Mathematical modelling of juxtacrine patterning

Spatial pattern formation is one of the key issues in developmental biology. Some patterns arising in early development have a very small spatial scale and a natural explanation is that they arise by direct cell—cell signalling in epithelia. This necessitates the use of a spatially discrete model, in contrast to the continuum-based approach of the widely studied Turing and mechanochemical models. In this work, we consider the pattern-forming potential of a model for juxtacrine communication, in which signalling molecules anchored in the cell membrane bind to and activate receptors on the surface of immediately neighbouring cells. The key assumption is that ligand and receptor production are both up-regulated by binding. By linear analysis, we show that conditions for pattern formation are dependent on the feedback functions of the model. We investigate the form of the pattern: specifically, we look at how the range of unstable wavenumbers varies with the parameter regime and find an estimate for the wavenumber associated with the fastest growing mode. A previous juxtacrine model for Delta-Notch signalling studied by Collier et al. (1996, J. Theor. Biol. 183, 429–446) only gives rise to patterning with a length scale of one or two cells, consistent with the fine-grained patterns seen in a number of developmental processes. However, there is evidence of longer range patterns in early development of the fruit fly Drosophila. The analysis we carry out predicts that patterns longer than one or two cell lengths are possible with our positive feedback mechanism, and numerical simulations confirm this. Our work shows that juxtacrine signalling provides a novel and robust mechanism for the generation of spatial patterns.     

Continuum limits of pattern formation in hexagonal-cell monolayers

Intercellular signalling is key in determining cell fate. In closely packed tissues such as epithelia, juxtacrine signalling is thought to be a mechanism for the generation of fine-grained spatial patterns in cell differentiation commonly observed in early development. Theoretical studies of such signalling processes have shown that negative feedback between receptor activation and ligand production is a robust mechanism for fine-grained pattern generation and that cell shape is an important factor in the resulting pattern type. It has previously been assumed that such patterns can be analysed only with discrete models since significant variation occurs over a lengthscale concomitant with an individual cell; however, considering a generic juxtacrine signalling model in square cells, in O’Dea and King (Math Biosci 231(2):172–185 2011), a systematic method for the derivation of a continuum model capturing such phenomena due to variations in a model parameter associated with signalling feedback strength was presented. Here, we extend this work to derive continuum models of the more complex fine-grained patterning in hexagonal cells, constructing individual models for the generation of patterns from the homogeneous state and for the transition between patterning modes. In addition, by considering patterning behaviour under the influence of simultaneous variation of feedback parameters, we construct a more general continuum representation, capturing the emergence of the patterning bifurcation structure. Comparison with the steady-state and dynamic behaviour of the underlying discrete system is made; in particular, we consider pattern-generating travelling waves and the competition between various stable patterning modes, through which we highlight an important deficiency in the ability of continuum representations to accommodate certain dynamics associated with discrete systems.     

Mutual inactivation of Notch receptors and ligands facilitates developmental patterning

Developmental patterning requires juxtacrine signaling in order to tightly coordinate the fates of neighboring cells. Recent work has shown that Notch and Delta, the canonical metazoan juxtacrine signaling receptor and ligand, mutually inactivate each other in the same cell. This cis-interaction generates mutually exclusive sending and receiving states in individual cells. It generally remains unclear, however, how this mutual inactivation and the resulting switching behavior can impact developmental patterning circuits. Here we address this question using mathematical modeling in the context of two canonical pattern formation processes: boundary formation and lateral inhibition. For boundary formation, in a model motivated by Drosophila wing vein patterning, we find that mutual inactivation allows sharp boundary formation across a broader range of parameters than models lacking mutual inactivation. This model with mutual inactivation also exhibits robustness to correlated gene expression perturbations. For lateral inhibition, we find that mutual inactivation speeds up patterning dynamics, relieves the need for cooperative regulatory interactions, and expands the range of parameter values that permit pattern formation, compared to canonical models. Furthermore, mutual inactivation enables a simple lateral inhibition circuit architecture which requires only a single downstream regulatory step. Both model systems show how mutual inactivation can facilitate robust fine-grained patterning processes that would be difficult to implement without it, by encoding a difference-promoting feedback within the signaling system itself. Together, these results provide a framework for analysis of more complex Notch-dependent developmental systems.     

Intra-membrane ligand diffusion and cell shape modulate juxtacrine patterning

A key problem in developmental biology is how pattern and planar polarity are transmitted in epithelial structures. Examples include Drosophila neuronal differentiation, ommatidia formation in the compound eye, and wing hair polarization. A key component for the generation of such patterns is direct cell-cell signalling by transmembrane ligands, called juxtacrine signalling. Previous models for this mode of communication have considered homogeneous distributions in the cell membrane, and the role of polarity has been largely ignored. In this paper we determine the role of inhomogeneous protein and receptor distributions in juxtacrine signalling. We explicitly include individual membrane segments, diffusive transport of proteins and receptors between these segments, and production terms with a combination of local and global responses to ligand binding. Our analysis shows that intra-membrane ligand transport is vital for the generation of long wavelength patterns. Moreover, with no ligand transport, there is no pattern formation for lateral induction, a process in which receptor activation up-regulates ligand production. Biased production of ligand also modulates patterning bifurcations and predicted wavelengths. In addition, biased ligand and receptor trafficking can lead to regular polarity across a lattice, in which each cell has the same orientation-directly analogous to patterns of hairs in the Drosophila wing. We confirm the trends in pattern wavelengths previously observed for patterns with cellular homogeneity-lateral inhibition tends to give short-range patterns, while lateral induction can give patterns with much longer wavelengths. Moreover, the original model can be recovered if intra-membrane bound receptor diffusion is included and rapid equilibriation between the sides is considered. Finally, we consider the role of irregular cell shapes and waves in such networks, including wave propagation past clones of non-signalling cells.     

Lateral induction by juxtacrine signaling is a new mechanism for pattern formation

Many signaling molecules in epithelia are now known to function in a membrane-bound form, binding to receptors on immediately neighbouring cells. This "juxtacrine" mode of communication has been well studied in the case of lateral inhibition, where ligand binding at the cell surface downregulates ligand and receptor expression, and is known to generate spatial patterns with a wavelength of exactly two cells. However, recent evidence shows that a number of juxtacrine signals can lead to the opposite phenomenon of lateral induction. Here, we use mathematical modeling to show that such positive feedback, in combination with juxtacrine communication, provides a novel mechanism for the generation of spatial patterns, with wavelengths that vary with parameters and can be many cell lengths.     

Capturing the Dynamics of a Hybrid Multiscale Cancer Model with a Continuum Model

Cancer is a complex disease involving processes at spatial scales from subcellular, like cell signalling, to tissue scale, such as vascular network formation. A number of multiscale models have been developed to study the dynamics that emerge from the coupling between the intracellular, cellular and tissue scales. Here, we develop a continuum partial differential equation model to capture the dynamics of a particular multiscale model (a hybrid cellular automaton with discrete cells, diffusible factors and an explicit vascular network). The purpose is to test under which circumstances such a continuum model gives equivalent predictions to the original multiscale model, in the knowledge that the system details are known, and differences in model results can be explained in terms of model features (rather than unknown experimental confounding factors). The continuum model qualitatively replicates the dynamics from the multiscale model, with certain discrepancies observed owing to the differences in the modelling of certain processes. The continuum model admits travelling wave solutions for normal tissue growth and tumour invasion, with similar behaviour observed in the multiscale model. However, the continuum model enables us to analyse the spatially homogeneous steady states of the system, and hence to analyse these waves in more detail. We show that the tumour microenvironmental effects from the multiscale model mean that tumour invasion exhibits a so-called pushed wave when the carrying capacity for tumour cell proliferation is less than the total cell density at the tumour wave front. These pushed waves of tumour invasion propagate by triggering apoptosis of normal cells at the wave front. Otherwise, numerical evidence suggests that the wave speed can be predicted from linear analysis about the normal tissue steady state.     

Oscillations and patterns in spatially discrete models for developmental intercellular signalling

We extend previous models for nearest neighbour ligand-receptor binding to include both lateral induction and inhibition of ligand and receptor production, and different geometries (strings of cells and hexagonal arrays, in addition to square arrays). We demonstrate the possibility of lateral inhibition giving patterns with a characteristic length scale of many cell diameters, when receptor production is included. In contrast, lateral induction combined with inhibition of receptor synthesis cannot give rise to a patterning instability under any circumstances. Interesting new dynamics include the analytical prediction and consequent numerical observation of spatiotemporal oscillations, this depends crucially on the production terms and on the relationship between the decay rates of ligand and free receptor. Our approach allows for a detailed comparison with the model for Delta-Notch interactions of Collier et al. [4], and we find that a formal reduction may be made only when the ligand receptor binding kinetics are very slow. Without such very slow receptor kinetics, spatial pattern formation via lateral inhibition in hexagonal cellular arrays requires significant activation of receptor production, a feature that is not apparent from previous analyses.Send offprint requests to:Markus R. Owen     

Delta-Notch signalling controls commitment to a secretory fate in the zebrafish intestine

The transparency of the juvenile zebrafish and its genetic advantages make it an attractive model for study of cell turnover in the gut. BrdU labelling shows that the gut epithelium is renewed in essentially the same way as in mammals: the villi are lined with non-dividing differentiated cells, while cell division is confined to the intervillus pockets. New cells produced in the pockets take about 4 days to migrate out to the tips of the villi, where they die. We have generated monoclonal antibodies to identify the absorptive and secretory cells in the epithelium, and we have used these antibodies to examine the part that Delta-Notch signalling plays in producing the diversity of intestinal cell types. Several Notch receptors and ligands are expressed in the gut. In particular, the Notch ligand DeltaD (Delta1 in the mouse) is expressed in cells of the secretory lineage. In an aei mutant, where DeltaD is defective, secretory cells are overproduced. In mind bomb (mib), where all Delta-Notch signalling is believed to be blocked, almost all the cells in the 3-day gut epithelium adopt a secretory character. Thus, secretory differentiation appears to be the default in the absence of Notch activation, and lateral inhibition mediated by Delta-Notch signalling is required to generate a balanced mixture of absorptive and secretory cells. These findings demonstrate the central role of Notch signalling in the gut stem-cell system and establish the zebrafish as a model for study of the mechanisms controlling renewal of gut epithelium.     

Restricted-range gradients and travelling fronts in a model of juxtacrine cell relay

Patterning events in development often depend on the transmission over a range of several cell diameters of signals emanating from a localized source. Experimental studies of such long-range signalling by members of the TGF-β family of growth factors suggests that a cell-relay mechanism in which cells signal only with their immediate neighbours (i.e., juxtacrine signalling) may be operating in some tissues. Here, this possibility is investigated through the analysis of a model of juxtacrine signalling. Depending on the strength of the signal relay between cells, a localized signal source can generate either stable gradients or travelling fronts of cell activation. Both of these behaviors could in principle be involved in the long-range transmission of signals and patterning of cell fates by cell relays. There are significant and surprising differences between the gradients generated by the mechanism studied here, and those generated by the diffusion of a morphogen. In particular, there is an upper limit on the distance over which any given level of cell activation can be attained in a relay-mediated gradient, irrespective of the strength of signal source.     

Effects of Growth and Mutation on Pattern Formation in Tissues

In many developing tissues, neighboring cells enter different developmental pathways, resulting in a fine-grained pattern of different cell states. The most common mechanism that generates such patterns is lateral inhibition, for example through Delta-Notch coupling. In this work, we simulate growth of tissues consisting of a hexagonal arrangement of cells laterally inhibiting their neighbors. We find that tissue growth by cell division and cell migration tends to produce ordered patterns, whereas lateral growth leads to disordered, patchy patterns. Ordered patterns are very robust to mutations (gene silencing or activation) in single cells. In contrast, mutation in a cell of a disordered tissue can produce a larger and more widespread perturbation of the pattern. In tissues where ordered and disordered patches coexist, the perturbations spread mostly at boundaries between patches. If cell division occurs on time scales faster than the degradation time, disordered patches will appear. Our work suggests that careful experimental characterization of the disorder in tissues could pinpoint where and how the tissue is susceptible to large-scale damage even from single cell mutations.     

Diffusion-driven instabilities and emerging spatial patterns in patchy landscapes

Spatial variation in population densities across a landscape is a feature of many ecological systems, from self-organised patterns on mussel beds to spatially restricted insect outbreaks. It occurs as a result of environmental variation in abiotic factors and/or biotic factors structuring the spatial distribution of populations. However the ways in which abiotic and biotic factors interact to determine the existence and nature of spatial patterns in population density remain poorly understood. Here we present a new approach to studying this question by analysing a predator–prey patch-model in a heterogenous landscape. We use analytical and numerical methods originally developed for studying nearest-neighbour (juxtacrine) signalling in epithelia to explore whether and under which conditions patterns emerge. We find that abiotic and biotic factors interact to promote pattern formation. In fact, we find a rich and highly complex array of coexisting stable patterns, located within an enormous number of unstable patterns. Our simulation results indicate that many of the stable patterns have appreciable basins of attraction, making them significant in applications. We are able to identify mechanisms for these patterns based on the classical ideas of long-range inhibition and short-range activation, whereby landscape heterogeneity can modulate the spatial scales at which these processes operate to structure the populations.     

A one-dimensional model of cell diffusion and aggregation,incorporating volume filling and cell-to-cell adhesion

We develop and analyse a discrete model of cell motility in one dimension which incorporates the effects of volume filling and cell-to-cell adhesion. The formal continuum limit of the model is a nonlinear diffusion equation with a diffusivity which can become negative if the adhesion coefficient is sufficiently large. This appears to be related to the presence of spatial oscillations and the development of plateaus (pattern formation) in numerical solutions of the discrete model. A combination of stability analysis of the discrete equations and steady-state analysis of the limiting PDE (and a higher-order correction thereof) can be used to shed light on these and other qualitative predictions of the model.      

The Dynamics of Turing Patterns for Morphogen-Regulated Growing Domains with Cellular Response Delays

Since its conception in 1952, the Turing paradigm for pattern formation has been the subject of numerous theoretical investigations. Experimentally, this mechanism was first demonstrated in chemical reactions over 20 years ago and, more recently, several examples of biological self-organisation have also been implicated as Turing systems. One criticism of the Turing model is its lack of robustness, not only with respect to fluctuations in the initial conditions, but also with respect to the inclusion of delays in critical feedback processes such as gene expression. In this work we investigate the possibilities for Turing patterns on growing domains where the morphogens additionally regulate domain growth, incorporating delays in the feedback between signalling and domain growth, as well as gene expression. We present results for the proto-typical Schnakenberg and Gierer–Meinhardt systems: exploring the dynamics of these systems suggests a reconsideration of the basic Turing mechanism for pattern formation on morphogen-regulated growing domains as well as highlighting when feedback delays on domain growth are important for pattern formation.     

FcgammaRIIb-mediated negative regulation of BCR signalling is associated with the recruitment of the MAPkinase-phosphatase, Pac-1, and the 3'-inositol phosphatase, PTEN

The low-affinity receptor for IgG, FcgammaRIIb, negatively regulates B cell antigen receptor (BCR)-mediated proliferative signalling. FcgammaRIIb has been reported to mediate this inhibition by uncoupling the BCR from the RasMAPkinase pathway. We now show that FcgammaRIIb-mediated negative feedback inhibition also correlates with induction of an Erk-associated phosphatase activity that reflects the rapid association of Erk and the MAPkinase phosphatase, Pac-1, and dephosphorylation and inactivation of ErkMAPkinase. This mechanism of abrogating ongoing ErkMAPkinase signalling therefore provides a rationale for rapid immune-complex-mediated feedback inhibition of active antigen-driven B cell responses. In addition, FcgammaRIIb signalling also induces the recruitment and activation of the 3'-inositol phosphatase, PTEN, which by antagonising PI 3kinase activity and inhibiting BCR-coupling to the anti-apoptotic kinase, Akt, provides an additional mechanism for FcgammaRIIb-mediated negative regulation of BCR-coupling to ErkMAPkinase, cell survival and proliferation.     

Power-law models of signal transduction pathways

The mathematical modelling of signal transduction pathways has become a valuable aid to understanding the complex interactions involved in intracellular signalling mechanisms. An important aspect of the mathematical modelling process is the selection of the model type and structure. Until recently, the convention has been to use a standard kinetic model, often with the Michaelis-Menten steady state assumption. However this model form, although valuable, is only one of a number of choices, and the aim of this article is to consider the mathematical structure and essential features of an alternative model form--the power-law model. Specifically, we analyse how power-law models can be applied to increase our understanding of signal transduction pathways when there may be limited prior information. We distinguish between two kinds of power law models: a) Detailed power-law models, as a tool for investigating pathways when the structure of protein-protein interactions is completely known, and; b) Simplified power-law models, for the analysis of systems with incomplete structural information or insufficient quantitative data for generating detailed models. If sufficient data of high quality are available, the advantage of detailed power-law models is that they are more realistic representations of non-homogenous or crowded cellular environments. The advantages of the simplified power-law model formulation are illustrated using some case studies in cell signalling. In particular, the investigation on the effects of signal inhibition and feedback loops and the validation of structural hypotheses are discussed.     

Membrane-bound macrophage colony-stimulating factor mediated auto-juxtacrine downregulates matrix metalloproteinase-9 release on J6-1 leukemic cell

Earlier studies indicate that J6-1 human leukemic cells proliferate and propagate via the membrane-bound macrophage colony-stimulating factor (M-CSF)-mediated auto-juxtacrine mechanism. Matrix metalloproteinases (MMPs) can modulate the activity of cell membrane molecules and influence many cellular behaviors. Therefore, we hypothesized that MMP may also be involved in the membrane-bound M-CSF-mediated juxtacrine mechanism. First, we investigated whether blocking of membrane-bound M-CSF by neutralizing antibody to M-CSF or M-CSF receptor and adding of exogenous M-CSF are able to influence MMP-9 release. Next, we determined whether MMP-9 participated in J6-1 cells proliferation and influence the shedding of membrane-bound M-CSF and its receptor. Current studies show that blockade of the interaction between membrane-bound M-CSF and M-CSF receptor by antibody to M-CSF or M-CSF receptor promotes MMP-9 release. Moreover, we demonstrated that because of M-CSF mediated juxtacrine, lack of MMP-9 promotes J6-1 cell proliferation, in which a decrease in the shedding of cell-surface M-CSFR is involved. Hence, we suggest that membrane-bound M-CSF inhibit MMP-9 release and down-regulated MMP-9 contribute to juxtacrine stimulating in leukemic cell growth.