Multiscale analysis of pattern formation via intercellular signalling

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.     

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.     

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.     

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.     

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.     

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.     

Cell communities and robustness in development

The robustness of patterning events in development is a key feature that must be accounted for in proposed models of these events. When considering explicitly cellular systems, robustness can be exhibited at different levels of organization. Consideration of two widespread patterning mechanisms suggests that robustness at the level of cell communities can result from variable development at the level of individual cells; models of these mechanisms show how interactions between participating cells guarantee community-level robustness. Cooperative interactions enhance homogeneity within communities of like cells and the sharpness of boundaries between communities of distinct cells, while competitive interactions amplify small inhomogeneities within communities of initially equivalent cells, resulting in fine-grained patterns of cell specialization.     

From Notch signaling to fine-grained patterning: Modeling meets experiments

Notch signaling is the canonical signaling pathway between neighboring cells. It plays an important role in fine-grained patterning processes such as the formation of checkerboard-like differentiation patterns and sharp boundaries between developing tissues. While detailed information about many of the genes and proteins involved have been identified, we still lack a quantitative mechanistic understanding of these processes. Here we discuss several recent studies that provide novel insights into Notch-dependent patterning by combining mathematical models with quantitative experimental results. Such approaches allow identification of mechanisms and design principles controlling how patterns are generated in a reproducible and robust manner.     

RA and FGF Signalling Are Required in the Zebrafish Otic Vesicle to Pattern and Maintain Ventral Otic Identities

During development of the zebrafish inner ear, regional patterning in the ventral half of the otic vesicle establishes zones of gene expression that correspond to neurogenic, sensory and non-neural cell fates. FGF and Retinoic acid (RA) signalling from surrounding tissues are known to have an early role in otic placode induction and otic axial patterning, but how external signalling cues are translated into intrinsic patterning during otic vesicle (OV) stages is not yet understood. FGF and RA signalling pathway members are expressed in and around the OV, suggesting important roles in later patterning or maintenance events. We have analysed the temporal requirement of FGF and RA signalling for otic development at stages after initial anteroposterior patterning has occurred. We show that high level FGF signalling acts to restrict sensory fates, whereas low levels favour sensory hair cell development; in addition, FGF is both required and sufficient to promote the expression of the non-neural marker otx1b in the OV. RA signalling has opposite roles: it promotes sensory fates, and restricts otx1b expression and the development of non-neural fates. This is surprisingly different from the earlier requirement for RA signalling in specification of non-neural fates via tbx1 expression, and highlights the shift in regulation that takes place between otic placode and vesicle stages in zebrafish. Both FGF and RA signalling are required for the development of the otic neurogenic domain and the generation of otic neuroblasts. In addition, our results indicate that FGF and RA signalling act in a feedback loop in the anterior OV, crucial for pattern refinement.     

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.     

Comparing a discrete and continuum model of the intestinal crypt

The integration of processes at different scales is a key problem in the modelling of cell populations. Owing to increased computational resources and the accumulation of data at the cellular and subcellular scales, the use of discrete, cell-level models, which are typically solved using numerical simulations, has become prominent. One of the merits of this approach is that important biological factors, such as cell heterogeneity and noise, can be easily incorporated. However, it can be difficult to efficiently draw generalizations from the simulation results, as, often, many simulation runs are required to investigate model behaviour in typically large parameter spaces. In some cases, discrete cell-level models can be coarse-grained, yielding continuum models whose analysis can lead to the development of insight into the underlying simulations. In this paper we apply such an approach to the case of a discrete model of cell dynamics in the intestinal crypt. An analysis of the resulting continuum model demonstrates that there is a limited region of parameter space within which steady-state (and hence biologically realistic) solutions exist. Continuum model predictions show good agreement with corresponding results from the underlying simulations and experimental data taken from murine intestinal crypts.     

An investigation of spatial signal transduction in cellular networks

ABSTRACT: BACKGROUND: Spatial signal transduction plays a vital role in many intracellular processes such as eukaryotic chemotaxis, polarity generation, cell division. Furthermore it is being increasingly realized that the spatial dimension to signalling may play an important role in other apparently purely temporal signal transduction processes. It is being recognized that a conceptual basis for studying spatial signal transduction in signalling networks is necessary. RESULTS: In this work we examine spatial signal transduction in a series of standard motifs/networks. These networks include coherent and incoherent feedforward, positive and negative feedback, cyclic motifs, monostable switches, bistable switches and negative feedback oscillators. In all these cases, the driving signal has spatial variation. For each network we consider two cases, one where all elements are essentially non diffusible, and the other where one of the network elements may be highly diffusible. A careful analysis of steady state signal transduction provides many insights into the behaviour of all these modules. While in the non-diffusible case for the most part, spatial signalling reflects the temporal signalling behaviour, in the diffusible cases, we see significant differences between spatial and temporal signalling characteristics. Our results demonstrate that the presence of diffusible elements in the networks provides important constraints and capabilities for signalling. CONCLUSIONS: Our results provide a systematic basis for understanding spatial signalling in networks and the role of diffusible elements therein. This provides many insights into the signal transduction capabilities and constraints in such networks and suggests ways in which cellular signalling and information processing is organized to conform to or bypass those constraints. It also provides a framework for starting to understand the organization and regulation of spatial signal transduction in individual processes.     

How far can a juxtacrine signal travel?

Juxtacrine signalling is the process of cell communication in which ligand and receptors are both anchored in the cell membrane. We develop three mathematical models for this process, involving different mathematical representations of the dynamics of membrane-bound ligand and free and bound receptors, within an epithelial sheet. We consider the dynamics of this system following a localized disturbance, such as would be provided by a source of ligand or by the generation of a free edge via wounding. We study the ability of the juxtacrine mechanism to transmit a signal away from this disturbance, and show analytically that the spatial half-life of the signal can in fact be arbitrarily large. This result is quite general, since we use a generic reaction kinetic scheme; the key assumption is that ligand and receptor production are both upregulated by binding. Moreover, the result applies to all three of our model formulations. We conclude by discussing applications of the result to the particular case of the transforming growth factor alpha binding to epidermal growth factor receptor in epidermal wound healing.     

Hybrid mathematical model of glioma progression

Objectives:  Gliomas are an important form of brain cancer, with high mortality rate. Mathematical models are often used to understand and predict their behaviour. However, using current modeling techniques one must choose between simulating individual cell behaviour and modeling tumours of clinically significant size.
Materials and Methods:  We propose a hybrid compartment-continuum-discrete model to simulate glioma growth and malignant cell invasion. The discrete portion of the model is capable of capturing intercellular interactions, including cell migration, intercellular communication, spatial cell population heterogeneity, phenotype differentiation, epigenetic events, proliferation, and apoptosis. Combining this with a compartment and continuum model allows clinically significant tumour sizes to be evaluated.
Results and Conclusions:  This model is used to perform multiple simulations to determine sensitivity to changes in important model parameters, specifically, the fundamental length parameter, necrotic cell degradation rate, rate of cell migration, and rate of phenotype transformation. Using these values, the model is able to simulate tumour growth and invasion behaviour, observed clinically. This mathematical model provides a means to simulate various tumour development scenarios, which may lead to a better understanding of how altering fundamental parameters can influence neoplastic progression.     

Effects of Intrinsic and Extrinsic Noise Can Accelerate Juxtacrine Pattern Formation

Epithelial pattern formation is an important phenomenon that, for example, has roles in embryogenesis, development and wound-healing. The ligand Epithelial Growth Factor (EGF) and its receptor EGF-R, constitute a system that forms lateral induction patterns by juxtacrine signalling—binding of membrane-bound ligands to receptors on neighbouring cells. Owen et al. developed a generic ordinary differential equation model of juxtacrine lateral induction that exhibits stable patterning under some conditions. The model predicts relatively slow pattern formation. We examine here the effects of both intrinsic and extrinsic cellular noise arising from the stochastic treatment of this model, and show that this noise could have an accelerating effect on the patterning process.     

A model of primitive streak initiation in the chick embryo

Initiation of the primitive streak in avian embryos provides a well-studied example of a pattern-forming event that displays a striking capacity for regulation. The mechanisms underlying the regulative properties are, however, poorly understood and are not easily accounted for by traditional models of pattern formation, such as reaction-diffusion models. In this paper, we propose a new activator-inhibitor model for streak initiation. We show that the model is consistent with experimental observations, both in its pattern-forming properties and in its ability to form these patterns on the correct time-scales for biologically realistic parameter values. A key component of the model is a travelling wave of inhibition. We present a mathematical analysis of the speed of such waves in both diffusive and juxtacrine relay systems. We use the streak initiation model to make testable predictions. By varying parameters of the model, two very different types of patterning can be obtained, suggesting that our model may be applicable to other processes in addition to streak initiation.     

Robustness of positional specification by the Hedgehog morphogen gradient

Spatial gradients of Hedgehog signalling play a central role in many patterning events during animal development, regulating cell fate determination and tissue growth in a variety of tissues and developmental stages. Experimental evidence suggests that many of the proteins responsible for regulating Hedgehog signalling and transport are themselves targets of Hedgehog signalling, leading to multiple levels of feedback within the system. We use mathematical modelling to analyse how these overlapping feedbacks combine to regulate patterning and potentially enhance robustness in the Drosophila wing imaginal disc. Our results predict that the regulation of Hedgehog transport and stability by glypicans, as well as multiple overlapping feedbacks in the Hedgehog response network, can combine to enhance the robustness of positional specification against variability in Hedgehog levels. We also discuss potential trade-offs between robustness and additional features of the Hedgehog gradient, such as signalling range and size regulation.     

Discrete Logic Modelling Optimization to Contextualize Prior Knowledge Networks Using PRUNET

High-throughput technologies have led to the generation of an increasing amount of data in different areas of biology. Datasets capturing the cell’s response to its intra- and extra-cellular microenvironment allows such data to be incorporated as signed and directed graphs or influence networks. These prior knowledge networks (PKNs) represent our current knowledge of the causality of cellular signal transduction. New signalling data is often examined and interpreted in conjunction with PKNs. However, different biological contexts, such as cell type or disease states, may have distinct variants of signalling pathways, resulting in the misinterpretation of new data. The identification of inconsistencies between measured data and signalling topologies, as well as the training of PKNs using context specific datasets (PKN contextualization), are necessary conditions to construct reliable, predictive models, which are current challenges in the systems biology of cell signalling. Here we present PRUNET, a user-friendly software tool designed to address the contextualization of a PKNs to specific experimental conditions. As the input, the algorithm takes a PKN and the expression profile of two given stable steady states or cellular phenotypes. The PKN is iteratively pruned using an evolutionary algorithm to perform an optimization process. This optimization rests in a match between predicted attractors in a discrete logic model (Boolean) and a Booleanized representation of the phenotypes, within a population of alternative subnetworks that evolves iteratively. We validated the algorithm applying PRUNET to four biological examples and using the resulting contextualized networks to predict missing expression values and to simulate well-characterized perturbations. PRUNET constitutes a tool for the automatic curation of a PKN to make it suitable for describing biological processes under particular experimental conditions. The general applicability of the implemented algorithm makes PRUNET suitable for a variety of biological processes, for instance cellular reprogramming or transitions between healthy and disease states.     

Dynamic determinations: patterning the cell behaviours that close the amphibian blastopore

We review the dynamic patterns of cell behaviours in the marginal zone of amphibians with a focus on how the progressive nature and the geometry of these behaviours drive blastopore closure. Mediolateral cell intercalation behaviour and epithelial-mesenchymal transition are used in different combinations in several species of amphibian to generate a conserved pattern of circumblastoporal hoop stresses. Although these cell behaviours are quite different and involve different germ layers and tissue organization, they are expressed in similar patterns. They are expressed progressively along presumptive lateral-medial and anterior-posterior axes of the body plan in highly ordered geometries of functional significance in the context of the biomechanics of blastopore closure, thereby accounting for the production of similar patterns of circumblastoporal forces. It is not the nature of the cell behaviour alone, but the context, the biomechanical connectivity and spatial and temporal pattern of its expression that determine specificity of morphogenic output during gastrulation and blastopore closure. Understanding the patterning of these dynamic features of cell behaviour is important and will require analysis of signalling at much greater spatial and temporal resolution than that has been typical in the analysis of patterning tissue differentiation.     

Discrete Element Framework for Modelling Extracellular Matrix,Deformable Cells and Subcellular Components

This paper presents a framework for modelling biological tissues based on discrete particles. Cell components (e.g. cell membranes, cell cytoskeleton, cell nucleus) and extracellular matrix (e.g. collagen) are represented using collections of particles. Simple particle to particle interaction laws are used to simulate and control complex physical interaction types (e.g. cell-cell adhesion via cadherins, integrin basement membrane attachment, cytoskeletal mechanical properties). Particles may be given the capacity to change their properties and behaviours in response to changes in the cellular microenvironment (e.g., in response to cell-cell signalling or mechanical loadings). Each particle is in effect an ‘agent’, meaning that the agent can sense local environmental information and respond according to pre-determined or stochastic events. The behaviour of the proposed framework is exemplified through several biological problems of ongoing interest. These examples illustrate how the modelling framework allows enormous flexibility for representing the mechanical behaviour of different tissues, and we argue this is a more intuitive approach than perhaps offered by traditional continuum methods. Because of this flexibility, we believe the discrete modelling framework provides an avenue for biologists and bioengineers to explore the behaviour of tissue systems in a computational laboratory.