where A i and a i stand for auxin concentrations in cell i and in the apoplast i , respectively, and space has been discretized ( Fig. S2 A ). E ij and I ij are the efflux and influx transport coefficients or permeabilities between cell i and apoplast j , which depend on the level of efflux and influx carriers, respectively. Influx and efflux permeabilities are detailed in SI Text . D represents the effective diffusion rate along the apoplast.ε is the ratio between the linear size of the apoplast and the linear size of cells. n ( i ) runs over the apoplast neighboring cell i , while N ( i ) runs over cells surrounding apoplast i . We included cell proliferation dynamics and considered that tissue cell proliferation occurs at a slower time scale than auxin dynamics. Cells divided at random and, as a first approximation, we assumed they reached their final growth very rapidly ( 47 ), expanding the provascular ring and preserving its circular shape (Fig. S9 ).