Downloaded by guest on September 24, 2021 Jo substrates on soft motility gliding bacterial for Mechanisms www.pnas.org/cgi/doi/10.1073/pnas.1914678116 known myxobacteria effect a an 9): of (8, rate stiffness substrate spreading has the it the on Indeed, depends that colony substrate. interaction reported its the and long of cell been nature A-motile the gliding a and between motility gliding behind ciples biochemistry, understood. adhesion, be must (hydrodynamics, etc.) interactions bacteria– emergent formation, to surface biofilm motility for com- responsible single-cell above-mentioned external behaviors from the collective any adventur- ranging explain without phenomena to (S)- and order plex occurring pili, social In “gliding” (7). IV motility: appendage or type of (A)-motility involving types ous “twitching” as 2 starvation such or myxobacteria, in reveals motility of bodies swarm xanthus, fruiting a into Myxococcus of Inspection aggregate presence (6). to the conditions underlying in or biofilms their nutrients with form to and of interaction ability complex motility their a its regulates myxobacteria, intrinsic and substrate of their case surface the between in a coupling instance, For of 5). presence (4, cells the well fundamentally duro- to are as growth, etc.) related formations, etc.) and streamers formation division, chemotaxis, (biofilm taxis, cell colonies sin- morphogenesis, multicellular both to as (motility, pertaining phenomena cells result, gle a As (1–4). solutions A elasto-capillary–hydrodynamics myxobacteria accurate an between media. toward interactions soft step in substrate-mediated proliferating and important bacteria friction an predictions of as theory the serve mea- with mechanisms here the The agreement reported find model. good elasto-capillary–hydrodynamic and our in stiffness from speeds varying isolated of gliding with substrates sured experiments agar perform leading on we bacterial theory, the cells our at gradient test in To pressure thrust localized edge. ridge a a a of creating of form thereby formation capillary the interface, the that to slime–substrate–air show lead the that we at considered however, be substrates, must soft effects propul- a very for thrust exerts On conjunction necessary the that sion. in provides pressure slime shape This bacterial bacte- of the length. with from flow bacterial arises motility the a along thrust of creating pressure the mechanisms deformations substrates, the distinct shape soft find 2 rial mildly we to force, On due net thrust. is zero gliding behavior under Defining translation slime 2-regime underneath. horizontal substrate the substrate the soft bacteria, as the the the and of of secretes, membrane function it the a between as elasto-capillary–hydrodynamic interactions the speed describes theory gliding Our 2-regime stiffness. the a predicts of that model behavior theoretical such a gliding of develop behind and principles example physical motility an the here as identify we myxobacteria organisms, Using remain understood. still exter- gliding poorly no behind a principles involves physical been gliding, The long as appendages. 2019) nal known has 30, motion, August prokaryotes review their certain for since (received mystery, of 2019 21, mechanism October motility approved and The MA, Cambridge, University, Harvard Weitz, A. David by Edited 94720 CA Berkeley, Laboratory, National Berkeley Lawrence India; Division, 500107, Hyderabad Research, Fundamental of a eateto hmcladBooeua niern,Uiest fClfri,Bree,C 94720; CA Berkeley, California, of University Engineering, Biomolecular and Chemical of Department lTchoufag el ntepeetwr,w ekt hdlgto h hsclprin- physical the on light shed to seek we work, present the In ¨ otbcei r on iigo ufcsrte hnin than rather surfaces on living cells, found prokaryotic are and bacteria eukaryotic of most range diverse the cross | ldn motility gliding a uhiaGhosh Pushpita , | mechanosensitivity b onrB Pogue B. Connor , c | eateto ilg,TxsAMUiest,CleeSain X783 and 77843; TX Station, College University, A&M Texas Biology, of Department lubrication c eynNan Beiyan , | doi:10.1073/pnas.1914678116/-/DCSupplemental at online information supporting contains article This 1 is ulse oebr2,2019. 25, November published First GitHub the in deposited been have simulations our for repository, used codes The deposition: Data the under Published Submission.y Direct PNAS a is article This interest.y K.K.M. competing and paper. no y B.N., the declare editing authors J.T., for The data; input analyzed provided P.G. K.K.M. and paper; and and the B.N., B.N., C.B.P., wrote J.T., J.T., research; research; designed performed K.K.M. K.K.M. and B.N., P.G., J.T., contributions: Author via slime their secreting of by model, case slime-extrusion glide the the cellu- myxobacteria as in the known that at that model, gliding suggests first say attempting behind The in mechanism to level. physical invoked lar here the repeatedly suffices explain are to It models organ- 21). 2 gliding myxobacteria, 20, different for (7, that mechanisms reviews isms these slime excellent of under are gradients, each There classified describe surface-tension waves. be traveling can forces, and level secretion, osmotic cellular categories: the at 4 prokaryotes of ing the analyzing models of level. class cellular environment. the the to at their bacteria belongs with theory interact our motion they such, gliding As how the and governing myxobacteria principles of physical rather to the but not identify (15–19) is focus mechanisms to our molecular Here, internal 13–15). the with (7, glide elucidate role concerned to their cell primarily and a is motors empowering in molecular former and The scale proteins, the (12). genes, on identifying cell models whole continuum the or scale of are molecular gliding the myxobacteria on explain either to have that undertaken approaches previously physical The between been unclear (11). secreted substrate layer remains the slime and thin stiffness cell a the of substrate gliding existence the the myxobacterial to due on of mainly mechanism dependency tissues the its in (10), and cells adhesion eukaryotic to of due mechanosensitivity physical the some for exist there models while However, mechanosensitivity. as owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To iiyo elsraigt h usrt ehnc,acommon a mechanics, sensi- substrate bacteria. the across the feature for to spreading explanation cell an of tivity provides and well- stiffness reproduces varying theory secretes, of Our speeds it underneath. measured slime capil- substrate the that and the bacteria, show viscous, and the elastic, we between by organisms, interactions mediated lary these is of gliding example single-cell myxobacteria form canonical Using to a conditions. aggregation as starvation or in nutrients bodies of prolif-fruiting presence external their the regulates of in it because eration aid crucial cycle is the developmental motility bacte- This their without pili. rod-shaped to or cilia, surfaces certain flagella, as on such of appendages translocate ability to the ria is motility Gliding Significance c otebs forkolde h hsclfre eidglid- behind forces physical the knowledge, our of best the To n rnh .Mandadapu K. Kranthi and , https://github.com/mandadapu-group/myxoglide.y PNAS | NSlicense.y PNAS eebr1,2019 10, December b etefrItricpiaySine,Tt Institute Tata Sciences, Interdisciplinary for Centre yoocsxanthus Myxococcus | a,d,1 . y o.116 vol. https://www.pnas.org/lookup/suppl/ | el nsrae of surfaces on cells d hmclSciences Chemical o 50 no. | 25087–25096

APPLIED PHYSICAL BIOPHYSICS AND SCIENCES COMPUTATIONAL BIOLOGY extrusion nozzles at the lagging pole, similar to a jet that propels A 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 rheolog- ical behaviors for the slime (26–28). However, these studies all B consider the substrate to be a rigid wall and are thus in essence unable to explain the mechanosensitivity feature of myxobac- teria 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 bacte- ria and the surface, namely elastic, capillary, and hydrodynamic interactions; 2) the interplay between these forces leading to a Fig. 1. (A) Epifluorescence and TIRF microscopic images of the surface nonzero gliding speed; and 3) how the speed depends on the of a single M. xanthus cell. The TIRF scan shows how the bacterial substrate softness (mechanosensitivity). surface has a quasiperiodic structure with a wavelength ∼1.2 µm. (B) Schematic description: a side view of a gliding bacterium (in gray) with An Elasto-Hydrodynamic Mechanism Governs Bacterial a sinusoidal basal shape that represents the bacterial surface undula- Gliding on Mildly Soft Substrates tions revealed by TIRF microscopy. The contact with the soft substrate Our model is built upon 2 essential features established through is lubricated by a thin film of slime. See the text for a description of the variables. previous experiments on myxobacteria. The first feature is the geometry of the cell’s basal surface that interacts with the sub- strate. As recently revealed through total internal reflection fluorescence (TIRF) microscopy experiments of M. xanthus cells substrate, we make the following assumptions. We consider small (24), this basal surface possesses an oscillatory structure that we shear rates of the exopolysaccharide (EPS) slime and hence treat approximate by a sinusoidal shape b(x, t) (Fig. 1). The TIRF it as a newtonian viscous fluid, despite its polymeric constitution experiments were carried out on cells expressing green fluo- (11). The good comparison of our model with experiments will rescence protein (GFP) in the cytoplasm and yet showed a justify a posteriori that the nonnewtonian rheology of the slime modulation of intensity with a period of L ∼ 1µm (24) (similar to has a second-order influence on the myxobacteria–substrate the periodicity of localization of MreB filaments; see refs. 19 and interaction. Furthermore, we neglect inertial effects given that 31). Given that GFP was evenly distributed in the cytoplasm, the for the characteristic speed V ∼ 2 µm/min (35), mean height TIRF images are thus evidence that the distance from the cyto- of the interstitial gap between the bacteria and the substrate plasm to the microscope glass is modulated. Images from atomic h0 ∼ 10 nm (11), and large viscosity of bacterial EPS µ ∼ 5 to force measurement also revealed that surfaces of single M. xan- 20 Pa.s (36, 37), the typical Reynolds number is Re  1 as in the thus cells display a helical pattern (32). Such shape deformation, locomotion of most microorganisms (38). Hence, we model the which arises from the assembly and aggregation of the motility dynamics of the slime using the Stokes equations (39). Given h complexes at the so-called “focal-adhesion sites,” may therefore the geometric ratio of the interstitial gap  := 0 ∼ 10−2  1, be a necessary condition for gliding (15, 24, 33). The second fea- L ture is that myxobacteria gliding is always accompanied by a trail we use the classical lubrication approximation (40, 41) for the of slime in the wake of the motile cells (11, 34). Using microscopy slime film and further simplify the Stokes equations. Moreover, with wet surface-enhanced ellipsometry contrast, a thin layer of ignoring the rotation of the myxobacteria along its long axis, we slime was observed underneath the basal cell surface of even confine the problem to 2 dimensions. Lastly, we account for the nonmotile mutants (11). Slime deposition was thus suggested to deformation δ(x, t) of the soft substrate, caused by the pres- be a general means for gliding organisms to adhere to and move sure in the lubricating slime (Fig. 1B). The total thickness of over surfaces. Recently, the stick–slip dynamics of twitching M. the slime layer is thus [h(x, t) − δ(x, t)], where h(x, t) = h0 + xanthus cells was also well explained by understanding the slime b(x, t). With these simplifications, the Stokes equations can be to function both as a glue and as a lubricant (35). Here, we cor- reduced to a modified form of the so-called Reynolds equa- roborate these findings and demonstrate that the slime acts as a tion (40, 41), governing the dynamics of the lubricating film of crucial agent that not only lubricates the motion of myxobacteria slime and given by (Materials and Methods and SI Appendix, but also creates a coupling between the cell and the deformable section 1.2) substrate. ∂ ∂  1 ∂p 1  The Slime Film Lubricates the Bacterial Gliding Motion. To model − (h − δ) + (h − δ)3 + V (h − δ) = 0, [1] the slime-mediated interaction between myxobacteria and their ∂t ∂x 12µ ∂x 2

25088 | www.pnas.org/cgi/doi/10.1073/pnas.1914678116 Tchoufag et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 etos2.1–2.5 ) sections film Appendix, the (SI the of by action framework, given the is this under pressure In surface by instant. substrate found the every of be at deformation problem can static solution quasistatic a the time-dependent solving make the half- and that elastic of deformations) assumption context purely (small the a in elasticity problem as linear the (44). substrate formulate magnitude we the of Furthermore, orders treat than space. more first smaller or remain we one to Therefore, by expected modulus is storage modulus the loss ( the concentrations high quencies, agar to of case moderate are the at which In gels complexes. (17), motor molecules the in AlgR components slow essential the of speed experimental of is Hz frequency frequency 0.32 disturbance the characteristic traveling on the the of Here, depend that material. effects the impor- elastic rep- of gel relative to excitation can Moreover, the viscous medium. where we of viscoelastic, semiinfinite tance myxobacteria, generally and a of are as centimeters substrates length substrate of typical the order much resent the are the dimensions than on both larger Since are (43). substrate respectively millimeters, the of Gliding Bacterial scales during Elastically Deforms Motion. Surface Substrate The deformation determine. to substrate out set the now we on which (nonlinearly) depends Eq. to According cofge al. et Tchoufag the of 1.3) that section Eq. as Appendix, ansatz, speed this same Using the surface. of bacterial waves traveling exhibits mation satisfies solutions wave Eq. traveling obtain of To respectively. of waves, values traveling negative and positive the Assuming speed of wave establish. slime-lubrication now the we Certainly, Eq. 24). by given 17, the equation, on (15, drive waves surface transverse to generates ventral that appear the structure bac- motors helical the a the with of externally, rotation translating frame viewed on reference disturbance When traveling the teria. a in to surface corresponding cell 24), station- the 23, almost 17, appear (15, when they velocity, ary bound- microscopy, low cell their regular to basal using were Due the substrate. observed complexes at the motor with arrive contact these they in of once ary, down Some slow 42). MreB to 24, by observed 19, provided 17, complexes scaffold (15, a motor on actin molecular trajectories report helical with experiments with correlated recent moving strongly several as is wave (24) traveling gliding a surface of cell propagation the the consider along we cell, the with ing Deformations. with Shape Compatible Bacterial Solutions Wave Traveling Admits Equation for. Slime solved The be to remains that bacteria the of velocity where e scnie ebaecryn ndrcinltraveling unidirectional a carrying membrane a consider us Let p esac o usrt-eomto edta also that field substrate-deformation a for search we 1, hr h siae aesedcrepnst the to corresponds speed wave estimated the where , nmn iutos h oiotladvria length vertical and horizontal the situations, many In (x ∂δ ∂ δ ∂ (x t , ∂ x t = , )  t x C stepesr ntesieand slime the in pressure the is C = ) ∂ ∂ xsi retdpstvl otergta nFg 1B, Fig. in as right the to positively oriented is axis x p uhta h shape the that such , ∂ ∂δ x (h h rsuedsrbto ntesielayer slime the in distribution pressure the 3, (1 hr easm httesbtaedefor- substrate the that assume we where , π − − G dissc rvln aesltosas solutions wave traveling such admits 1, δ ν ) ) 3 ∂ Z ∂ 6 + h −`/2 t `/2 ntefaeo eeec translat- reference of frame the In = µ(V C C p f (x = ∂ ∂ − orsodt etadright and left to correspond h x 0 C , . h 2C t /L (x ln ) 1 )(h , ≥ a erwitna (SI as rewritten be can ≈ t |x ) n tsc fre- such at and 1%) 0. − satisfies − x 32µm δ 0 V x )  0 | 0 = stegliding the is dx .s −1 0 . . = /1µm δ (x , [3] [2] [4] t ), en h ldn oino yoatraa h ellrlevel. cellular the at myxobacteria of motion gliding the Eqs. define i.e., constraints, drag-free and lift-free The otrsbtae.Uigteaoedmninesprmtr and of parameters defining dimensionless values above the Using larger substrates. softer increasingly that such Here, rvdsa qain hc utb netdt banthe obtain to inverted be must which speed gliding equation, an the Eq. In for provides condition. condition global boundary second a the consequently, and, Appendix, (SI lubri- 4) reads the section In condition vanish. lift-free vanish must this pressure must approximation, film cation the cell from the force internal lift on the the the that In forces law. from external second result Newton’s the to gliding according of for sum forces the Stokes Sec- the motors, the pressure. in that inertia (zero) leading and without atmospheric the regime move the myxobacteria at to that pressure given drops slime ond, cell the the that of assume edge we words, other soft- the Eq. of values different at parameter Forces. pressure ness Drag slime and the Lift obtain Zero to order under Occurs Motion Gliding Bacterial and (Materials form substrate dimensionless Methods the a in and recast film be slime can the deformation for equations governing the `/L, softness the as known 47) (46, also lubrication as is recast the be it of and Therefore, parameter variations gliding. on during deformability gap substrate slime the the of to film. the due of thickness substrate mean the the to pressure of deformation characteristic slime the of Here, thickness yield the can to nondimensionalize This layer to 3). the used section of be Appendix, (SI then scale scale deformation pressure characteristic characteristic we the problem, reads end, that the that substrate in note To play forms. first dimensionless at their we forces in equations elastic the and rewrite Single viscous a the by Number. pare Softness Parametrized The Be Variable: Can Nondimensional Model Elasto-Hydrodynamic The assumed Eq. condition model no-slip lubrication slime the the in with earlier hori- contradiction slime–substrate the the in in at velocity vertical motion interface, slip a A the imply 2.5). in would direction section occurs zontal Appendix, surface (SI substrate only the direction of deformation the which for substrates, pressible ratio (45), Poisson substrate and the on point displacement and zero the sets that stant o ie elshape, cell given a For G 3 − ` disauiu ouin efis set first We solution. unique a admits η = Z stelnt fteetr atra nEq. In bacteria. entire the of length the is −n ∆/h = and E p ˆ n /2 = ( + /2(1 /2 eto 3). section Appendix, SI p

V (h /P PNAS 0 p ˆ η eto 4) section Appendix, (SI (1 = ∆ ∂ sadmninesnme htcmae the compares that number dimensionless a is ∂ − , eseiy2budr odtossc that such conditions boundary 2 specify we , ν ν x b ˆ ˆ x oeta ersrc u nlsst incom- to analysis our restrict we that Note . ˆ ) δ | + = ) stesermdlsfrYugmodulus Young for modulus shear the is = Z eebr1,2019 10, December −n 2 1 x − n η h /L, ∂ ∂ /2 0 /2 = ν x p ˆ ˆ  x η ˆ )PL/G p ˆ  h (1 ˆ ˆ t (ˆ ieto,teda-recondition drag-free the direction, setal atrsteinfluence the captures essentially h ˆ − x = − ν Gh − , η t ˆ t 1 = ν δ /(LC ˆ η )dˆ 0 )PL where ,  δ ˆ where ,  x nodrt nuethat ensure to order in /2, + 0, = | , h ˆ ), z ˆ o.116 vol. η 1. − V ieto,ti implies this direction, ˆ V ˆ P eoeincreasingly denote η h ˆ δ ˆ = = = nodrt com- to order In p ˆ ! 6 V (n µCL/h | h and dˆ /h 4, /C x o 50 no. /2, 0 = 0 x together 7, , and , 0 ˆ t δ ˆ In 0. = ) 0 sacon- a is 2 . = | sthe is δ/ 25089 n 3 [7] [5] [6] ∆. In E = is

APPLIED PHYSICAL BIOPHYSICS AND SCIENCES COMPUTATIONAL BIOLOGY Note that having imposed the shape of the bacteria, we can ignore the zero-torque condition, as commonly seen in the stud- ies 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 pˆ(n/2, ˆt) = 0, admits a unique solution q(ˆx, ˆt) := [ˆp(ˆx, ˆt), δˆ(ˆx, ˆt), Vˆ (ˆt)]T for a given softness number η and a given geometry of the basal cell shape. In the case of myxobac- teria, images of myxobacteria obtained with TIRF microscopy (Fig. 1A) indicate that the basal cell geometry can be approx- imated, in 2 dimensions, by a sinusoidal shape. Therefore, throughout this study, we consider sinusoidal basal shapes of the  2π  form b(x, t) = A sin L (x + Ct) , corresponding to the dimen- sionless height hˆ(ˆx, ˆt) = 1 + Aˆ sin 2π(ˆx + ˆt), where the ampli-

tude is nondimensionalized by h0. As such, it must satisfy Aˆ ∈ [0, 1] for a thin film to exist between the cell and the substrate. Fig. 2. Time-averaged gliding speed as function of the softness parameter We solve the problem numerically for a given η and hˆ(ˆx, ˆt) for different amplitudes of the bacterial shape. Here, we consider a bacterial and obtain q(ˆx, ˆt) at different instants. Due to the time period- shape with 5 wavelengths, i.e., n = 5. The gliding speed increases with the substrate stiffness and with the amplitude of the bacterial shape. icity of the bacterial membrane, we expect q(ˆx, ˆt) to be periodic, hence the gliding speed as well (SI Appendix, section 5.4). A net gliding motion only then exists when the mean velocity, averaged balance that must be satisfied for gliding to occur. Defined by Eq. over a wave period, is nonzero. Therefore, the results hereafter 7, this force balance involves 3 contributions, I1(ˆx, ˆt), I2(ˆx, ˆt), will be presented in their time-averaged form, expressed by the and I3(ˆx, ˆt), given by notation h·i. ∂bˆ 1 ∂pˆ   −Vˆ First Result: Elasto-Hydrodynamics Dictates That the Gliding Speed I1 = −pˆ , I2 = − hˆ − ηδˆ , I3 = . [9] Decreases to Zero as Substrates Become Softer. Fig. 2 shows the ∂xˆ 2 ∂xˆ hˆ − ηδˆ average gliding velocity as a function of the softness parameter for different amplitudes of the basal cell shape. Clearly, the glid- Firstly, I1 represents the action of the slime pressure on the bac- ing velocity decreases with the softness parameter η. For very teria induced by variations of the bacterial geometry. Hence, it small values of η, the substrate is a very stiff solid and we recover vanishes for bacteria with spatially uniform shapes. Secondly, I2 the prediction of the speed of a periodic wavy sheet (Taylor’s denotes the effect of pressure variations along the bacteria and swimmer; see ref. 48) near a rigid wall. As one could expect, would vanish in the absence of pressure gradients. Lastly, I3 con- the precision of this agreement improves with n, since the bacte- stitutes the resistance experienced by the bacteria as it must shear rial length increases (L being fixed) and hence its approximation the slime to achieve gliding. by a periodic wavy sheet (SI Appendix, section 5.5). Indeed, it is We show in Fig. 3 the time-averaged spatial distribution of known that in the lubrication regime near a rigid wall (1/η → ∞), the components I1, I2, and I3 along the length of the bacte- small amplitude wavy membranes have a swimming velocity ria, for 3 values of the softness parameter η ∈ {0.001, 5, 1000}. given by (50, 52): For the input parameters corresponding to Fig. 3 A, B, and C, we provide Movies S1, S2, and S3, respectively (also see SI D E 3 Appendix, section 5.7), illustrating the dynamics of the bacte- Vˆη=0 = , [8] 2 + 1/Aˆ 2 rial membrane, the slime pressure field, and the deformation of the substrate surface. Fig. 3 shows, as expected, that hI3i is which corresponds to the asymptotic values of the small-η always negative and contributes to the friction experienced by hI i plateaus in Fig. 2. Since Aˆ = A/h0, increasing the dimensionless the bacteria. Next, regarding the component 2 , its integral amplitude is equivalent to decreasing the mean gap h0. There- over the bacterial basal shape is negative and thus constitutes fore, we recover the well-known result that the locomotion speed an additional contribution to friction. Lastly, Fig. 3 shows the of a wavy sheet increases as it approaches a rigid boundary (52). term hI1i is positive over the length of the bacteria and provides In our context, this conclusion is equivalent to stating that on the necessary thrust balancing the Poiseuille and Couette fric- stiff but elastically deformable substrates, the gliding motility tions from hI2i and hI3i, respectively. Fig. 3 therefore reveals speed increases with the deformation amplitude of myxobacte- the decrease of the gliding speed with the softness number η rial membrane. However, as we increase the softness parameter, is correlated to that of the local thrust generation along the the gliding velocity transitions from a nonzero value for very bacteria. This can be explained in the following manner. For stiff substrates to zero in the limit of very soft substrates. For all small values of η, the substrate is very stiff and remains almost amplitudes Aˆ, Fig. 2 shows that substantial deviations from the unperturbed by the oscillations of the bacterial surface (Movie SI Appendix rigid wall solution occur when the substrate number η ≈ 1, which S1 and , section 5.7). This implies the thickness of the corresponds to a substrate whose deformation is comparable to gap between the bacterial surface and the substrate oscillates and that of the slime thickness. induces a lubricating flow of slime exerting on the cell-pressure oscillations in phase with the bacterial shape (SI Appendix, sec- Second Result: Bacterial Gliding on Mildly Soft Substrates Requires tion 5.6). The resulting flow of slime exerts the thrust hI1(ˆx)i That Energy Be Converted from Shape Deformations to Viscous Slime along the bacteria, which leads to a nonzero gliding speed hVˆ i. Flow. The vanishing nature of the gliding speed in the limit Therefore, this mechanism of thrust generation, whereby the η → ∞ can be understood by analyzing the horizontal force shape oscillations are converted into a lubrication flow of slime,

25090 | www.pnas.org/cgi/doi/10.1073/pnas.1914678116 Tchoufag et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 e u odsrb.Floigtewr yLmt(8 and (58) deforma- Limat substrate 6.2) and by the 6.1 Eq. that sections work by show given the we formerly (59), tion, Following Limat describe. and edge, Dervaux to now leading out we as the set framework, impor- elasto-capillary–hydrodynamic at an are condition within effects at capillary zero-pressure the difference curvature p the If pressure a cell. then a creates the of tant, ridge inducing edge a we interface, leading such Here, the slime–air (53–57). of the solids growth of soft the of that literature consider the in documented Appendix, of Model. (SI balance Elasto-Capillary–Hydrodynamic the line by triple determined substrate–slime–air “ridge” shape 6.3). the section a a substrate, at takes inter- soft slime–air tensions which the the from 4), fact, In generate, (Fig. bacteria. critical can the tension particularly of facial is edge leading substrate deformation the the at substrate longer of the no balance creating can elasto-capillary in effects ignored surface-tension be substrates, soft very For Capillary for Account Must Effects Substrates Soft Very softness on Motility the a of values Such large 8). for pure Fig. a down parameter. as (see breaks substrate half-space the substrates modeling elastic soft that that shows extremely qua- feature so remarkable be on increases, to even softness speed glide the gliding the as show siconstant instead which experiments, limit the speed gliding the There- (as well. as zero as terms zero gradient pressure-dependent pressure to the vanishing fore, decays a to thus leading bacteria thereby the along distribution instan- that 5.7 and so and 5.6 surface compliant bacterial very oscillating is η the substrate it to when the such, conforms substrates, limit, As taneously soft this substrate. very in the for fact, of sustained deformation be no cannot to little requires η i.3. Fig. cofge al. et Tchoufag ˆ AB δ (n ˆ = oee,ti large-η this However, ≈ δ 0 vr tf usrt)(A), substrate) stiff (very 0.001 /2, (x b ˆ sections Appendix, (SI computations our by supported as , , ieaeae itiuino h contributions the of distribution Time-averaged ˆ t t olne od.Ti rsuems eobtained be must pressure This holds. longer no 0, = ) ) ∼ η ≈ ∞. → 1/η and (1 π ntelmto eysf usrts saresult, a As substrates. soft very of limit the in ) − .Gvnthat Given S3). Movie G ν h V ) ˆ Z i −`/2 ftefrefe atratnst eoin zero to tends bacteria force-free the of `/2 eairi o orbrtdb our by corroborated not is behavior p o ae h om( form the takes now 4, (x η 0 = , t ,and (B), substrate) soft (mildly 5 ln ) hI |x ailr igsaewell- are ridges Capillary 1 p ˆ (ˆ − x ∼ )i x δ ˆ 0 x | and ≈ 1 + b ˆ .xanthus M. ` /η I s 1 hI /π , h pressure the , I 2 2 IAppendix, SI (ˆ δ ˆ and , dx x (ˆ η x )i In ∞. → 0 , . ˆ t η edto tend I 3 The ). ∞, → otehrzna oc aac o ifrn ausof values different for balance force horizontal the to [10] cells η = 00(eysf usrt)(C substrate) soft (very 1000 lm,adarsbtaeitrae.Setetx o ecito fthe of description a for substrate– text slime–air, the the See at interfaces. variables. tensions is air–substrate surface ridge and A the shape. slime, by ventral sinusoidal balanced a and with gray) formed (in bacterium gliding a lem: 4. Fig. substrate number the elasto-capillarity of the values substrates, all stiff for very For effects softness. capillary and elastic both function The form at the pressure in the obtain edge and we leading deformations, 6.3 small the interface sections of Appendix, limit slime–air the (SI the In pressure 6.4). across leading-edge law the Laplace’s find to use and (59) ridge stresses. capillary) (resp. elastic by governed are deformations of values large) (resp. where Rewriting number substrate. elasto-capillarity the the of obtain as over bulk defined we dominate form, the dimensionless interface in in stresses slime–substrate elastic the the at stresses and illary (45), substrate the on Eq. In sacplaynme oprn icu oitrailfre at by forces defined interfacial and bacteria to the viscous of comparing tip the number Lastly, capillary 61). a (56, thickness is interfacial slime–air and the tensions, of measure interfacial slime–substrate and slime–air the relation, utemr,w a rt h oc aac ttecapillary the at balance force the write can we Furthermore, γ 10, ceai ecito fteeat-ailr–yrdnmcprob- elasto-capillary–hydrodynamic the of description Schematic s  stesiesbtaesraetnin hrfr,small Therefore, tension. surface slime–substrate the is x stelbiainparameter, lubrication the is 1 P sacntn htst h eodslcmn point displacement zero the sets that constant a is PNAS enn h edn-depesr,acut for accounts pressure, leading-edge the defining , .Hr,teiptprmtr are parameters input the Here, ). | ξ C eebr1,2019 10, December ξ = orsodt usrtswoesurface whose substrates to correspond ` L s Ca ` s = salnt cl eo hc cap- which below scale length a is p ˆ = 2γ (n µC s )= /2) γ (1 GL . − η | h otesprmtr are parameters softness The . ν P R ) o.116 vol. (, , = R γ/γ ˆ , A a ˆ s | , = ξ stertoof ratio the is o 50 no. 5and 0.25 , nthis In Ca). | a ˆ n 25091 [11] [12] = sa is Ca ` 5. s

APPLIED PHYSICAL BIOPHYSICS AND SCIENCES COMPUTATIONAL BIOLOGY ˆ (1 − α) ξ → 0 and we recover pˆ(n/2, t) ≈ 0, as previously assumed in Vˆη=∞(ˆt) ≈ 2 − ∆ˆp + α, [13] the elasto-hydrodynamic model void of capillary effects. Like- β wise, the substrate deformation converges to that of a purely elastic half-space (Materials and Methods and SI Appendix, where ∆ˆp is the pressure difference between the leading and section 6.5). trailing edges of the cell, while α and β are factors related to For extremely soft substrates, the substrate deformation is the total slime layer thickness (hˆ − ηδˆ). Since the leading-edge dominated by the slime–substrate interfacial tension so that the pressure is inversely proportional to the capillary number, so is elasto-capillary length is much larger than the bacterial length, the asymptotic velocity Vˆη=∞, according to Eq. 13. In fact, a i.e., ξL  nL. Using this information, we find in particular that more detailed analysis of the governing equations in the limit the tip pressure is negative and inversely proportional to the cap- ˆ 1/3 −1/3 of very soft substrates predicts hlc i = hlc /h0i ∼ R Ca and illary number (SI Appendix, section 6.5). In other words, when hVˆ i ∼  n−1Ca−1 (SI Appendix, section 7). In other words, Ca is small, there exists at the leading edge a strong localized- R while the gliding speed is independent of the bacterial length pressure sink, as hypothesized. This pressure pulls a ridge with a in an elasticity-dominated regime, the gliding speed hVˆ i scales characteristic height (∆r ) and extent (lc ) as indicated in Fig. 4. with the bacterial length as 1/n when capillary effects dominate. The length scale lc hence represents a threshold below which the This scaling law is obtained by balancing the localized capillary localized capillary pressure is significant. Assuming lc  L, the force-free condition in the gliding direction reduces to the force at the slime–air interface and the viscous friction over the entire bacteria. Although the thrust results from the capillary- balance between I2 and I3, that is, between the Poiseuille- like and Couette-like forces. This conclusion is supported by induced pressure at the leading edge and is thus essentially the simulations of the full elasto-capillary–hydrodynamic equa- independent of bacterial length, the friction coefficient, given by ˆ tions as shown in Fig. 5, which shows the contributions of I3/V , depends on the cell geometry and increases with n. Hence, I1, I2, and I3 to the horizontal force balance. For the input the resulting velocity decreases with n. The scaling laws given parameters corresponding to Fig. 5 A, B, and C, we provide above are confirmed by computations of the full elasto-capillary– Movies S4, S5, and S6, respectively (also see SI Appendix, sec- hydrodynamic equations, as shown in Figs. 6 and 7. As expected, −1/3 −1 tion 8.7), illustrating the dynamics of the bacterial membrane, we find hˆlc i ∼ Ca and hVˆ i ∼ Ca when the softness num- the slime pressure field, and the substrate surface deformation ber is very large. The contrast between the elasticity-dominated during motion. and the capillarity-dominated regimes is emphasized in Fig. 7, which shows that hVˆ i is independent on the wavenumber for very Third Result. Bacterial gliding on very soft substrates is mediated stiff (small η) substrates, while hVˆ i ∼ 1/n for very soft (large η) by a localized strong pressure gradient near the leading edge of substrates. the cell The comparison between Figs. 3A and 5A shows that for very The Elasto-Capillary–Hydrodynamic Model Predicts the stiff substrates, the full solution reduces to that obtained by con- Experimental Gliding Speeds of M. xanthus Cells on Agar sidering only the elasto-hydrodynamic interactions. This conclu- Substrates of Varying Stiffness sion is also supported by comparing the instantaneous dynamics To test the validity of our theory, we cultured M. xanthus cells of bacterial gliding on stiff substrates with and without consid- in a rich casitone yeast extract (CYE) medium, spotted cell eration of capillary effects (Movies S1 and S4 and SI Appendix, suspensions on agar gel pads, and measured the average glid- sections 5.7 and 8.7). However, as the softness increases, Fig. 5 ing speed of A-motile cells on gels of different concentrations of B and C shows the growing effect of the leading-edge pressure. agar, corresponding to different stiffnesses (Materials and Meth- While I3 still contributes to a Couette-like friction, the term I2 is ods). A good fit of collected data obtained by various authors now responsible for a nonzero thrust, largely due to the localized (62–64) shows that the increase of the shear modulus (G) with strong pressure gradient at the tip of the myxobacteria (Movie agar concentration (Cagar) can be approximated by the empir- 2 S6 and SI Appendix, section 8.7). In the capillarity-dominated ical law: G ≈ 20(Cagar − 0.1) kPa. Such power laws are more regime (n  ξ → ∞), we derive an asymptotic expression for the rigorously derived in the percolation theory for gels (65). After gliding speed on very soft substrates Vˆη=∞, as (SI Appendix, recording and postprocessing the myxobacteria motion for a section 6.5) sample of m = 40 cells, we obtained the mean and SE of the

A B C

Fig. 5. Time-averaged distribution of the contributions I1, I2, and I3 to the horizontal force balance for different values of η. The softness parameters are η = 0.001 (very stiff substrate) (A), η = 5 (mildly soft substrate) (B), and η = 1000 (very soft substrate) (C). Here, the input parameters are Aˆ = 0.25, n = 5, ˆa ≈ 3.14 × 10−3, and Ca ≈ 1.67 × 10−3.

25092 | www.pnas.org/cgi/doi/10.1073/pnas.1914678116 Tchoufag et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 ˆ cofge al. et Tchoufag h edn deweetesoeo h usrt rtvnse,i.e., vanishes, first substrate the a the of of values slope different the for where (B), edge speed gliding leading the the Here, and parameter. (A) softness ridge the of 6. Fig. ilgiigsedicesscnieal.I re oillustrate to order In considerably. increases of shows concentration speed 8 agar gliding Fig. critical rial speeds. a gliding beyond higher substrate that to based the leading pressure slime As a increases, the from thrust bacteria. concentration, a the thrust, the with to gliding stiffer over interface becomes the pressure slime–air of slime the nature toward distributed the there gradient in Thus, into pressure bacteria. switch localized converted the gradual rather along a be pressure is slime to sub- the the shape The causes of and bacterial bulk. variations deform the the to its in of harder at becomes ones oscillations it effects elastic stiffer, of capillary being favor strate and in stiffer decrease prediction gets surface Eq. the substrate by follows the given velocity increases, speed whose localized asymptotic bacteria effects the the capillary of of to tip due the substrate is at low motion at gliding i.e., the concentrations, stiffness, agar agar of low function a At as speed concentration. gliding average the of behavior regime show use 1. we Table data when in experimental model given our values The of parameter prediction 8. the the Fig. with agreement in good shown a as speed gliding d d ≈ h B A ˆ δ x xeietldt nFg ofimteeitneo 2- a of existence the confirm 8 Fig. in data Experimental ˆ i 3.14  ˆ x cln ihtecplaynme ftehrzna eghscale length horizontal the of number capillary the with Scaling × = 10 n 2 −3 − ˆ and , l c  = R .Hr,teiptprmtr are parameters input the Here, 0. ' 0.1. l c sabtaiydfie stedsac from distance the as defined arbitrarily is steconcentration the As 13. ∼  myxobacte- 5%, = 0.008, A ˆ = 0.25, o ifrn iesols atra egh.Hr,teiptprmtr are parameters input the  Here, lengths. bacterial dimensionless different for 7. Fig. number softness of C substrates between on race η a gliding showing cells animation an 2 8.7), in section show Appendix, we SI feature, mechanosensitive this gemn ihtevleue normodel. reasonable our in in 70), used (69, value mN/m the 10 with as low agreement as values surface to the down reduce tension can lipopolysaccharides that evidence mental air–water the (∼ of of that estimate than interface lower our much is for which tension, made interfacial be can argument ilar value: polysaccharide– estimated our that with tension. agreement as previously good low air–substrate as observed be the could been tensions of substrate has that than it lower Indeed, be surface slime–substrate could the tension that slime indicating bacterial surfactants, that 60 show to contains (∼50 experiments substrate However, agar 66). an (60, and mN/m) air between interface the sur- of slime–substrate that the Concerning 37). (36, tension Pa.s as face 20 reported con- to been 5 the as have to high on viscosities similar depending polysaccharides, fluid, Indeed, of in viscous sug- centration fluids. rich very 1 concentrated a is Table EPS is slime in other slime to their viscosity myxobacterial apply of that that estimate gests also believed Our 1 is (11). Table it polysaccharides in as listed cells, characteristics myxobacteria slime argue these can one that Nevertheless, slime. flu- myxobacterial polysaccharide than of rather (µ) types ids, generic viscosity on conducted as experiments such to (γ slime, the tensions Regarding the surface 26). to and (25, specific nm slime 25 parameters whose and material 10 motility, between gliding ranges with thickness bacteria Further- another bacteria. Flexibacter, (11). the of of underneath beneath choice thickness underestimation not our more, slime an and mean cells likely actual motile very the of is for wake it reported the Therefore, nm) in (5 found thickness is slime the begin with, To myxobacteria. underneath directly measured properties concerning values reported thus of range the with agreement h 2 = 0 agar sdi oprsnt u xeietldt r ngood in are data experimental our to comparison in used ) al hw htms fteprmtrvle (`, values parameter the of most that shows 1 Table orsodn oaa concentrations agar to corresponding 463, = 0.008, 2 el.Hwvr hr slc faalbedt o slime for data available of lack is there However, cells. ≈ ieaeae ldn pe safnto ftesfns parameter softness the of function a as speed gliding Time-averaged respectively. 1%, A ˆ = 0.25, γ 70 s PNAS u siaei al sfudt eaothalf about be to found is 1 Table in estimate our , Nm e.6) eea el hr sexperi- is there well, as Here 67). ref. mN/m; R ≈ h | 0 ,and 0.1, eebr1,2019 10, December 10 = s , γ ,tesucslse nTbe1refer 1 Table in listed sources the ), mi ossetwt ausue for used values with consistent is nm Ca ≈ 1.67 ∼ × 35 | 10 o.116 vol. −3 Nm(7,wihi in is which (67), mN/m γ s . oi S7 Movie 30 = .xanthus M. C γ | agar h slime–air the , Nm sim- A mN/m. o 50 no. η h 1 1 0 7 = hc is which , ≈ as see (also L, 7% .xan- M. | . 9 A, 25093 cells and and C ,

APPLIED PHYSICAL BIOPHYSICS AND SCIENCES COMPUTATIONAL BIOLOGY We find that satisfying these constraints can lead to different bal- ances 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 Cagar ≤ 3%, the motility thrust is balanced by the fric- tion I3 exerted by the slime on the cell, and consequently, it V was calculated as (in dimensional form) T ≈ 2RµR ` dx ∼ 0 h − δ µV 2R` , where R ' 250 nm is the radius of the rod-shaped h0 bacteria. Using the parameters given above, along with V ≈ 1 µm/min, we obtain T ≈ 196 pN. This value is in good agree- ment with other experimental and computational estimates (∼50 to 150 pN) of the propulsive force of single A-motile cells (30, 72) and S-motile cells (73, 74) (which move at approximately the same speed). Fig. 8. Mean velocity of A-motile M. xanthus cells as a function of In conclusion, the speed–stiffness relationship investigated in the concentration of agar in the substrate. The experimental data (sym- this work improves our understanding of friction and substrate- bols) are reported in terms of the mean values with an error bar mediated interaction between bacteria in a swarm of cells pro- corresponding to the SE, i.e., the uncertainty on the estimate of the mean. The dark green line corresponds to our simulations (see text for liferating in soft media (75). A crucial next step would be parameters) of the elasto-capillary–hydrodynamic problem. The blue dash- to consider the actual shape of the cell and its modification dot line corresponds to the case of a pure elastic substrate, while the under the torque exerted both by the outside slime pressure red dash line is the asymptotic solution given by Eq. 13 for a very and by the inner aggregation of motor complexes at the so- soft substrate. 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 mem- To assess the robustness of our model to the parameter val- brane of the cell. These membrane deformations may be due ues listed in Table 1, we investigated how 20% modifications to the differences in the distance between the cell and the sub- of these numbers influence our predicted speed–stiffness curve strate in the bound and unbound states (76). Tackling such (SI Appendix, section 9). Our sensitivity analysis shows that these problems will require determining stable 3-dimensional defor- variations of the parameter values not only result in minor quan- mations of the bacterial curved surface using membrane mechan- titative changes of the V = f (Cagar) curve, but they do not alter ics (77) and could give insight in the shape–motility coupling the (experimentally observed) existence of 2 regimes for glid- of other rod-shaped cells. This will also enable direct con- ing cells of soft substrates. Therefore, the emergence of gliding nections with physical aspects of force transduction from the motility mechanisms from the interplay between elastic, capil- bacterial motors. lary, and viscous forces is a rather robust theory, independent of the values of the model parameters. 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 Discussion and 2 for details). The slime being confined to a thin layer between the bac- We have presented a model for the gliding of single myxobac- terial membrane and the substrate, its governing equations are obtained teria cells and their underlying substrate. It appears that the within the lubrication approximation. In the reference frame of the bacte- dynamics of motor complexes can be modeled as a traveling ria, the slime velocity component in the (horizontal) gliding direction reads wave, while the secreted slime serves as a lubricating film medi- (see SI Appendix, section 1.2 for derivation) ating the cell-substrate coupling. Our analysis shows that the mechanosensitivity of myxobacteria to their substrate results 1 ∂p z − h ux (x, z, t) = (z − h)(z − δ)+ V , [14] from their need to glide under drag-free and lift-free constraints. 2µ ∂x h − δ

Table 1. List and values of the parameters used for comparing the elasto-capillary–hydrodynamic model to experiments Parameters symbols Definition Model values Experimentally reported values Refs.

` Bacterial total length 5 µm 4 to 10 µm 24, 32, 71 L Wavelength of membrane deformation 1 µm 0.7 to 1.5 µm 17, 24 A Amplitude of membrane deformation 3 nm 3 to 5 nm 32 C Speed of traveling motors 320 nm/s 100 to 1,000 nm/s 17

h0 thickness of slime film 10 nm 5 to 25 nm 11, 26 µ viscosity of slime film 47 Pa.s 5 to 20 Pa.s 36, 37 a thickness of slime–air interface 1 nm O (nm) 56, 57 γs slime–substrate surface tension 30 mN/m 35 to 60 mN/m 60, 66, 67 γ slime–air surface tension 4.8 mN/m ∼ 10 mN/m 69, 70

25094 | www.pnas.org/cgi/doi/10.1073/pnas.1914678116 Tchoufag et al. Downloaded by guest on September 24, 2021 Downloaded by guest on September 24, 2021 olwn oie enlseuto (see (41): equation derivation) Reynolds modified following (z surface substrate Eq. ing with n h oenn qain around 8.1). equations and governing 5.1 solution guess the sections gliding a ing with Appendix, bacterial starting (SI in the consists pressure algorithm deformation, This leading-edge substrate cubic the the the and field, by speed, seen pressure readily slime as the equation nonlinear Eq. in a power is bacteria the beneath Treatment. Numerical by obtained limit are the problem taking elasto-hydrodynamic the for equations governing i.e., form, dimensionless in ansatz wave traveling the use we that Note Written bacteria. the Appendix, of (SI edge respectively leading read the equations at these 6.1–6.4): sections form, balance equa- dimensionless an force 1) conditions, in a drag-free by and 3) governed lift-free the also 2) and deformation, is substrate the substrates for soft tion Eq. on gliding to myxobacterial addition In fluid, that incompressible such an are as components treated velocity is its slime where the Moreover, speed. gliding rial where h usrpsrfrigt h etnieainse.Nw sinn to assigning Now, step. iteration Newton the to referring subscripts the increment an find to system cofge al. et Tchoufag .M ots .Kie,Mxccu el epn oeatcfre ntersubstrate. their in forces elastic to respond cells Myxococcus Kaiser, D. Fontes, M. 9. of systems motility two The Zusman, R. myxobacteria. D. the Shi, in W. motility gliding 8. of mystery the Uncovering Zusman, R. D. Nan, B. 7. P J. Moraleda-Munoz, A. Garcia-Bravo, E. motility. Marcos-Torres, J. swarming F. Munoz-Dorado, bacterial J. to 6. guide field A Kearns, B. Lappin-Scott, D. M. 5. H. Korber, Persat R. D. Caldwell, A. E. D. 4. Lewandowski, Z. Costerton, W. J. biofouling. 3. and biofilms on interfaces Bacterial of Fletcher, Influence Zehnder, M. J. A. 2. Norde, W. Lyklema, J. Loosdrecht, Van C. M. 1. ∂ ∂ h ˆ ˆ x al cd c.U.S.A. Sci. Acad. Natl. surfaces. (1993). various on advantages selective Genet. Rev. Annu. together. surviving (2016). and 781 feeding, killing, Moving, Myxobacteria: (2010). 634–644 (2015). biofilms. Microbial (1994). activity. microbial orwieEq. rewrite to , u z p (x 14 stesiepressure, slime the is , z − − , noEq. into δ ˆ t hrfr,w s nieaieNwo ehdt obtain to method Newton iterative an use we Therefore, 17a. ( ∂ h etclcmoeto h lm eoiy ysubstitut- By velocity. slime the of component vertical the ) h ehnclwrdo bacteria. of world mechanical The al., et Z ∂ ˆ x t −n/2 , ξ (h ˆ n/2 t ) → − irbo.Rev. Microbiol. −   13 (2011). 21–39 45, nu e.Microbiol. Rev. Annu. 16 =

fEq. of 0 2 δ 15 π 1 0285 (1999). 8052–8057 96, h lsocplayhdoyai rbe of problem elasto-capillary–hydrodynamic the 16, s ) ∂ δ p h enlseuto oenn h icu slime viscous the governing equation Reynolds The ˆ ∂ sEq. as + Z otebceilmmrn (z membrane bacterial the to ) ˆ x ∂ ∂ n nertn h eutn qainfo the from equation resulting the integrating and 1 −n/2 b ˆ ˆ x  n/2 ∂ + p(n/2, ∂ ˆ ∂ ∂ x +  δ p 17 ˆ ˆ x  δ R nteasneo ailrt fet,the effects, capillarity of absence the In 17a. q ∂ p( ˆ 1 2 ( ∂ 12µ h ˆ h usrt eomto,and deformation, substrate the u ihwhich with eto 6.5). section Appendix, (SI above 58 (1990). 75–87 54, ˆ x x ∂ ∂ 2 1 x ξ − 0 ˆ t ˆ a p ˆ ˆ x , ) + ˆ t η ln = ∂  ∂ ln ) δ ˆ p ∂ x h ˆ ∂ )  − q 1–4 (1995). 711–745 49, 3 u (h rc al cd c.U.S.A. Sci. Acad. Natl. Proc. − z 0 1 + z h(n/2) ˆ

+ − η n netn h eutn linear resulting the inverting and ˆ x = 6( δ ˆ − q δ  2 0, V ˆ Z ξ ) ur pn Biotechnol. Opin. Curr. 0 yoocsxanthus Myxococcus ˆ a + ˆ x 3 −n/2 eto 1.2 section Appendix, SI − /Ca sudtdinto updated is 0  n/2 + −

h ˆ 2)( + 2 η 2 1 − − V ˆ δ ˆ ξ/π p( ˆ h V ˆ (n/2) η 2 (h − ˆ x δ ˆ   ,
! q ˆ t η − = a.Rv Microbiol. Rev. Nat. 1/2 d )d 0 δ ˆ d ˆ x × δ ,w banthe obtain we h), ) n hnlineariz- then and Cell ˆ ˆ x x  0 . ) rn.Microbiol. Front.  = = = = = 0, 0, 0, 0, q hwdifferent show 988–997 161, 0. 3378–3382 90, V 1 = h bacte- the 302–306 5, q 0 ∂ + ∂ [17b] [17d] [17e] [17a] [17c] erez, Proc. h ´ ˆ ˆ t [15] [16] δ for q q, = 7, 8, 1 6 .Xn,F a,R er,G se,Tru–pe eainhpo h atra flagellar bacterial the Nan of B. relationship Torque–speed 17. Oster, G. Berry, R. Bai, F. Xing, J. 16. Faure, M. L. over movement cell 15. for mechanisms Multiple motility: gliding Bacterial McBride, J. M. 14. in motility gliding of Genetics Kaiser, D. Hodgkin, J. glide. myxobacteria 13. How Oster, G. Kaiser, D. Hoiczyk, Wet-surface- E. Wolgemuth, Theodoly, C. O. 12. Mignot, T. Mouhamar, F. Valignat, M.-P. Ducret, mechanosensitivity adhesion A. cell of 11. principles based Physically Nicolas, A. Ladoux, B. 10. ehd eslenmrcly o ie auso h e fparame- of set the of values given func- for shape (η numerically, of given ters solve choice are we problem the system. discrete method, form, the linear of weak discrete verification basis the in convergence resulting a regarding the the and in details tions, requires inverting problem the the that and of com- of All functions, the solver unknown discretizing numeri- shape element form, the this of weak finite expanding a implemented a domain, in We putational equations (79), governing (78). FreeFem++ the method rewriting via element method finite cal the using tolerance a than smaller is increment to the set of we norm that the until repeated is tion of role the eia cecso h I ne rn 0G190.JT n ...are K.K.M. and J.T. R01GM110066. R01GM129000. General Grant Grant NIH of under by Institute thanks NIH supported National J.T. the the of by the manuscript. Sciences supported of Medical are aspects the B.N. elasto-capillarity on and on C.B.P. comments problem. helpful comments for Duprat helpful Camille Dr. and discussions 40 ifying of velocities ACKNOWLEDGMENTS. the concentration, agar ImageJ. given camera using a MRm calculated AxioCam for were cells ZEISS and, a were intervals, and videos 20-s microscope time-lapse with at AXIO 20-min interact ZEISS not a concentration, did with agar cells recorded each where point For the colli- another. to cell one diluted of interference was potential culture avoid cell to sions, Moreover, pili. of synthe- absence the to twitching to of difficult a interference used very For potential we agar. becomes motility, avoid To of it substrates. concentrations 7%, homogeneous moderate than size to higher small concentrations containing agar an pads of on agar velocities nm) CYE Gliding 160 100 objective. size, a with (pixel TIRF microscope C9100-23B Eclipse-Ti camera Nikon EM-CCD inverted X2 ImagEM matsu Method. Experimental repository, of aim val- our higher in hand, well other serve the not On of do capillarity. ues and of importance edge the leading emphasizing the at effects capillary in of shown presence as hand, the of in value 7) this arbitrary, (Fig. number somewhat capillary parameter the softness chose we the effects, of capillary bacterial function the a as and velocity deformation, substrate the pressure, speed. slime gliding the of ues instants time different al ,w n ahrcoevalue: close of rather estimates a myxobacteria. find best on we experiments 1, our Table for using relevant compromise. by also good Indeed, a is is value motility of this gliding on value Nevertheless, capillarity the of Therefore, effects the convergence. in illustrate mesh discussed on as 8.3 numerically, section resolve accurately to challenging as scales h ierzdeutosotie tec etnieainaesolved are iteration Newton each at obtained equations linearized The h oe sdfrorsmltoshv endpstdi h GitHub the in deposited been have simulations our for used codes The the and 5) (Fig. play at forces the of importance the illustrate to order In oiiyb oigi eia trajectories. helical (2013). E1513 in moving by motility motor. complexes. surfaces. movement. control systems (1979). gene Two terales): (2002). 369–377 adhesion major a as (2012). slime motility. surface identifies bacterial during microscopy factor contrast ellipsometric enhanced tissues. in sn h aforementioned the Using . 8.1–8.3 and 5.1–5.3 sections Appendix, SI Ca , Ca, lglasao ooosfnto smtr o yoatra gliding myxobacterial for motors as function homologs stator Flagella al., et rc al cd c.U.S.A. Sci. Acad. Natl. Proc. ∼ ol edt oecnndeat-ailr ig hs width whose ridge elasto-capillary confined more a to lead would nu e.Microbiol. Rev. Annu. https://github.com/mandadapu-group/myxoglide. e.Po.Phys. Prog. 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