Benjamin J. Binder

Neural crest regionalisation for enteric nervous system formation: Implications for Hirschsprung's disease and stem cell therapy

Midbrain, hindbrain and vagal neural crest (NC) produced abundant enteric nervous system (ENS) in co-grafted aneural hindgut and midgut, using chick-quail chorio-allantoic membrane grafts, forming complete myenteric and submucosal plexuses. This ability dropped suddenly in cervical and thoracic NC levels, furnishing an incomplete ENS in one or both plexuses. Typically, one plexus was favoured over the other. This deficiency was not caused by lower initial trunk NC number, yet overloading the initial number decreased the deficiency. No qualitative difference in neuronal and glial differentiation between cranial and trunk levels was observed. All levels formed HuC/D+ve, NOS+ve, ChAT+ve, and TH-ve enteric neurons with SoxE+ve, GFAP+ve, and BFABP+ve glial cells. We mathematically modelled a proliferative difference between NC populations, with a plexus preference hierarchy, in the context of intestinal growth. High proliferation achieved an outcome similar to cranial NC, while low proliferation described the trunk NC outcome of incomplete primary plexus and even more deficient secondary plexus. We conclude that cranial NC, relative to trunk NC, has a positionally-determined proliferation advantage favouring ENS formation. This has important implications for proposed NC stem cell therapy for Hirschsprungs disease, since such cells may need to be optimised for positional identity.

Forced solitary waves and fronts past submerged obstacles

Herein, an efficient numerical method is presented to describe the flow of a liquid in an open channel with various types of bottom configurations. The method is developed for steady two-dimensional potential free surface flows. The resulting nonlinear problem is solved numerically by boundary integral equation methods. In addition weakly nonlinear solutions are derived. New solutions which complement those of Dias and Vanden-Broeck [J. Fluid Mech. 59, 93-102 (2004)] are presented. Furthermore some solutions for channel flows past dips in the bottom are discussed.

Experimental and Modelling Investigation of Monolayer Development with Clustering

Standard differential equation-based models of collective cell behaviour, such as the logistic growth model, invoke a mean-field assumption which is equivalent to assuming that individuals within the population interact with each other in proportion to the average population density. Implementing such assumptions implies that the dynamics of the system are unaffected by spatial structure, such as the formation of patches or clusters within the population. Recent theoretical developments have introduced a class of models, known as moment dynamics models, which aim to account for the dynamics of individuals, pairs of individuals, triplets of individuals, and so on. Such models enable us to describe the dynamics of populations with clustering, however, little progress has been made with regard to applying moment dynamics models to experimental data. Here, we report new experimental results describing the formation of a monolayer of cells using two different cell types: 3T3 fibroblast cells and MDA MB 231 breast cancer cells. Our analysis indicates that the 3T3 fibroblast cells are relatively motile and we observe that the 3T3 fibroblast monolayer forms without clustering. Alternatively, the MDA MB 231 cells are less motile and we observe that the MDA MB 231 monolayer formation is associated with significant clustering. We calibrate a moment dynamics model and a standard mean-field model to both data sets. Our results indicate that the mean-field and moment dynamics models provide similar descriptions of the 3T3 fibroblast monolayer formation whereas these two models give very different predictions for the MDA MD 231 monolayer formation. These outcomes indicate that standard mean-field models of collective cell behaviour are not always appropriate and that care ought to be exercised when implementing such a model.

Distinguishing between mechanisms of cell aggregation using pair-correlation functions

Many cell types form clumps or aggregates when cultured in vitro through a variety of mechanisms including rapid cell proliferation, chemotaxis, or direct cell-to-cell contact. In this paper we develop an agent-based model to explore the formation of aggregates in cultures where cells are initially distributed uniformly, at random, on a two-dimensional substrate. Our model includes unbiased random cell motion, together with two mechanisms which can produce cell aggregates: (i) rapid cell proliferation and (ii) a biased cell motility mechanism where cells can sense other cells within a finite range, and will tend to move towards areas with higher numbers of cells. We then introduce a pair-correlation function which allows us to quantify aspects of the spatial patterns produced by our agent-based model. In particular, these pair-correlation functions are able to detect differences between domains populated uniformly at random (i.e. at the exclusion complete spatial randomness (ECSR) state) and those where the proliferation and biased motion rules have been employed - even when such differences are not obvious to the naked eye. The pair-correlation function can also detect the emergence of a characteristic inter-aggregate distance which occurs when the biased motion mechanism is dominant, and is not observed when cell proliferation is the main mechanism of aggregate formation. This suggests that applying the pair-correlation function to experimental images of cell aggregates may provide information about the mechanism associated with observed aggregates. As a proof of concept, we perform such analysis for images of cancer cell aggregates, which are known to be associated with rapid proliferation. The results of our analysis are consistent with the predictions of the proliferation-based simulations, which supports the potential usefulness of pair correlation functions for providing insight into the mechanisms of aggregate formation.

Exclusion processes on a growing domain

A discrete model provides a useful framework for experimentalists to understand the interactions between growing tissues and other biological mechanisms. A cellular automata (CA) model with domain growth, cell motility and cell proliferation, based on cellular exclusion processes, is developed here. Average densities can be defined from the CA model and a continuum representation can be determined. The domain growth mechanism in the CA model gives rise to a Fokker-Planck equation in the corresponding continuum model, with a diffusive and a convective term. Deterministic continuum models derived from conservation laws do not include this diffusive term. The new diffusive term arises because of the stochasticity inherited from the CA mechanism for domain growth. We extend the models to multiple species and investigate the influence of the flux terms arising from the exclusion processes. The averaged CA agent densities are well approximated by the solution of nonlinear advection-diffusion equations, provided that the relative size of the proliferation processes to the diffusion processes is sufficiently small. This dual approach provides an understanding of the microscopic and macroscopic scales in a developmental process.

Cell lineage tracing in the developing enteric nervous system: superstars revealed by experiment and simulation

Cell lineage tracing is a powerful tool for understanding how proliferation and differentiation of individual cells contribute to population behaviour. In the developing enteric nervous system (ENS), enteric neural crest (ENC) cells move and undergo massive population expansion by cell division within self-growing mesenchymal tissue. We show that single ENC cells labelled to follow clonality in the intestine reveal extraordinary and unpredictable variation in number and position of descendant cells, even though ENS development is highly predictable at the population level. We use an agent-based model to simulate ENC colonization and obtain agent lineage tracing data, which we analyse using econometric data analysis tools. In all realizations, a small proportion of identical initial agents accounts for a substantial proportion of the total final agent population. We term these individuals superstars. Their existence is consistent across individual realizations and is robust to changes in model parameters. This inequality of outcome is amplified at elevated proliferation rate. The experiments and model suggest that stochastic competition for resources is an important concept when understanding biological processes which feature high levels of cell proliferation. The results have implications for cell-fate processes in the ENS.

Assessing the role of spatial correlations during collective cell spreading.

Spreading cell fronts are essential features of development, repair and disease processes. Many mathematical models used to describe the motion of cell fronts, such as Fishers equation, invoke a mean–field assumption which implies that there is no spatial structure, such as cell clustering, present. Here, we examine the presence of spatial structure using a combination of in vitro circular barrier assays, discrete random walk simulations and pair correlation functions. In particular, we analyse discrete simulation data using pair correlation functions to show that spatial structure can form in a spreading population of cells either through sufficiently strong cell–to–cell adhesion or sufficiently rapid cell proliferation. We analyse images from a circular barrier assay describing the spreading of a population of MM127 melanoma cells using the same pair correlation functions. Our results indicate that the spreading melanoma cell populations remain very close to spatially uniform, suggesting that the strength of cell–to–cell adhesion and the rate of cell proliferation are both sufficiently small so as not to induce any spatial patterning in the spreading populations.

Free surface flows past surfboards and sluice gates

Free surface potential flows past surfboards and sluice gates are considered. The problem is solved numerically by boundary integral equation methods. In addition weakly nonlinear solutions are presented. It is shown among the six possible types of steady flows, only three exist. The physical relevance of these solutions is discussed in terms of the radiation condition (which requires that there is no energy coming from infinity). In particular, it is shown there are no steady subcritical flows which satisfy the radiation condition. Similarly there are no solutions for the flow under a sluice gate.

Free surface flow past topography : a beyond-all-orders approach

The problem of steady subcritical free surface flow past a submerged inclined step is considered. The asymptotic limit of small Froude number is treated, with particular emphasis on the e!ect that changing the angle of the step face has on the surface waves. As demonstrated by Chapman & Vanden-Broeck, (2006) Exponential asymptotics and gravity waves. J. Fluid Mech. 567, 299‐326, the divergence of a power series expansion in powers of the square of the Froude number is caused by singularities in the analytic continuation of the free surface; for an inclined step, these singularities may correspond to either the corners or stagnation points of the step, or both, depending on the angle of inclination. Stokes lines emanate from these singularities, and exponentially small waves are switched on at the point the Stokes lines intersect with the free surface. Our results suggest that for a certain range of step angles, two wavetrains are switched on, but the exponentially subdominant one is switched on first, leading to an intermediate wavetrain not previously noted. We extend these ideas to the problem of flow over a submerged bump or trench, again with inclined sides. This time there may be two, three or four active Stokes lines, depending on the inclination angles. We demonstrate how to construct a base topography such that wave contributions from separate Stokes lines are of equal magnitude but opposite phase, thus cancelling out. Our asymptotic results are complemented by numerical solutions to the fully nonlinear equations.

Spectral analysis of pair-correlation bandwidth: application to cell biology images

Images from cell biology experiments often indicate the presence of cell clustering, which can provide insight into the mechanisms driving the collective cell behaviour. Pair-correlation functions provide quantitative information about the presence, or absence, of clustering in a spatial distribution of cells. This is because the pair-correlation function describes the ratio of the abundance of pairs of cells, separated by a particular distance, relative to a randomly distributed reference population. Pair-correlation functions are often presented as a kernel density estimate where the frequency of pairs of objects are grouped using a particular bandwidth (or bin width), Δ>0. The choice of bandwidth has a dramatic impact: choosing Δ too large produces a pair-correlation function that contains insufficient information, whereas choosing Δ too small produces a pair-correlation signal dominated by fluctuations. Presently, there is little guidance available regarding how to make an objective choice of Δ. We present a new technique to choose Δ by analysing the power spectrum of the discrete Fourier transform of the pair-correlation function. Using synthetic simulation data, we confirm that our approach allows us to objectively choose Δ such that the appropriately binned pair-correlation function captures known features in uniform and clustered synthetic images. We also apply our technique to images from two different cell biology assays. The first assay corresponds to an approximately uniform distribution of cells, while the second assay involves a time series of images of a cell population which forms aggregates over time. The appropriately binned pair-correlation function allows us to make quantitative inferences about the average aggregate size, as well as quantifying how the average aggregate size changes with time.