Pattern-generating travelling waves in a discrete multicellular system with lateral inhibition

On a one-dimensional string of cells, the juxtacrine signalling model for Delta–Notch lateral inhibition by 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 interaction, J. Theoret. Biol. 183 (1996) 429–446] exhibits a predominant alternating pattern of cells expressing either Delta or Notch, as well as many aperiodic patterns. Despite this multistationarity, in the idealised situation of no noise, travelling waves invading the unstable, homogeneous state only generate the predominant alternating pattern in their wake all over the lattice. However, this robustness is totally lost in the presence of stochastic noise because the invaded, initial state is unstable.Using linear approximations around the initial, homogeneous state and around the final, patterned state, we are able to derive analytically all essential properties of the wave: the shape of the wave front, the unique, alternating pattern generated by the wave, and the asymptotic speed of the wave front. We show that the asymptotic wave speed equals the theoretical minimum wave speed. The latter agrees extremely well with the value estimated from numerical simulations. Thus, in this system travelling waves are pulled by the leading edge of the front.