Oscillations and patterns in spatially discrete models for developmental intercellular signalling

Abstract

Abstract. 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.

Multiscale analysis of pattern formation via intercellular signalling

“…While the period-two pattern is distinguished by its dominance, the consideration of other patterning modes is nonetheless significant in related cell-signalling studies; for instance, in arrays of hexagonal cells, the fine-grained patterning regime corresponds to patterns containing three cells rather than two. In addition, microscopic patterns of considerably longer wavelength can be observed in developing tissues and Webb & Owen (2004b) have shown that such patterns may be generated in related models of lateral inhibition. Multiscale analyses of such considerations will be presented elsewhere.…”

“…The positive parameters k, h, a and b determine the feedback strength. The value of the exponent k expresses whether Delta-Notch binding is monovalent (k = 1) or cooperative (k 2); examples of such binding phenomena are given in Webb & Owen (2004b). We note that, In vivo, Delta-Notch binding and the resulting Delta production will be dependent on the cell's biochemical and biophysical environment.…”

“…Collier et al (1996) 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 & Owen (2004b) 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 (1996).…”

A Bio-Inspired Distributed Clustering Algorithm for Wireless Sensor Networks

“…Theoretical biologists and mathematicians have successfully modeled the Delta-Notch signaling process by sets of coupled ordinary differential equations (ODEs) [5,13,15,23]. We consider the simple model in [5], where for the i th cell, ni denotes the levels of Notch activity, and di denotes the levels of Delta activity.…”

Continuum limits of pattern formation in hexagonal-cell monolayers

“…Lateral induction (positive feedback between ligandreceptor binding and subsequent ligand production) was accommodated and the range over which juxtacrine signals may be transmitted was studied; Wearing and Sherratt (2001) performed a comprehensive nonlinear analysis of this model, highlighting that linear analysis alone is unable to predict the model's behaviour for large numbers of cells. Webb and Owen (2004) 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 and that cell shape is a crucial determining factor in the patterns produced.…”