Fig. 3

Mathematical modeling predicts a mechanism for consistent collective migration of diverse phenotypes. a Collective migration of diverse phenotypes at the same speed is made possible by the spontaneous spatial ordering of individual phenotypes within the band such that each individual’s chemotactic ability is matched to the local gradient steepness \(\frac{{\mathrm{d}}f}{{\mathrm{d}}z}\). The proportion of better performers (larger χ, lower TB; red) should be enriched where the gradient is shallower (front), whereas the proportion of weaker performers (smaller χ, higher TB; blue) should be enriched where the gradient signal is steeper (back). The position where the perceived gradient steepness is maximum (dashed border of the gray regions) determines the highest tumble bias able to travel with the band. Cells in the gray region slowly fall out of the band. b Simulated density profiles of cells migrating in 200 μM aspartate (the same simulation is shown in Fig. 2def, blue dashed) show sorting based on tumble bias. c Chemotactic coefficient χ(z) (blue) defined by the phenotype whose density profile peaks at position z and perceived gradient steepness \(\frac{{\mathrm{d}}f}{{\mathrm{d}}z}\) (black). Red symbols correspond to the location of the peak cell density for individual phenotypes. d Spatial sorting enables consistent migration velocity for traveling phenotypes. The migration velocity, \(\chi \left( z \right)\frac{{\mathrm{d}}f}{{\mathrm{d}}z}\), of the phenotype whose density profile peaks at position z gradually decreases toward the back of the band until the gray region is reached. In the gray region the migration velocity falls off more rapidly, preventing the high TB phenotypes located there from staying in the band