Time to stop; agent-based modelling of chemoaffinity with competition

In the classic Chemoaffinity theory, the retinotectal axon projection is thought to use pairs of orthogonal signalling gradients in the retina to specify the eventual location of synapses made on the surface of the tectum/superior colliculus. Similar orthogonal gradients in the tectum provide a coordinate system which allows the axons to match their prespecified destination with the correct location. Although the Ephrins have been shown to guide axons toward their destination, there has yet to emerge a complete account of the local interactions which halt the axonal growth cones in the correct locations to recreate the topography of the retinal cells. The model of Simpson and Goodhill (2011) provides an account of the basic topographic arrangement of cells on the tectum, as well as reproducing well known surgical and genetic manipulation experiments. However, it suffers from the absence of a local chemotactic guidance mechanism. Instead, each agent in their model is given instantaneous knowledge of the vector that would move it toward its pre specified destination. In addition to the globally supervised chemoaffinity term, Simpson and Goodhill (2011) introduced a competitive interaction for space between growth cone agents and a receptor-ligand axon-axon interaction in order to account for the full set of experimental manipulations. Here, we propose the replacement of the chemoaffinity term with a gradient following model consisting of axonal growth cone agents which carry receptor molecule expression determined by their somas location of origin on the retina. Growth cones move on the simulated tectum guided by two pairs of opposing, orthogonal signalling molecules representing the Ephrin ligands. We show that with only the chemoaffinity term and a receptor-ligand based axon-axon interaction term (meaning that all growth cone interactions are by receptor-ligand signalling), a full range of experimental manipulations to the retinotectal system can be reproduced. Furthermore, we show that the observation that competition is not and essential requirement for axons to find their way (Gosse et al., 2008) is also accounted for by the model, due to the opposing influences of signalling gradient pairs. Finally, we demonstrate that, assuming exponentially varying receptor expression in the retina, ligand expression should either be exponential if the receptor-ligand signal induces repulsion (i.e. gradient descent) or logarithmic if the signal induces attraction (gradient ascent). Thus, we find that a model analogous to the one we presented in James et al. (2020) that accounts for murine barrel patterning is also a candidate mechanism for the arrangement of the more continuous retinotectal system.