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Tip-driven growth processes underlie the development of many plants. To date,tip-driven growth processes have been modelled as an elongating path or seriesof segments without taking into account lateral expansion during elongation.Instead, models of growth often introduce an explicit thickness by expandingthe area around the completed elongated path. Modelling expansion in this waycan lead to contradictions in the physical plausibility of the resultingsurface and to uncertainty about how the object reached certain regions ofspace. Here, we introduce "fiber walks" as a self-avoiding random walk modelfor tip-driven growth processes that includes lateral expansion. In 2D, thefiber walk takes place on a square lattice and the space occupied by the fiberis modelled as a lateral contraction of the lattice. This contractioninfluences the possible follow-up steps of the fiber walk. The boundary of thearea consumed by the contraction is derived as the dual of the lattice facesadjacent to the fiber. We show that fiber walks generate fibers that havewell-defined curvatures, enable the identification of the process underlyingthe occupancy of physical space. Hence, fiber walks provide a base from whichto model both the extension and expansion of physical biological objects withfinite thickness.

The Fiber Walk: A Model of Tip-Driven Growth with Lateral Expansion