thin fluid layer, and lubrication theory is used to derive evolution equations for the film height and the surface and cross‐sectionally averaged bulk surfactant concentrations. A special case of the governing equations describes the Marangoni flow induced by a locally hot region of the layer. It is shown that in comparison to the spreading of insoluble surfactant, transient desorption of surfactant from the monolayer to the bulk causes the spreading rate to diminish, although once the bulk and surface concentrations are locally in equilibrium, film deformations are more severe, with a sharp pulse in the film height created just upstream of the leading edge of the surfactant distribution.

The motion of a viscous drop through a cylindrical tube

Liquid of viscosity mu moves slowly through a cylindrical tube of radius R under the action of a pressure gradient. An immiscible force-free drop having viscosity lambdamu almost fills the tube; surface tension between the liquids is gamma. The drop moves relative to the tube walls with steady velocity U, so that both the capillary number Ca = muU/gamma and the Reynolds number are small. A thin film of uniform thickness epsilonR is formed between the drop and the wall. It is shown that Brethertons (1961) scaling epsilon proportional to Ca-2/3 is appropriate for all values of lambda, but with a coefficient of order unity that depends weakly on both lambda and Ca. The coefficient is determined using lubrication theory for the thin film coupled to a novel two-dimensional boundary-integral representation of the internal flow. It is found that as lambda increases from zero, the film thickness increases by a factor 4(2/3) to a plateau value when Ca-1/3 much less than lambda much less than Ca-2/3 and then falls by a factor 2(2/3) as lambda --> infinity. The multi-region asymptotic structure of the flow is also discussed.

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.

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.

Crypt dynamics and colorectal cancer: advances in mathematical modelling

Abstract.   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.

Growth-induced hormone dilution can explain the dynamics of plant root cell elongation

In the elongation zone of the Arabidopsis thaliana plant root, cells undergo rapid elongation, increasing their length by ∼10-fold over 5 h while maintaining a constant radius. Although progress is being made in understanding how this growth is regulated, little consideration has been given as to how cell elongation affects the distribution of the key regulating hormones. Using a multiscale mathematical model and measurements of growth dynamics, we investigate the distribution of the hormone gibberellin in the root elongation zone. The model quantifies how rapid cell expansion causes gibberellin to dilute, creating a significant gradient in gibberellin levels. By incorporating the gibberellin signaling network, we simulate how gibberellin dilution affects the downstream components, including the growth-repressing DELLA proteins. We predict a gradient in DELLA that provides an explanation of the reduction in growth exhibited as cells move toward the end of the elongation zone. These results are validated at the molecular level by comparing predicted mRNA levels with transcriptomic data. To explore the dynamics further, we simulate perturbed systems in which gibberellin levels are reduced, considering both genetically modified and chemically treated roots. By modeling these cases, we predict how these perturbations affect gibberellin and DELLA levels and thereby provide insight into their altered growth dynamics.

High-frequency self-excited oscillations in a collapsible-channel flow

High-Reynolds-number asymptotics and numerical simulations are used to describe two-dimensional, unsteady, pressure-driven flow in a finite-length channel, one wall of which contains a section of membrane under longitudinal tension. Asymptotic predictions of stability boundaries for small-amplitude, high-frequency, self-excited oscillations are derived in the limit of large membrane tension. The oscillations are closely related to normal modes of the system, which have a frequency set by a balance between membrane tension and the inertia of the fluid in the entire channel. Oscillations can grow by extracting kinetic energy from the mean Poiseuille flow faster than it is lost to viscous dissipation. Direct numerical simulations, based on a fully coupled finite-element discretization of the equations of large-displacement elasticity and the Navier–Stokes equations, support the predicted stability boundaries, and are used to explore larger-amplitude oscillations at lower tensions. These are characterized by vigorous axial sloshing motions superimposed on the mean flow, with transient secondary instabilities being generated both upstream and downstream of the collapsible segment.

Chaotic Oscillations in a Simple Collapsible-Tube Model

A steady flow through a segment of externally pressurized, collapsible tube can become unstable to a wide variety of self-excited oscillations of the internal flow and tube walls. A simple, one-dimensional model of the conventional laboratory apparatus, which has been shown previously to predict steady flows and multiple modes of oscillation, is investigated numerically here. Large amplitude oscillations are shown to have a relaxation structure, and the nonlinear interaction between different modes is shown to give rise to quasiperiodic and apparently aperiodic behavior. These predictions are shown to compare favorably with experimental observations.