Topotaxis of active Brownian particles Koen Schakenraad,1, 2 Linda Ravazzano,1, 3 Niladri Sarkar,1 Joeri A.J. Wondergem,4 Roeland M.H. Merks,2, 5 and Luca Giomi1, ∗ 1Instituut-Lorentz, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands 2Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands 3Center for Complexity and Biosystems, Department of Physics, University of Milan, Via Celoria 16, 20133, Milano, Italy 4Kamerlingh Onnes-Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands 5Institute of Biology, Leiden University, P.O. Box 9505, 2300 RA Leiden, The Netherlands Recent experimental studies have demonstrated that cellular motion can be directed by topograph- ical gradients, such as those resulting from spatial variations in the features of a micropatterned substrate. This phenomenon, known as topotaxis, is especially prominent among cells persistently crawling within a spatially varying distribution of cell-sized obstacles. In this article we introduce a toy model of topotaxis based on active Brownian particles constrained to move in a lattice of obsta- cles, with space-dependent lattice spacing. Using numerical simulations and analytical arguments, we demonstrate that topographical gradients introduce a spatial modulation of the particles' per- sistence, leading to directed motion toward regions of higher persistence. Our results demonstrate that persistent motion alone is sufficient to drive topotaxis and could serve as a starting point for more detailed studies on self-propelled particles and cells. I. INTRODUCTION quently force the cells to move around them. If the ob- stacles' density smoothly varies across the substrate, the Whether in vitro or in vivo, cellular motion is of- cells have been shown to perform topotaxis: the topo- ten biased by directional cues from the cell's micro- graphical gradient serves as a directional cue for the cells environment. Chemotaxis, i.e., the ability of cells to move to move toward the regions of lower obstacle density. in response to chemical gradients, is the best known ex- Although the precise biophysical or biochemical prin- ample of this functionality and plays a crucial role in ciples behind topotaxis are presently unknown, its oc- many aspects of biological organization in both prokary- currence for cells performing amoeboid migration sug- otes and eukaryotes [1,2]. Yet, it has become increasingly gests the possibility of cell-type-indenpendent mecha- evident that, in addition to chemical cues, mechanical nisms that, separately from the cell's mechanosensing cues may also play a fundamental role in dictating how machinery, provide a generic route to the emergence of cells explore the surrounding space. Haptotaxis (i.e., di- directed motion. In this article we explore this hypothe- rected motion driven by gradients in the local density of sis. Using active Brownian particles (ABPs) constrained adhesion sites) and durotaxis (i.e., directed motion driven to move within a lattice of obstacles, we demonstrate that by gradients in the stiffness of the surrounding extracel- topotaxis can result solely from the spatial modulation lular matrix) are well studied examples of taxa driven by of persistence resulting from the interaction between the mechanical cues [3{5]. particles and the obstacles. In vivo, cells crawl through topographically intricate ABPs represent a simple stochastic model for self- environments, such as the extracellular matrix, blood propelled particles, such as active Janus particles [16], and lymphatic vessels, other cells, etc., that can signifi- and cell motility on flat substrates [17]. ABPs perform cantly influence migration strategies [6{9]. For instance, persistent self-propelled motion in the direction of the it has been shown that local anisotropy in the underly- particle orientation in combination with rotational dif- ing substrate, in the form of adhesive ratchets [10{12] fusion of this orientation. The motion of active parti- or three-dimensional structures on the subcellular scale cles has been explored in several complex geometries, in- [10, 13, 14], can lead to directed motion even in the ab- cluding convex [18, 19] and nonconvex [20] confinements, arXiv:1908.06078v1 [cond-mat.soft] 16 Aug 2019 sence of chemical stimuli. More recently, Wondergem mazes [21], walls of funnels [22], interactions with asym- and coworkers demonstrated directed migration of single metric [23, 24] and chiral [25] passive objects, porous cells using a spatial gradient in the density of cell-sized topographies [26] and random obstacle lattices [27{30]. topographical features [15]. In these experiments, highly For a review, see Refs. [31, 32]. Because of the non- motile, persistently migrating cells (i.e., cells perform- equilibrium nature of active particles, local asymme- ing amoeboid migration) move on a substrate in between tries in the environment can be leveraged to create a microfabricated pillars that act as obstacles and conse- drift; these particles have been demonstrated to perform chemotaxis [33, 34], durotaxis [35], and phototaxis [36]. Furthermore, topographical cues, such as those obtained in the presence of arrays of asymmetric posts [37, 38] ∗ Corresponding author: [email protected] and ratchets consisting of asymmetric potentials [39{41] 2 or asymmetric channels [42{45], have been shown to pro- timescales larger than the persistence time, t τp, ABPs 2 2 duce a directional bias in the motion of active particles diffuse, i.e., j∆r(t)j = 4Dt, with D = v0τp=2 the dif- reminiscent of those observed for cells. fusion coefficient. From τp, one can define a persistence The paper is organized as follows: in Sec. II we present length, lp = v0τp, as the typical distance travelled by a our model for ABPs and their interaction with obstacles. particle before loosing memory of its previous orienta- In Sec. IIIA we show that, in the presence of a gradient tion. Consistently, the autocorrelation function of the in the obstacle density, ABPs drift, on average, in the di- velocity v = dr=dt (v = v0p in free space) is given by: rection of lower density. The speed of this net drift, here 2 −∆t/τp referred to as topotactic velocity, increases as a function hv(t + ∆t) · v(t)i = v0e : (3) of both the density gradient and the persistence length of the ABPs. In Sec. IIIB (numerically) and Sec. IIIC Our ABPs roam within a two-dimensional array of cir- (analytically) we study ABPs in regular obstacle lattices cular obstacles of radius Ro. Following Refs. [19, 20], the and demonstrate that the origin of topotaxis of active interactions between particles and obstacles are modeled particles can be found in the altered persistence length via a force of the form: ( of the particles in the presence of obstacles. − v0 (p · N) N if j∆r j ≤ R; F = µ o (4) 0 otherwise ; II. THE MODEL where N is a unit vector normal to the obstacle surface, j∆roj is the distance between the obstacle center and the Our model of ABPs consists of disks of radius Rp self- particle center, and the effective obstacle radius R is the propelling at constant speed v0 along the unit vector p = sum of the obstacle and the particle radii: R = Ro + Rp. (cos θ; sin θ) and subject to rotational white noise. The Eq. (4) describes a frictionless hard wall force that can- dynamics of the particles is governed by the following cels the velocity component normal to the obstacle sur- overdamped equations: face whenever the particle would penetrate the obstacle, and vanishes otherwise. We stress that the wall force dr does not influence the intrinsic direction of motion p. = v p + µF ; (1a) dt 0 Thus, a particle slides along an obstacle until either the dθ p obstacle wall becomes tangential to p or rotational diffu- = 2D ξ ; (1b) dt r sion causes the particle to rotate away. This is consistent with experimental observations on self-propelled colloids where r = r(t) is the position of the particle, t is time, [18] as well as various types of cells [48, 49]. For details and µ is a mobility coefficient. The force F = F (r) on the numerical implementation of Eqs. (1) and (4), embodies the interactions between the particles and the see Appendix A. In the following Sections, we measure obstacles. ξ = ξ(t) is a random variable with zero times in units of the the persistence time, i.e., t~ = t/τp, 0 mean, i.e., hξ(t)i = 0, and time-correlation hξ(t)ξ(t )i = and lengths in units of the effective obstacle radius, i.e., 0 δ(t − t ). The extent of rotational diffusion is quanti- `~= `=R. fied by the rotational diffusion coefficient Dr, whereas translational diffusion is neglected under the assumption of large P´ecletnumber: Pe 1. Overall, this set-up III. RESULTS provides a reasonable toy model for highly motile cells such as those used in experimental studies of topotaxis The motion of ABPs in different lattices of obstacles [15, 46, 47]. For a study on the influence of the P´eclet is visualized in Fig.1. Each panel shows 20 simulated number on the motion of ABPs around obstacles, see, for ~ trajectories with persistence length lp = 5. Figs.1a,b example, Ref. [28]. show regular square lattices with dimensionless center-to- In free space, (i.e., F = 0), ABPs described by Eqs. center obstacle spacings of d~= 2:5 and d~= 4 respectively. (1) perform a persistent random walk (PRW) with mean In Fig.1, the obstacles are graphically represented as displacement h∆r(t)i = 0 and mean squared displace- disks of radius R and the ABPs as point particles. To ment: avoid biasing the statistics of the particle trajectories, t ABPs start at a random location inside the unit cell of 2 2 2 −t/τp j∆r(t)j = 2v0τp + e − 1 ; (2) the regular square lattice (Fig.1c) at t~= 0 with random τp orientation.