Scaling relations for auxin waves. (English) Zbl 1505.34069
Summary: We analyze an ‘up-the-gradient’ model for the formation of transport channels of the phytohormone auxin, through auxin-mediated polarization of the PIN1 auxin transporter. We show that this model admits a family of travelling wave solutions that is parameterized by the height of the auxin-pulse. We uncover scaling relations for the speed and width of these waves and verify these rigorous results with numerical computations. In addition, we provide explicit expressions for the leading-order wave profiles, which allows the influence of the biological parameters in the problem to be readily identified. Our proofs are based on a generalization of the scaling principle developed by Friesecke and Pego to construct pulse solutions to the classic Fermi-Pasta-Ulam-Tsingou model, which describes a one-dimensional chain of coupled nonlinear springs.
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 34A33 | Ordinary lattice differential equations |
| 92C37 | Cell biology |
| 34B40 | Boundary value problems on infinite intervals for ordinary differential equations |
| 35C07 | Traveling wave solutions |
Keywords:
travelling waves; polar auxin transport; up-the-gradient models; scaling limits; cross-diffusion; lattice differential equationsReferences:
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