# Mechanisms for bacterial gliding motility on soft substrates

^{a}Department of Chemical and Biomolecular Engineering, University of California, Berkeley, CA 94720;^{b}Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research, Hyderabad 500107, India;^{c}Department of Biology, Texas A&M University, College Station, TX 77843;^{d}Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720

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Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved October 21, 2019 (received for review August 30, 2019)

## Significance

Gliding motility is the ability of certain rod-shaped bacteria to translocate on surfaces without the aid of external appendages such as flagella, cilia, or pili. This motility is crucial to their developmental cycle because it regulates their proliferation in the presence of nutrients or aggregation to form fruiting bodies in starvation conditions. Using myxobacteria as a canonical example of these organisms, we show that single-cell gliding is mediated by elastic, viscous, and capillary interactions between the bacteria, the slime it secretes, and the substrate underneath. Our theory reproduces well-measured speeds of *Myxococcus xanthus* cells on surfaces of varying stiffness and provides an explanation for the sensitivity of cell spreading to the substrate mechanics, a common feature across bacteria.

## Abstract

The motility mechanism of certain prokaryotes has long been a mystery, since their motion, known as gliding, involves no external appendages. The physical principles behind gliding still remain poorly understood. Using myxobacteria as an example of such organisms, we identify here the physical principles behind gliding motility and develop a theoretical model that predicts a 2-regime behavior of the gliding speed as a function of the substrate stiffness. Our theory describes the elasto-capillary–hydrodynamic interactions between the membrane of the bacteria, the slime it secretes, and the soft substrate underneath. Defining gliding as the horizontal translation under zero net force, we find the 2-regime behavior is due to 2 distinct mechanisms of motility thrust. On mildly soft substrates, the thrust arises from bacterial shape deformations creating a flow of slime that exerts a pressure along the bacterial length. This pressure in conjunction with the bacterial shape provides the necessary thrust for propulsion. On very soft substrates, however, we show that capillary effects must be considered that lead to the formation of a ridge at the slime–substrate–air interface, thereby creating a thrust in the form of a localized pressure gradient at the bacterial leading edge. To test our theory, we perform experiments with isolated cells on agar substrates of varying stiffness and find the measured gliding speeds in good agreement with the predictions from our elasto-capillary–hydrodynamic model. The mechanisms reported here serve as an important step toward an accurate theory of friction and substrate-mediated interactions between bacteria proliferating in soft media.

Across the diverse range of eukaryotic and prokaryotic cells, most bacteria are found living on surfaces rather than in solutions (1⇓⇓–4). As a result, phenomena pertaining to both single cells (motility, morphogenesis, cell division, etc.) as well as multicellular colonies (biofilm formation and growth, durotaxis, chemotaxis, streamers formations, etc.) are fundamentally related to the presence of a surface and its interaction with cells (4, 5). For instance, in the case of myxobacteria, a complex coupling between their intrinsic motility and their underlying substrate regulates their ability to form biofilms in the presence of nutrients or to aggregate into fruiting bodies in starvation conditions (6). Inspection of a swarm of myxobacteria, such as *Myxococcus xanthus*, reveals 2 types of motility: social (S)-motility or “twitching” involving type IV pili, and adventurous (A)-motility or “gliding” occurring without any external appendage (7). In order to explain the above-mentioned complex phenomena ranging from single-cell motility to emergent collective behaviors responsible for biofilm formation, bacteria–surface interactions (hydrodynamics, adhesion, biochemistry, etc.) must be understood.

In the present work, we seek to shed light on the physical principles behind gliding motility and the nature of the interaction between a gliding A-motile cell and its substrate. Indeed, it has been long reported that the spreading rate of a myxobacteria colony depends on the substrate stiffness (8, 9): an effect known as mechanosensitivity. However, while there exist some physical models for the mechanosensitivity of eukaryotic cells in tissues due to adhesion (10), the mechanism of myxobacterial gliding and its dependency on the substrate stiffness remains unclear mainly due to the existence of a thin slime layer secreted between the cell and the substrate (11). The physical approaches that have been previously undertaken to explain myxobacteria gliding are either on the molecular scale or continuum models on the scale of the whole cell (12). The former is primarily concerned with identifying genes, proteins, and molecular motors and their role in empowering a cell to glide (7, 13⇓–15). Here, our focus is not to elucidate the internal molecular mechanisms (15⇓⇓⇓–19) but rather to identify the physical principles governing the gliding motion of myxobacteria and how they interact with their environment. As such, our theory belongs to the class of models analyzing the bacteria at the cellular level.

To the best of our knowledge, the physical forces behind gliding of prokaryotes at the cellular level can be classified under 4 categories: osmotic forces, surface-tension gradients, slime secretion, and traveling waves. There are excellent reviews that describe each of these mechanisms for different gliding organisms (7, 20, 21). It suffices here to say that in the case of myxobacteria, 2 models are repeatedly invoked in attempting to explain the physical mechanism behind gliding at the cellular level. The first model, known as the slime-extrusion model, suggests that myxobacteria glide by secreting their slime via extrusion nozzles at the lagging pole, similar to a jet that propels through fuel ejection (12, 22). However, recent experiments have shown that the thrust-generating complex motors are distributed all along the cell rather than being concentrated at the lagging pole (23, 24). Moreover, slime was still found to be secreted underneath the surface of *M. xanthus* mutant cells, which are nonmotile, thereby showing that the production of slime does not necessarily lead to bacterial gliding (11). For these reasons and others (for a review, see ref. 7), the slime-extrusion model has now been disproved and is obsolete.

The second model, built at the scale of the entire cell, is based on waves propagating on the bacterial surface and the slime acting as a thin lubricating fluid. First developed for another gliding organism, namely Flexibacter (25), this model has been applied to myxobacteria with various complex rheological behaviors for the slime (26⇓–28). However, these studies all consider the substrate to be a rigid wall and are thus in essence unable to explain the mechanosensitivity feature of myxobacteria reported by many experiments (8, 9, 29). Moreover, all of the aforementioned studies as well as recent agent-based models (24, 30) prescribe the bacterial speed and, therefore, do not explain the physical mechanisms leading to gliding and self-propulsion. In this work, we focus on uncovering the physical mechanisms behind the gliding motion of myxobacteria without any appendages on soft and deformable substrates. In doing so, we identify 1) the physical nature of the forces between the bacteria and the surface, namely elastic, capillary, and hydrodynamic interactions; 2) the interplay between these forces leading to a nonzero gliding speed; and 3) how the speed depends on the substrate softness (mechanosensitivity).

## An Elasto-Hydrodynamic Mechanism Governs Bacterial Gliding on Mildly Soft Substrates

Our model is built upon 2 essential features established through previous experiments on myxobacteria. The first feature is the geometry of the cell’s basal surface that interacts with the substrate. As recently revealed through total internal reflection fluorescence (TIRF) microscopy experiments of *M. xanthus* cells (24), this basal surface possesses an oscillatory structure that we approximate by a sinusoidal shape *M. xanthus* cells display a helical pattern (32). Such shape deformation, which arises from the assembly and aggregation of the motility complexes at the so-called “focal-adhesion sites,” may therefore be a necessary condition for gliding (15, 24, 33). The second feature is that myxobacteria gliding is always accompanied by a trail of slime in the wake of the motile cells (11, 34). Using microscopy with wet surface-enhanced ellipsometry contrast, a thin layer of slime was observed underneath the basal cell surface of even nonmotile mutants (11). Slime deposition was thus suggested to be a general means for gliding organisms to adhere to and move over surfaces. Recently, the stick–slip dynamics of twitching *M. xanthus* cells was also well explained by understanding the slime to function both as a glue and as a lubricant (35). Here, we corroborate these findings and demonstrate that the slime acts as a crucial agent that not only lubricates the motion of myxobacteria but also creates a coupling between the cell and the deformable substrate.

### The Slime Film Lubricates the Bacterial Gliding Motion.

To model the slime-mediated interaction between myxobacteria and their substrate, we make the following assumptions. We consider small shear rates of the exopolysaccharide (EPS) slime and hence treat it as a newtonian viscous fluid, despite its polymeric constitution (11). The good comparison of our model with experiments will justify a posteriori that the nonnewtonian rheology of the slime has a second-order influence on the myxobacteria–substrate interaction. Furthermore, we neglect inertial effects given that for the characteristic speed *B*). The total thickness of the slime layer is thus *Materials and Methods* and *SI Appendix*, section 1.2)

### The Slime Equation Admits Traveling Wave Solutions Compatible with Bacterial Shape Deformations.

In the frame of reference translating with the cell, we consider the propagation of a traveling wave along the cell surface (24) as several recent experiments report gliding is strongly correlated with molecular motor complexes moving with helical trajectories on a scaffold provided by MreB actin (15, 17, 19, 24, 42). Some of these motor complexes were observed to slow down once they arrive at the basal cell boundary, in contact with the substrate. Due to their low velocity, when observed using regular microscopy, they appear almost stationary (15, 17, 23, 24), corresponding to a traveling disturbance on the cell surface in the reference frame translating with the bacteria. When viewed externally, the motors appear to drive the rotation of a helical structure that generates transverse waves on the ventral surface (15, 17, 24). Certainly, the slime-lubrication equation, given by Eq. **1**, admits such traveling wave solutions as we now establish.

Let us consider a membrane carrying a unidirectional traveling wave of speed C, such that the shape *x* axis is oriented positively to the right as in Fig. 1*B*, positive and negative values of C correspond to left and right traveling waves, respectively. To obtain traveling wave solutions of Eq. **1**, we search for a substrate-deformation field that also satisfies **1** can be rewritten as (*SI Appendix*, section 1.3)**3**, the pressure distribution in the slime layer depends (nonlinearly) on the substrate deformation

### The Substrate Surface Deforms Elastically during Bacterial Gliding Motion.

In many situations, the horizontal and vertical length scales of the substrate are on the order of centimeters and millimeters, respectively (43). Since both dimensions are much larger than the typical length of myxobacteria, we can represent the substrate as a semiinfinite medium. Moreover, gel substrates are generally viscoelastic, where the relative importance of viscous to elastic effects depend on the frequency of excitation of the material. Here, the characteristic frequency is that of the traveling disturbance *SI Appendix*, sections 2.1–2.5)**4**, *SI Appendix*, section 2.5). A motion in the horizontal direction would imply a slip velocity at the slime–substrate interface, in contradiction with the no-slip condition assumed earlier in the slime lubrication model Eq. **1**.

### The Elasto-Hydrodynamic Model Can Be Parametrized by a Single Nondimensional Variable: The Softness Number.

In order to compare the viscous and elastic forces at play in the problem, we rewrite the equations in their dimensionless forms. To that end, we first note that the characteristic deformation scale of the substrate reads *SI Appendix*, section 3). This can then be used to nondimensionalize the thickness of the slime layer to yield

For a given cell shape, η essentially captures the influence of the substrate deformability on variations of the lubrication gap during gliding. Therefore, it is also known as the softness parameter and be recast as (46, 47)*Materials and Methods* and *SI Appendix*, section 3).

### Bacterial Gliding Motion Occurs under Zero Lift and Drag Forces.

In order to obtain the slime pressure at different values of the softness parameter η, we specify 2 boundary conditions such that Eq. **3** admits a unique solution. We first set *SI Appendix*, section 4)**3** is a global condition. In the *SI Appendix*, section 4)**6** and **7**, together define the gliding motion of myxobacteria at the cellular level. Note that having imposed the shape of the bacteria, we can ignore the zero-torque condition, as commonly seen in the studies of swimming sheets (48⇓–50). An alternative would consist in solving for the bacterial membrane shape under the requirement that its bending and tensile stresses balance those due to the slime and those imposed by the internal motors (51). However, in such an approach, one must postulate the form of the torque applied by the traveling motors.

The system of coupled Eqs. **3**, **6**, and **7**, along with the condition *A*) indicate that the basal cell geometry can be approximated, in 2 dimensions, by a sinusoidal shape. Therefore, throughout this study, we consider sinusoidal basal shapes of the form

We solve the problem numerically for a given η and *SI Appendix*, section 5.4). A net gliding motion only then exists when the mean velocity, averaged over a wave period, is nonzero. Therefore, the results hereafter will be presented in their time-averaged form, expressed by the notation ⋅.

### First Result: Elasto-Hydrodynamics Dictates That the Gliding Speed Decreases to Zero as Substrates Become Softer.

Fig. 2 shows the average gliding velocity as a function of the softness parameter for different amplitudes of the basal cell shape. Clearly, the gliding velocity decreases with the softness parameter η. For very small values of η, the substrate is a very stiff solid and we recover the prediction of the speed of a periodic wavy sheet (Taylor’s swimmer; see ref. 48) near a rigid wall. As one could expect, the precision of this agreement improves with n, since the bacterial length increases (L being fixed) and hence its approximation by a periodic wavy sheet (*SI Appendix*, section 5.5). Indeed, it is known that in the lubrication regime near a rigid wall (

### Second Result: Bacterial Gliding on Mildly Soft Substrates Requires That Energy Be Converted from Shape Deformations to Viscous Slime Flow.

The vanishing nature of the gliding speed in the limit **7**, this force balance involves 3 contributions,

We show in Fig. 3 the time-averaged spatial distribution of the components *A*, *B*, and *C*, we provide Movies S1, S2, and S3, respectively (also see *SI Appendix*, section 5.7), illustrating the dynamics of the bacterial membrane, the slime pressure field, and the deformation of the substrate surface. Fig. 3 shows, as expected, that *SI Appendix*, section 5.7). This implies the thickness of the gap between the bacterial surface and the substrate oscillates and induces a lubricating flow of slime exerting on the cell-pressure oscillations in phase with the bacterial shape (*SI Appendix*, section 5.6). The resulting flow of slime exerts the thrust *SI Appendix*, sections 5.6 and 5.7 and Movie S3). Given that

However, this large-η behavior is not corroborated by our experiments, which instead show the gliding speed to be quasiconstant as the softness increases, so that *M. xanthus* cells glide even on extremely soft substrates (see Fig. 8). Such a remarkable feature shows that modeling the substrate as a pure elastic half-space breaks down for large values of the softness parameter.

## Motility on Very Soft Substrates Must Account for Capillary Effects

For very soft substrates, surface-tension effects can no longer be ignored in creating the substrate deformation *SI Appendix*, section 6.3).

### Elasto-Capillary–Hydrodynamic Model.

Capillary ridges are well-documented in the literature of soft solids (53⇓⇓⇓–57). Here, we consider that the growth of such a ridge creates a curvature of the slime–air interface, inducing a pressure difference at the leading edge of the cell. If the capillary effects are important, then the zero-pressure condition at the leading edge, **4**, now takes the form (*SI Appendix*, sections 6.1 and 6.2)**10**,

Furthermore, we can write the force balance at the capillary ridge (59) and use Laplace’s law across the slime–air interface to find the leading-edge pressure (*SI Appendix*, sections 6.3 and 6.4). In the limit of small deformations, we obtain the pressure at the leading edge in the form *Ca* is a capillary number comparing viscous to interfacial forces at the tip of the bacteria and defined by*Materials and Methods* and *SI Appendix*, section 6.5).

For extremely soft substrates, the substrate deformation is dominated by the slime–substrate interfacial tension so that the elasto-capillary length is much larger than the bacterial length, i.e., *SI Appendix*, section 6.5). In other words, when *Ca* is small, there exists at the leading edge a strong localized-pressure sink, as hypothesized. This pressure pulls a ridge with a characteristic height (*A*, *B*, and *C*, we provide Movies S4, S5, and S6, respectively (also see *SI Appendix*, section 8.7), illustrating the dynamics of the bacterial membrane, the slime pressure field, and the substrate surface deformation during motion.

### Third Result.

Bacterial gliding on very soft substrates is mediated by a localized strong pressure gradient near the leading edge of the cell

The comparison between Figs. 3*A* and 5*A* shows that for very stiff substrates, the full solution reduces to that obtained by considering only the elasto-hydrodynamic interactions. This conclusion is also supported by comparing the instantaneous dynamics of bacterial gliding on stiff substrates with and without consideration of capillary effects (Movies S1 and S4 and *SI Appendix*, sections 5.7 and 8.7). However, as the softness increases, Fig. 5 *B* and *C* shows the growing effect of the leading-edge pressure. While *SI Appendix*, section 8.7). In the capillarity-dominated regime (*SI Appendix*, section 6.5)**13**. In fact, a more detailed analysis of the governing equations in the limit of very soft substrates predicts *SI Appendix*, section 7). In other words, while the gliding speed is independent of the bacterial length in an elasticity-dominated regime, the gliding speed

## The Elasto-Capillary–Hydrodynamic Model Predicts the Experimental Gliding Speeds of *M. xanthus* Cells on Agar Substrates of Varying Stiffness

To test the validity of our theory, we cultured *M. xanthus* cells in a rich casitone yeast extract (CYE) medium, spotted cell suspensions on agar gel pads, and measured the average gliding speed of A-motile cells on gels of different concentrations of agar, corresponding to different stiffnesses (*Materials and Methods*). A good fit of collected data obtained by various authors (62⇓–64) shows that the increase of the shear modulus (G) with agar concentration (

Experimental data in Fig. 8 confirm the existence of a 2-regime behavior of the average gliding speed as a function of agar concentration. At low agar concentrations, i.e., at low substrate stiffness, the gliding motion is due to capillary effects localized at the tip of the bacteria whose velocity follows the prediction of the asymptotic speed given by Eq. **13**. As the concentration increases, the substrate gets stiffer and capillary effects at its surface decrease in favor of elastic ones in the bulk. The substrate being stiffer, it becomes harder to deform and causes the oscillations of the bacterial shape to be rather converted into variations of the slime pressure along the bacteria. Thus, there is a gradual switch in the nature of the gliding thrust, from a localized pressure gradient toward the slime–air interface to a distributed slime pressure over the bacteria. As the substrate becomes stiffer with the concentration, the slime pressure based thrust increases, leading to higher gliding speeds. Fig. 8 shows that beyond a critical agar concentration of *SI Appendix*, section 8.7), an animation showing a race between 2 cells gliding on substrates of softness number

Table 1 shows that most of the parameter values (ℓ, L, A, C, *M. xanthus* cells. However, there is lack of available data for slime properties measured directly underneath myxobacteria. To begin with, the slime thickness (5 nm) reported for *M. xanthus* cells is found in the wake of motile cells and not underneath (11). Therefore, it is very likely an underestimation of *Flexibacter*, another bacteria with gliding motility, whose slime thickness ranges between 10 and 25 nm (25, 26). Regarding the material parameters specific to the slime, such as viscosity (μ) and surface tensions (

To assess the robustness of our model to the parameter values listed in Table 1, we investigated how *SI Appendix*, section 9). Our sensitivity analysis shows that these variations of the parameter values not only result in minor quantitative changes of the

## Discussion

We have presented a model for the gliding of single myxobacteria cells and their underlying substrate. It appears that the dynamics of motor complexes can be modeled as a traveling wave, while the secreted slime serves as a lubricating film mediating the cell-substrate coupling. Our analysis shows that the mechanosensitivity of myxobacteria to their substrate results from their need to glide under drag-free and lift-free constraints. We find that satisfying these constraints can lead to different balances between the viscous, capillary and elastic forces depending on the substrate stiffness. This leads to a 2-regime behavior of the gliding velocity. On very soft substrates, the motility thrust is due to the existence of a localized capillary-induced pressure gradient toward the slime–air interface. However, for much stiffer substrates, it originates from oscillations of the slime pressure in phase with the shape deformations over the bacteria length.

As a final calculation, we can estimate the thrust T required for the propulsion of a single myxobacteria cell. On substrates with

In conclusion, the speed–stiffness relationship investigated in this work improves our understanding of friction and substrate-mediated interaction between bacteria in a swarm of cells proliferating in soft media (75). A crucial next step would be to consider the actual shape of the cell and its modification under the torque exerted both by the outside slime pressure and by the inner aggregation of motor complexes at the so-called focal-adhesion sites (15, 23). It is possible that binding and unbinding of focal-adhesion complexes to the substrate may produce a traveling wave disturbance on the outer membrane of the cell. These membrane deformations may be due to the differences in the distance between the cell and the substrate in the bound and unbound states (76). Tackling such problems will require determining stable 3-dimensional deformations of the bacterial curved surface using membrane mechanics (77) and could give insight in the shape–motility coupling of other rod-shaped cells. This will also enable direct connections with physical aspects of force transduction from the bacterial motors.

## Materials and Methods

### Governing Equations.

We model the bacterial slime as a newtonian viscous fluid and the substrate as a linear elastic solid (see *SI Appendix*, sections 1 and 2 for details). The slime being confined to a thin layer between the bacterial membrane and the substrate, its governing equations are obtained within the lubrication approximation. In the reference frame of the bacteria, the slime velocity component in the (horizontal) gliding direction reads (see *SI Appendix*, section 1.2 for derivation)**14** into Eq. **15** and integrating the resulting equation from the substrate surface (*SI Appendix*, section 1.2 for derivation) (41):**16**, the elasto-capillary–hydrodynamic problem of myxobacterial gliding on soft substrates is also governed by 1) an equation for the substrate deformation, 2) the lift-free and drag-free conditions, and 3) a force balance at the leading edge of the bacteria. Written in dimensionless form, these equations read respectively (*SI Appendix*, sections 6.1–6.4):**16** as Eq. **17a**. In the absence of capillarity effects, the governing equations for the elasto-hydrodynamic problem are obtained by taking the limit **17** above (*SI Appendix*, section 6.5).

### Numerical Treatment.

The Reynolds equation governing the viscous slime beneath the bacteria is a nonlinear equation as readily seen by the cubic power in Eq. **17a**. Therefore, we use an iterative Newton method to obtain the slime pressure field, the substrate deformation, the bacterial gliding speed, and the leading-edge pressure (*SI Appendix*, sections 5.1 and 8.1). This algorithm consists in starting with a guess solution *SI Appendix*, sections 5.1 and 8.1).

The linearized equations obtained at each Newton iteration are solved using the finite element method (78). We implemented this numerical method via FreeFem++ (79), a finite element solver that requires rewriting the governing equations in a weak form, discretizing the computational domain, expanding the unknown of the problem in a basis of shape functions, and inverting the resulting discrete linear system. All of the details regarding the weak form, the choice of shape functions, and the convergence verification of the discrete problem are given in *SI Appendix*, sections 5.1–5.3 and 8.1–8.3. Using the aforementioned method, we solve numerically, for given values of the set of parameters (η,

In order to illustrate the importance of the forces at play (Fig. 5) and the velocity as a function of the softness parameter (Fig. 7) in the presence of capillary effects, we chose the capillary number *SI Appendix*, Fig. S16, smaller values of *B*. Such highly confined ridges are challenging to accurately resolve numerically, as discussed in *SI Appendix*, section 8.3 on mesh convergence. Therefore, the value of *M. xanthus* parameters, listed in Table 1, we find a rather close value:

The codes used for our simulations have been deposited in the GitHub repository, https://github.com/mandadapu-group/myxoglide.

### Experimental Method.

TIRF microscopy images were captured using a Hamamatsu ImagEM X2 EM-CCD camera C9100-23B (pixel size, 160 nm) on an inverted Nikon Eclipse-Ti microscope with a *M. xanthus* cells were measured on CYE agar pads containing small to moderate concentrations of agar. For agar concentrations higher than

## Acknowledgments

We thank Amaresh Sahu and Yannick Omar for clarifying discussions and helpful comments on the manuscript. J.T. thanks Dr. Camille Duprat for helpful comments on elasto-capillarity aspects of the problem. C.B.P. and B.N. are supported by the National Institute of General Medical Sciences of the NIH under Grant R01GM129000. J.T. and K.K.M. are supported by NIH Grant R01GM110066.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: kranthi{at}berkeley.edu.

Author contributions: J.T., P.G., B.N., and K.K.M. designed research; J.T., C.B.P., B.N., and K.K.M. performed research; J.T., B.N., and K.K.M. analyzed data; J.T., B.N., and K.K.M. wrote the paper; and P.G. provided input for editing the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

Data deposition: The codes used for our simulations have been deposited in the GitHub repository, https://github.com/mandadapu-group/myxoglide.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1914678116/-/DCSupplemental.

Published under the PNAS license.

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- An Elasto-Hydrodynamic Mechanism Governs Bacterial Gliding on Mildly Soft Substrates
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