Multiscale analysis of pattern formation via intercellular signalling
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.
Section snippets
Formulation
In Collier et al. [6], the activity of a signalling protein, Delta, and its receptor, Notch, is studied. This interaction is known to be important in early animal development. The crucial aspect of the feedback loop considered is that high Delta expression in a cell downregulates Delta expression in its neighbours via the Notch receptor. This mechanism, known as lateral inhibition, is a fundamental cell-fate control mechanism [21], creating fine-grained patterns in developing tissue that
Model formulation
In this section, we employ a multiscale method to derive systematically a continuum model based upon the discrete system (1a), (1b) and capable of describing the dominant fine-grained (period-two) patterning phenomena examined in Section 2.2. Fig. 1 suggests that introducing appropriate spatial variation of one (or more) of the model parameters can result in patterning being induced in certain regions of the domain only, and this generalisation from a uniform choice of parameters will be
Simulations
We now present numerical solutions of (26) on the domain 0 < x < L in which we choose A(x) = 2x/L − 1 to illustrate how macroscale spatial variation in feedback strength dictates the emergence of fine-grained patterning in the domain. We compare these to corresponding numerical simulations of the full discrete system as well as those predicted by considering the local feedback strength, in which case a pitchfork bifurcation is located at x = L/2 (A = 0). Since we expect macroscale variation over the
Pattern formation in a two-dimensional array of cells
In the preceding section, we have thoroughly analysed the fine-grained pattern forming behaviour of a string of cells, deriving a homogenised model which faithfully represents such behaviour. We now consider the behaviour of a two-dimensional array of square cells. Collier et al. [6] showed that in two-dimensional arrays of hexagonal cells labelled (i, j), patterns of period three in i + j are possible; however, numerical simulations of the discrete system (1a), (1b) in square cells indicate that
Discussion
In this paper, we have employed a multiscale technique to construct a continuum model which is capable of describing the fine-grained patterning which arises in certain parameter regimes associated with signalling feedback strength within the Delta–Notch signalling model of Collier et al. [6]. While many authors have considered the derivation of continuous models from an underlying discrete systems (indeed, Roussel and Roussel [29] consider a similar signalling model to that analysed here
Acknowledgments
This research was undertaken at The University of Nottingham; the authors gratefully acknowledge funding from BBSRC and EPSRC for this work (BB/D008522/1). JRK also acknowledges the support of the Royal Society and Wolfson Foundation. All bifurcation diagrams were generated in XPPAUT v5.91.
References (37)
- et al.
Pattern formation by lateral inhibition with feedback: a math. Model of Delta–Notch intercellular signalling
J. Theor. Biol.
(1996) - et al.
A dual function of the notch gene in Drosophila sensillum development
Dev. Biol.
(1990) - et al.
Do morphogen gradients arise by diffusion?
Dev. Cell
(2002) - et al.
Stimulation of human epidermal differentiation by Delta–Notch signalling at the boundaries of stem-cell clusters
Curr. Biol.
(2000) - et al.
Reactivation of Delta–Notch signaling after injury: complementary expression patterns of ligand and receptor in dental pulp
Experim. Cell Res.
(1999) Waves and propagation failure in discrete space models with nonlinear coupling and feedback
Phys. D: Nonlin. Phenomena
(2002)- et al.
Mathematical modelling of juxtacrine cell signalling
Math. Biosci.
(1998) - et al.
Pattern-generating travelling waves in a discrete multicellular system with lateral inhibition
Phys. D: Nonlin. Phenomena
(2007) - et al.
Reaction–diffusion models of development with state-dependent chemical diffusion coefficients
Prog. Biophys. Molecular Biol.
(2004) On two types of moving front in quasilinear diffusion
Math. Biosci.
(1976)
Intra-membrane ligand diffusion and cell shape modulate juxtacrine patterning
J. Theor. Biol.
Positional information and the spatial pattern of cellular differentiation
J. Theor. Biol.
Notch signalling regulates stem cell numbers in vitro and in vivo
Nature
Delta–Notch signaling and lateral inhibition in zebrafish spinal cord development
BMC Dev. Biol
Poroelasticity equations derived from microstructure
J. Acous. Soc. Am.
Early neurogenesis in Drosophila melanogaster
Dev. Dros. Mel.
Production and auto-induction of transforming growth factor-α in human keratinocytes
Nature
Notch-mediated restoration of regenerative potential to aged muscle
Science
Cited by (21)
Coupling dynamics of 2D Notch-Delta signalling
2023, Mathematical BiosciencesCoordination of local and long range signaling modulates developmental patterning
2021, Journal of Theoretical BiologyCitation Excerpt :Consequently, while we use these gene names to identify model variables for convenience, the analyses and conclusions in this study are not limited to processes only involving these three signaling pathways. To investigate patterning dynamics close to the bifurcation point, we derive a continuum PDE model that explicitly incorporates both local and long range signaling, following a similar procedure as O’Dea and King (2011, 2012). In particular, we wish to understand how small changes in local or long range signaling can influence the type and extent of patterning across an array of cells.
Microstructural Influences on Growth and Transport in Biological Tissue-A Multiscale Description
2017, Modeling of Microscale Transport in Biological ProcessesPattern formation in discrete cell tissues under long range filopodia-based direct cell to cell contact
2016, Mathematical BiosciencesCitation Excerpt :Juxtacrine signalling was incorporated via a cell’s receptor activity depending on the ligand activity of its nearest neighbours, and the model was shown to reproduce the fine-grained patterning of a cell sheet. This framework has subsequently been adapted and extended in various directions, from more detailed analytical explorations to refined representation of the molecular interactions, or specific modelling of particular instances of pattern formation: as examples, we refer to [2,3,8,13,17–21,26,28,29,31,32]. On initial reading, the “juxtacrine” labelling of Delta–Notch interactions suggests strictly local communication, such that a cell can only signal to its nearest neighbours.
Diffusion-driven instabilities and emerging spatial patterns in patchy landscapes
2015, Ecological ComplexityCitation Excerpt :We implement habitat heterogeneity through patch attributes and movement bias. A series of papers explores pattern formation in epithelia where cell–cell interaction is dominated by nearest-neighbour (juxtacrine) signalling (Owen and Sherratt, 1998; Owen et al., 2000; Webb and Owen, 2004a; O’Dea and King, 2011, 2013; Wearing et al., 2000; Wearing and Sherratt, 2001). In these works, all cells have equal properties (i.e. there is no spatial variation), and interaction between neighbouring cells is non-linear.
Pattern selection by dynamical biochemical signals
2015, Biophysical JournalCitation Excerpt :Thus, a ligand-expressing cell drives lateral inhibition to its neighbors, i.e., it inhibits the expression of the ligand in its adjacent cells. When all cells interact with each other through lateral inhibition, spontaneous symmetry breaking (driving the periodic patterning, arising from small variability between cells) can occur (25,38–40). We also took into account that the protein ligand can impede Notch signaling by binding to the Notch receptor within the same cell, an action known as “cis-inhibition” (19,41–47).