The fiber walk: a model of tip-driven growth with lateral expansion

Abstract

Tip-driven growth processes underlie the development of many plants. To date, tip-driven growth processes have been modeled as an elongating path or series of segments, without taking into account lateral expansion during elongation. Instead, models of growth often introduce an explicit thickness by expanding the area around the completed elongated path. Modeling expansion in this way can lead to contradictions in the physical plausibility of the resulting surface and to uncertainty about how the object reached certain regions of space. Here, we introduce fiber walks as a self-avoiding random walk model for tip-driven growth processes that includes lateral expansion. In 2D, the fiber walk takes place on a square lattice and the space occupied by the fiber is modeled as a lateral contraction of the lattice. This contraction influences the possible subsequent steps of the fiber walk. The boundary of the area consumed by the contraction is derived as the dual of the lattice faces adjacent to the fiber. We show that fiber walks generate fibers that have well-defined curvatures, and thus enable the identification of the process underlying the occupancy of physical space. Hence, fiber walks provide a base from which to model both the extension and expansion of physical biological objects with finite thickness.

Figure 1. Example of contradictions arising from decoupling elongation and expansion of a SAW.

(A) SAW of 16 steps beginning at the origin (yellow triangle) with current position marked via the green triangle. (B) Thickened SAW - note the curvature singularity of the resulting surface. (C) SAW thickened by of an edge length - multiple regions in space are “associated” with different steps, leading to an identifiability problem, which holds for all thickening greater than or equal to of an edge length.

Figure 2. A growing fiber (green) and its self-avoiding edges (purple).

The example shows a fiber on a lattice as defined in Def. 1 and the vertex positions are assigned to each vertex.

Figure 3. Random choice of a follow up step.

A step (green) shown on a 2D lattice. The possible follow-up steps are represented as a dotted line.

Figure 4. Fiber walk on a 2D lattice.

Elongation occurs as the first step which is chosen randomly between the edges incident to the origin (compare Fig. 3). Here the chosen edge to reach is shown in green. Selection of vertices incident to the walk from the side correspond to the expansion. The vertices selected to be merged with are shown in grey. Contraction of the selected vertices and its result after merging the selected vertices to . Another step, including elongation and expansion, is shown as a second step on the lattice. The second step reaching uses the same color schema as before.

Figure 5. The lattice contraction of the fiber walk.

Left: The fiber (green) on a lattice and its edges selected for contraction. Right: The contracted edges which are possible follow up steps of length 1, and 2. In both figures the edge labels involved in the contraction and their direction, indicated as arrows are shown.

Figure 6. Edge length classes of the fiber walk.

Case 2a shows a self-avoiding edge (purple) of length and Case 2b shows a self-avoiding edge of length 3. The fiber on a lattice is colored green and its edges selected for contraction are shown in grey. The dotted green line denotes an unknown fiber walk that is not affecting the given configuration. In both figures the edge labels involved in the contraction and their direction indicated as arrows are shown.

Figure 7. The fiber walk boundary.

The boundary (blue) of the fiber walk is derived from the face dual of the lattice (red) in 2D. The shown configuration corresponds to the example given in Fig. 4. Here, dotted line segments denote non-unique edges, which do not belong to the boundary.

Figure 8. Recovering the surface.

(left) The original lattice with grey edges and orange vertices. The walk is shown in green and self-avoidend edges are shown in purple. The black line denotes the half-edge length of edges incident to the walk. (right) The intermediate lattice consisting of the black half edge line and the grey edges incident to the green walk. Small orange vertices are placed at half-edge distance while the bigger orange vertices are the original vertices belonging to the walk.

Figure 9. Computed comparison of SAW and fiber walk.

(A) A growing SAW in 2D and (B) a Fiber Walk in 2D. All walk edges are colored in green, the lattice is shown in (grey) and the self avoiding edges are colored in purple for all images. (C) A growing SAW in 2D and (D) a fiber walk 3D.

Figure 10. Avoiding right angles.

The four cases of right angle configurations of the fiber walk. Configuration 1 shows possible right angle configurations containing a self-avoiding edge. Configuration 2 shows the possible configurations of right angles without self-avoiding edges.

Figure 11. A computed fiber walk example with boundary.

(A)The illustrated fiber walk (green) is shown on a grey lattice, with two locations marked where curvature singularities are avoided. (B) The intermediate lattice (orange). (C) The filled boundary (brown) smoothed with a B-Spline in 2D.

Figure 12. Stopping times of SAW and fiber walk.

Comparison of the growing SAW (blue) and fiber walk (red) stopping times. The figure shows the computed walk length at termination of 100.000 single walks.

Figure 13. Scaling of the fiber walk and SAW.

The summarized overview of the MSD scaling of the growing SAW (blue) and the fiber walk (green) is shown. For both, the fit line in the scaling region is shown in red.