Chemotactic smoothing of collective migration
Tapomoy Bhattacharjee, Daniel B. Amchin, Ricard Alert, J. A. Ott, Sujit S. Datta
CChemotactic smoothing of collective migration
Tapomoy Bhattacharjee, ∗ Daniel B. Amchin, ∗ Ricard Alert,
3, 4, ∗ J. A. Ott, and Sujit S. Datta † The Andlinger Center for Energy and the Environment, Princeton University, Princeton, NJ, 08544, USA Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ, 08544, USA Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ 08544, USA Princeton Center for Theoretical Science, Princeton University, Princeton, NJ 08544, USA (Dated: January 13, 2021)Collective migration—the directed, coordinated motion of many self-propelled agents—is a fascinating emer-gent behavior exhibited by active matter that has key functional implications for biological systems. Extensivestudies have elucidated the different ways in which this phenomenon may arise. Nevertheless, how collectivemigration can persist when a population is confronted with perturbations, which inevitably arise in complexsettings, is poorly understood. Here, by combining experiments and simulations, we describe a mechanismby which collectively migrating populations smooth out large-scale perturbations in their overall morphology,enabling their constituents to continue to migrate together. We focus on the canonical example of chemotacticmigration of
Escherichia coli , in which fronts of cells move via directed motion, or chemotaxis, in responseto a self-generated nutrient gradient. We identify two distinct modes in which chemotaxis influences the mor-phology of the population: cells in different locations along a front migrate at different velocities due to spatialvariations in (i) the local nutrient gradient and in (ii) the ability of cells to sense and respond to the local nutrientgradient. While the first mode is destabilizing, the second mode is stabilizing and dominates, ultimately drivingsmoothing of the overall population and enabling continued collective migration. This process is autonomous,arising without any external intervention; instead, it is a population-scale consequence of the manner in whichindividual cells transduce external signals. Our findings thus provide insights to predict, and potentially control,the collective migration and morphology of cell populations and diverse other forms of active matter.
The flocking of birds, schooling of fish, herding of animals,and procession of human crowds are all familiar examplesof collective migration. This phenomenon also manifests atsmaller scales, such as in populations of cells and dispersionsof synthetic self-propelled particles. In addition to being a fas-cinating example of emergent behavior, collective migrationcan be critically important—enabling populations to followcues that would be undetectable to isolated individuals , es-cape from harmful conditions and colonize new terrain , andcoexist . Thus, diverse studies have sought to understand themechanisms by which collective migration can arise.Less well understood, however, is how collective migra-tion persists after a population is confronted with perturba-tions. These can be external, stemming from heterogeneitiesin the environment , or internal, stemming from differ-ences in the behavior of individuals . Mechanisms bywhich such perturbations can disrupt collective migrationare well documented. Indeed, in some cases, perturbationscan abolish coordinated motion throughout the populationentirely . In other cases, perturbations couple to theactive motion of the population to destabilize its leading edge,producing large-scale disruptions to its morphology . In-deed, for one of the simplest cases of collective migration— via chemotaxis, the biased motion of cells up a chemi-cal gradient—morphological instabilities can occur due tothe disruptive influence of hydrodynamic or chemical-mediated interactions between cells. By contrast, mech-anisms by which migrating populations can withstand pertur-bations have scarcely been examined. ∗ These authors contributed equally to this work. † [email protected] Here, we demonstrate a mechanism by which collectivelymigrating populations of
E. coli autonomously smooth outlarge-scale perturbations in their overall morphology. Weshow that chemotaxis in response to a self-generated nutrientgradient provides both the driving force for collective migra-tion and the primary smoothing mechanism for these bacterialpopulations. Using experiments on 3D printed populationswith defined morphologies, we characterize the dependenceof this active smoothing on the wavelength of the perturba-tion and on the ability of cells to migrate. Furthermore, us-ing continuum simulations, we show that the limited ability ofcells to sense and respond to a nutrient gradient causes themto migrate at different velocities at different positions along afront—ultimately driving smoothing of the overall populationand enabling continued collective migration. Our work thusreveals how cellular signal transduction enables a populationto withstand large-scale perturbations, and provides a frame-work to predict and control chemotactic smoothing for activematter in general.
RESULTSChemotactic smoothing is regulated byperturbation wavelength and cellular motility
To experimentally investigate the collective migration of
E.coli populations, we confine them within porous media oftunable properties , as schematized in Figs. 1A and 1Band detailed in the Materials and Methods. The media arecomposed of hydrogel particles that are swollen in a de-fined rich liquid medium with L -serine as the primary nutrientand chemoattractant. We enclose the particles at prescribed a r X i v : . [ phy s i c s . b i o - ph ] J a n Figure 1 Experiments reveal that migrating
E. coli populations autonomously smooth large-scale morphological perturbations. ( A )Schematic of an undulated population (green cylinder) 3D-printed within a porous medium made of jammed hydrogel particles (gray). Eachundulated cylinder requires ∼ s to print, two orders of magnitude shorter than the duration between successive 3D confocal image stacks, ∼ min. The surrounding medium fluidizes as cells are injected into the pore space, and then rapidly re-jams around the dense-packed cells.( B ) Two-dimensional xy slice through the mid-plane of the population. The starting morphology of the 3D-printed population has undulationwavelength λ and amplitude A , as defined by the undulated path traced out by the injection nozzle. The cells subsequently swim through thepores between hydrogel particles, with mean pore size ξ . The population thereby migrates outward in a coherent front that eventuallysmooths; we track the radial position of the leading edge of the front R f and the undulation amplitude A over time t . ( C )-( E ) Bottom-up ( xy plane) projections of cellular fluorescence intensity measured using 3D confocal image stacks. Images show sections of three initiallyundulated populations in three different porous media, each at three different times (superimposed white, yellow, cyan), as the cells migrateradially outward. A pixel corresponds to ∼ cell, and the images only show a magnified view of the overall population. Panels ( C )-( D )demonstrate the influence of varying the undulation wavelength, keeping the mean pore size the same; increasing λ slows smoothing. Panels( C ) and ( E ) demonstrate the influence of varying the pore size, keeping the undulation wavelength the same; increasing ξ hastens smoothing.( F ) For each experiment shown in ( C )-( E ), the undulation amplitude A , normalized by its initial value A , decays exponentially with the time ∆ t elapsed from the initiation of smoothing at t = t . Fitting the data (symbols) with an exponential decay (red lines) yields the smoothingtime τ for each experiment. ( G ) Smoothing time τ measured in experiments increases with increasing undulation wavelength λ anddecreasing medium mean pore size ξ , which enables cells to migrate more easily. Error bars reflect the uncertainty in determining theinitiation time t from the exponential fit of the data. jammed packing fractions in transparent chambers. Becausethe hydrogel is highly swollen, it is freely permeable to oxy-gen and nutrient. However, while the particles do not hinderexposure of bacteria to these chemical signals, the cells can-not penetrate the individual particles, and are instead forcedto swim through the interparticle pores (Fig. 1B). Varying thehydrogel particle packing density thus enables us to tune poresize and thereby modulate cellular migration without alteringthe nutrient field . Specifically, we vary the mean poresize ξ between . µ m and . µ m, causing cellular migra-tion through the pore space to be more and less hindered, re-spectively, without deforming the solid matrix . Moreover,the packings are transparent, enabling the morphologies ofthe migrating populations to be tracked in the xy plane usingconfocal fluorescence microscopy (Fig. 1A); to this end, weuse cells that constitutively express green fluorescent proteinthroughout their cytoplasm.A key feature of the hydrogel packings is that they are yield-stress solids; thus, an injection micronozzle can move alonga prescribed path inside each medium by locally rearrangingthe particles, gently extruding densely-packed cells into theinterstitial space (Figs. 1A and 1B). The particles then rapidlyre-densify around the newly-introduced cells, re-forming ajammed solid matrix that supports the cells in place with mini-mal alteration to the overall pore structure . This approachis therefore a form of 3D printing that enables the initial mor-phology of each bacterial population to be defined within theporous medium. The cells subsequently swim through thepores between particles, migrating outward through the porespace. For example, as we showed previously , cells of E.coli initially 3D printed in densely-packed straight cylinderscollectively migrate radially outward in flat, coherent fronts.These fronts form and propagate via chemotaxis: the cellscontinually consume surrounding nutrient, generating a localgradient that they in turn bias their motion along . As thisfront of cells migrates, it propagates the local nutrient gradi-ent with it through continued consumption, thereby sustain-ing collective migration. In the absence of nutrient, migratingfronts do not form at all .To test how perturbations in the overall morphology of thepopulation influence its subsequent migration, we 3D printdensely-packed E. coli in 1 cm-long cylinders with spatially-periodic undulations as perturbations prescribed along the x direction (Fig. 1B). Each population is embedded deep withina defined porous medium; an initial population morphologyis schematized at time t = 0 in Fig. 1B, with the undula-tion wavelength and amplitude denoted by λ and A , respec-tively. An experimental realization with A ( t = 0) ≈ µ m, λ ≈ . mm, and ξ = 1 . µ m is shown in white in Fig. 1C,which shows an xy cross section through the midplane of thepopulation. After 3D printing, the outer periphery of the pop-ulation spreads slowly, hindered by cell-cell collisions in thepore space, as the population establishes a steep gradient ofnutrient through consumption . Then, this periphery sponta-neously organizes into a ∼ µ m-wide front of cells thatcollectively migrates outward (yellow in Fig. 1C). The undu-lated morphology of this front initially retains that of the ini-tial population. Strikingly, however, the front autonomously smooths out these large-scale undulations as it continues topropagate (Movie S1). We characterize this behavior by track-ing the decay of the undulation amplitude, normalized by itsinitial value A ≡ A (∆ t = 0) , as a function of time elapsedfrom the initiation of smoothing, ∆ t (green circles in Fig. 1F).The normalized amplitude decays exponentially (red line inFig. 1F), with a characteristic time scale τ ≈ . h, and thepopulation eventually continues to migrate as a completely flatfront (cyan in Fig. 1C).We observe similar behavior when the wavelength λ is in-creased to . mm (Fig. 1D, Movie S2) or when the pore size ξ is increased to . µ m (Fig. 1E, Movie S3); however, the dy-namics of front smoothing are altered in both cases. Specifi-cally, increasing the undulation wavelength slows smoothing,increasing τ by a factor of ≈ to reach τ ≈ . h (greensquares in Fig. 1F). Conversely, increasing the pore size—which enables cells to migrate through the pore space moreeasily—greatly hastens smoothing, decreasing τ by more thana factor of ≈ to become τ ≈ . h (blue circles in Fig. 1F).This behavior is consistent across multiple experiments withvarying λ and ξ , as summarized in Fig. 1G. Our experimentsthus indicate that the smoothing of collective migration is reg-ulated by both the undulation wavelength and the ease withwhich cells migrate. A continuum model of chemotactic migration recapitulatesthe spatio-temporal features of smoothing
To gain further insight into the processes underlyingsmoothing, we use the classic Keller–Segel model of chemo-tactic migration to investigate the dynamics of undulatedpopulations. Variants of this model can successfully capturethe key features of chemotactic migration of flat
E. coli frontsin bulk liquid and in porous media ; we therefore hy-pothesize that it can also help identify the essential physics ofsmoothing.To this end, we consider a two-dimensional (2D) represen-tation of the population in the xy plane for simplicity, with r ≡ ( x, y ) , and model the evolution of the nutrient concen-tration c ( r , t ) and number density of bacteria b ( r , t ) using thecoupled equations: ∂ t c = D c ∇ c − bκg ( c ) , (1) ∂ t b = − ∇ · J b + bγg ( c ) , J b = − D b ∇ b + bχ ∇ f ( c ) . (2)Equation (1) relates changes in c to nutrient diffusion andconsumption by the bacteria; D c is the nutrient diffusion co-efficient, κ is the maximal consumption rate per cell, and g ( c ) = c/ ( c + c ) describes the influence of nutrient avail-ability relative to the characteristic concentration c throughMichaelis-Menten kinetics. Equation (2) relates changes in b to the bacterial flux J b , which arises from their undirected anddirected motion, and net cell proliferation with a maximal rate γ . In the absence of a nutrient gradient, bacteria move in anunbiased random walk ; thus, undirected motion is diffusiveover large length and time scales, with an effective diffusion Figure 2 Continuum model captures the essential features of the smoothing of migrating bacterial populations. ( A )-( C ) Simulationscorresponding to experiments reported in Figs. 1C and 1E, respectively, performed by numerically solving Eqs. (1) and (2) in two dimensions( xy plane). Images show the calculated cellular signal (details in Materials and Methods) for three initially undulated populations in threedifferent porous media, each at three different times (superimposed white, yellow, cyan), as the cells migrate outward. Panels ( A )-( B )demonstrate the influence of varying the undulation wavelength, keeping the mean pore size the same; as in the experiments, increasing λ slows smoothing. Panels ( A ) and ( C ) demonstrate the influence of varying the pore size, keeping the undulation wavelength the same; as inthe experiments, increasing ξ , incorporated in the model by using larger values of the diffusion and chemotactic coefficients as obtaineddirectly from experiments, hastens smoothing. ( D ) For each simulation shown in ( A )-( C ), the undulation amplitude A ,normalized by itsinitial value A , decays exponentially with the time ∆ t elapsed from the initiation of smoothing at t = t as in the experiments. Fitting thedata (symbols) with an exponential decay (red lines) again yields the smoothing time τ for each simulation. ( E ) Smoothing time τ obtainedfrom the simulations increases with increasing undulation wavelength λ and decreasing medium mean pore size ξ , as in the experiments.Error bars reflect the uncertainty in determining the initiation time t from the exponential fit of the data. coefficient D b whose value depends on both cellular activityand confinement in the pore space . In the presence ofthe local nutrient gradient established through consumption,bacteria perform chemotaxis, biasing this random walk ; thefunction f ( c ) ≡ log [(1 + c/c − ) / (1 + c/c + )] describes theability of the bacteria to logarithmically sense nutrient withcharacteristic concentrations c − and c + , and the chemo-tactic coefficient χ describes their ability to then bias theirmotion in response to the sensed nutrient gradient . Thechemotactic velocity is thus given by v ch ≡ χ ∇ f ( c ) , wheresimilar to D b , the value of χ depends on both intrinsic cellularproperties and pore-scale confinement . Together, Eqs. (1)and (2) provide a continuum model of chemotactic migrationthat has thus far been successfully used to describe unper-turbed E. coli populations .To simulate the chemotactic migration of perturbed pop-ulations, we numerically solve Eqs. (1) and (2) using undu-lated morphologies as initial conditions for b , similar to those explored in the experiments. The simulations employ val-ues for all parameters based on direct measurements, as de-tailed in the Materials and Methods. Although we do not ex-pect perfect quantitative agreement between the experimentsand simulations due to their difference in dimensionality andthe simplified treatment of cell-cell interactions, the simulatedfronts form, collectively migrate, and smooth in a mannerthat is remarkably similar to the experiments. Three exam-ples are shown in Figs. 2C to 2E (Movies S4 to S6), corre-sponding to the experiments shown in Figs. 1C to 1E (MoviesS1 to S3). Similar to the experiments, the outer periphery ofeach population first spreads slowly, then spontaneously orga-nizes into an outward-migrating front that eventually smooths.We again find that the normalized undulation amplitude de-cays exponentially over time, as shown in Fig. 2D. As in theexperiments, increasing the undulation wavelength λ slowssmoothing; compare Fig. 2B to Fig. 2A. Also as in the exper-iments, increasing the pore size ξ , which increases the migra- Figure 3 Chemotaxis is the primary driver of morphological smoothing.
Images show the same simulation as in Fig. 2A, which serves asan exemplary case, but with either ( A ) diffusive cell motion, ( B ) cell proliferation, or ( C ) cell chemotaxis knocked out by setting thediffusivity D b , proliferation rate γ , or chemotactic coefficient χ to zero, respectively. Simulated bacterial fronts lacking diffusion orproliferation still smooth, as shown in ( A )-( B ), but simulated fronts lacking chemotaxis do not smooth, as shown in ( C )—demonstrating thatchemotaxis is necessary and sufficient for the observed morphological smoothing. tion parameters D b and χ , greatly hastens smoothing; com-pare Fig. 2C to Fig. 2A. This variation of the smoothing timescale τ obtained from simulations with λ and ξ is summa-rized in Fig. 2E. We observe the same behavior as in the ex-periments, with the absolute values of τ agreeing to within afactor of ∼ . This agreement confirms that the continuumKeller-Segel model recapitulates the essential spatio-temporalfeatures of smoothing seen in the experiments. Chemotaxis is the primary driver of front smoothing
The simulations provide a way to directly assess the relativeimportance of cellular diffusion, chemotaxis, and cell prolif-eration to front smoothing. To this end, we perform the samesimulation as in Fig. 2A, but with each of the correspondingthree terms in Eq. (2) knocked out, and determine the result-ing impact on collective migration. This procedure enables usto determine the factors necessary for smoothing.While diffusion typically causes spatial inhomogeneities tosmooth out, we do not expect it to play an appreciable rolein the front smoothing observed here: the characteristic timescale over which undulations of wavelength λ ≈ mm dif-fusively smooth is ∼ λ /D b ≈ to h, up to three or-ders of magnitude larger than the smoothing time τ measuredin experiments and simulations. We therefore expect that theundirected motion of bacteria is much too slow to contributeto front smoothing. The simulations for λ = 0 . mm and ξ = 1 . µ m confirm this expectation: setting D b = 0 yieldsfronts that still smooth over a time scale τ ∼ h similar to thefull simulations (Fig. 3A).Another possible mechanism of front smoothing is differ-ences in bacterial proliferation at different locations along thefront periphery—for example, the front would smooth if cellsin concave regions were able to proliferate faster than thosein convex regions. However, differential proliferation typi-cally destabilizes bacterial communities, as shown previouslyboth experimentally and theoretically . Furthermore, even if proliferation were to help smooth the overall population, weagain expect this hypothetical mechanism to be too slow to ap-preciably contribute: the shortest time scale over which cellsall growing exponentially at a maximal rate γ ∼ h − spreadover the length scale A ≈ µ m by growing end-to-end is γ − log ( A /l cell ) ∼ h, where l cell ≈ µ m is the cell bodylength. This time scale is over an order of magnitude largerthan the τ measured in experiments and simulations. The sim-ulations again confirm our expectation: setting γ = 0 yieldsfronts that still smooth over a time scale τ ∼ h similar to thefull simulations (Fig. 3B).These findings leave chemotaxis as the remaining possi-ble mechanism of front smoothing. The simulations con-firm this expectation: setting χ = 0 yields a population thatslowly spreads via diffusion and proliferation, but that doesnot form collectively migrating fronts at all (Fig. 3C). There-fore, chemotaxis is both necessary and sufficient for the ob-served front smoothing. Distinct modes by which chemotaxis impacts front morphology
How exactly does chemotaxis smooth bacterial fronts?To address this question, we examine the spatially-varyingchemotactic velocity v c = χ ∇ f ( c ) , which quantifies howrapidly different regions of the population migrate via chemo-taxis. To gain intuition for the determinants of v c , we recastthis expression in terms of the nutrient gradient: v c = χf (cid:48) ( c ) (cid:124) (cid:123)(cid:122) (cid:125) Response function ∇ c (cid:124)(cid:123)(cid:122)(cid:125) Forcing . (3)As in linear response theory, the chemotactic velocity can beviewed as the bacterial response to the driving force given bythe nutrient gradient, ∇ c , modulated by the chemotactic re-sponse function χf (cid:48) ( c ) . Thus, variations in chemotactic ve-locity along the leading edge of the front, which specify howthe overall front morphology evolves, are determined by the Figure 4 Chemotaxis alters the morphology of migrating bacterial fronts in two distinct ways. ( A ) Magnified view of a migratingbacterial front from the simulation shown in Fig. 2A at time t = 41 min as a representative example. To illustrate the spatially-varyingnutrient levels, we show the contours of constant nutrient concentration c = c + and c = c − in magenta and cyan, respectively; theserepresent characteristic upper and lower limits of sensing. The contours are spaced closer at the leading edge of the convex peak ( y/λ = 0 . )than the concave valley ( y/λ = 0 ), indicating that the magnitude of the local nutrient gradient is larger at peaks than at valleys. The nutrientconcentration itself, which increases monotonically with increasing x , is also larger at the peak than at the valley. ( B ) Top and bottom panelsshow the variation of the nutrient sensing function f ( c ) and chemotactic response function f (cid:48) ( c ) , respectively, with nutrient concentration c .Because sensing saturates at high nutrient concentrations, chemotactic response is weaker at higher c (peaks) than at lower c (valleys). ( C )Top panel shows the x component of the nutrient gradient ∂ x c (red, left axis) and the response function f (cid:48) (blue, right axis), and bottom panelshows the x component of the chemotactic velocity v c ,x = χf (cid:48) ∂ x c computed from these quantities, evaluated at different lateral positions y along the leading edge of the front in ( A ). While the driving force of chemotaxis represented by ∂ x c is smaller at the valley, the chemotacticresponse χf (cid:48) is larger at the valley and dominates in setting v c ,x : valleys move out faster than peaks, eventually catching up to them andsmoothing out the undulations. ( D ) For all simulations (Fig. 2E), the smoothing time τ determined by analyzing the decay of large-scaleundulations (Fig. 2D) is set by the time τ (cid:48) needed for valleys to catch up to peaks estimated using their different x -component chemotacticvelocities. combined effect of variations in the nutrient gradient and thechemotactic response function. We therefore examine each ofthese modes by which chemotaxis influences front morphol-ogy in turn.We first consider the nutrient gradient, which is the typicalfocus of chemotaxis studies. Our simulations, which numer-ically solve the coupled system of Eqs. (1) and (2), directlyyield the spatially-varying nutrient field c and therefore ∇ c . Asnapshot from the representative example of Fig. 2A is shownin Fig. 4A, with the contours of c = c − and c = c + indicatedby the cyan and magenta lines, respectively. The contours are spaced closer at the convex “peaks” (e.g., at y/λ = 0 . )than at the concave “valleys” (e.g., at y/λ = 0 ) along theleading edge of the front. Thus, the magnitude of the drivingforce given by ∇ c is larger at the peaks. We confirm this ex-pectation by directly quantifying the nutrient gradient alongthe leading edge, focusing on the component ∂ x c in the over-all front propagation direction ( x ) for simplicity, as shown bythe orange symbols in Fig. 4C; as expected, this driving forceis stronger at the peaks. This spatial variation in the driv-ing force promotes faster outward chemotactic migration atthe peaks than at the valleys, amplifying front undulations—in opposition to our observation that the migrating populationself-smooths. Variations in the local nutrient gradient alongthe leading edge of the front do not contribute to smoothing;rather, they oppose it.We next turn to the chemotactic response function, whichcharacterizes cellular signal transduction. Because χ is a con-stant for each porous medium , spatial variations in the re-sponse function are set by variations in f (cid:48) ( c ) . The sensingfunction f ( c ) is plotted in the upper panel of Fig. 4B. It varieslinearly as ∼ c (1 /c − − /c + ) for c (cid:28) c − and saturates at log ( c + /c − ) for c (cid:29) c + ; the characteristic concentrations c − and c + represent the dissociation constants of the nutrient forthe inactive and active conformations of the cell-surface re-ceptors, respectively . The response function χf (cid:48) ( c ) therefore decreases strongly as c increases above c + , whichaccordingly is often referred to as an upper limit of sens-ing (Fig. 4B, lower panel). That is, because high nutrientconcentrations saturate cell-surface receptors, the chemotac-tic response function decreases with nutrient concentration.Inspection of the nutrient field indicates that nutrient concen-trations are larger at the peaks than at the valleys along theleading edge of the front (Fig. 4A). Thus, the chemotactic re-sponse of cells is weaker at peaks than at valleys, as shownby the points in Fig. 4B, yielding slower outward chemotac-tic migration at peaks than at valleys and thereby reducing theamplitude of front undulations. Variations in the chemotacticresponse along the leading edge of the front promote smooth-ing, unlike variations in the nutrient gradient. Spatial variations in chemotactic responsedrive morphological smoothing
We therefore hypothesize that the stabilizing effect of thechemotactic response (Fig. 4C, blue) dominates over thedestabilizing influence of the nutrient gradient (Fig. 4C, red),leading to smoothing. Computation of the spatially-varyingchemotactic velocity at the leading edge of the front usingEq. (3), focusing on the x velocity component v c ,x ≈ χf (cid:48) ∂ x c for simplicity, supports this hypothesis: cells at concave re-gions migrate outward faster than those at convex regions(Fig. 4C, lower panel). To further test this hypothesis, weassess the influence of varying c + ; we expect that reducingthis upper limit weakens chemotactic response not just at thepeaks, but also the valleys, thereby slowing smoothing. Whiletuning solely c + is challenging in the experiments, this can bereadily done in the simulation—yielding slower smoothing,as expected (Fig. S1). As a final test of our hypothesis, foreach simulation shown in Fig. 2, we determine the differencebetween the chemotactic velocities of the valleys and peaks,approximated by ∆ v c ,x ≈ χ [( f (cid:48) ∂ x c ) valley − ( f (cid:48) ∂ x c ) peak ] , asa function of time ∆ t . If smoothing is indeed due to varia-tions of the chemotactic velocity along the leading edge, thenthe smoothing time τ determined by analyzing the decay oflarge-scale undulations (Figs. 2D and 2E) should be approxi-mately given by the time τ (cid:48) at which valleys catch up to peaks, i.e. , (cid:82) τ (cid:48) ∆ v c ,x d∆ t ≈ A . The τ (cid:48) thus obtained is shownfor all of our simulations of varying λ and ξ in Fig. 4D. We find excellent agreement in all cases between τ (cid:48) and τ , shownon the vertical and horizontal axes respectively—confirmingthat smoothing is indeed determined by spatial variations inchemotactic velocity. DISCUSSION
By combining experiments and simulations, this work elu-cidates a mechanism by which collectively migrating popula-tions can smooth out large-scale perturbations in their overallmorphology. We focus on the canonical example of chemo-tactic migration, in which coherent fronts of cells move in re-sponse to a self-generated nutrient gradient. Over the pasthalf century, extensive studies have focused on the migrationof unperturbed flat fronts ; our work now demonstrateshow perturbed fronts smooth out.The 3D printing platform provides a unique way to tune theshape of the initial perturbation, as well as the extent to whichcellular migration is hindered. Our experiments using this ap-proach reveal that the dynamics of smoothing are regulated byboth the undulation wavelength and the ease with which cellsmigrate. The continuum simulations recapitulate the essen-tial features of this behavior and shed light on the underlyingmechanism; while studies of chemotaxis typically focus onthe role of the nutrient gradient in driving cellular migration,our work highlights the distinct and pivotal role played bythe chemotactic response function in regulating migration andlarge-scale population morphology. In particular, we find thateven though cells in peaks of an undulated front experience astronger driving force given by the local nutrient gradient, thehigher nutrient levels they are exposed to saturate their cell-surface receptors, and hence they exhibit a weaker chemotac-tic response than cells in valleys. That is, while variations inthe nutrient gradient along the leading edge of a front act toamplify undulations, variations in the ability of cells to senseand respond to this gradient dominate and instead smooth outthe undulation.Our work thus reveals how chemotaxis in response to a self-generated nutrient gradient can enable a migrating populationto withstand large-scale perturbations, providing a counter-point to previous studies investigating the ability of pertur-bations to instead disrupt collective migration .The chemotactic smoothing process is autonomous, aris-ing without any external intervention; instead, it is apopulation-scale consequence of the limitations in cellu-lar signal transduction—motivating future studies of otherpopulation-scale effects that may emerge from individual be-haviors. By demonstrating how chemotaxis drives smooth-ing, our work contributes a new factor to be considered indescriptions of morphogenesis, which thus far have focusedon the role of other factors—such as differential proliferation,intercellular mechanics, substrate interactions, and osmoticstresses —in regulating the overall morphology of abacterial population. Finally, because many other active sys-tems such as other prokaryotes, cancer cells, white blood cells,amoeba, enzymes, chemically-sensitive colloidal microswim-mers, and chemical robots also migrate via chemotaxis,this mechanism of smoothing could broadly manifest in di-verse forms of active matter.
ACKNOWLEDGMENTS
It is a pleasure to acknowledge Tommy Angelini for provid-ing microgel polymers; Bob Austin for providing fluorescent
E. coli ; and Stas Shvartsman, Howard Stone, Sankaran Sun-daresan, and Ned Wingreen for stimulating discussions. Thiswork was supported by NSF grant CBET-1941716, the ProjectX Innovation fund, a distinguished postdoctoral fellowshipfrom the Andlinger Center for Energy and the Environmentat Princeton University to T.B., the Eric and Wendy SchmidtTransformative Technology Fund at Princeton, the PrincetonCatalysis Initiative, and in part by funding from the Prince-ton Center for Complex Materials, a Materials Research Sci-ence and Engineering Center supported by NSF grant DMR-2011750. This material is also based upon work supportedby the National Science Foundation Graduate Research Fel-lowship Program (to J.A.O.) under Grant No. DGE-1656466.Any opinions, findings, and conclusions or recommendationsexpressed in this material are those of the authors and do notnecessarily reflect the views of the National Science Founda-tion. R.A. acknowledges support from the Human FrontierScience Program (LT000475/2018-C).
AUTHOR CONTRIBUTIONS
T.B. and S.S.D. designed the experiments; T.B. performedall experiments with assistance from J.A.O.; D.B.A., J.A.O.,and S.S.D. designed the numerical simulations; D.B.A. per-formed all numerical simulations with assistance from J.A.O.;R.A. performed all theoretical calculations through discus-sions with S.S.D.; T.B., D.B.A., R.A., and S.S.D. analyzedthe data; S.S.D. designed and supervised the overall project.All authors discussed the results and implications and wrotethe manuscript.
COMPETING INTERESTS
The experimental platform used to 3D print and image bac-terial communities in this publication is the subject of a patentapplication filed by Princeton University on behalf of T.B. andS.S.D.
DATA AVAILABILITY
All data are available from the authors upon request.
CODE AVAILABILITY
All codes are available from the authors upon request.
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We prepare 3D porous media by dispersing dry granules ofcrosslinked acrylic acid/alkyl acrylate copolymers (Carbomer980, Ashland) in liquid EZ Rich, a defined rich medium for
E. coli . The components to prepare the EZ Rich are pur-chased from Teknova Inc., autoclaved prior to use, and aremixed following manufacturer directions; specifically, the liq-uid medium is an aqueous solution of 10X MOPS Mixture(M2101), 10X ACGU solution (M2103), 5X Supplement EZsolution (M2104), 20% glucose solution (G0520), 0.132 Mpotassium phosphate dibasic solution (M2102), and ultrapuremilli-Q water at volume fractions of 10%, 10%, 20%, 1%,1%, and 58%, respectively. We ensure homogeneous disper-sions of swollen hydrogel particles by mixing each dispersionfor at least 2 h at 1600 rpm using magnetic stirring, and ad-just the pH to 7.4 by adding 10 N NaOH to ensure optimalcell viability. The hydrogel granules swell considerably, re-sulting in a jammed medium made of ∼ to µ m diame-ter swollen hydrogel particles with ∼ polydispersity andwith an individual mesh size of ∼ to nm, as we es-tablished previously , which enables small molecules (e.g.,amino acids, glucose, oxygen) to freely diffuse throughout themedium.Tuning the mass fraction of dispersed hydrogel particlesenables the sizes of the pores between particles to be pre-cisely tuned. We measure the smallest local pore dimen-sion by tracking the diffusion of 200 nm-diameter fluorescenttracers through the pore space, as we detailed in a previouspaper . This previous paper shows the full pore size distribu-tions thereby measured for porous media prepared in an iden-tical manner to those used here; in this present paper, we onlydescribe each medium using the mean pore size ξ , for simplic-ity.
3D printing bacterial populations
Prior to each experiment, we prepare an overnight cultureof
E. coli
W3110 in LB media at 30 ◦ C. We then incubate a1% solution of this culture in fresh LB media for 3 h until theoptical density reaches ∼ . , and then resuspend the cells inliquid EZ Rich to a concentration of . × cells/mL. Wethen use this suspension as the inoculum that is 3D printedinto a porous medium using a pulled glass capillary with a ∼ to µ m-wide opening as an injection nozzle. Eachporous medium has a large volume of 4 mL and is confinedin a transparent-walled glass-bottom petri dish 35 mm in di-ameter and 10 mm in height; in each experiment, the injec-tion nozzle is mounted on a motorized translation stage thattraces out a programmed two-dimensional undulating pathwithin the porous medium, at least ∼ to µ m awayfrom any boundaries, at a constant speed of 1 mm/s. As theinjection nozzle moves through the medium, it locally rear-ranges the hydrogel packing and gently extrudes the cell sus-pension into the interstitial space using a flow-controlled sy- ringe pump at 50 µ L/hr, which corresponds to a gentle shearrate of ∼ to s − at the tip of the injection nozzle. As thenozzle continues to move, the surrounding hydrogel particlesrapidly densify around the newly-introduced cells, re-forminga jammed solid matrix that compresses the cellular sus-pension until the cells are close-packed to an approximatedensity of . × cells/mL. This protocol thus resultsin a 3D-printed bacterial population having a defined initialamplitude and wavelength. Moreover, as we showed in ourprevious work , this process does not appreciably alter theproperties of the hydrogel packing and is sufficiently gentle tomaintain the viability and motility of the cells. Imaging bacteria within porous media
Because the 3D-printed undulated cylinders of dense-packed cells are ∼ cm long, each printing process requires ∼ s. After 3D printing, the top surface of the porousmedium is sealed with a thin layer of 1 to 2 mL of paraffin oilto minimize evaporation while allowing unimpeded oxygendiffusion. We then commence imaging within a few minutesafter printing. Once an undulated population is 3D printed, itmaintains its shape until cells start to move outward throughthe pore space. The time needed to print each cylinder is twoorders of magnitude shorter than the duration between succes-sive 3D confocal image stacks. Moreover, the 3D printing isfast enough to be considered as instantaneous when comparedwith the speed of bacterial migration. Thus, the imaging issufficiently fast to capture the front propagation dynamics. Toimage how the distribution of cells evolves over time, we usea Nikon A1R+ inverted laser-scanning confocal microscopemaintained at ± ◦ C. In each experiment, we acquire verti-cal stacks of planar fluorescence images separated by . µ malong the vertical ( z ) direction, successively every to minutes for up to h. We then produce a maximum inten-sity projection from each stack at every time frame with thelogarithm of fluorescent intensities displayed at every pixel;examples are shown in Fig. 1. Characterizing experimental front dynamics
We use each maximum intensity projection at each timepoint to manually measure the time-dependent amplitude ( A )and radial location of the front ( R f ) as defined in Fig. 1B,identifying the edges of the front as the positions at which thefluorescent signal from cells matches the background noise.As we showed in our previous work , due to the initiallyhigh cell density in the population, inter-cell collisions limitoutward migration of the population; a coherent outward-propagating front only forms after at least ∼ h. Here, we donot focus on these initial transient dynamics, but instead ex-amine the long-time smoothing behavior of undulated fronts.We do this by tracking the decay of the time-dependent undu-lation amplitude over time, as shown in Fig. 1F; we identifythe time t at which smoothing is initiated as the earliest timeat which the error associated with an exponential fit to the de-2cay of A ( t ) is minimized. The initial value A is then givenby A ( t ) . Details of continuum model
To mathematically model the dynamics of bacterial fronts,we use a continuum description of chemotactic migration thatwe previously showed captures the essential dynamical fea-tures of flat fronts . This model extends previous work on theclassic Keller-Segel model to the case of dense pop-ulations in porous media. In particular, we consider a 2D rep-resentation of the population in the xy plane for simplicity anddescribe the evolution of the nutrient concentration c ( r , t ) andnumber density of bacteria b ( r , t ) using the coupled Eqs. (1)and (2). Nutrient diffusion and consumption.
The media used inour experiments have L -serine as the most abundant nutrientsource and chemoattractant . E. coli consume this amino acidfirst and respond to it most strongly as a chemoattractantcompared to other components of the media . Further-more, the nutrient levels of our liquid medium are nearly twoorders of magnitude larger than the levels under which E. coli excrete appreciable amounts of their own chemoattractant and generate strikingly different front behavior than thosethat arise in our experiments; however, the nutrient levels weuse are sufficiently low to avoid toxicity associated with ex-tremely large levels of L -serine . Thus, given all of thesereasons, we focus on L -serine as the primary nutrient sourceand attractant, described by the scalar field c ( r , t ) . Equa-tion 1 then relates changes in c to nutrient diffusion andconsumption by the bacteria. The nutrient diffusion coeffi-cient D c = 800 µ m /s is given by previous measurementsin bulk liquid; we treat nutrient diffusion as being unhin-dered by the highly-swollen hydrogel matrix due to its largeinternal mesh size. The maximal consumption rate per cell κ = 1 . × − mM(cell/mL) − s − is chosen based on pre-vious measurements , and g ( c ) = c/ ( c + c ) describes theinfluence of nutrient availability relative to the characteristicconcentration c = 1 µ M through Michaelis-Menten kinet-ics, as established previously . These values yield sim-ulated fronts that we have previously validated against exper-iments in porous media for the unperturbed case . Bacterial diffusion and chemotaxis.
The bacterial flux J b as included in Eq. (2) arises from the undirected and di-rected motion of cells, i.e., diffusion − D b ∇ b and chemo-taxis bχ ∇ f ( c ) , respectively. The value of the activecellular diffusion coefficient D b decreases with increas-ing pore-scale confinement ; as validated in our previ-ous work for porous media identical to those used here, D b = 2 . , . , and . µ m /s for porous mediawith ξ = 2 . , . , and . µ m, respectively. We de-scribe cellular chemotaxis using the sensing function f ( c ) ≡ log [(1 + c/c − ) / (1 + c/c + )] and the chemotactic coefficient χ , as established previously . The characteristic concentra-tions c − = 1 µ M and c + = 30 µ M represent the dissociationconstants of the nutrient for the inactive and active conforma-tions of the cell-surface receptors, respectively . Similar to D b , the value of the active chemotactic coeffi-cient χ decreases with increasing pore-scale confinement ;by matching the long-time front propagation speed in our sim-ulations with the experiments, we obtain χ = 145 , , and µ m /s for porous media with ξ = 2 . , . , and . µ m,respectively. Although heterogeneity in D b and χ may bepresent within each population itself , we focus our anal-ysis on the influence of pore size by assuming a constantvalue of both for each simulation. Finally, we note that themotility parameters D b and χ reflect the ability of cells tomove through the pore space via an unbiased or biased ran-dom walk with mean step length l whose value depends onpore-scale confinement and possible cell-cell collisions in thepore space. For the case of sufficiently dilute cells in porousmedia, l is set by the geometry of the pore space, as we pre-viously established ; in particular, l ≈ l c , the mean lengthof chords, or straight paths that fit in the pore space . How-ever, when the cells are sufficiently dense, as arises in the ex-periments explored here, cell-cell collisions truncate l . Wemodel this by considering the mean separation between cells l cell ≈ (cid:16) f πb (cid:17) / − d , where f is the volume fraction of thepore space between hydrogel particles, b is the local bacterialnumber density, and d ≈ µ m is the characteristic size of acell; for simplicity, when l cell < l c , we assume that cell-cellcollisions truncate the mean step length l and set its value to l cell . That is, wherever b is so large that l cell < l c , we multiplythe values of both D b and χ used in Eq. (2) by the correctionfactor ( l cell /l c ) that accounts for the truncated l due to cell-cell collisions. Moreover, wherever b is even so large that thiscorrection factor is less than zero—i.e. cells are jammed—weset both D b and χ to zero. Based on our experimental char-acterization of pore space structure we use f = 0 . , . ,and . , and l c = 4 . , . , and . µ m, for porous mediawith ξ = 2 . , . , and . µ m, respectively. Bacterial proliferation.
Changes in b can also arise fromnet cell proliferation, as described in Eq. (2). In particular, wedescribe net cell proliferation with the maximal rate per cell γ multiplied by the Michaelis-Menten function g ( c ) that againdescribes describes the influence of nutrient availability i.e. itquantifies the reduction in proliferation rate when nutrient issparse. We directly measured γ ≡ ln 2 /τ previously, where τ = 60 min is the mean cell division time in a porous mediumfor our experimental conditions. We note that because c and b are coupled in our model, we do not require an additional“carrying capacity” of the population to be included, as is of-ten done ; we track nutrient deprivation directly through theradially-symmetric nutrient field c ( r , t ) . Implementation of numerical simulations
While the experimental geometry is three dimensional, inprevious work , we found that radial and out of plane effectsdo not need to be considered to capture the essential featuresof bacterial front formation and migration. Thus, for simplic-ity, we use a 2D representation. In the x direction (coordinatesdefined in Figs. 2 and 4), no flux boundary conditions are used3at the walls of the simulated region for both field variables b and c . In the y direction, no flux boundary conditions are usedafter one wavelength of the undulation, peak to peak, whichcomprises a single repeatable unit. The initial cylindrical dis-tribution of cells 3D printed in the experiments has a diam-eter of ∼ µ m; so, in the x dimension of the numericalsimulations, we use a Gaussian with a µ m full width athalf maximum for the initial bacteria distribution b ( x, t = 0) ,with a peak value that matches the 3D printed cell density inthe experiments, . × cells/mL. We vary the center x position of the Gaussian distribution sinusoidally along y to reproduce a given experimental wavelength and amplitude.Experimental wavelengths were measured directly from con-focal images and rounded to the nearest µ m. The initialcondition of nutrient is c = 10 mM everywhere, character-istic of the liquid media used in the experiments. The initialnutrient concentration is likely lower within the experimentalpopulation initially due to nutrient consumption during the 3Dprinting process; however, we expect this discrepancy to playa negligible role as nutrient deprivation occurs rapidly in thesimulations.As previously detailed , while the periphery of a 3Dprinted bacterial population forms a propagating front, cells inthe inner region remain fixed and eventually lose fluorescencebecause they are nutrient-limited. Specifically, the fluores-cence intensity of this fixed inner population remains constantover an initial duration τ delay = 2 h, and then exponentiallydecreases with a decay time scale τ starve = 29 . min. To facil-itate comparison to the experiments, our simulations incorpo-rate this feature to represent the cellular signal, which is theanalog of the fluorescence measured in experiments, in Figs. 2and 4. We do this by multiplying the cellular density obtainedby solving Eq. (2) by a correction factor that incorporates thehistory of nutrient depletion. Specifically, wherever c ( r (cid:48) , t (cid:48) ) drops below a threshold value, for all times t > t (cid:48) + τ delay , wemultiply the cellular density b ( r (cid:48) , t ) by e − ( t − t (cid:48) ) /τ starve , where t (cid:48) is the time at which the position r (cid:48) became nutrient-depleted.To numerically solve the continuum model, we usean Adams-Bashforth-Moulton predictor corrector method where the order of the predictor and corrector are 3 and 2,respectively. Since the predictor corrector method requirespast time points to inform future steps, the starting time pointsmust be found with another method; we choose the Shanksstarter of order 6 . For the first and second derivatives inspace, we use finite difference equations with central differ-ence forms in 2D. Time steps of the simulations are . s andspatial resolution is µ m. Because the experimental cham-bers are . cm in diameter, we use a distance of . × µ mfor the size of the entire simulated system in the x directionwith the cells initially situated in the center. Our previouswork demonstrated that the choice of discretization does notappreciably influence the results in numerical simulations offlat fronts; furthermore, our new results for the simulationsperformed here (Fig. S2) indicate that our choice of discretiza-tion used is sufficiently finely-resolved such that the results innumerical simulations of undulated fronts are not appreciablyinfluenced by discretization. Characterizing simulated front dynamics
For the analysis shown in Fig. 2, the leading edge is definedas the locus of positions at which b falls below a thresholdvalue equal to − times the maximum cell density of theinitial bacterial distribution, as in . For the analysis shownin Fig. 4, to more accurately track the leading edge of thefront, we define it as the locus of positions at which b fallsbelow a threshold value specific to each condition tested; thethreshold is . cells per µ m for the prototypical case of ξ = 1 . µ m and λ = 0 . mm shown in Figs. 4A to 4C, aswell as all simulations for ξ = 2 . µ m; . cells per µ m for simulations for ξ = 1 . µ m and λ = 2 . and . mm;and . cells per µ m for simulations for ξ = 1 . µ m and λ = 0 . mm.4 Supplementary Information
SUPPLEMENTARY FIGURESFigure S1 Effect of reduced sensing.
To investigate the influence of varying the upper limit of sensing c + , we repeat the simulation for theprototypical case of ξ = 1 . µ m and λ = 0 . mm but with c + lowered by a factor of . Consistent with our expectation, we find thatreducing this upper limit weakens chemotactic response not just at the peaks, but also the valleys, thereby slowing smoothing. Image ispresented as in Fig. 2A. Figure S2 Convergence of the numerical simulations.
To assess the influence of discretization, we repeat the simulation for theprototypical case of ξ = 1 . µ m and λ = 0 . mm with different choices of the spatial discretization ∆ x and measure the smoothing time τ .In all cases we find qualitatively similar results, although the dynamics vary; however, as shown by the green data points, the dynamics do notappreciably change for discretization smaller than ≈ µ m, which is the value used in the main text simulations, as indicated by the blue star. SUPPLEMENTARY MOVIES
Movie S1: Experiment probing chemotactic smoothing for λ = 0 . mm, ξ = 1 . µ m. Movie shows the maximum intensityfluorescence projection (bottom up view) of migration from a 3D-printed undulated cylinder of close-packed
E. coli . The cellscollectively migrate outward in a front that autonomously smooths out the large-scale undulations as it continues to propagate.
Movie S2: Experiment probing chemotactic smoothing for λ = 3 . mm, ξ = 1 . µ m. Movie shows the maximum intensityfluorescence projection (bottom up view) of migration from a 3D-printed undulated cylinder of close-packed
E. coli . Movie S3: Experiment probing chemotactic smoothing for λ = 0 . mm, ξ = 2 . µ m. Movie shows the maximum intensityfluorescence projection (bottom up view) of migration from a 3D-printed undulated cylinder of close-packed
E. coli . Movie S4: Simulation probing chemotactic smoothing for λ = 0 . mm, ξ = 1 . µ m. Movie shows the calculated cellularfluorescence signal of cells migrating from an undulated stripe of close-packed
E. coli similar to Movie S1. As in the experiments,the cells collectively migrate outward in a front that autonomously smooths out the large-scale undulations as it continues topropagate.
Movie S5: Simulation probing chemotactic smoothing for λ = 3 . mm, ξ = 1 . µ m. Movie shows the calculated cellularsignal fluorescence signal of cells migrating from an undulated stripe of close-packed
E. coli similar to Movie S2.
Movie S6: Simulation probing chemotactic smoothing for λ = 0 . mm, ξ = 2 . µ m. Movie shows the calculated cellularfluorescence signal of cells migrating from an undulated stripe of close-packed