Hydrodynamic Effects on the Motility of Crawling Eukaryotic Cells
HHydrodynamic Effects on the Motility of Crawling Eukaryotic Cells
Melissa H. Mai a and Brian A. Camley ab Eukaryotic cell motility is crucial during development, wound healing, the immune response, and cancer metastasis.Some eukaryotic cells can swim, but cells more commonly adhere to and crawl along the extracellular matrix. We studythe relationship between hydrodynamics and adhesion that describe whether a cell is swimming, crawling, or combiningthese motions. Our simple model of a cell, based on the three-sphere swimmer, is capable of both swimming andcrawling. As cell-matrix adhesion strength increases, the influence of hydrodynamics on migration diminish. Cells withsignificant adhesion can crawl with speeds much larger than their nonadherent, swimming counterparts. We predictthat, while most eukaryotic cells are in the strong-adhesion limit, increasing environment viscosity or decreasing cell-matrix adhesion could lead to significant hydrodynamic effects even in crawling cells. Signatures of hydrodynamiceffects include dependence of cell speed on the medium viscosity or the presence of a nearby substrate and thepresence of interactions between noncontacting cells. These signatures will be suppressed at large adhesion strengths,but even strongly adherent cells will generate relevant fluid flows that will advect nearby passive particles and swimmers.
Introduction
Throughout development, wound healing, and cancer metasta-sis , eukaryotic cells crawl while adherent to the extracellularmatrix . An increasing amount of evidence shows that eukary-otic cells without strong adhesion can also swim ; this is dis-tinct from other mechanisms of adhesion-independent cell mo-tion, e.g. “chimneying” or osmotic engines . In particu-lar, Aoun and coworkers have observed lymphocytes directly tran-sitioning between crawling on adhesive and swimming over non-adhesive regions of substrate, showing that cells may exploit bothstrategies depending on their environment . We use a minimalmodel incorporating both hydrodynamics and regulated substrateadhesion to understand what happens when cells are intermedi-ate between swimming and crawling.At the micron length scales typical for eukaryotic cells, theymust swim through fluids at low Reynolds number, where iner-tial forces become irrelevant. A low-Reynolds number swimmeris strongly constrained by the linearity and reversibility of theStokes equations and can only achieve a net displacement if itsmotion is defined by a nonreciprocal, or time-irreversible, cycle ofconformational changes . As a result, a microswimmer needsat least two degrees of freedom to achieve a productive cycle ofmotion – the so-called “Scallop Theorem.”Eukaryotic cells crawl on substrates by cycles of extending pro-trusions at the cell front and contracting the cell body at therear (Fig. 1a). Forward protrusions attach through complexesof adhesion proteins, and contractions are aided by the mo-tor protein myosin through rupturing the adhesive bonds in therear . Since cell-substrate adhesion may be regulated to dif-fer between the cell’s back and front, crawling cells can violatethe Scallop Theorem, allowing minimal models of cells as dimersto crawl .To capture swimming motion, our model must have at least twodegrees of freedom. We adapt the classical three-sphere swim-mer , describing our cell as three beads connected by twoarms. These arms extend and contract around a mean arm length a Department of Biophysics, Johns Hopkins University, Baltimore, Maryland b Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Mary-land
Fig. 1
The motion sequence of our three sphere crawler (b) is chosento resemble lamellipodial migration (a): the adhesion of each bead (bluehashes) depends on the current motion to model the maturation and rup-ture of adhesive contacts. The crawler’s arms deform with prescribed ve-locities W ± L (leading arm) and W ± T (trailing arm). In an expansion phase,the cell arms expand from length L − ∆ L / to length L + ∆ L / , and incontraction vice versa; the geometric parameters L and ∆ L and otherparameters are listed in Table S1. L in a nonreciprocal sequence with prescribed distortion velocities(Fig. 1b). Significant work has been done to characterize theo-retical three-sphere swimmers , including their interactionswith walls and swimmer-swimmer interactions . Three-sphere swimmers have even been built experimentally with opti-cal tweezers .Here our approach is to use a minimal model, neglecting manydetails of biochemistry and cell shape which have been studiedextensively for crawling cells and more recently also forswimming ones (and in a very recent example, a transi-tion between confined crawling and swimming ). In our three-sphere crawler, we describe adhesion to the surface by introduc-ing an adhesive drag force, and we examine the relationships be-tween adhesion and the hydrodynamics of swimming and crawl-ing cells. Finally, we examine the hydrodynamic interactionsamong multiple crawlers and swimmers. Model and Methods
We describe our cell with the minimal structure of three beads,representing the tail, body, and head of the cell (labeled 1, 2, and3 in Fig. 1). We prescribe the relative motion of the head and a r X i v : . [ q - b i o . CB ] A ug ail of the cell to match the stereotypical cycle of protrusion andretraction, as shown in Fig. 1. This differs from earlier models ofcell crawling that generally prescribe the forces driving cell mo-tion, then find the resulting cell velocities . Instead, we setthe velocities of the cell head and tail relative to the body andthen solve for the forces that obey physical constraints of zero netinternal force and torque (see below); this is a more typical ap-proach for modeling swimming , and allows our model to limitback to the classical three-sphere swimmer at zero adhesion.We also include friction-like adhesion forces between the celland substrate, the strength of which will control whether thecell’s motion is primarily driven by swimming or crawling. Cell-substrate adhesion is tightly regulated, and so we choose theseadhesion strengths to depend on the cycle of the motion (Ta-ble 1), allowing the cell to crawl. These motions are also cho-sen to be nonreciprocal , so that in the absence of adhesiveforce, the cell may still swim. Below, we describe how we solvethe Stokes equations that describe fluid flow, how we model cell-substrate adhesion, the physical constraints on the cell, and thetime-stepping algorithm we use to evolve the cell’s motion. Hydrodynamic model of forces and motion
We describe cell motion in a fluid environment by relating theforces applied to the model’s three beads to their velocities. Cellsmove in a low Reynolds number environment where viscous dragforces dominate fluid motion and inertial forces become irrele-vant . In this regime, the fluid flow surrounding a motile cellis described by the time-independent Stokes equations for incom-pressible fluids, which describe the velocity of a point at r in afluid with pressure p subject to force density f ( r ) : η ∇ v ( r ) = ∇ p ( r ) − f ( r ) (1) ∇ · v ( r ) = (2)If the force density is a tightly-localized point, with f ( r ) = F δ ( r ) ,the Stokes equation can be solved as v ( r ) = ↔↔↔ G ( r ) · F (3)where ↔↔↔ G ( r ) is the Green’s function of the Stokes equations, knownas the Oseen tensor. In components, this equation is v α ( r ) = G αβ ( r ) F β . We assume Einstein summation here and throughoutthe paper. In an unbounded, three-dimensional fluid, G αβ ( r ) = πη (cid:18) δ αβ r + r α r β r (cid:19) (unbounded fluid) (4)where α , β = x , y , z are the Cartesian coordinates. Generalizationsof this Oseen tensor can be made to different boundary condi-tions . Eq. 4 diverges as r → , suggesting the velocity ofa point subject to a point force is ill-defined. We handle thisthrough the method of regularized Stokeslets , smearing thepoint force over a scale ε . In this approach, we assume that theforce distribution over a bead is f ( r ) = F φ ε ( r ) , where φ ε ( r ) is a radially-symmetric “blob” that integrates to one. In this case, v ( r ) = ↔↔↔ G ( r ; ε ) · F (5)where now the regularized response G i j ( r ; ε ) remains finite as r → . G i j ( r ; ε ) depends on the choice of φ ε ( r ) ; several variants arediscussed in .By the linearity of the Stokes equations, the velocity in responseto many regularized forces is a superposition of these solutions v ( r ) = ∑ n ↔↔↔ G ( r − R n ; ε ) · F ( R n ) (6)If the forces are known, the velocity of bead m is v ( R m ) = ∑ n ↔↔↔ G ( R m − R n ; ε ) · F ( R n ) (7)If there are n b blobs in our system (composed of one or manycells, with three blobs per cell), Eq. 7 can be thought of as a set of n b linear equations giving the bead velocity components in termsof the force components, V = ˆ M F (8)where ˆ M is a n b × n b mobility matrix defining the hydrodynamicinteractions among the spheres.We will treat two hydrodynamic geometries in this paper: 1)cells with no hydrodynamic obstruction, and 2) cells near a solidsurface, when we will use the regularized Stokeslet solution ofAinley et al. , which creates the response to a point force neara no-slip wall from a superposition of higher-order solutions tothe Stokes equations, defined in the Supplemental Material .This is a regularization of the solutions by Blake , which hasbeen previously used to study the behavior of swimmers nearwalls . For cells away from a solid substrate (Case 1),we simply take the regularized Stokeslet of in the limit of cellsfar from the wall. Adhesion forces
To model the effect of protein-mediated cell adhesion to a sub-strate or fiber, we introduce an adhesive force, F adh , by the sub-strate on each bead in the form of a frictional drag: F adh ( t ) = − ξξξ ( t ) ◦ V (9)where ξξξ ( t ) is a n b × column vector defining the adhesive fric-tion coefficients for each bead in each direction depending on thecurrent motion (and therefore time: see Table 1). The symbol ◦ represents Hadamard (elementwise) multiplication: the com-ponents of the force are F adh i = − ξ i ( t ) V i . Here i is a generalizedindex going over both bead and dimension, i.e. i = x , y , z , x ··· .This linear form is appropriate in the low-speed limit of motionover the substrate . We choose the drag force to be tangen-tial to the substrate, i.e. the x and y components for each beadare equal to each other, but the z component is set to zero. This isirrelevant in practice, since we will assume that an adherent cellis constrained to not move in the z direction.Because this frictional drag force is linear in the velocity, we able 1 The adhesive friction, ξ i , for the trailing (1), center (2), and leading (3) beads are qualitatively described for each motion. In the simulationsused for this work, ξ high = ξ and ξ low = . ξ , where ξ is the global adhesion parameter. Finally, the deformation velocities for the leading ( W L ) andtrailing ( W T ) arms are, for simplicity, chosen to be ± W or for each motion. Motion ξ ξ ξ W L W T Trailing arm extension High High Low + W Leading arm extension High High Low + W Trailing arm contraction Low High High − W Leading arm contraction Low High High − W can derive a simple form for the velocity even in the presence ofthis additional drag. Assuming that the total force in Eq. 8 is com-posed of cell-internal forces F int and the cell-substrate friction, i.e. F = F int + F adh ( t ) , we find V = ˆ M (cid:16) F int + F adh ( t ) (cid:17) = ˆ M F int − ˆ M ( ξξξ ( t ) ◦ V ) which implies I V + ˆ M ( ξξξ ( t ) ◦ V ) = ˆ M F int (10)where I is the identity matrix. The term ˆ M ( ξξξ ( t ) ◦ V ) is just amatrix multiplying V : (cid:2) ˆ M ( ξξξ ( t ) ◦ V ) (cid:3) i = M i j ( ξ j V j ) ≡ (cid:2) ˆ Ξ V (cid:3) i (11)where Ξ i j = M i j ξ j (12)As a result, Equation (10) becomes V = (cid:0) I + ˆ Ξ (cid:1) − ˆ M F int (13)We define the modified mobility matrix, ˆ M , such that ˆ M = (cid:0) I + ˆ Ξ (cid:1) − ˆ M (14) V = ˆ M F int . (15)The value of Eq. 15 is that we can now directly relate the veloci-ties and the internal forces, without needing to handle the adhe-sion forces explicitly. This is useful because some of our physicalconstraints apply only to the internal forces – such as the require-ment that each cell cannot exert a net internal force on itself.Depending on the phase of the cell’s motion (Table 1), wechoose the components of ξξξ ( t ) to be either ξ high = ξ or ξ low = . ξ , where ξ is the overall scale of the adhesion. For exam-ple, during leading edge extension, adhesion at the front is lowsince the focal contacts have not yet matured, yet the adhesion inthe rear is strong. During trailing edge contraction, the ruptureof focal contacts and targeted disassembly causes the adhesionon the rear bead to be lower, while other parts of the cell aremore strongly bound to the surface. We only model the switch-ing between “high” and “low” adhesion strengths during the cy-cle – intermediate values can also be used but require furtherparametrization and produce qualitatively similar results. Constraints
Defining a cell’s motion via Eq. 15 requires knowledge of all theinternal forces, which can be found by applying the necessaryconstraints on the cell’s motion. We have three important sets ofconstraints: 1) the pattern of extension of the cell front and back,2) the linear geometry of the three-bead cell, where we apply theappoach of , and 3) no unphysical forces or torques required onthe cell. We enforce the linearity constraint as a minimal modelfor the cell’s internal resistance to deformation. These constraintswill be different for adherent cells (those attached to a surface)and non-adherent cells (those just swimming near a surface). Constraints for non-adherent cells
For non-adherent cells ( ξ = ), F = F int , and our model is just athree-bead swimmer, as in e.g. . For a fully and uniquely deter-mined system, defining the nine components of F per cell (threefor each bead) requires nine constraints.We require that the cell is not generating a net internal force(“force-free” ), providing three independent constraints: ∑ n = F int n = (16)The cell also cannot generate a net internal torque: ∑ n = , , ( R n − R ) × F int n = (17)where, for convenience, the reference point for calculating thetorques is set as the center sphere’s position. Due to the symmetryof this system and the rigid-body constraints discussed later, thecomponent of the torque along the cell’s axis is always zero, soonly two components of the torque constraint are independentand enforced. When simulating multiple cells at a time, each cellis required to be individually force- and torque-free.To keep track of the cell’s orientation, we define a rotated setof orthonormal basis vectors as in (Fig. 2): ˆ α = ( sin θ cos φ , sin θ sin φ , cos θ ) T (18) ˆ β = ( cos θ cos φ , cos θ sin φ , − sin θ ) T (19) ˆ γ = ( − sin φ , cos φ , ) T (20)where θ and φ are the polar and azimuthal angles, respectively.Here, ˆ α is the cell migration direction, and ˆ β and ˆ γ are two con-venient vectors normal to the cell’s direction. Hence in this basis,as discussed above, only the projections of the torque onto ˆ β and ˆ γ are explicitly constrained. ig. 2 The cell’s orientation defines an orthonormal basis { ˆ α , ˆ β , ˆ γ } (green) depending on the angles θ and φ . The final four constraints arise from fixing the deformation ofthe arms and the rigid body constraint. In defining the motionof the cell, we choose the deformation velocity of the leadingarm (connecting bead 3 and 2) to be W L and that of the trail-ing (connecting 2 and 1) to be W T , which both depend on thephase of motion. The motion is along the axis of the crawler, soprojections of the relative velocities onto the principal orientationvector ˆ α should be equal to the deformation velocities: ( V − V ) · ˆ α = W L (21) ( V − V ) · ˆ α = W T (22)The projections onto the other two orientation vectors enforcethe rigid body constraint of no internal bending – the projectionsof the change in the orientation and length of both arms onto ˆ β should be equal and opposite. Specifically, L − L ( V − V ) · ˆ β = − L − T ( V − V ) · ˆ β (23)The same constraint is applied to the projection onto ˆ γ : L − L ( V − V ) · ˆ γ = − L − T ( V − V ) · ˆ γ (24)These constraints are linear equations for the velocity, eventhough our earlier constraints are linear equations for the forces F . We can convert these to linear equations for F via Eq. 15. (SeeSupplemental Material for detailed explanation). Adherent cells
We assume that an adherent cell ( ξ > ) does not move awayfrom the substrate – it is strongly attached. Instead of explicitlymodeling an attachment force, we handle this by constraining the z -directional velocity for each bead to be zero: V n · ˆ z = n = , , (25)A crawler will thus stay at a fixed distance away from the sub-strate ( z = ); we choose the crawler to be at height z = a , i.e.with the spheres resting on the surface. Again, this constrainton the velocity can be converted into a constraint on the forcesthrough Eq. 15. Including these three additional constraints re-quires relaxation of three constraints from the non-adherent case.This avoids mathematical overdetermination of the system andphysical redundancy of constraints for an adherent cell. We relax the z -component of the force-free condition, as theremust be some vertical force keeping the cell bound to the surface.Additionally, we remove the now-redundant constraint of Eq. 23,since attachment to the surface mandates fixing of the polar angleto θ = π / . Finally, we remove the projection of the torque-freecondition onto ˆ γ due to the relaxation of the force-free condition’s z -component. The constraint matrix
Once all of the constraint equations have been written in terms ofthe forces F int , we will have an equation of the form ˆ C F int = d (26)where ˆ C and d for both adherent and non-adherent cases are ex-plicitly defined in the Supplemental Material. We solve Eq. 26using LU factorization (MATLAB’s linsolve ). Threshold force
In our simulation, we prescribe the relative motion of the frontand back of the cell and solve for the forces needed to move atthis speed. However, the internal force required increases withincreasing adhesion, yet a cell can only exert a finite amount offorce. We apply a simple limitation on the internal force exertedon any bead. If the prescribed deformation of the arms requiresan internal force F req of a magnitude greater than a thresholdforce F thresh , the cell exerts only its maximal force, resulting in alinear scaling of all internal forces: F int = F req · F thresh max | F req | (27)This constraint is enforced after defining the matrix in Eq. 26.Since the only nonzero terms of d refer to the deformation ve-locity constraints (Eq. 21 and 22), any linear scaling of F int stillsatisfies all other constraints. As a result, the scaled forces con-tinue to obey all the necessary physics of the system but reducethe cell’s overall center-of-mass velocity.If the determined velocities are scaled down in this manner, weadapt the time step used as ∆ t (cid:48) = ∆ t · max | F req | F thresh . This allows thesame deformation length to occur during each iteration, reducingthe computational cost of the simulation. Algorithm
We employ a time-stepping algorithm to numerically solve theproblem, outlined below. The parameters used for the simulationsare presented in the Supplemental Material. The mobility tensors,forces, and velocities are reevaluated at each step.1. Determine which arm is extending and/or contracting, anddefine the appropriate adhesion strength (Table 1)2. Calculate the modified mobility matrix, ˆ M
3. Construct the constraint matrix, ˆ C
4. Find required forces (Eq. 26), scale by F thresh if needed5. Calculate velocities V via Eq. 15 . Update the configuration via Euler’s method with a definedtime step ∆ t : R ( t + ∆ t ) = R ( t ) + V ∆ t Parameter setting
Throughout this paper, we will use convenient units of mean cellarm length L = , arm speeds W = . , and fluid viscosity η = .To map between our simulation units and experimental measure-ments for different cells, we must have estimates for these dif-ferent numbers, as well as for the friction coefficient ξ and thethreshold force F thresh . Fibroblasts on nanofibers have a protru-sive velocity of order . µ m/s , so in this context, our units ofvelocity can be interpreted as µ m/s; the maximum velocities oforder . in simulation units correspond to speeds of ∼ µ m/hr,consistent with .If we assume η = − Pa s is the viscosity of water and L = µ m (order of magnitude correct for fibroblasts , thoughthey can be very long in narrow confinement or on fibers), oursimulation unit of force corresponds to − Pa s × µ m s − × µ m = . pN. We expect these maximum forces to be on theorder of nanonewtons, so this suggests F thresh ≈ − in sim-ulation units. Similarly, one simulation unit of the drag ξ is − Pa s × µ m = . µ m Pa s = − nN / ( µ m/s ) .The only remaining variable to be set is the drag coefficient ξ :this is a difficult parameter to estimate, and in general we willvary ξ over a broad range and see what consequences follow. Wemake an initial, rough estimate by using data from traction stressexperiments on keratocytes. Ref. found a linear relationshipbetween actin velocity v and substrate stress σ of the form σ = kv + σ , with k ∼ . − kPa / ( µ m/s ) . We estimate ξ as the productof k with the contact area of one section of the cell, A ≈ µ m ,or ξ ∼ nN / ( µ m / s ) . This suggests that in our simulation units,strongly adherent cells will have a friction coefficient of ξ ≈ .When we are below the threshold force, the dynamics of ourcrawler will be largely controlled by the relative importance ofhydrodynamic flow and adhesion. We characterize this with theunitless parameter ξ / πη L . Strongly adherent cells will have ξ / πη L ≈ . Cells with weaker adhesion (e.g.
Dictyostelium amoebae or cells on less adhesive substrates) or cells in moreviscous environments will have a stronger relative importance ofhydrodynamics.The specific parameters used in each figure are presented eitherin the figure or in Tables S1-S3.
Results
Biphasic dependence of migration speed on adhesionstrength
A typical velocity profile of a migrating three-sphere cell is shownin Fig. 3. The model captures the biphasic dependence on ad-hesion strength that has been observed experimentally . Aweakly-adherent cell slips along the surface, essentially swim-ming. Movement in this limit is slow – low Reynolds numberswimming is typically quite inefficient . As adhesion increases,the cell can better grip the surface and drag itself along, withvelocity increasing until a plateau at a value roughly sixty timesgreater than the swimming speed. At sufficiently high adhesion,to maintain its motion, the cell would have to exceed the thresh-old force F thresh . Constrained by this maximal force, the cell mustslow down and eventually stop moving.The position of the stalling transition at high adhesion can bemodulated via changing F thresh . In the case of a single crawlerwhere the forces are dominated by adhesion, we can exactly solvefor the motion using an approach similar to that of . We find,for the modulation of adhesion strengths outlined in Table 1, thatthe center-of-mass velocity is given by v cm = W − α + α F ∗ < F thresh F thresh F thresh + F ∗ α + α F ∗ ≤ F thresh ≤ F ∗ + α + α F thresh F ∗ F thresh < α + α F ∗ (28)where α = ξ low / ξ high ( α = . in our simulations), and F ∗ = Fig. 3
A typical velocity profile for a three-sphere cell over ξ / πη L exhibits a biphasic dependence on adhesion strength. At low adhesion, the cellexhibits a slipping, swimming behavior. As adhesion increases, the cell is better able to grip the surface and crawl, until the adhesion becomes toostrong, which leads to arrested migration. Decreasing (dotted, circles) and raising (dashed, triangles) the threshold force F thresh moves the turningpoint linearly in the appropriate direction. Profiles were generated for F thresh = , , and . Other parameters are as in Table S1. ig. 4 Substrate hydrodynamic effects. (a) At low adhesion, wall-inducedhydrodynamic drag (solid, z / a = ) slows a cell relative to its motion on ornear a thin fiber (dashed, z / a > ), but high-adhesion motion is unaf-fected by substrate hydrodynamics. (b) Center-of-mass velocity, scaledby the uninhibited (fiber) velocity, at different adhesive strengths corre-sponding to the colors in (a), as a function of cell height above the wall z , scaled by the bead radius a . Flow fields for cells at different distancesabove the substrate are shown in Fig. S4. Parameters are listed in TableS1. Fig. 5
Time-averaged flow fields for a swimmer, ξ / πη L = (a) anda crawler, ξ / πη L = (b) with respect to the cell’s center of masson a wall located at z = . The color of the arrows corresponds to themagnitude of the velocity with respect to the cell’s average center-of-mass velocity. The field shown is the flow field in the xz -plane, throughthe cell’s axis. The inset shows that the fluid velocity vanishes near thewall to obey no-slip boundary conditions. Parameters are listed in TableS1. W ξ high ( + α ) + α is a characteristic force. (Details of derivation arein the Supplemental Material.) This result neglects all hydrody-namic interactions, but successfully describes the plateau in v cm and subsequent arrest.We can see from Eq. 28 that the plateau velocity W − α + α de-pends only on the speed of protrusion W and the ratio betweenthe high and low levels of adhesion. Unsurprisingly, when thereis no difference between the adhesion at the front and the backof the cell ( α = ξ low / ξ high = ), the cell cannot crawl via ad-hesion. We can also identify the critical adhesion strength atwhich the cell begins to stall, the point at which F ∗ = F thresh ,or ξ thresh = + α + α F thresh / W . Substrate hydrodynamics
Because fluid cannot penetrate the substrate or slip past it, thesubstrate alters the hydrodynamic flow near the cell, alteringswimming patterns, with attraction or repulsion depending onorientation and distance from the wall . Does the presenceof a wall alter crawling speeds? We calculated velocity profiles fora cell crawling on a planar substrate or on an isolated fiber (Fig. 4a). We describe the fiber as infinitely thin with negligiblehydrodynamic effects – this would correspond to the limit of be-ing infinitely far away from a supporting wall. We see that cellson substrates crawl more slowly than those on fibers – but onlywhen ξ is sufficiently small. At large ξ , these hydrodynamic dragdistinctions are negligible.This substrate-induced drag would also be present for a cellcrawling along a fiber suspended above a substrate, as in the ex-periments of , and might provide an experimental signature ofhydrodynamics-dependent motility. We show that the speed of acell on a fiber depends on distance from the substrate (Fig. 4b).The influence of the wall depends on adhesion strength and thedistance from the wall, vanishing almost entirely when the cell is10 a above the surface, and with this distance becoming smaller athigher adhesions. Streamlines for crawlers on fibers at differingheights from the substrate are shown in Fig. S4. Crawlers generate fluid flow
While we have shown so far that hydrodynamic effects do notdetermine the cell’s migration speed in the high adhesion limit,this does not mean that a crawling cell does not interact with itssurrounding fluid. We calculate the flow field v ( r ) around a cell,averaged over five full motion cycles. We measure this flow fieldas a function of distance from the the cell’s center of mass, finding v on a grid of points defined around the cell’s center of mass usingEq. 6. The time-averaged flow fields for a nonadherent cell ( ξ = ) and a strongly adherent crawler ( ξ / πη L = ) on a wall areshown in Fig. 5.While the swimmer produces a velocity field similar to a forcequadrupole, the crawler behaves as three individual Stokesletsin the near-field limit and as a single Stokeslet far away; thisis particularly apparent when we simulate crawlers on fibers farfrom substrates (Figs. S1-S3). The critical difference between thecrawler and the swimmer is that, because the crawler can exertforce on a substrate, it can create a force monopole on the sur-rounding fluid without a net internal force, creating longer-rangeresponses in flow than the three-sphere swimmer . Interactions between adherent cells
To quantify hydrodynamic interactions between cells, we studythe trajectories of cells crawling toward each other on initiallyantiparallel paths separated by a distance of . L . We initiallyset the cell protrusion cycles to be in phase. In the low adhesionlimit, the two cells briefly revolve around each other before es-caping and continuing on straight tracks, but at a different angle(Fig. 6a, blue). Conversely, in the high adhesion limit, the twocrawling cells move directly past one another on their originalpaths without any angular deflection (Fig. 6a, purple). ig. 6 Hydrodynamic interactions of in-phase, antiparallel migrating cells that are adherent to a substrate (at height z = a ). (a) Trajectories of twoantiparallel cells at low (blue, ξ / πη L = − ) and high (purple, ξ / πη L = ) adhesion. (b) The azimuthal angle of the + x -oriented cell as the twocells pass one another. A schematic of the two cells are depicted at the corresponding positions in the trajectory. (c) Net deflection of the + x -orientedcell as a function of adhesion shows loss of hydrodynamic interactions at the high-adhesion limit. Parameters are listed in Table S2. Deflection, or scattering, was measured in terms of the netangular displacement of the azimuthal angle ( ∆ φ ) over the pe-riod of interaction (Fig. 6b). This is motivated by the com-putational work of , who studied hydrodynamic interac-tions for pure swimmers. Consistent with our observations in thesingle-cell case, hydrodynamic scattering effects vanish rapidlywith strengthening adhesion, suggesting that strongly-adherentcrawling cells can no longer feel each other through the fluid(Fig. 6c). Two approaching swimming or weakly-adherent cellsinteract with each other through perturbations of the fluid, thencontinue forward along a new, fixed trajectory once they are suf-ficiently far enough apart to no longer influence each other. Bycontrast, strongly adherent cells remain on their original paths,unaffected by and seemingly ignorant of the proximity of anothercell.In Fig. 6 we have assumed that the cells’ motion cycles are inphase, but hydrodynamic interactions will also depend on the rel-ative phase between the crawlers’ motions (Fig. S5). We note thatin these cases, the angular displacement ∆ φ may be a misleadingmetric for hydrodynamic interactions, as cells can oscillate butremain on their original trajectories.The hydrodynamic interactions of the weakly-adherent three-sphere crawlers in Fig. 6 are similar to those observed for three-sphere swimmers by . However, even in the limit of true swim-ming at zero adhesion, we do not see the large-angle scatteringevents reported in that paper. We believe this distinction arisesfrom a subtle difference between our numerical methods, indi-cating that these events may be more dependent on numericaldetails than immediately apparent. Crawlers transiently perturb nearby swimmers
Finally, we examine the motion of a cell swimming near a walland assess its motion under the influence of crawlers on the wall.This is motivated by tumor cell migration and adhesion to bloodvessel walls, where the hydrodynamic effects of interstitial flow,matrix geometries, and existing epithelial cells may be signifi-cant . For these simulations, we choose crawlers and swim-mers to have initial directions within the xz plane, allowing for a Fig. 7
Trajectory snapshots of a swimmer (blue) above a wall underthe influence of crawlers (purple, ξ / πη L = ) moving in the same(a) or opposite (b) directions. Two crawlers have already passed, cor-responding to the bumps in the swimmer’s path. Lateral dragging fromthe crawlers causes deviation from the isolated swimmer’s path (gray).Sphere radii have been reduced for visualization. Movies are available inthe SI. Parameters are listed in Table S3. simpler analysis with the cells remaining in this plane.The trajectories of a swimmer in the presence of cells crawlingbelow it are depicted in Fig. 7, with cells crawling in the samedirection as the swimmer in a), and opposing the swimmer in b).We see significant deviations from the motion of a swimmer in theabsence of crawlers (gray). The swimmer experiences longitudi-nal bumps and lateral advection in its trajectory corresponding tothe passage of a cell crawling underneath. This behavior is consis-tent with the flow fields shown in Fig. 5–the approaching crawlerpushes the fluid up and forward to repel the swimmer but pullsthe fluid back down as it passes. Discussion
Our simple three-sphere crawler model makes contact betweenlow-Reynolds number swimming and cell crawling, allowing usto determine the relative prominence of hydrodynamic effects nd adhesion-driven motion in adherent cell motility. Sufficientlyhigh adhesion strength ( ξ / πη L (cid:29) ) will suppress any hydrody-namic effects on a single cell’s motion or on two adherent cells,though even strongly adherent cells still generate significant flowsaround them that can alter the motion of nearby passive particlesor swimming cells. However, depending on which signature ofhydrodynamic flow is being observed, the level of adhesion re-quired to suppress it varies. For instance, the hydrodynamic draginduced by a substrate significantly reduces a cell’s velocity un-til ξ / πη L ≈ . Hydrodynamic interactions between cells areexpected to be more sensitive to adhesion strength, with suppres-sion observed for adhesion above ξ / πη L ≈ − .Our simulations suggest several potential experimental tests forthe presence of hydrodynamic effects in crawling cells. First,we note that cells crawling on a fiber will have their motilityreduced by the presence of a wall at sufficiently low adhesionstrengths (Fig. 4). This effect could be observed in experi-ments on fibers . Secondly, hydrodynamic interactions be-tween weakly adherent cells can be observed (Fig. 6). Third, wenote that we predict that increasing fluid viscosity can slow themotion of weakly adherent cells (Fig. 3). This is in contrast to thelimit of freely swimming cells where, holding the shape dynam-ics constant, changing viscosity will not change swimming speed– and also the limit of strongly adherent crawlers, where viscos-ity can be neglected. (Swimmer speed also depends on viscositywhen the swimmer’s forces, rather than motion, are prescribed,but this arises from a fundamentally different reason .) Depend-ing on the experiment, cell type, and viscogen, increased viscosityhas been seen to both increase cell speed and decrease it ;however, we emphasize that interpreting experiments with in-creased viscosity can be difficult due to the different effectiveviscosities at different scales and the effect of external viscosityon receptor dynamics . We should also note that our plots, suchas Fig. 3, which describe velocities as a function of ξ / πη L , showthe dependence when ξ is varied, holding F thresh constant in sim-ulation units (i.e. holding η = constant). If η is varied, F thresh should be constant in real units, not simulation units, and v cm willnot increase with increasing η .We see qualitative, but not quantitative, agreement with exper-iments varying the degree of adhesive coating on the substrate,with our model predicting a slower speed for swimming than forcrawling. This is consistent with, e.g., Aoun et al. , who seesurface-adjacent but nonadherent swimming cells moving with alower speed than crawling, adherent cells, and the foundationalexperiments of Barry and Bretscher . Similarly, calculations byBae and Bodenschatz demonstrate that, if there is no retrogradecell surface flow, swimming by cell protrusions may be a factor often slower than crawling with the same set of shape dynamics .We see a reduction by a factor of around 60 in our model, as wehave included fewer details of shape dynamics. However, theseresults are all broadly consistent with the emerging consensusthat the flow of the cell surface is a primary driver of eukaryoticcell swimming . As our results do not include membrane flow,we do not expect quantitative agreement. We also note that othermechanisms have been suggested to explain the non-monotonicvelocity-adhesion curve, including cell shape changes with adhe- sive wetting and links between adhesivity and protrusion ; wehave not addressed either of these aspects.Our coarse-grained, minimal model provides intuition for ex-periments in which the apparent distinction whether cells areswimming or crawling is ambiguous, because hydrodynamic ef-fects may alter a crawling cell’s speed. The model suggests thataverage speed in different conditions, including different viscosi-ties and hydrodynamic geometries, as well as intercellular inter-actions may be used at least as a qualitative metric to charac-terize the extent of hydrodynamic effects in motility. Moreover,further improvement and inspection of this model may be ableto describe how crawling cells may attract or repel nearby parti-cles or swimmers in the context of problems in collective motil-ity, cancer metastasis, and biofilm dynamics. In particular, wenote that have recently shown that nutrient transport towardthe surface can be a consequence of active swimmers near a sur-face; our results provide a more microscopic view of this problemand how it relates to crawling eukaryotic cells. Extensions of ourmodel could also be made to study mixing induced by eukaryoticcell crawling, as has been done for ciliary carpets . In addition,as the dynamics of swimmers in non-Newtonian and viscoelasticenvironments has proved to be a fertile area , it is a natu-ral question what effect these mechanical features will have oncrawling cells in biological complex fluids. Author Contributions
MHM developed all the code and carried out all simulations.MHM and BAC designed the research, analyzed data, and wrotethe article.
Acknowledgments
MHM acknowledges support from Johns Hopkins Universitythrough the Provost’s Undergraduate Research Award (PURA).We would like to thank Gwynn Elfring for useful comments ona draft of the manuscript, and Yun Chen and Matthew Pittmanfor valuable conversations and references on viscosity-dependentmotility.
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10 | 1–19 upplementary Material
Supplementary Movie Captions • Movie 1: Anti-parallel cells at low adhesion, corresponding with the parameters of Fig. 6a. • Movie 2: Anti-parallel cells at high adhesion, corresponding with the parameters of Fig. 6b. • Movie 3 and 4: Swimmer with four crawlers in the same (3) or opposite (4) direction with a superimposed isolated trajectory.Parameters correspond to those of Fig. 7.
Supplementary Figures
Fig. S1
Far-field time-averaged flow fields for a swimmer ( ξ / πη L = ) (a) and a crawler ( ξ / πη L = ) (b) moving to the right far from a wall. Fig. S2
Instantaneous streamlines (blue) and flow field (gray) around a swimmer ( ξ / πη L = ) far from a wall through its axis during each phase ofmotion. The total force on each bead is illustrated by the red arrows. The flow field around each phase resembles that of a positive (a-b) or negative(c-d) force dipole flow, corresponding to extension or contraction of the cell. ig. S3 Instantaneous streamlines (blue) and flow field (gray) around a crawler ( ξ / πη L = ) on a fiber far from a wall through its axis during eachphase of motion. The total force on each bead is illustrated by the red arrows. During trailing arm extension (a), the trailing and center bead must bothexert a large internal force to overcome strong adhesion, creating a flow field that roughly resembles a positive force dipole. (b-c) During leading armextension (b) and trailing arm contraction (c), the beads for the trailing and leading arms, respectively, exert little force due to high adhesion and a zerodeformation velocity. As a result, the flow field behaves as that of a Stokeslet around the mobile bead. During leading arm contraction (d), since theleading and center beads are moving in spite of strong adhesion, the flow field again roughly resembles a force dipole, though now negative. Fig. S4
Instantaneous streamlines (blue) and flow field (gray) around a cell at very low adhesion ( ξ / πη L = − ) on a fiber at varying heights duringeach phase of motion. The total force on each bead is illustrated by the red arrows. The substrate surface at z = is shown in the thick black line. (a) z = a . The cell is just hovering over the surface and is in the first region of increasing velocity in Fig. 4b. (b) z = a . The cell is high enough off the wallto allow for flow underneath it, but close enough to still be affected by the wall. The cell exists in the intermediate plateau in the velocity profile of Fig.4b. (c) z = a . The cell is high enough off the wall to create vortices underneath, and velocity again is increasing. (d) z > a . The cell is far enoughaway from the wall to no longer be affected by its hydrodynamics. Note that in these figures, as in Fig. 5 above, local streamlines can be misleading;no-slip boundary conditions are obeyed at z = 0.
12 | 1–19 ig. S5 (a) Trajectories for two antiparallel cells, out of phase by half a motion cycle, at low (blue) and high (purple) adhesion. The − x oriented cell isshown in yellow. (b) Angle φ of the + x -oriented cell, shown as a function of the cell’s center of mass position (large x corresponding to post-interaction).(c) Net angular deflection ( ∆ φ ) of the + x (square) and − x (circle) oriented cells as a function of ξ / πη L is both small and non-monotonic with adhesionstrength, suggesting that sometimes another metric must be used to characterize hydrodynamic interactions. (d) RMS deflection, calculated per cycleover the period of interaction, defined as the time over which the cells are within a certain distance of each other (in this case d = L ), exhibits monotonicbehavior over ξ / πη L and may be a useful metric of hydrodynamic interactions when net deflection is insufficient. Analytical results in the large-adhesion limit
In the large-adhesion limit, we can neglect hydrodynamics and Eq. 15 is equivalent to: V = ξξξ ( t ) ◦ F int , (S1)i.e. the motion of one bead is only controlled by the force on that bead and the friction coefficient on the bead. This allows us to simplycompute the velocity of a single cell in the high-adhesion limit. To do this, we simplify to one dimension, following the approach of ,and find the internal forces that satisfy V − V = F int / ξ − F int / ξ = W L (S2) V − V = F int / ξ − F int / ξ = W T (S3) F int + F int + F int = . (S4)This can be done analytically due to the simplicity of the model, though we do not write it explicitly here. These forces then determine v cm ( t ) = ( V + V + V ) = − W L µ µ + W T µ µ + W L ( µ + µ ) µ − W T µ ( µ + µ ) µ µ + µ ( µ + µ ) (S5)where µ i = / ξ i is a mobility for bead i . This gives the center-of-mass velocity at a given instant, and depends on W L and W T as well as ξ i ( t ) for each bead. In addition, the forces determine the maximum required force max | F req | ; we scale the internal forces as in the maintext if max | F req | exceeds F thresh .We can then compute the time average of v cm ( t ) over one whole cycle. During each phase of the cycle, the center of mass velocity isconstant, so this time average is merely v cm = T trail-ext + T lead-ext + T trail-cont + T lead-cont × (S6) [ T trail-ext v trail-ext cm + T lead-ext v lead-ext cm + T trail-cont v trail-cont cm + T lead-cont v lead-cont cm ] where the velocities for each phase are worked out by choosing the appropriate values of ξ , ξ , and ξ and W L and W T from Table1. Note that for working out the time T of each phase, if the arm is contracting with a constant rate W , this time is merely ∆ L / W ;however, if the force is above the threshold, then the contraction will be slower, taking a time ∆ LW × max | F req | F thresh . Computing the average,and simplifying, yields Eq. 28 in the main text. We have found computer algebra systems ( Mathematica ) useful for keeping track of thespecial cases for when the force exceeds the threshold.We find that this analytic result captures our full simulations very well in the large-adhesion limit (Fig. S6).
Fig. S6
Comparison between full simulation and the high-friction asymptotic result of Eq. 28. F thresh = ; all other parameters are as in Table S1. B The regularized Blake tensor
We apply the results of to compute how the fluid velocity at point R m depends on a force F n exerted at point R n , in the presence ofa substrate with a no-slip boundary condition at z = (with a normal vector of ˆ e z ). This regularized Green’s function generalizes the
14 | 1–19 imple expressions given in the main paper (Eq. 4 and Eq. 7). The result of is: V ( R m , R n ) = [ F n H ( r ∗ n ) + ( F n · r ∗ n ) r ∗ n H ( r ∗ n )] − [ F n H ( r n ) + ( F n · r n ) r n H ( r n )] − h n [ g n D ( r n ) + ( g n · r n ) r n D ( r n )]+ h n (cid:20) H (cid:48) ( r n ) r n + H ( r n ) (cid:21) [( F n × ˆ e z ) × r n ]+ h n (cid:20) ( r n g nz + g n z n ) H ( r n ) + ( g n · r n ) (cid:18) ˆ e z H (cid:48) ( r n ) r n + r n z n H (cid:48) ( r n ) r n (cid:19)(cid:21) (S7)We have defined two relative distances: first, the explicit distance coordinate between the two points, r ∗ n = R m − R n , and the imagedistance coordinate between the target sphere and the image of the force-generating sphere, r n = R m − R n + h n ˆ e z . Here, h is thedistance of a sphere from the surface, g n = ( F n · ˆ e z ) ˆ e z − F n = ( − F nx , − F ny , F nz ) T and z n is the z -component of r n . Note that z n refers tothe relative coordinate r n and is not equivalent to h n , which is the z -component of the absolute coordinate R n . The full derivation forEq. S7 is available in Reference , but we note that the expression above corrects a sign error in its fourth bracketed term, which refersto the image rotlets.Eq. S7 is still well-defined when R m = R n . The forces are smoothed over the sphere volumes using four scalar regularization, or "blob,"functions: H ( r ) = π ( r + ε ) / + ε π ( r + ε ) / (S8) H ( r ) = π ( r + ε ) / (S9) D ( r ) = π ( r + ε ) / − ε π ( r + ε ) / (S10) D ( r ) = − π ( r + ε ) / (S11)Here ε defines the width of these functions and, therefore, the characteristic length over which to smooth the forces. Thus we set ε equal to the sphere radius a so that the sphere becomes a ball of finite force density instead of a singular point force.The form Eq. S7 is useful only if the forces on each sphere are already known. Since we must also calculate the individual forcesin addition to the velocities through Eq. 10, the expression in Eq. S7 can be rearranged into a more functional form, which is a 3 × n acting on sphere m , ˆ s n → m , that satisfies the relationshipdefined in Eq. 7. We define the mobility submatrices: η ˆ s n → m = H ( r ) − hz x ( h − z ) y ( h − z ) hx hy z ( h − z ) + H ( r ∗ ) x ∗ x ∗ y ∗ x ∗ z ∗ x ∗ y ∗ y ∗ y ∗ z ∗ x ∗ z ∗ y ∗ z ∗ z ∗ + (cid:20) h D ( r ) − H ( r ) − h (cid:18) H (cid:48) ( r ) r (cid:19) z (cid:21) x xy − xzxy y − yzxz yz − z + h D ( r ) − + h (cid:18) H (cid:48) ( r ) r (cid:19) z − + ( H ( r ∗ ) − H ( r )) I (S12)in which the subscripted n is implied on all relevant terms. The standard mobility matrix, ˆ M , is then an arrangement of these mobility ubmatrices: ˆ M = ˆ s → ... ˆ s N → ... ... ... ˆ s → N ... ˆ s N → N (S13)To solve the system with friction given by Eq. 15, the modified mobility matrix, ˆ M , is numerically calculated. The submatrices, ˆ S , arethen defined such that (cid:0) + ˆ Ξ (cid:1) − ˆ M = ˆ M = ˆ S → ... ˆ S N → ... ... ... ˆ S → N ... ˆ S N → N . (S14)which can now be used for all calculations. For example, to rewrite the constraints presented in Eq. 21-24, W L = (cid:20) ∑ n (cid:16) ˆ S n → − ˆ S n → (cid:17) F n (cid:21) · ˆ α (S15) W T = (cid:20) ∑ n (cid:16) ˆ S n → − ˆ S n → (cid:17) F n (cid:21) · ˆ α (S16) = (cid:26) ∑ n (cid:104) L T (cid:16) ˆ S n → − ˆ S n → (cid:17) − L L (cid:16) ˆ S n → − ˆ S n → (cid:17)(cid:105) F n (cid:27) · ˆ β (S17) = (cid:26) ∑ n (cid:104) L T (cid:16) ˆ S n → − ˆ S n → (cid:17) − L L (cid:16) ˆ S n → − ˆ S n → (cid:17)(cid:105) F n (cid:27) · ˆ γ (S18)
16 | 1–19
Constraint matrices
The constraint matrices are explicitly defined here, using the same notation as before, and where δ L = L T − L L . ˆ C s w i m d s w i m = T L i β i T L i β i T L i β i T T i β i T T i β i T T i β i T L i γ i T L i γ i T L i γ i T T i γ i T T i γ i T T i γ i (cid:16) ˆ S → − ˆ S → (cid:17) · ˆ α (cid:16) ˆ S → − ˆ S → (cid:17) · ˆ α (cid:16) ˆ S → − ˆ S → (cid:17) · ˆ α (cid:16) ˆ S → − ˆ S → (cid:17) · ˆ α (cid:16) ˆ S → − ˆ S → (cid:17) · ˆ α (cid:16) ˆ S → − ˆ S → (cid:17) · ˆ α (cid:104) L T ˆ S → + L L ˆ S → − δ L ˆ S → (cid:105) · ˆ β (cid:104) L T ˆ S → + L L ˆ S → − δ L ˆ S → (cid:105) · ˆ β (cid:104) L T ˆ S → + L L ˆ S → − δ L ˆ S → (cid:105) · ˆ β (cid:104) L T ˆ S → + L L ˆ S → − δ L ˆ S → (cid:105) · ˆ γ (cid:104) L T ˆ S → + L L ˆ S → − δ L ˆ S → (cid:105) · ˆ γ (cid:104) L T ˆ S → + L L ˆ S → − δ L ˆ S → (cid:105) · ˆ γ W L W T iiiiiii vvv i v ii v iiii x ( S19 ) ˆ C c r a w l d c r a w l = T L i β i T L i β i T L i β i T T i β i T T i β i T T i β i (cid:16) ˆ S → − ˆ S → (cid:17) · ˆ α (cid:16) ˆ S → − ˆ S → (cid:17) · ˆ α (cid:16) ˆ S → − ˆ S → (cid:17) · ˆ α (cid:16) ˆ S → − ˆ S → (cid:17) · ˆ α (cid:16) ˆ S → − ˆ S → (cid:17) · ˆ α (cid:16) ˆ S → − ˆ S → (cid:17) · ˆ α (cid:104) L T ˆ S → + L L ˆ S → − δ L ˆ S → (cid:105) · ˆ γ (cid:104) L T ˆ S → + L L ˆ S → − δ L ˆ S → (cid:105) · ˆ γ (cid:104) L T ˆ S → + L L ˆ S → − δ L ˆ S → (cid:105) · ˆ γ ˆ S → · ˆ e z ˆ S → · ˆ e z ˆ S → · ˆ e z ˆ S → · ˆ e z ˆ S → · ˆ e z ˆ S → · ˆ e z ˆ S → · ˆ e z ˆ S → · ˆ e z ˆ S → · ˆ e z W L W T iiiiiii vvv i v ii v iiii x ( S20 ) hen we define the constraints that keep the cell torque-free, we use ˆ T L = z L − y L − z L x L y L − x L ˆ T T = z T − y T − z T x T y T − x T where x L indicates the displacement of the leading arm in the x direction, etc.The rows of each constraint matrix correspond to the following constraints:For swimming (Eq. S19),i-iii: Force-free conditionsiv-v: Torque-free conditions (projection onto ˆ β and ˆ γ )vi: Leading arm deformation velocityvii: Trailing arm deformation velocityviii-ix: Rigid body conditions with projections of the deformation velocities onto ˆ β and ˆ γ , respectivelyFor crawling (Eq. S20),i-ii: Force-free conditionsiii: z -component ( ˆ β -projection) of the torque-free conditioniv: Leading arm deformation velocityv: Trailing arm deformation velocityvi: Rigid body condition with projection of the deformation velocities onto ˆ γ vii-ix: Zero z -directional velocities for each sphere
18 | 1–19
Parameters
Below are the parameters used for the simulations discussed in this project. Parameters are given in the simulation units discussed inthe Methods of the main paper. Table S1 provides all the parameters for the standard simulation. Tables S2-S3 refer to their respectivesimulations discussed above. Parameters not listed in Tables S2-S3 are unchanged from the standard parameter values given in TableS1.
Table S1
Parameters for the standard simulation
Parameter Symbol ValueMean arm length L a R com ( t = ) ( , , a ) Deformation magnitude ∆ L ± . Deformation velocities W + L , W − L , W + T , W − T ± . Polar angle θ π / Azimuthal angle φ ξ high ξ Low adhesion scale ξ low ξ Viscosity η F thresh Time step ∆ t − Table S2
Parameters for the anti-aligned pair
Parameter Cell 1 Cell 2Initial center of mass, R com ( t = ) (-2, -0.2, 0.1) (2, 0.2, 0.1)Azimuthal Angle, φ π Global adhesion (swim), ξ − · π Global adhesion (crawl), ξ · π Table S3
Parameters for the swimmer among multiple crawlers. Crawlers are generated every 90 cycles and removed from the system once thedistance between the swimmer and crawler is sufficiently large (after around 60 cycles) to reduce computational load. R c , ± xcom ( t gen ) refers to the center ofmass of each crawler at the time of its generation. Parameter Value R scom ( t = ) (0, 0, 1)Time between crawlers 90 cycles R c , + xcom ( t gen ) (-3, 0, 0.1) R c , − xcom ( t gen ) (8, 0, 0.1)Crawler deformation velocity, W c ± . Crawler deformation magnitude ∆ L c ± . Azimuthal Angle, φ π Global adhesion (swim), ξ ξ · π F thresh5