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The authors have declared that no competing interests exist.

Conceived and designed the experiments: AB GT. Performed the experiments: AB. Analyzed the data: AB. Contributed reagents/materials/analysis tools: AB JSW GT. Wrote the paper: AB JSW GT.

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

The growth and development of biological organisms involves processes at multiple scales, including regulation of the timing and location of specific developmental programs. Despite their structural complexity, many prototypical processes involved in development have been identified. One such process is the notion of tip-driven growth in plants

In SAWs and derived models, the elongation and expansion steps are decoupled. The consequences of such decoupling is that the resulting shape may be non-physical. In

(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

Inspired by the growth of the taproot in plants in particular and by growth processes in biology in general, we introduce here the fiber walk: a SAW model that includes a notion of lateral expansion of the path. We consider the simplest case of a randomly growing line that thickens while elongating into a random direction. We examine the fiber walk in 2D and 3D, and note that although plant roots self-evidently grow in 3D, a number of experimental studies restrict root growth to 2D or quasi-2D (e.g.,

We introduce the formal notion of the fiber in

A lattice in

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

We aim at defining the fiber as a non-branching and loop free graph

In the following two definitions we introduce the fiber as a subset of possible directed path graphs. First we define self-avoiding edges to limit ourself to cycle-free directed path graphs and as a measurable characteristic of self-avoidance of the walk. Secondly, we use self-avoiding edges to obtain a stopping criteria of the fiber.

Given self-avoiding edges we can define the stopping configuration of a fiber as a vertex that is only incident to self-avoiding edges.

In the following we describe the process that constructs the fiber. The process of constructing a fiber is called a fiber walk. The fiber walk is a sequence of steps, each from a vertex

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

Def. 4–5 are equivalent to the notion of a growing self-avoiding random walk (

In our fiber walk each elongation step causes a contraction on the lattice, which is a prerequisite to reconstruct the expanded boundary locally at each time step. The contraction of the lattice at each step is performed as a set of merges between vertices adjacent to the last reached vertex on the lattice. A merge is the union of two adjacent vertices

Elongation occurs as the first step which is chosen randomly between the edges incident to the origin

Each merge is said to cause a new edge to exist on the lattice, because two previously distant vertices are newly adjacent after each merge. Therefore, a number of merges are associated with each edge.

In the following we define an initial edge labeling based on the vertex positions introduced in Def. 1. After that we give the mechanism to recalculate labels after each contraction step.

For example, the labels for all 3D-directions are:

The recalculation of the labels considers the merge of two vertices

We identify edges for contraction by an entry in the

Edges selected for contraction, are said to be incident from the side and reflect the directions of lateral expansion. In practice, we select all edges that do not have an entry in their edge label that indicates a movement opposite to the current growth direction and is different from the elongation step taken in the current growth direction. Recall that edges belonging to the edge set of the fiber are self-avoiding edges.

The contraction results in edges of different lengths in the edge set of the fiber and of self-avoiding edges. In the following we show that a fiber has three edge length classes and self-avoiding edges have five edge length classes in 2D by investigating the edge set of the lattice, but the proofing scheme applies to higher dimensions as well. We make use of two prerequisites; first we consider two cases of merging results: (1) merges not resulting in self-avoiding edges (

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,

Case 2a shows a self-avoiding edge (purple) of length

Proof 1 exhausts all combinations of the introduced prerequisites. It is trivial, that each of the three edge length classes of the fiber are possible edge length classes of self-avoiding edges, because the fiber walk has a non-contracting edge that can connect to each of the edge length classes of the fiber and to the fiber directly. This occurs if

In the following we note that a face is bounded by edges that connect vertices. Our central interest is to define the spatial expansion of the fiber walk as represented by a boundary. We introduce the fiber walk boundary for simplicity in 2D, but the principle extends naturally to higher dimensions. Based on Def. 3, we can distinguish three kinds of faces, which share an edge or a vertex with the walk: 1.)

We first introduce the basic definition of the boundary by assuming that no self-avoiding edges are present on the lattice. In a second step we will extend this notion to self-avoiding edges.

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

As a final component of our setting we extend Def. 8 to achieve validity in the presence of self-avoiding edges based on an intermediate lattice. The intermediate lattice defines the faces adjacent to the walk by placing vertices at a certain fraction of the edge length of incident edges to the walk. This essential definition, as we see later on, is depicted in

(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.

Def. 9 defines a maximal distance to compensate for self-avoiding edges of length

Def. 9 refines the lattice such that Def. 8 applies to all faces incident to the fiber. We finalize this section with actual computations of the introduced fiber walk in 2D and 3D (

(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.

In the introduction we gave 2D examples of singularities on the boundary if only the symmetric expansion around the vertices of the walk is considered. It is trivial to observe that each vertex contributes to the boundary wherever a vertex of the fiber has an incident edge belonging to the edge set of the lattice. On an uncontracted lattice, a vertex does not contribute to the boundary on both sides of the walk at locations where the walk turns

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.

Let

Configuration 1 demonstrates the case where our boundary construction guarantees that the expansion around

(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.

Each step of a fiber walk elongates the fiber and expands its boundary, which defines a region occupied by each step on the lattice. The fiber walk contains three edge length classes in 2D of length

Fiber walks generate self-avoiding edges. Self-avoiding edges correspond to directions of growth that are inaccessible as a direct result of the spatial expansion of the fiber. There are five length classes of self-avoiding edges for a fiber grown on a 2D square lattice:

The stopping time describes the probability of a walk to generate a stopping configuration (compare Def. 4). We give here the stopping times for the 2D growing 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.

From the previous paragraph it follows that we need an indication of how the fiber walk expansion affects the growth of the fiber. We compare the end-to-end distance per step to evaluate how far the growing SAW and the fiber walk are on average away from the origin. The classic property to characterize this behavior is the scaling obtained as the average Euclidean mean-square-displacement (MSD) from the origin as a function of the number of steps on the lattice

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.

In alignment with the scaling behaviour of the self-avoiding edges, the number of merges defines the amount of occupied space on the lattice. Nevertheless, the unique characteristic of the fiber walk is the contraction, which defines the boundary describing the spatial expansion of the fiber walk. Here we give the scaling of the number of contractions for the fiber walk as a measure for the for the size of the boundary (See

In this paper we introduced the fiber walk, which is a growing SAW that includes lateral expansion. The expansion of the fiber walk is modeled as a local contraction around the last vertex reached on the lattice. We have shown that the fiber walk process constitutes a mechanism by which physical space is reserved, yet does not imply the expansion of the object into this space. We have found that the expansion lets a walk diverge initially further away from the origin before entering the scaling region and that the fiber walk takes, on average, fewer steps until termination than the SAW. Self-avoidance as proposed in our model causes encapsulated regions on the lattice that inhibit further exploration by the fiber unless a branching mechanism is added.

One benefit of modeling the contraction is that local object thickness larger than the step size is possible. Although we have shown only the difference between modeling elongation (alone) and elongation with expansion, our fiber walk is capable to perform more than one contraction while growing. A practical variation of the principal fiber walk would be to assign probabilities to the selection of walk edges. Such probabilities should allow the simulation of more ballistic walk behaviour. As far as we are aware, the connection between self-avoidance and contraction in SAW-derived models has not been established in the study of random walks in biology (see

Future work on the fiber walks will be targeted towards the creation and simulation of physical networks including the spatial interactions between several fiber that walk on the same lattice. In our opinion the fiber walk has the potential to parallel geometrically the apical tip-driven growth of individual branches in many plants at the apical meristem. Morphogens within the apical meristem regulate the extension of individual branches

Many such networks face the problem of being below the maximum resolution of sensing technologies in non-laboratory conditions (e.g. roots in real soil). The fiber walk enables us to develop localized descriptors or measures that may serve as the basis for models in efforts to identify adaptive growth rules in complex organisms. Finally, the current definition and implement of the fiber walk best describe the growth of sessile objects. However, extensions to the model could also include the possibility of movement of edges within the fiber and further coupling of physical mechanisms of growth influenced by surface properties. The method is available within the supporting material of this paper, for download on the web pages of the authors and on git hub (

The results of this paper were produced with the python program that we provide together with the paper. This software is suitable for developers to use as a template to integrate our fiber walk into their software. An end user should follow the requirements provided in the “readme” file to execute the program.

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The authors thank Daniel I. Goldman for helpful suggestions on the manuscript.