^{1}

^{2}

^{2}

^{3}

^{4}

^{1}

^{5}

^{6}

^{2}

^{*}

Conceived and designed the experiments: MC BM FB FTA SM. Performed the experiments: DC SM. Analyzed the data: MC TV. Contributed reagents/materials/analysis tools: MC DC SM. Wrote the paper: MC FTA SM.

The authors have declared that no competing interests exist.

Interactions between individuals that are guided by simple rules can generate swarming behavior. Swarming behavior has been observed in many groups of organisms, including humans, and recent research has revealed that plants also demonstrate social behavior based on mutual interaction with other individuals. However, this behavior has not previously been analyzed in the context of swarming. Here, we show that roots can be influenced by their neighbors to induce a tendency to align the directions of their growth. In the apparently noisy patterns formed by growing roots, episodic alignments are observed as the roots grow close to each other. These events are incompatible with the statistics of purely random growth. We present experimental results and a theoretical model that describes the growth of maize roots in terms of swarming.

To exploit soil resources optimally, plants have developed intricate root systems that are characterized by complex patterns and based on the coordinated group behavior of the growing root apices

A. An image showing root growth in one of 10 experiments. B. Decision-making by the root apex: the movement of the root in the two opposite directions before committing to a growth direction. C. The alignment of one root with others based on distance, i.e., without physically contacting neighbors. D. An example of collective behavior: a group of roots chooses the same growth direction.

We generated random growing paths by considering an assembly of independent particles separated initially by distance

In the model of non-random growth, we consider the root apex or tip as a moving particle, and we treat the length of the root as the temporal history of the particle. Each particle is described by a velocity vector, the orientation of which denotes the direction in which the root apex moves. We assume that each seed produces only one main root and no collateral roots. This simplification is reasonable because the main root provides the longest and clearest history of interaction with its neighbors. The roots can interact with their neighbors at any point along their lengths. We assume that each root grows at a slightly different speed, giving rise to different root lengths. While this feature is clearly observed in experiments, thus motivating its inclusion in the theoretical model, we remark that this is not crucial in determining the growing-roots patterns. We have introduced the variation in speeds in accordance to experimental data which revealed that the variability in root lengths is always observed. Moreover, each plant is assumed to interact with neighboring plants causing spatial attraction or repulsion depending on the individual root responses. As mentioned before electric fields may be a possible mechanism of such interaction. These attractive and repulsive forces become effective when a certain distance separates the roots and leads them to grow closer or far away to together. As two roots approach each other, the mechanisms of direction adjustment (alignment) would switch on. The root apices adjust their direction as they detect their neighbors within a certain radius

A. The force of attraction,

We calculate the velocity vectors associated with root growth rate and direction for 10 experiments and analogously for 10 realizations of the numerical experiments. In numerical simulations, approximately the same number of seeds and temporal data points are considered as in the experiment. The sample growing paths from the experiment, the model of non-random growth and the model of random growth are shown in

The parameter values used in B are:

As described in the previous paragraph, the asymmetry in the heights of the bimodal distributions may indicate the existence of a preferred direction of growth or episodic alignments of directions between neighboring roots. To investigate, we calculated the spatial correlations between velocity vectors as a function of the distances between the roots. More precisely, we defined a radius

In the non-random model, the mean probability distribution of the velocities tends very slowly to a symmetric shape, and large fluctuations in the differences between the heights of the bimodal peaks are observed between individual experiments. In the random-growth model, different patterns are observed. The effect of noise makes the distributions symmetrical, and consequently, the fluctuations in the differences of the heights of the bimodal distributions are smaller. However, when interactive forces are present, the distributions become asymmetric and the fluctuations larger. These fluctuations arise from the alignment of growing roots and some type of synchronization of growth directions. The results of the experiment and non-random model differ qualitatively from those of the random-growth model, and the experimental results differ slightly from the predictions of the non-random model. This difference may be related to the fact that the non-random model considers only a few of the most essential ingredients that contribute to non-random growth. Other elements which have been neglected for the sake of simplicity contribute to root interactions and behavior: for instance, inhomogeneities in water distribution, initial distances between the seeds, or angle of the support on which the roots are growing. Despite these facts the proposed model allowed us to qualitatively reproduce the observed growing root patterns by making use of few basic ingredients.

Concerning the swarming behavior in roots, one of the advantages could be the efficient chemical modification of the soil in their vicinity. This would allow the maintenance of specific root micro-niche which is optimal for their physiological performances, in particular, the extraction of the essential nutrients from the soil and the defense against pathogens.

In conclusion, the experiments revealed that the qualitative features of root growth are well explained by a model of swarm behavior. The main insight gained in this study is that the root apices act as decision-making centers, giving rise to correlations in the growth patterns. We have identified a few key ingredients allowing us to explain and reproduce qualitatively the observed phenomenology, in particular, the angle adjustment and the attractive and repulsive interactions. Repulsive forces have been considered in the description of the polymer brushes

Maize seeds were germinated at 24

(EPS)

(EPS)

(MOV)

(MOV)

MC wishes to thank T. Vicsek for his hospitality at Eötvös Loránd University.