The emergence and transient behaviour of collective motion in active filament systems

Most living systems, ranging from animal flocks, self-motile microorganisms to the cytoskeleton rely on self-organization processes to perform their own specific function. Despite its importance, the general understanding of how individual active constituents initiate the intriguing pattern formation phenomena on all these different length scales still remains elusive. Here, using a high density actomyosin motility assay system, we show that the observed collective motion arises from a seeding process driven by enhanced acute angle collisions. Once a critical size is reached, the clusters coarsen into high and low density phases each with fixed filament concentrations. The steady state is defined by a balance of collision induced randomization and alignment effects of the filaments by multi-filament collisions within ordered clusters.

where alignment (in this condition ) can be seen for approximately 36% of the total collision events.

A. Obtaining binary collision statistics from experiment
To determine the incoming angle ( ) and outgoing angle ( ) for the collision events, the microscopy images were first transformed into binary images. From these binary images, the filaments were identified by skeletonization, using Matlab with a standard library "bwmorph". Then, the coordinates of the filament contour were extracted by use of a cubic spline fit. The obtained coordinates were used to determine , and , , hence , [ Supplementary Fig. 1a].
A collision is detected when the images of two filaments intersect [ Supplementary Fig. 1b]. The incoming angle is obtained from and , 1 frame (0.13sec) before the detected collision event.
The end of a collision is detected when the filaments cease to intersect. Here, only binary collisions are studied and all collisions involving more than 3 filaments were discarded.

B. Boltzmann scattering cylinder for slender rods
A Boltzmann scattering cylinder for rods describes the frequency in which collisions between propelled rods of constant speed occur with respect to the relative angle and can be derived from geometric considerations [1,2]. In the experiments, constant speed of the filament motion is ensured by applying high ATP concentrations. The Boltzmann scattering cylinder for rods as a function of the incoming angle is given by where is the rod length, is the rod diameter and ⁄ denotes the aspect ratio of the filament.
For actin filaments in the experiments, , so we can apply the slender rod assumption, .
Thus for slender rods, Eq. S1 becomes The incoming angle statistics is compared to the functional form of the Boltzmann scattering cylinder for slender rods (see main text Fig. 2c) which, as can be inferred by Eq. S2, depends solely on the incoming angle . Filament properties such as , and only sets the magnitude of .

C. Angular correlation
To imitate the existence of locally aligned filaments or seeds in the framework of binary collisions, angular correlations are implemented. Such angular correlations can be emulated by the function ⁄ , where is a free parameter that determines the strength of orientational correlations [3,4]. It is a measure of how strong the incoming angle probability deviates from the Boltzmann scattering cylinder that represents no angular correlations. Hence, ⁄ . For , , so correlations are absent and resembles molecular chaos. Whereas, large -values correspond to small, locally aligned filament seeds. Here, smaller incoming angle increases its probability, hence more polar alignment.
Using the experimentally obtained (see Fig. 2c in main text) and Eq. S2 (black solid line in Fig. 2c), is derived. For the dilute case, for all incoming angles , which results in and hence no angular correlations (molecular chaos). For the high filament density case, a large -value can be seen (increase in for small incoming angles), indicating the birth of angular correlations within the system.

A. Identification of clusters
To identify clusters, raw images [ Supplementary Fig. 2a] were first treated with a Gaussian blur filter (radius: 2 pixels) [ Supplementary Fig. 2b], following the procedure in Ref. [5]. The filtering process made the fluorescence signal profile smooth, making the identification of the cluster border simpler when applying an intensity cut-off to obtain a binary image of clusters [ Supplementary Fig. 2c]. Here, a same cut-off value was used for all images.

B. Filament density within clusters
To determine the filament density within clusters [ Fig. 3(e), inset], the conservation of total number of filaments was assumed. This is only valid in the first hour after the initiation of actin filament motion by adding ATP since after this period, although slowly, actin filaments start to dissociate from the HMM due to depletion of ATP. The above assumption gives .

A. Determining
Incoming angle is obtained with respect to the filament that is collided by the incoming single filament and path length is determined as the length until the incoming filament aligns with the surrounding filaments [ Supplementary Fig. 3]. The trajectory for which is measured, is obtain via hand by tracking the filament head. Filaments that were not persuaded into the clusters were discarded from the results, though such event is not common.

B. Calculating alignment enhancement
In this section, for the sake of convenience, we write the filament densities used in Fig. 2 as for the dilute case, for the pre-cluster high density case between 0-3 minutes and between 6-9 minutes.
Defining alignment as outgoing angle , from Fig. 2b the necessary incoming angle range to achieve this is approximately , , and for , and , respectively. Associating these values to Fig. 2c, approximately 8%, 27%, and 36% of the total collisions allow for alignment for , and , respectively. Supplementary Fig. 4 shows Fig. 2c as a cumulative probability. Take as an example. Collision events with correspond to 36% of the whole collision events, as is depicted by the black broken lines in Supplementary Fig. 4. Hence, the alignment is enhanced by a factor of approximately 3 and 4 for and , respectively.