Topotaxis of Active Brownian Particles

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 topographical 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 obstacles, with space-dependent lattice spacing. Using numerical simulations and analytical arguments, we demonstrate that topographical gradients introduce a spatial modulation of the particles’ persistence, 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 obstacles’ 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., |∆r(t)| = 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 |∆r | ≤ R, F = µ o (4) 0 otherwise , II. THE MODEL where N is a unit vector normal to the obstacle surface, |∆ro| 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 Pécletnumber: 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 Péclet 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 |∆r(t)| = 2v0τp + e − 1 , (2) the regular square lattice (Fig.1c) at t˜= 0 with random τp orientation. All trajectories are shown for a total time of where ∆r(t) = r(t) − r(0) and h· · · i represents an aver- t˜ = 3. Comparing the spreading of the active particles age over ξ (see, e.g., Ref. [32]). The constant τp = 1/Dr, in Fig.1a with that in Fig.1b, we observe that the commonly referred to as persistence time, quantifies the more dense the obstacle lattice is, the more it hinders typical timescale over which a particle tends to move the diffusion of the active particles. We will quantify along the same direction. Thus, over timescales shorter this later. than the persistence time, t τp, ABPs move ballis- To study topotaxis, we define an irregular square lat- 2 2 tically with speed v0: |∆r(t)| ≈ (v0t) , while over tice comprising a linear gradient of the obstacle spacing in 3

FIG. 2. The emergence of topotaxis in density gradient lattices. (a) hx˜i = hxi /R as a function of time t˜ = t/τp for several values of the density gradient r, with d˜ = d/R = 5

and ˜lp = v0τp/R = 5. (b) Topotactic velocity, defined in the main text, in the x direction as a function of the density FIG. 1. Simulated trajectories of 20 active Brownian par- gradient r based on the data in (a). (c) hx˜i = hxi /R as a ˜ ticles (ABPs) with persistence length ˜lp = v0τp/R = 5 in function of time t = t/τp for several values of the persistence different lattices of obstacles. The obstacles are graphically length ˜lp = v0τp/R, with d˜ = d/R = 5 and r = 0.07. (d) represented as disks of radius R and the ABPs as point par- Topotactic velocity in the x direction as a function of the ticles. (a,b) ABPs in regular square lattices of obstacles with persistence length ˜lp based on the data in (c). Data in (a) center-to-center obstacle spacings d˜ = d/R = 2.5 and d˜ = 4 and (c) represent averages over 106 particles. Error bars in respectively. The particles start at t˜= 0 at a random location (b) and (d) are given by the standard error of hxi (t)/(R t) at in the unit cell of the lattice and are simulated for a total time t = 30 τp. of t˜ = 3. (c) The unit cell of the regular square lattice show- ing the starting points (crosses) of the 20 trajectories in (b). (d) ABPs in a lattice with a linear gradient in obstacle spac- on the left and right, see Appendix B. ing, quantified by a dimensionless parameter r = 0.07 (see Appendix B), and d˜= 5 at the center of the gradient region. The particles start at t˜ = 0 in the origin and are simulated for a total time of t˜= 5. A. The emergence of topotaxis

To quantify topotaxis, we measure the average x and the positive x−direction. The latter is quantified in terms y coordinates, hx˜i and hy˜i, as a function of time for 106 of a dimensionless parameter r representing the rate at particles. The results are given for five values of the di- which the obstacle spacing decreases as x increases. Thus mensionless density gradient r in Fig.2a and Fig.6a r = 0 corresponds to a regular square lattice, whereas (Appendix C) for x and y respectively. The emergence large r values corresponds to rapidly decreasing obstacle of topotaxis is clear from Fig.2a: the active particles spacing. Fig.1d shows this lattice for r = 0.07 with move, on average, in the positive x−direction, hence in 20 particle trajectories, starting in the origin at t˜ = 0 the direction of lower obstacle density. As expected by with a random orientation, plotted for a simulation time the symmetry of the lattice, there is no net motion in the of t˜ = 5, where d˜ = 5 represents the obstacle spacing y direction independently of the value of r (Fig.6a, Ap- in the center of the gradient region. The gradient re- pendix C). To further quantify topotaxis, we define the gion has a finite width (not visible in Fig.1d) and is topotactic velocity as the average velocity in the positive flanked by regular square lattices to the left, with lattice x direction in a time interval ∆t, vtop = h∆xi /∆t, and ˜ ˜ ˜ spacing dmin = 2.1, and to the right, with lattice spacing evaluate it between t = 0 and t = 30. Figs.2a and2b ˜ ˜ ˜ dmax = 2d − dmin. The minimal and maximal obstacle- show thatv ˜top is approximatively constant in time and ˜ ˜ to-obstacle distances (dmin and dmax, respectively) do proportional to the density gradient r. not depend on the steepness of the gradient, and con- Next, we investigate the effect of the intrinsic mo- sequently the width of the gradient region decreases for tion of the ABPs on topotaxis. This intrinsic motion is steeper gradients (larger r). For a detailed description of characterized by the persistence length lp = v0τp, which both the regular and gradient lattices as well as an im- uniquely determines the statistics of the particle trajec- age of the gradient lattice including the regular lattices tory in free space. Namely, if two types of ABPs have 4 different v0 and τp, but the same lp, their trajectories and blue curves is slightly smaller than that of the black have the same statistical properties, even though faster curve. This indicates that ABPs in obstacle lattices do particles move along these trajectories in a shorter time. not move purely balistically, because obstacles prevent Fig.2c shows hx˜i as a function of time for five values of them from moving in straight lines. ˜ lp. The speed of topotaxis is again approximately con- The slope of the curves at timescales larger than the ˜ persistence time, on the other hand, is independent of stant in time and increases with lp (Fig.2d). This trend partly results from the fact that increasing the persis- the presence of obstacles and equal to 1 (see inset). In other words, even though the motion of the ABPs is hin- tence length corresponds either to an increment in v0 or dered by the obstacles at all timescales, the long-time τp, both resulting into an increase ofv ˜0. However, Fig. motion remains diffusive, as was also observed for ABPs 2d shows thatv ˜top increases faster than linear as a func- ˜ in random obstacle lattices of low density [28]. Fitting tion of lp, suggesting an additional effect caused by the obstacle lattice. As we will see in Sec.IIIB, this effect the MSD at long times allows one to define an effective is caused by the fact that the lattice hinders ABPs with diffusion coefficient Deff , namely: large persistence lengths more than ABPs with smaller 2 |∆r(t)| −−−→ 4Deff t. (5) persistence lengths. Finally, we note that there is no net tτp motion in the y direction irrespective of the persistence length, as expected by symmetry (Fig.6b, Appendix C). Fig.3b shows the velocity autocorrelation function (VACF) on a semilog plot as a function of the time in- ˜ terval ∆t˜ = ∆t/τp for lp = 10. The d → ∞ curve again B. The physical origin of topotaxis represents the theoretical curve in free space and shows exponential decay [Eq. (3)]. Interestingly, in the presence of increasing obstacle densities, hence for smaller lattice The observed occurrence of topotaxis of ABPs is intu- spacings d˜, the velocity autocorrelation decreases but re- itive, as particles migrate in the direction where there is mains, to good approximation, exponential. From this more available space. However, the mechanism by which numerical evidence, we conclude that the average mo- ABPs are guided toward the less crowded regions is not tion of ABPs in a two-dimensional square lattice can be obvious from the results in SectionIIIA. To gain more in- described as a persistent random walk with an effective sight into the physical origin of topotaxis, we investigate velocity v and an effective persistence time τ [28], how particle motility depends on the local obstacle spac- eff eff ing. In doing so, we take inspiration from recent works hv(t + ∆t) · v(t)i = v2 e−∆t/τeff . (6) [35, 50, 51] that have shown, in the context of durotaxis, eff that persistent random walkers, moving in a spatial gra- Fig.4 shows Deff , τeff and veff , normalized by their free dient of a position-dependent persistence length, show an space values, as a function of the obstacle spacing d˜ for average drift toward the region with larger persistence. ˜ three values of the free space persistence length lp. Start- As is the case in our system (Sec.IIIA), this effect is ing with the effective diffusion coefficient (Fig.4a), we stronger in the presence of larger gradients [35, 50]. In or- observe that, for every value of the persistence length, the der to understand whether or not such a space-dependent effective diffusion coefficient Deff increases as a function persistence might explain the observed topotactic mo- of d˜until it approaches the free space diffusion coefficient tion, we study and characterize the motion of ABPs in D for large d˜. This is consistent with what we observed regular square lattices. To do so, we measure the mean 2 in Figs.1a,b and3: ABPs on low density lattices spread squared displacement (MSD) |∆r(t)| as a function of out more than ABPs on high density lattices. Moreover, time and the velocity autocorrelation function (VACF) the effective diffusion coefficient deviates more from its hv(t + ∆t) · v(t)i as a function of the time interval ∆t ˜ ˜ free space value for large lp than it does for small lp. 4 ˜ for 10 particles for various lattice spacings d and persis- This is intuitive because more persistent particles tend ˜ tence lengths lp. to move longer along the same direction and therefore ˜ Fig.3a shows a log-log plot of the MSD for lp = 10. are hindered more in their motion by the obstacle lat- The curve with d˜ → ∞ (black) represents the theoret- tice. ical MSD in free space [Eq. (2)] and exhibits the well- The effective persistence time τeff (Fig.4b) and the known crossover from the ballistic regime (slope equal effective velocity veff (Fig.4c), both extracted from the to 2) to the diffusive regime (slope equal to 1) around velocity autocorrelation function [Eq. (6)], show a sim- ˜ t˜= 1 (t = τp). The hindrance of the obstacles is evident ilar trend: they increase as a function of d until they from the data obtained in regular lattices with d˜ = 4 approach their free space values at high d˜, and they de- ˜ (red curve) and d˜ = 2.5 (blue curve), as the MSD is viate more from their free space values for large lp than ˜ smaller than the MSD in free space at all times (see also they do for small lp. These data show that the obstacles the inset). Moreover, the MSD is smaller for the smaller cause the ABPs, on average, to move slower and turn lattice spacing, as we already observed qualitatively in more quickly. The decreased effective velocity, with re- Figs.1a,b. The hindrance also manifests itself in the spect to free space, is intuitive given the interactions be- short-time (t < τp) regime, where the slope of the red tween particles and obstacles [Eq. (4)], which slow down 5

FIG. 3. Regular square lattices of obstacles modify the effective parameters of the persistent random walk. (a) Dimen- sionless mean square displacement |∆r˜|2 = |∆r|2 /R2 as a function of dimensionless time t˜= t/τp for ˜lp = v0τp/R = 10 in free space (d → ∞, black line) and in the presence of square lattices with obstacle spacings d˜ = d/R = 2.5 (blue) and d˜ = 4 (red). The effective diffusion coefficient Deff is obtained from a linear fit to the long-time (t > τp) regime of the log-log data. (b) Dimensionless velocity autocorrelation function hv˜(t + ∆t) · v˜(t)i = hv(t + ∆t) · v(t)i τ 2/R2 as FIG. 4. Effective parameters of the persistent random walk in p regular lattices of obstacles. (a) Normalized effective diffusion a function of ∆t˜ = ∆t/τ for ˜l = v τ /R = 10 in free space p p 0 p coefficient D /D, obtained from the mean squared displace- (d → ∞) and in the presence of square lattices with obstacle eff ment [Eq. (5)], as a function of the normalized obstacle spac- spacings d˜ = 2.5 and d˜ = 4. The effective velocity v and eff ing d˜ = d/R for three values of the normalized persistence effective persistence time τeff are obtained from a exponen- ˜ ˜ tial fit to the autocorrelation function. MSD and VACF data length lp = v0τp/R. Inset shows Deff for lp = 5 obtained 4 via the mean squared displacement (MSD) and via the veloc- represent averages over 10 particles. 2 ity autocorrelation function (VACF), using Deff = veff τeff /2. (b) Normalized effective persistence time τeff /τp, (c) normalized effective velocity veff /v0, both obtained from the velocity the ABPs. The decreased effective persistence time, on autocorrelation function [Eq. (6)], and (d) normalized effec- the other hand, is less obvious as one could imagine the tive persistence length leff /lp = veff τeff /(v0τp) as functions of periodic obstacle lattice to guide ABPs along straight the normalized obstacle spacing d˜ = d/R for three values of ˜ lines. Apparently, this potential guiding mechanism is the normalized persistence length lp = v0τp/R. Data points outcompeted by the fact that encounters of ABPs with represent the average of 10 independent measurements from individual obstacles at shorter timescales cause them to MSD or VACF data (Fig.3). The error bars show the corresponding standard deviations. turn more quickly than in free space. In Sec.IIIC we will study these short-timescale interactions in greater detail. Combining the effective persistence time (Fig.4b) and ˜ and lp = 10 is negligible, indicating that the free space the effective velocity (Fig.4c) gives the effective persis- persistence length ˜l affects particle motion only when it tence length l = v τ and the effective diffusion co- p eff eff eff is comparable with the lattice spacing. efficient D = v2 τ /2. The inset of Fig.4a shows the eff eff eff These observations, combined with those in Refs. effective diffusion coefficient for ˜l = 5, calculated both p [35, 50, 51] which demonstrate a net flux of persistent by using the effective persistence time and effective ve- random walkers toward regions of larger persistence, ul- locity from the velocity autocorrelation function (VACF) timately explain the origin of topotaxis in our system. and by a direct measurement from the mean squared dis- ABPs migrate, on average, toward regions of higher per- placement (MSD). The excellent agreement between D eff sistence, hence to regions of lower obstacle density. More- measured at short and long timescales (using the VACF over, the dependence of the effective persistence length and MSD respectively) is another indication that the mo- l on the free space persistence length ˜l (Fig.4d) justi- tion of the ABPs in regular square obstacle lattices can eff p fies the superlinear increase of the topotactic velocityv ˜top indeed be considered to be an effective persistent random ˜ walk. as a function of lp (Fig.2d). In addition to the normal speed-up due to the higher persistence, more persistent The effective persistence length leff = veff τeff is plot- particles experience a larger gradient in persistence. ted in Fig.4d. As anticipated, the effective persistence length increases with increasing lattice spacing d˜, consistent with findings of ABPs in random obstacle lattices [28] and a model of persistently moving cells in a tissue C. Fokker-Planck equation for regular lattices of stationary cells [52]. This effect increases with the free space persistence length, as the motion of less persistent As we explained in Sec.IIIB, topotaxis in our model of particles is randomized before they can reach an obsta- ABPs crucially relies on the fact that, even when trapped ˜ cle. Furthermore, the difference between data with lp = 5 in an array of obstacles, ABPs still behave as persistent 6 random walkers. The physical origin of this behavior is, component of the force experienced by a particle mov- however, less clear from the numerical simulations. In ing in the positive x−direction (y−direction), becomes this Section, we rationalize this observation using some more negative as the particle moves away from the ori- simple analytical arguments. The probability distribu- gin. This allows us to write tion function P = P (r, θ, t) of the position and orien- ∂Fx tation of an ABP, whose dynamics is governed by Eqs. hp · F i| = (0) hr · pi , (13) tτp ∂x (1), evolves in time based on the following Fokker-Planck equation: and by inserting Eq. (13) into Eq. (11) we find: ∂hr · pi ∂F 2 x ∂P ∂ P = v0 − Dr − µ (0) hr · pi . (14) = −v p · ∇P − µ∇ · (P F ) + D , (7) ∂t ∂x ∂t 0 R ∂θ2 tτp Solving Eq. (14) yields: subject, at all times, to the normalization constraint: v0 hr · pi = 1 − e−Dr,eff t , (15) Z tτ D dr dθ P (r, θ, t) = 1 , (8) p r,eff with Dr,eff = Dr − µ ∂xFx(0) > Dr. By comparing Eq. (15) with its free space equivalent [Eq. (12)], we identify with dr = dx dy. Eq. (7) cannot be solved exactly, but D as an increased effective rotational diffusion coeffi- useful insights can be obtained by calculating the rate of r,eff cient. This implies a decreased effective persistence time, change of the mean squared displacement |∆r|2 . Here 2 2 2 2 consistent with the data in Fig.4b. The above analysis we assume r(0) = 0, which yields |∆r| = |r| = x +y , shows that, to first order, the observed decrease in effec- and tive persistence time simply results from the short-time ∂ |r|2 Z ∂P (r, θ, t) interactions, within a unit cell of the lattice, that cause = dr dθ |r|2 (9) the particles to turn more quickly. Finally, substituting ∂t ∂t Eq. (15) in Eq. (10) and taking again the first order Upon substituting Eq. (7) in Eq. (9) and integrating by Taylor expansion for F allows one to solve Eq. (10) ex- parts, we obtain: actly. Expanding this exact solution at the second order in time yields: 2 ∂h|r| i 2 2 2 = 2v hr · pi + 2µhr · F i. (10) h|r| i = v0t , (16) ∂t 0 tτp which is the standard ballistic regime of the mean Analogously, the term hr · pi evolves accordingly to: squared displacement. Hence, the decreased effective velocity observed in Fig.4c originates from interactions ∂hr · pi = v0 − DRhr · pi + µhp · F i . (11) with obstacles at larger timescales (t ∼ τp, see also Fig. ∂t 3b). In free space (F = 0), Eqs. (10) and (11) can be solved In the long timescale (t 1/Dr = τp) the particles exactly to find: reach a diffusive steady state, thus ∂thr · pi = 0. Hence, solving Eq. (11) for hr · pi and substituting in Eq. (10) v 0 −DRt yields: hr · piF =0 = 1 − e , (12) DR 2 2 ∂h|r| i 2v0 µ µDr = 1 + hp · F i + hr · F i , 2 ∂t D v v2 and the mean squared displacement h|r| i given by Eq. tτp r 0 0 (2). A generic nonzero F compromises the closure of (17) the equations, thus making the problem intractable with As the long time behavior is diffusive, the expression on exact methods. Nevertheless, it is possible to use some the right-hand side of Eq. (17) is constant and equal simplifying assumptions to obtain intuitive results about to 4Deff . Now, according to Eq. (4), µ(p · F ) = 0 if 2 Deff and τeff (Fig.4) at short ( t 1/Dr = τp) and long |∆ro| > R, and µ(p · F ) = −v0(p · N) otherwise. Thus, (t 1/Dr = τp) timescales. hp·F i < 0. This term shows that diffusion is slowed down At short timescales, we can assume a particle to be because the obstacle force F always slows down the par- still relatively close to its initial position r(0) = 0. Thus ticles (but never accelerates them). Moreover, for more one can expand the force in Eq. (11) at the linear or- dense obstacle lattices, particles interact with obstacles der in r, i.e., F (r) ≈ F (0) + ∇F (0) · r. Evidently, more often, which explains the observed dependence of such an expansion is ill-defined for discontinuous forces Deff on the obstacle spacing in Fig.4a. Analogously, such as that given by Eq. (4). However, one can imag- since particles move in an open space and, on average, ine to smoothen the force (for instance using a trun- away from the center, hr·F i < 0 (i.e., the repulsion forces cated Fourier expansion), without alterning the quali- due to the obstacles are directed more often toward the tative picture. By the symmetry of the obstacle lattice, origin than toward infinity, further slowing down diffu- F (0) = 0, ∂yFx(0) = ∂xFy(0) = 0, and the constant sion). Thus Deff < D, consistent with our numerical ∂xFx(0) = ∂yFy(0) < 0, as the horizontal (vertical) simulations (Figs.3 and4a). 7

IV. DISCUSSION AND CONCLUSIONS cellular Potts model [52, 63], which allow explicitly taking into account effects such as the resistance of cells against In this article we investigated topotaxis, i.e., directed deformations or adhesion between cells and obstacles. motion driven by topographical gradients, in a toy model of ABPs constrained to move within a two-dimensional array of obstacles of smoothly varying density. We found ACKNOWLEDGMENTS that ABPs migrate preferentially toward regions of lower density with a velocity that increases with the gradient This work was supported by funds from the in the lattice spacing and with the particles’ persistence Netherlands Organization for Scientific Research length. In our model, the origin of topotaxis crucially (NWO/OCW), as part of the Frontiers of Nanoscience relies on the fact that, even when moving in a lattice of program (L.G.), the Netherlands Organization for Sci- obstacles, ABPs still behave as persistent random walk- entific Research (NWO-ENW) within the Innovational ers, but with renormalized transport coefficients: τeff and Research Incentives Scheme (R.M.H.M.; Vici 2017, No. veff . As these depend on the topography of the substrate, 865.17.004), and the Leiden/Huygens fellowship (K.S.). here quantified in terms of lattice spacing, topographical gradients result into spatially varying persistence in the motion of the particles, which in turn drives directed mo- Appendix A: Numerical methods tion toward regions of larger persistence [35, 50, 51]. We note that the motion we report here, just like the durotac- We numerically generate particle trajectories that per- tic motion described in Refs. [35, 50], is perhaps better form a persistent random walk by discretizing the equa- described as a “kinesis” than as a “taxis”, because the tions of motion as follows [35]: a particle starts at posi- underlying mechanism of transport is a nondirectional tion r0 at t = 0, after which the particle is moved by a change in behavior induced by a purely positional cue. distance v0∆t in a random initial direction −π < θ1 < π, This in contrast to the true directional bias underlying, such that the new position is r1 = r0 + v0∆t p(θ1). For for instance, chemotaxis of E. coli [53] which leads to all subsequent time steps, the angle at time step n, θn, significantly more efficient transport [51]. is updated by adding a small deviation angle to the an- Several questions remain open to future investigation. gle of the previous time step, θn = θn−1 + δθ. Here, For instance, how is the picture affected by translational −π < δθ < π is extracted randomly from a Gaussian 2 diffusion? Is topotaxis robust against competing direc- distribution with mean 0 and variance σ = 2∆t/τp us- tional cues, such as chemotaxis [15]? How sensitive is ing the Box-Muller transform. The new position of the the performance of topotaxis with respect to the obsta- particle, rn, is then found by rn = rn−1 + v0∆t p(θn), cles’ shape [18, 37, 38], the type of motion (e.g., per- with rn−1 the position at the previous time step. sistent random walk, run-and-tumble, Lévywalk, etc. If the update step moves the particle into an obstacle, [18, 21, 26, 54–56]), and the details of particle-obstacle however, the particle-obstacle force [Eq. (4)] is triggered. interactions [29, 37, 57–59]? Another interesting setting In that case, the normal component of the attempted of the problem could be obtained by considering random displacement is subtracted, and the actual displacement arrangements of obstacles, where, unlike in the lattices is given by the tangential component of the attempted studied here, particles can be trapped into convex-shaped displacement, rn = rn−1 + v0∆t p(θn) · T T , with features that can significantly alter their motion [26, 28]. T the tangent unit vector of the obstacle surface at the Finally, although here we demonstrated that topotaxis point of the surface closest to rn−1. We choose the time can be solely driven by the interplay between topographi- step ∆t such that it is much smaller than the persistence cal gradients and persistent random motion, whether this time, ∆t τp, and such that every displacement is much is sufficient to explain cellular topotaxis remains an open smaller than the obstacle radius, v0∆t R. In all re- problem. A quantitative comparison between our nu- ported simulations we have used ∆t = 0.01τp. merical data and experiments on highly motile cells [15] shows, in fact, discrepancies that could be ascribed to the enormously more complex interactions between cells and Appendix B: Obstacle lattices their environment. Specifically, the topotactic velocity in our simulations is in the order of 1% of the intrinsic par- We define a regular square lattice of obstacles with the ticle speed (Fig.2), whereas in the experiments on cells coordinates of the centers of the obstacles given by this ratio is approximately 5%, provided that the obstacles are not spaced further apart than the cell size [15]. d x(n, m) = nd + (B1a) In order to better understand this surprising efficiency, 2 the topotactic response of several types of persistently d moving cells, such as amoeba [60], cancer cells [61], or y(n, m) = md + (B1b) 2 leukocytes [62], could be compared. On the theoretical side, we are currently addressing the problem using more where n, m ∈ Z are the obstacle numbers and d is the dis- biologically-realistic models of cell motility based on the tance between the centers of two neighboring obstacles. 8

FIG. 6. There is no average drift in the y direction in density gradient lattices. (a) hy˜i = hyi /R as a function of time t˜ =

t/τp for ﬁve values of the density gradient r, with d˜= d/R = 5 and ˜lp = v0τp/R = 5. (b) hy˜i = hyi /R as a function of time t˜= t/τp for ﬁve values of the persistence length ˜lp = v0τp/R, with d˜= d/R = 5 and r = 0.07.

between two adjacent pairs of obstacles, divided by the distance between those two pairs, FIG. 5. Snapshot of the gradient lattice as described in Ap- x(n + 1) − x(n) − x(n) − x(n − 1) pendix B. The gradient region is characterized by r = 0.15 and = er − 1, (B3) d˜ = d/R = 5. The obstacles are graphically represented as x(n) − x(n − 1) disks of radius R. The lattice spacing varies from d˜min = 2.1 to d˜max = 7.9 over the x range [xmin, xmax] = [−19, 19]. The which is independent of n, as required for a linear gra- gradient region is flanked by a regular square lattice with dient. In the limit of r → 0, Eqs. (B2) reduce to the d˜ = d˜min on the left (x < xmin) and by a regular square lat- regular square lattice given in Eqs. (B1). tice with d˜= d˜max on the right (x > xmax). Only the first two The gradient lattice is cut off on the left side at columns of both (infinitely large) regular lattices are shown. xmin < 0, where the vertical distance between two neighboring obstacles, y(n, m) − y(n, m − 1), would otherwise become smaller than a minimal distance dmin = 2.1R. The term d/2 is added to make sure that the origin of At the first column of obstacles for which this is the the coordinate system is in the middle of four obstacles. case, the vertical coordinates [Eq. (B2b)] are replaced An illustration of this lattice is given in Figs.1a,b. by y(n, m) = mdmin + dmin/2. To the left of this tran- We define an irregular square lattice with a linear gra- sition column (x < xmin), a regular obstacle lattice with dient of the obstacle spacing in the positive x direction. spacing dmin is placed such that the transition column is The gradient region has a finite width, is centered in the part of this regular lattice. origin, and is flanked by regular square lattices to the left On the right side the gradient lattice is cut off at and to the right. The coordinates of the centers of the xmax = −xmin. To the right of this cut-off (x > xmax), obstacles in the gradient region are given by a regular obstacle lattice with spacing dmax = 2d − dmin is placed such that the horizontal distance between the d rn d rightmost column of the gradient lattice and the leftmost x(n, m) = −r (e − 1) + (B2a) 1 − e 2 column of the regular lattice is equal to dmax. Thus, the 1 y(n, m) = d(m + )ern (B2b) gradient lattice connects two regular square lattices of 2 lattice spacings dmin and dmax. The width of the gradient region, 2xmax, then depends on the gradient parameter where n, m ∈ Z are again the obstacle numbers, d is the r. For an illustration of the gradient lattice for r = 0.15 distance between the centers of obstacles with (n, m) = and d˜= d/R = 5, see Fig.5. (0, 0) and (n, m) = (−1, 0) (i.e., the lattice spacing in the origin), and r is a dimensionless number that quantifies the gradient in the obstacle spacing. Appendix C: Average motion in y Eq. (B2) represents an obstacle lattice where the lattice spacing depends exponentially on the horizontal ob- We plot hy˜i (t˜) of 106 particles moving in a density stacle number n, such that x(n, m) − x(n − 1, m) = dern gradient lattice with d˜ = 5, starting in the origin with a and y(n, m)−y(n, m−1) = dern. This exponential gradi- random orientation, for several values of the dimension- ent in the obstacle spacing, as a function of the obstacle less density gradient r in Fig.6a, and for several values ˜ number n, leads to a linear gradient in the obstacle spac- of the persistence length lp in Fig.6b. As expected, there ing as a function of the horizontal coordinate x. This can is no average drift in the y direction. The fluctuations in be seen by calculating the difference in obstacle distance hyi are in the order of 10% of the effective radius R. 9

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