Enhanced propagation of motile bacteria on surfaces due to forward scattering

How motile bacteria move near a surface is a problem of fundamental biophysical interest and is key to the emergence of several phenomena of biological, ecological and medical relevance, including biofilm formation. Solid boundaries can strongly influence a cell’s propulsion mechanism, thus leading many flagellated bacteria to describe long circular trajectories stably entrapped by the surface. Experimental studies on near-surface bacterial motility have, however, neglected the fact that real environments have typical microstructures varying on the scale of the cells’ motion. Here, we show that micro-obstacles influence the propagation of peritrichously flagellated bacteria on a flat surface in a non-monotonic way. Instead of hindering it, an optimal, relatively low obstacle density can significantly enhance cells’ propagation on surfaces due to individual forward-scattering events. This finding provides insight on the emerging dynamics of chiral active matter in complex environments and inspires possible routes to control microbial ecology in natural habitats.

Microorganisms live in natural environments that present, to different extents, physical, chemical and biological complexity [1,2]. This heterogeneity influences all aspects of microbial life and ecology in a wide range of habitats, from marine ecosytems [3] to biological hosts [4]. For example, flow and surface topology can trigger or disrupt quorum sensing in bacterial communities [5][6][7] as can shape dynamics of microbial competition in biofilms [8].
To enhance their fitness within such complexity, several bacteria species, e.g. Escherichia coli bacteria [9], are motile, which is key in promoting many biologically relevant processes, such as the formation of colonies and biofilms on surfaces [1,2,10].
Even though natural bacterial habitats present characteristic features that vary on a spatial scale comparable to that of the cells' motion [7,8], experimental studies of near-surface swimming have surprisingly focused on smooth surfaces devoid of this natural complexity. Nonetheless, for far-from-equilibrium self-propelling particles, such as motile bacteria, both individual and collective motion dynamics can depend on environmental factors in non-intuitive ways, as recently shown for microscopic non-chiral active particles numerically [31][32][33][34][35] and experimentally [36][37][38].
Here we show that the motion of individual E. coli cells swimming near a surface is strongly influenced by the presence of randomly distributed micro-obstacles of comparable size. Counterintuitively, at low obstacle densities, the bacterial cells diffuse ≈ 50% more efficiently than on a smooth surface. The interaction with the obstacles can, in fact, rectify the cells' near-surface motion chirality over distances orders-of-magnitude longer than the typical cell size. This behaviour is fundamentally different from that of non-chiral active colloids cruising through random obstacles with a fixed motion strategy, which instead get more localised for increasing obstacle densities [31,39,40]. For chiral bacteria, the expected behaviour is only observed at higher densities. We develop, and verify numerically, a microscopic understanding of the transition between enhanced surface propagation and localisation by identifying two types of cell-obstacle interactions, namely forward scattering events and head-on collisions.
To identify how the spatial heterogeneity on surfaces influences the propagation of bacteria, we recorded trajectories of motile E. coli cells swimming over a glass surface with different densities of fixed obstacles in a quasi-2D geometry (Methods). By controlling the salt concentration in the colloidal dispersion used to prepare the sample chamber, we were able to produce surfaces with different densities ρ (defined as fractional surface coverage) of fixed obstacles in the range 0% ≤ ρ ≤ 12% (Methods). E. coli bacteria are peritrichously flagellated prokaryotic cells that swim through an alternation of run and tumble events [9].
Consistent with previously reported sizes after cell division [9], the typical bacterial cell in our experiments was 2.6 ± 0.7 µm long and 1.2 ± 0.4 µm wide (estimated from microscopy images). When swimming near a surface, E. coli cells move in long circular trajectories [14,17,18], which are typically stably entrapped by the surface [14,27,28,30,41]. We estimated the average translational and angular speeds of the motile cells in our experiments to be v = 10.96 ± 4.27 µm s −1 and Ω = 0.81 ± 0.54 rad s −1 , respectively ( Fig. S1 and Methods). The 10 s-long trajectories in Fig. 1a, along with Fig. S1b, highlight the experimental spread in Ω, which spans from 0 rad s −1 (non-chiral) to 2.5 rad s −1 (strongly chiral), due to both intercell variability and distance variations of the cells from the two surfaces of the sample chamber.
When the bacterial cells swim near a surface with a complex microstructure as in Fig.   1b, interactions with the fixed obstacles become unavoidable. These interactions can significantly affect a cell's propagation over the surface. For example, the trajectory in Fig.   1b frequently slows down or stops near the obstacles which can sterically impede the cell's progression until its direction of motion changes to point away from them. To quantify the influence of these interactions on the cells' motion as a function of ρ, we initially considered how efficiently the bacteria can propagate through a circular area of radius R = 25 µm ( Fig. 1b and Methods), i.e. whose radius is one order of magnitude longer than the typical cell's length. For all cells that propagate through any such area at a given ρ, we can assign an average effective propagation distance L eff ∈ [0, 2R] as a function of the obstacle density ( Fig. 1c and Methods). This quantity measures the average distance run by the cells when crossing the circular area: independently of the actual path taken by each trajectory  within the corresponding area, the two limit values of L eff respectively represent the cases where all cells exit from where they entered or at the diametrically opposite point. Fig.  1c shows that, without obstacles (ρ = 0%), L eff ≈ R. This value has a purely geometrical meaning as it closely corresponds to the length (≈ 24 µm) of the radical segment at the intersection between the circular area and the average circular trajectory (with radius R EC = v Ω = 13.7 µm) of the E. coli cells propagating within it when entering perpendicularly to the area perimeter. Counterintuitively, instead of hindering propagation as for non-chiral active particles [40], a slight increase in ρ (2% ≤ ρ ≤ 8%) allows bacterial cells to propagate over longer distances than on a smooth surface (with an ≈ 20% peak enhancement at ρ = 2%). The more intuitive behaviour, where L eff decreases for increasing ρ, is only observed at higher obstacle densities (ρ > 8%).
The previous result suggests that a few micro-obstacles have a beneficial effect on the capability of chiral bacteria to swim over large distances near surfaces, and only become detrimental at high densities. To account for differences in the time spent by the bacteria within an area for different obstacle densities, we also calculated the cells' normalised average effective speed V eff as a function of ρ ( Fig. 1d and Methods). This quantity shows a similar trend to L eff . Initially, for 2% ≤ ρ ≤ 4%, the cells propagate faster than on a smooth surface due to the increase in L eff (with an ≈ 12% peak enhancement at ρ = 2%). However, unlike L eff , V eff at ρ = 6% is already comparable with the value at ρ = 0% and rapidly decreases thereafter, as more frequent encounters with the obstacles increasingly prolong the cells' residence time within the area. These variations in V eff with ρ are also reflected in the spatial distribution of the cells on the surface (Fig. 1e): while at low obstacle densities (ρ = 2%) this distribution is basically uniform in space as for ρ = 0%, it becomes more heterogenous at higher obstacle densities, as localisation hot spots start to emerge in the proximity of the obstacles.
By analysing typical trajectories (Fig. 2a-e), we can qualitatively appreciate how cellobstacle interactions are directly responsible for the observed trends in L eff and V eff . As shown by the probability distributions of the change in effective propagation direction ∆θ eff the cells, therefore, predominantly propagate backward (∆Θ eff > 90 • in Fig. 2f). At low obstacle densities (ρ = 2% and ρ = 4%), sporadic cell-obstacle interactions are sufficient to rectify the cells' motion chirality (Fig. 2b), thus effectively making them propagate forward Fig. 2f), consistently with the observed enhancement in L eff and V eff (Fig.  1). While both L eff and ∆Θ eff point towards a minor rectification of the bacterial chirality for ρ = 6% and ρ = 8%, V eff is comparable with the value on the smooth surface as a consequence of an increased residence time due to cells stopping at the obstacles (Fig. 2c).
For even higher densities ( Fig. 2d-e), more frequent encounters with the obstacles increase the chances of cells turning backward and exiting near their entrance point, as also shown by ∆Θ eff , once again, becoming comparable to the value on a smooth surface (Fig. 2f); L eff and V eff are however significantly reduced with respect to the values for ρ = 0% as cell-obstacle interactions physically hinder cell propagation on the surface in space and time.
By visually inspecting the trajectories in Figs. 2 and S2, we can identify two repeated types of cell-obstacle interactions, which we respectively named "forward scattering" and "collisions" (Fig. 3a- We can quantify these observations by calculating three quantities during a cell-obstacle interaction: the relative change in speedṽ = v int vrun (Fig. 3c), where v int and v run are the average cell's speed during the interaction and the preceding run phase, the change ∆θ int in the cell's direction of motion pre-and post-interaction (Fig. 3d), and the interaction duration t int (Fig. 3e).
For collisions,ṽ is almost uniformly distributed in the range [0, 1] ( ṽ ≈ 0.61), ∆θ int shows a preference for cells leaving the obstacles in the opposite direction from that of approach, and t int follows a Poissonian distribution with a characteristic time (λ ≈ 1.33 s) comparable to the characteristic time of the E. coli cells' tumbling rate [9]. In a collision, therefore, the bacteria tend to stop at the obstacle until a tumble event points them away from it, thus validating the decrease in V eff at high ρ ( Fig. 1) as jointly due to a decrease in the cells' propagation distance L eff and an increase in their residence time due to the presence of the obstacles. This type of interaction becomes increasingly detrimental at higher obstacle densities as collisions become more probable (Fig. S3f), also because of colloids forming larger clusters (Figs. 2 and S2).
Contrarily, for forward scattering,ṽ follows a Gaussian distribution centred at ṽ ≈ 1, ∆θ int is strongly peaked forward, and the cells quickly leave the obstacles as t int follows a negative exponential distribution with a characteristic time (λ = 0.29 s) comparable to the time needed for the average cell to travel a distance equal to one obstacle's diameter. In a forward scattering event, therefore, the cells' speed and directionality are, on average, not significantly influenced by the obstacle during the interaction [42]. However, when leaving the obstacle, the cells' motion properties change: while the average translational speed ( v = 11.52 ± 3.87 µm s −1 ) only mildly increases with respect to the value at ρ = 0%, the cells' average angular speed Ω is significantly reduced, i.e., on average, the cells' motion becomes significantly less chiral. Fig. 3f shows the decorrelation of the cell's direction of motion θ over time calculated as where ... represents an ensemble average and t 0 is the first instant following the end of a cell-obstacle interaction (Methods). By fitting Eq. 1 to the function f (τ ) = cos(Ωτ )e −τ /τ 0 (Methods), we can indeed appreciate how, after forward scattering, the cells' average angular speed Ω is reduced to Ω 0 = 0.62 rad s −1 from Ω ∞ = 0.81 rad s −1 at ρ = 0% without, nevertheless, affecting the cell's motion persistence time (τ 0 ≈ 3.5 s in both cases). We thus hypothesise that this rectification of chirality during forward-scattering events is the microscopic reason behind the increase in L eff and V eff observed in Fig. 1 at small ρ, when this type of interaction is indeed predominant (Fig. S3f). Practically, this chirality rectification is due to an average increase of the cells' distance from the closest surface because of the hydrodynamic torque exerted on the cells swimming near the obstacles ( Fig. S4a-b) [14]. It is important to note that this is an average behaviour as, depending on which side the cells pass the obstacle, not all forward-scattering events will lead to such a change in height (Fig.   S4c). Interestingly, after collisions, the cells behave similarly to those swimming without obstacles (Fig. 3f), thus further confirming that, during collisions, the bacteria tend to stop at the obstacles before restarting their motion on the surface. Fig. 3g shows how Ω changes as the cells move away from the obstacles, gradually restabilising at Ω ∞ from Ω 0 following the exponential trend where t n is the n-th instant following the end of a forward-scattering event and τ Ω = 0.93 s (as fitted from the experimental data). In fact, as the cell changes its height, it approaches the sample chamber's other surface where it gets entrapped again (after a wobbling period [30]) until another forward scattering event induces a new change in height (Fig. S4).
In our experimental configuration, the effect of a forward scattering event on the cell's motion is, therefore, over after the cell has moved away from the obstacle by a distance To test the relative importance of forward-scattering events and collisions in determining the non-monotonic trends of L eff and V eff with increasing ρ, we considered a simple particlebased model that includes the two types of cell-obstacle interactions (Methods). Briefly, cells are modelled as chiral active particles, where the angular speed Ω depends on the distance to the closest obstacle (forward scattering) and the direction of motion is changed at random when the particle speed drops significantly (collision). Fig. 4a shows the good quantitative agreement between the experimental and simulated values of L eff , V eff and ∆Θ eff . In particular, the simulated distributions of the change in effective propagation direction ∆θ eff (Fig. 4b) confirm that the enhancement in V eff at low obstacle densities is due to the rectification of the active particles' chirality by the repulsive interaction with the obstacles, as previously hypothesised. Interestingly, the experimental behaviour in Figs. 1 and 2 is qualitatively preserved even when only considering forward-scattering events and excluding collisions (Fig. 4c): a few obstacles enhance the particles' propagation with respect to a smooth surface before hindering it at higher densities; however, without the further penalisation introduced by collisions, significant localisation effects only appear at higher obstacle densities than they would when collisions are considered. These numerical results, therefore, show how forward scattering is the primary mechanism of particle-obstacle interaction behind the non-monotonic trends of L eff and V eff with increasing ρ, with collisions mainly influencing this behaviour quantitatively rather than qualitatively. Finally, Fig. 5 shows how the behaviour observed in Figs. 1 and 2 is preserved over large propagation distances, both in experiments and simulations. The enhancement of the average effective propagation speed V eff at low obstacle densities can be observed across all areas whose diameter is larger than the average radius of curvature R EC of the chiral bacterial cells (Fig. 5a-b). For very small areas indeed (R = 5 µm, i.e. 2R < R EC ), cells propagate better in the absence of obstacles since these, like for non-chiral active colloids [40], disrupt their motion which is mainly directed forward (Figs. 5c and S5a). However, when R = 10 µm (2R > R EC ), the values of V eff at ρ = 0% and at ρ = 2% become comparable (Fig. 5a). For increasing R values, a clear peak in V eff can be observed around ρ = 2% (Fig. 5a-b) due to the rectification of the cells' chirality by the obstacles as shown by the persistent minimum in ∆Θ eff (Figs. 5c): even for R = 50 µm (i.e. when the area is approximately two orders of magnitude bigger than the typical cell's size), V eff at ρ = 2% is ≈ 20% higher than at ρ = 0% and the distribution of ∆θ eff is more uniform than at any other ρ value where these distributions are peaked backward (Fig. S5b). This long-range enhancement in cells' propagation due to a few obstacles is also confirmed by the higher Our results demonstrate the critical role played by surface defects on the near-surface swimming of bacterial cells. In particular, we show how cells' propagation near surfaces is significantly enhanced by individual forward scattering events due to a few microscopic obstacles of comparable size. The intuitive behaviour, where obstacles hinder propagation rather than enhancing it [31,39,40], is only recovered at higher obstacle densities due to cells' head-on collisions with the obstacles. Our results are corroborated by a numerical model based on chiral active Brownian particles cruising through random obstacles that confirms the universality of the experimentally observed behaviour, which is independent of the microscopic nature of the self-propulsion mechanism and of the repulsive interaction between particles and obstacles. Interestingly, for E. coli cells, as a consequence of a hydrodynamic torque, forward scattering events on the obstacles also lead the cells' trajectory to leave the surface, thus potentially offering a way to reduce escape times when swimming near it [23][24][25][26]. We envisage our results will help understand the individual and collective behaviour of chiral active matter in complex and crowded environments at all length scales [16]: examples include other microorganisms, such as microalgae and sperm cells [43,44], and macroscopic robotic swarms [45]. Beyond these fundamental interests, our findings can help design microfluidic devices to sort and rectify chiral active matter [16,18,[46][47][48].
Similarly, microstructured surfaces can be employed to better understand the emergence of bacterial social behaviours in natural habitats and to devise engineered materials to control and prevent bacterial adhesion to surfaces.  salt crystals and colloids that did not strongly adhere were washed away with DI water before drying the slide with nitrogen gas. Following this protocol, we were able to produce random distributions of fixed obstacles with different density values, 0% ≤ ρ ≤ 12%, on the same surface, where ρ is the fractional surface coverage of the colloids in a given region of interest (typically circular with radius R in our experiments). Finally, 10 µL of the bacterial suspension was deposited on the glass slide, which was subsequently sealed with the clean coverslip to form a chamber with spacing provided by the same colloidal particles. The size of the polystyrene microparticles was indeed chosen to guarantee, after sealing the chamber, a quasi-2D geometry for the bacteria to move in without the possibility of squeezing through the remaining gaps between two colloids in contact.

Experimental setup
All experimental observations were performed on a homemade inverted bright-field microscope enclosed in a custom-made environmental chamber (Okolab) with temperature control (T = 22 ± 0.5 • C). The microscope was mounted on a floated optical table for vibration dampening. The bacteria were tracked by digital video microscopy using the image projected by a microscope objective (x 20, NA = 0.5, Nikon CFI Plan Fluor) on a monochrome CMOS camera (1280 x 1024 pixels, Thorlabs DCC1545M) at 10 f.p.s. [50]. The magnification of our imaging path allowed us to achieve a conversion of 0.22 µm per pixel, corresponding to a field of view of ≈ 280 x 225 µm 2 . The incoherent illumination for the tracking of the bacteria was provided by a red LED (λ = 660 nm, Thorlabs M660L3-C2) employed in a Köhler configuration to control and improve coherence and contrast of the illumination at the sample plane. The typical duration of an experiment was ≈ 60 min before bacteria motility started to decrease considerably. In total, we recorded and analysed over 3500 individual bacterial trajectories of variable duration.

Estimation of the E. coli cells' average translational and angular speeds
We estimated the average translational speed, v , and the average angular speed, Ω , of the bacteria cells by taking an average of the individual speeds of 85 trajectories obtained on a smooth surface, i.e. for ρ = 0% (Fig. S1). To determine v , we first calculated the probability distribution of the instantaneous speed v for each trajectory, as exemplified in Fig. S1a. This distribution typically shows two peaks which we were respectively able to predominantly assign to a cell's tumble phase and its run phase, so that, by thresholding at the local minimum between the two peaks, the average translational speed of each trajectory could be estimated from the speed values associated with the run phase. To do so, we first segmented each trajectory in runs separated by tumbles (inset in Fig. S1a) following the procedure detailed in [51]. Briefly, after smoothing each trajectory with a running average over 5 time steps, the duration of individual tumbles was determined based on two dimensionless thresholds (α = 0.7 and β = 2), which were respectively used to determine sufficiently large local variations in instantaneous speed v and direction of motion θ. The numerical values of these two thresholds were validated against several trajectories by visual inspection. Similarly, to estimate Ω , we first calculated an angular speed Ω for each trajectory independently (Fig. S1b) and then averaged these values over all 85 trajectories. In analogy to the estimation of the persistence length of a polymer [52], each Ω was determined from the decorrelation of the cell's direction of motion θ over time fitting the following expression to the function f (τ ) = cos(Ωτ )e −τ /τ 0 cos(∆θ(τ )) = cos(|θ(t + τ ) − θ(t)|) where ∆θ is the angle between the tangents to the trajectory at times t + τ and t, the bar represents a time average, and τ 0 is the trajectory's persistence time. The direction of motion therefore decorrelates following an exponential decay, which is modulated by a cosine function when Ω = 0. Fig. S1b shows exemplary fits to the experimental data for three different values of Ω.

Estimation of the E. coli cells' average effective propagation distance, normalised speed and direction of motion
To calculate the average effective propagation quantities (L eff , V eff and ∆Θ eff ) of the bacterial cells, we first divided the entire field of view of all acquired experimental videos into M circular areas of radius R with centres on a square lattice of periodicity R. For example, for R = 25 µm as in Fig. 1b, M = 80 in our field of view. For statistics, based on its calculated obstacle density value, each circular area was then mapped on a discrete ρ scale with a 2 ± 0.6% separation step, and its trajectories were used to calculate the average effective propagation quantities of the corresponding ρ value on this scale. We excluded from the analysis all the trajectories (≤ 5% at any ρ) that did not exit a circular area after entering it and, to avoid biasing our results with extremely short trajectories, those that predominantly moved along the area perimeter, i.e. those that penetrated ≤ 10% of the area diameter without interacting with any obstacle. After smoothing with a running average over 5 time steps, we assigned an effective propagation distance eff = P out − P in to each of the remaining trajectories, where P in and P out are the trajectory's entrance and exit points respectively (Fig. 1b). This distance can take any value between 0 (the cell exits from where it entered) and 2R (the cell exits at the diametrically opposite point from where it entered). By averaging eff over all trajectories propagating through all circular areas of same ρ, we calculated the average effective propagation distance at different obstacle densities as L eff = eff . The normalised average effective propagation speed V eff as a function of ρ was instead calculated as V eff = eff v eff t eff , where, for a single cell, v eff and t eff are respectively its average translational speed when in run phase and its time of residence within the circular area. The normalisation by v eff makes different trajectories directly comparable, thus accounting for the fact that the intercell variability in translational speed can influence residence times. Finally, the average change in effective propagation direction as a function of ρ was calculated as ∆Θ eff = ∆θ eff = |θ(t out ) − θ(t in )| , where ∆θ eff is the angle between the tangents to a cell's trajectory when exiting and entering a circular area respectively.

Classification of cell-obstacle interactions as forward scattering or collisions
In order to distinguish between forward scattering and collisions, we first identified all cell-obstacle interactions along each trajectory. To simplify our analysis, we considered an interaction to take place only while there was a degree of overlap between the area occupied by an obstacle and the area occupied by the average cell body (centred along the trajectory and aligned with its direction of motion). Collisions were then identified out of this pool of interactions in analogy to the procedure for determining tumbles on a cell's trajectory as in Fig. S1a [51]. Briefly, after smoothing each trajectory with a running average over 5 time steps, individual collision events were selected based on two concomitant dimensionless thresholds (α = 0.7 and β = 2), which were respectively used to determine sufficiently large local variations in instantaneous speed v and direction of motion θ during the cell interaction with the obstacle with respect to the values preceding it (Fig. S3). All remaining interactions were classified as forward scattering events. We determined that, following this protocol, ≤ 10% of all interactions were wrongly attributed based on the visual inspection of 140 cell-obstacle interactions selected at random.

Numerical Model
We consider a numerical model where identical spherical active particles of radius d/2 move inside a two-dimensional square box of side B = R + 4.5 µm with periodic boundary conditions, where R is the variable radius of a circular area in the box centre. Within the circular area, we place uniformly distributed circular obstacles with variable densities ρ at random. The obstacles have the same size as the active particles. The trajectory of the i-th particle is then obtained by solving the following Langevin equation in the overdamped regime using the second-order stochastic Runge-Kutta numerical scheme [53] x i (t) = where x i (t) andû i (t) are respectively the active particle's position and direction of motion at time t, v i is its speed and γ is its friction coefficient in water. The direction of the particle's self-propulsion is defined by the unitary vectorû i (t) = [cos(θ i (t)), sin(θ i (t))], where θ i (t) is the particle's rotational degree of freedom given bẏ where Ω i and τ d are the active particle's angular speed and rotational diffusion time, and ξ i is a white noise process [54]. For simplicity, we describe the cell-obstacle interaction as a superposition of three contributions: a repulsive interaction, forward scattering and random reorientations upon collision. We modelled the first by introducing a repulsive force F i (r i , t) in the equation of motion. This force depends on the particle's distance r i from the nearest obstacle as wherer i is the unitary vector in the direction connecting the centres of the particle and the closest obstacle. This function was chosen to reproduce a strong (local) repulsive interaction between particle and obstacle, i.e. to mimic a hardcore potential. The exponential term ensures that the force does not increase too abruptly when approaching the obstacle. To model forward scattering (the second contribution), we introduced a position dependent angular speed Ω i given by where Ω i ∞ corresponds to the value of the particle's angular speed in the absence of obstacles and is a constant that sets a length scale for the interaction. Finally, any time the particle's speed drops below v i /100, a uniformly generated random angle ∈ [π/2, 3π/2] is added to θ i to better reproduce the experimental case of collisions (the third contribution). The values for the parameters in the simulations were chosen to closely reproduce the experimental values:

Calculation of the E. coli cells' average mean square displacement
For a given value of ρ, the calculation of the average mean square displacement (MSD) was performed as an ensemble average according to MSD(τ ) = MSD i (τ ) , where MSD i (τ ) = |x i (t + τ ) − x i (t)| 2 is the MSD of the i-th cell calculated from its trajectory x i (t) as a time average. The MSDs from simulations were calculated from individual trajectories whose translational and angular speeds were drawn from two Gaussian distributions respectively centred at v and Ω and with standard deviations that match the experimental ones.

Data Availability
Data and resources in support of the findings of this study are available from the corresponding authors upon reasonable request.