Fig. 4

Numerical simulations of self-propelled interacting chemotactic rods in 2D. a Schematic representation of the simulated steric interaction between the rods, where interpenetration (\(\delta\)) results in Hertzian-like repulsion (\({F}_{{\rm{el}}}\)) and friction (\({F}_{{\rm{fr}}}\)), and the fluid flow generated by the hydrodynamic force dipole, here for channel height \(h=30\ \upmu\)m. For other heights, see Supplementary Fig. 6. b Snapshot of a simulation output (length \(L=4\ \upmu\)m, \(h=30\ \upmu\)m, \(\Phi =0.244\)), showing collective motion of packs of rods. The green shading represents the gradient. c Flow structure factor \(E(q)\) for indicated conditions, exhibiting a growing maximum at a \({q}_{{\rm{str}}}\) depending only on the channel height \(h\). Darkening shading of a given colour indicates increasing area fraction in the range \(0.002-0.25\). d Maximum subtracted of its low-density value, \(\Delta E({q}_{{\rm{str}}})\), as a function of cell area fraction. (Inset) Typical vortex size \(\pi /{q}_{{\rm{str}}}\) as a function of the channel height. Note the difference in scale compared to Fig. 2c. e Chemotactic coefficient as a function of the cell area fraction. Dotted lines represent its value in absence of interactions. f, Chemotactic coefficient, normalized to its low-density value, as a function of the maximum of the flow structure factor \(\Delta E({q}_{{\rm{str}}})\). g Time autocorrelations of the cells velocity, the colour coding is the same as in (c). h Chemotactic coefficient as a function of the decorrelation time \({\tau }_{{\rm{dec}}}\), defined by \({C}_{v}({\tau }_{{\rm{dec}}})=0.5\), and fit according to Eq. (4). d–f, h, Conditions are as indicated in panel d and error bars represent SEM on at least 3 runs, totalizing at least 1000 cells.