Multiscale analysis of pattern formation via intercellular signalling

Abstract

Lateral inhibition, a juxtacrine signalling mechanism by which a cell adopting a particular fate inhibits neighbouring cells from doing likewise, has been shown to be a robust mechanism for the formation of fine-grained spatial patterns (in which adjacent cells in developing tissues diverge to achieve contrasting states of differentiation), provided that there is sufficiently strong feedback. The fine-grained nature of these patterns poses problems for analysis via traditional continuum methods since these require that significant variation takes place only over lengthscales much larger than an individual cell and such systems have therefore been investigated primarily using discrete methods. Here, however, we apply a multiscale method to derive systematically a continuum model from the discrete Delta–Notch signalling model of Collier et al. (J.R. Collier, N.A.M. Monk, P.K. Maini, J.H. Lewis, Pattern formation by lateral inhibition with feedback: a mathematical model of Delta–Notch intercellular signalling, J. Theor. Biol., 183, 1996, 429–446) under particular assumptions on the parameters, which we use to analyse the generation of fine-grained patterns. We show that, on the macroscale, the contact-dependent juxtacrine signalling interaction manifests itself as linear diffusion, motivating the use of reaction–diffusion-based models for such cell-signalling systems. We also analyse the travelling-wave behaviour of our system, obtaining good quantitative agreement with the discrete system.

Introduction

The development of spatial organisation is fundamental to the construction of multicellular organisms: starting from a homogeneous initial state, a collection of cells must have the ability to arrange themselves in a certain configuration to enable the creation of specialised body parts. This spatial organisation impacts upon cell differentiation and the determination of cell fate (i.e. the adoption of a particular programme of gene activation). Cell fates are regulated by a variety of mechanisms, typically mediated by the production and transport of certain signalling molecules that bind to specific sites on the cell membrane. In many developing tissues fine-grained patterning is generated, in which neighbouring cells diverge to achieve different cell fates. Lateral inhibition, a juxtacrine signalling mechanism (in which the signalling molecule is anchored in the cell membrane and acts only on neighbouring cells) is one process by which such patterns are generated and is well-documented in flies, fishes, worms and invertebrates, and has been observed in the developing nervous system in particular [6], [13], [2]. In all of these systems the transmembrane proteins Notch and Delta (or their homologues) have been identified as mediators of the interaction. Within the context of regenerative medicine, Delta–Notch signalling has been shown to regulate cell fate in stem-cell clusters, with high Delta expression preventing differentiation and improving cohesiveness [19], maintaining regenerative potential in muscle cells [7] and regulating stem cell numbers both in vitro and in vivo [1]. Many other ligand-receptor interactions and subsequent intracellular signalling cascades have been identified as cell fate determination mechanisms and subsequently studied in the mathematical biology literature; examples include the binding of cyclic AMP to Dictyostelium cells [30], [8], Transforming and Epidermal Growth Factor binding in keratinocytes [5], [9], [36] and Wnt binding in a variety of stem cell types [22], [20]. However, in this paper, we concentrate on the well-understood Delta–Notch juxtacrine signalling system, which provides an ideal model system for the type of analysis which we seek to pursue and, after Collier et al. [6], we neglect any intracellular detail.

Due to its importance in the understanding of tissue formation, the development of spatial patterning in response to biologically-relevant mechanisms has been widely studied. Inspired by the study of Turing [32], who showed that reaction and diffusion of chemicals can produce spatial patterning in chemical concentration that consequently determines cell fate, reaction–diffusion systems have been employed in modelling a variety of biological systems. For instance, Painter et al. [25] showed that patterned solutions of striking similarity to animal coat patterns can be generated by such a model. An alternative mechanism was proposed by Wolpert [37] in which the level of a morphogen (whose gradient is established via diffusion from a fixed source) determines cell fate; Lander et al. [17] have provided evidence that diffusive transport and ligand-receptor interaction is able to set up appropriate morphogen gradients. Most such studies employ a continuum approach (though it is noteworthy that Turing [32] exploited both discrete and continuous formulations); however, a number of relatively recent studies have employed discrete models to analyse signalling processes at the cell scale. Collier et al. [6] presented the first model of juxtacrine signalling, considering the activity of a protein, Delta, and its receptor, Notch, showing that lateral inhibition is able to generate fine-grained spatial patterns. The model was formulated in terms of ordinary differential equations (ODEs) representing Delta–Notch signalling activity in each cell; Webb and Owen [35] extended this model, considering the dynamics of ligand and free and bound receptors in systems of varying geometry (strings and square or hexagonal arrays), showing that lateral inhibition can produce patterns with a lengthscale of many cell diameters. In the limit of slow ligand-receptor binding, this model is equivalent to that of Collier et al. [6]. Webb and Owen [34] investigated the effects of cell polarity, demonstrating that coherent arrays of polarised cells (analogous to the polarity of bristles and hairs in Drosophila) can be produced.

The analysis of discrete models can be highly numerical in nature, and realistic simulations demand large numbers of cells; this is presumably in part why much research has concentrated until recently on continuous reaction–diffusion models. However, as well as exchanging matter (by diffusion and active transport) cells interact by exchanging signals via specific signalling mechanisms. Indeed, Plahte [26] and Plahte and Øyehaug [27] remark that it is unclear that diffusion is a meaningful approximation for these interactions, especially when considering juxtacrine signalling. Nevertheless, Roussel and Roussel [29] demonstrated that such a mechanism may be represented by a diffusion process within a continuum formulation, provided that the signalling-molecule concentration varies slowly in comparison to the length of a cell – this result, however, is not applicable in the current context in which neighbouring cells can exhibit very different behaviour. In this paper, we employ a multiscale technique to derive a continuum model of juxtacrine intercellular signalling (based on the Delta–Notch model of Collier et al. [6]) which is nevertheless able to capture fine-grained patterns with significant variation in signalling molecule concentration between adjacent cells. The resulting continuum model allows representation of the generation of microscale patterns in a tissue in response to macroscale variation in cell signalling (e.g. that induced by tissue-level chemical or physical stimulation). Our reduced model takes the form of a semilinear partial differential equation and is therefore amenable to well-established methods of both analytic and numerical analysis and we demonstrate that, on the macroscale, the interaction between adjacent cells does in fact manifest itself as linear diffusion, despite the short-range variation in signal concentration. We employ this model to investigate the development of fine-grained patterns in response to spatial variation of the strength of the nearest-neighbour feedback mechanism. Comparison with simulations of the discrete system shows that, in the appropriate parameter regimes, the continuum model faithfully reproduces the behaviour of the underlying discrete system. Quantitative comparison between the continuum and discrete models is made by analysing their travelling-wave behaviour, in particular.

The method of multiple-scale expansions for partial differential equations is well-developed (see for example, Kevorkian and Cole [16]) and widely used to derive models for a variety of physical and biological problems. Representative examples include Burridge and Keller [3], in which a two-scale homogenisation technique was employed to derive the equations of linear poroelasticity for a porous material, while, in a biological context, Goel et al. [12] have derived a diffusion-type equation for calcium dynamics within the cell cytoplasm, which was viewed as constituting the endoplasmic reticulum (ER) surrounded by cytosol: the homogenised diffusion coefficients were found to be dependent upon the underlying geometry of the ER, and an example calculation for a specific geometry was given. Such techniques have also been employed to derive continuum representations of collective movement of adherent cells by, for instance, Turner et al. [33] and Fozard et al. [11]. We remark that in many studies which exploit a homogenisation technique to derive a continuum model from an underlying discontinuous system, local periodicity is in effect imposed by the geometry of the physical system considered, such as the repeating ER/cytosol pattern in Goel et al. [12] or the individual material pore in Burridge and Keller [3]. In contrast, in the Delta–Notch signalling system considered here, periodicity is an emergent property of the nearest-neighbour interactions between identical cells rather than being enforced a priori. This periodicity can be exploited by judicious choice of variables to obviate the complications that arise in many homogenisation problems.

The structure of this paper is as follows. In Section 2, a short summary of the juxtacrine signalling model of Collier et al. [6] is given and its pattern-forming behaviour analysed: the patterning bifurcation structure in small populations of cells is demonstrated and the range of patterns exhibited by the model discussed. A continuum model for Delta–Notch signalling in a line of cells based upon this underlying discrete system is then derived in Section 3. In Section 4, numerical solutions to the continuum equations are presented and compared to simulations of the discrete system; pattern-generating travelling waves are considered, as well as competition between stable patterns. In Appendix A, an indication of the (surprisingly wide) range of parameter values for which our continuum limit applies is provided via comparison of the speed of a linearly-selected travelling wave in the continuous and discrete systems. In Section 5, a two-dimensional continuum model is derived by considering an array of square cells and comparisons to the underlying discrete system are again made. A discussion, together with suggestions for further work, is given in Section 6. The analysis is generalised somewhat in Appendix B, illustrating how the range of applicability of our continuum approach may be significantly extended (albeit in a rather artificial context), with a non-linear diffusion term resulting in the relevant limit.