ABCB-mediated auxin transport in outer root tissues regulates lateral root spacing in Arabidopsis

Root branching is an important strategy to explore efficiently large volumes of soil. To economize this process, lateral roots (LR) are formed along the growing root at discrete positions that are instructed by oscillating auxin signals derived from the lateral root cap (LRC). This assumes that auxin moves from the LRC across multiple layers to accumulate in the pericycle. Here, we identified, using gene silencing and CRISPR based approaches, a group of five genetically linked, closely related ABCBs that control LR spacing by modulating the amplitude of the auxin oscillation. The transporters localize to the plasma membrane and reveal significant auxin export activity. These ABCBs are mainly expressed in the LRC and epidermis where they contribute to auxin transport towards the root oscillation zone. Our findings highlight the importance of auxin transport in the outer tissues of the root meristem to regulate LR spacing.

TWO-PHASE ASYMPTOTICS IN THE FORM OF AIRY FUNCTIONS IN THE WAVE BREAKING PROBLEM FOR THE I-BURGERS EQUATION

We consider wave breaking problems for the Burgers equation with a small “imaginary viscosity,” which in fact plays the role of small dispersion. Although this equation has no apparent physical meaning, the problem in question is an interesting analog of the famous Gurevich-Pitaevsky problem on the onset of an oscillation zone as the breaking of a simple wave occurs for the Korteweg-de Vries equation. In contrast to the latter, the solution of the i-Burgers equation in the oscillation zone can be described explicitly and has a two-phase structure. This was indicated more than 25 years ago in (Dobrokhotov et al., 1992), where the solution was constructed in the form of a function of Maslov’s canonical operator. Now we use the recent results in (Dobrokhotov, Nazaikinskii, 2018) to present the solutions in the more efficient form of uniform asymptotics represented as the logarithmic derivative of the Airy function of a composite argument. The research was supported by the Ministry of Science and Higher Education of the Russian Federation (project no. AAAA-A17-117021310377-1).