In micro-swimmer suspensions locomotion necessarily generates fluid motion,
and it is known that such flows can lead to collective behavior from unbiased
swimming. We examine the complementary problem of how chemotaxis is affected by
self-generated flows. A kinetic theory coupling run-and-tumble chemotaxis to
the flows of collective swimming shows separate branches of chemotactic and
hydrodynamic instabilities for isotropic suspensions, the first driving
aggregation, the second producing increased orientational order in suspensions
of "pushers" and maximal disorder in suspensions of "pullers". Nonlinear
simulations show that hydrodynamic interactions can limit and modify
chemotactically-driven aggregation dynamics. In puller suspensions the dynamics
form aggregates that are mutually-repelling due to the non-trivial flows. In
pusher suspensions chemotactic aggregation can lead to destabilizing flows that
fragment the regions of aggregation.Comment: 4 page

Enkeleida Lushi

H. C. Berg

M. J. Schnitzer

Michael J. Shelley

Raymond E. Goldstein

Physical Review E

English

arXiv

Crossref

Collective chemotactic dynamics in the presence of self-generated fluid flows

Lushi, E

Goldstein, RE

Shelley, MJ

Spiral - Imperial College Digital Repository

Collective Chemotactic Dynamics in the Presence of Self-Generated Fluid FlowsEnkeleida Lushi1,2, Raymond E. Goldstein3, Michael J. Shelley11Courant Institute of Mathematical Sciences, New York University, NY, USA2Department of Mathematics, Imperial College London, London, UK3Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UKIn micro-swimmer suspensions locomotion necessarily generates fluid motion, and it is knownthat such flows can lead to collective behavior from unbiased swimming. We examine the comple-mentary problem of how chemotaxis is affected by self-generated flows. A kinetic theory couplingrun-and-tumble chemotaxis to the flows of collective swimming shows separate branches of chemo-tactic and hydrodynamic instabilities for isotropic suspensions, the first driving aggregation, thesecond producing increased orientational order in suspensions of “pushers” and maximal disorder insuspensions of “pullers”. Nonlinear simulations show that hydrodynamic interactions can limit andmodify chemotactically-driven aggregation dynamics. In puller suspensions the dynamics form ag-gregates that are mutually-repelling due to the non-trivial flows. In pusher suspensions chemotacticaggregation can lead to destabilizing flows that fragment the regions of aggregation.PACS numbers: 87.17.Jj, 87.18.Hf, 47.63.Gd, 05.20.DdA growing body of experimental work has establishedthat suspensions of motile microorganisms can developcomplex large-scale patterns of collective swimming atsufficiently high concentration [1, 2]. This behavior gen-erally occurs in the absence of directional cues for swim-ming, purely as a consequence of steric and hydrody-namic interactions between the cells. Yet, there are manycircumstances in which cells exhibit chemotaxis, directedmotion in response to chemical gradients, and this pro-cess by itself can lead to complex spatio-temporal patternformation [3]. As swimmer-generated flows may also ad-vect any chemoattractant field, it is natural then to askhow self-generated fluid flows in suspensions of microor-ganisms affect modes of communication [4] and aggrega-tion. Here we present an analysis of this issue and suggestpotential realizations of this pattern-forming system.Chemotactic focusing of cell concentration has beenstudied using the classical Keller-Segel (KS) model [5]and in theories incorporating the run-and-tumble (RT)[6] dynamics of bacterial motion [7, 8]. We extend awell-known kinetic model for modulated RT dynamicsto include flows produced by the active stresses due toswimming. A simpler version of our model is consid-ered in [7] to study swimmer transport and rotation ina given background shear flow. Without RT dynam-ics, our model reduces to one for active suspensions [9]which captures the large-scale flows seen in experiments[1, 10] and illuminates the effect of propulsion mechanism(pusher vs. puller) on large-scale dynamics and stability.When swimmers produce a chemoattractant leading toaggregation, the self-generated flows can have a large ef-fect; pushers create complex flows that can bound growthin organism density, while pullers show limited patterncoarsening and isolated aggreggates repelling due to non-trivial flows. Merging the RT and active suspension mod-els is seamless as both are kinetic theories with conforma-tion variables the particle position and orientation [11].Consider a suspension of swimmers at local concentra-tion Φ(x, t), each of which moves at a constant speedU in a run-and-tumble dynamics. They move in a fluidof local velocity u(x, t) and produce a chemoattractantof concentration C(x, t) and molecular diffusivity DC .We choose rescalings based on the swimmer contributionto the fluid stress tensor (below), with a characteristiclength ` = l/φ, where l is the swimmer size and φ ≡ l3〈Φ〉is the effective mean volume fraction in suspension. Scal-ing time by `/U , C evolves asCt + u ·∇C = Pe−1∇2C − β1C + β2Φ , (1)where β1 and β2 are rate constants for chemoattractantself-degradation and production, and the Péclet numberPe = U`/DC measures the strength of diffusion to ad-vection on the intrinsic scale `. Collective swimming maygenerate coherence on larger scales with higher speeds,increasing the importance of advection. Without advec-tion this is the KS model [5]. In the case of E. coli [12]gives U ∼ 25 µm/s, l ∼ 5 µm, φ ∼ 0.1 at a cell concen-tration of 109 cm−3, so ` ∼ 50 µm. With DC ∼ 5× 10−6cm2/s we obtain Pe ' 2.5, β1 ' 0.008 and β2 ' 0.004.For faster-swimming organisms, such as marine bacteria[13], this intrinsic Péclet number can reach O(10− 20).The configuration of swimmers is given by a distribu-tion function Ψ(x,p, t) of the center of mass position xand orientation p satisfying the Fokker-Planck equationΨt =−[λ(DtC)Ψ−14π∫dp′λ(DtC)Ψ(x,p′, t)]−∇x · (Ψẋ)−∇p · (Ψṗ) , (2)where the local swimmer concentration is Φ(x, t) =∫dpΨ(x,p, t). The bracketed term in (2) describes theeffect of RT chemotaxis based on a swimming dynamicsof straight runs and modulated reorientations (tumbles)where λ(DtC) is a tumbling frequency, the probability ofarXiv:1210.4332v1  [physics.bio-ph]  16 Oct 20122FIG. 1: (color online). Chemotactic suspensions at large times, t = 3000. Swimmer concentration Φ (top) and mean directionn (bottom) for (i) chemotactic neutral swimmers (u = 0), and (ii) Φ (top), fluid streamlines and velocity field u (bottom) forchemotactic pullers. (iii,iv) Φ (top), streamlines and u (bottom) for chemotactic pushers. Parameters β1 = β2 = 0.25, Pe = 20,γ = 1, DT = DR = 0.025. Parameters λ0 and χ are indicated in Figs. 2c,d for cases (i-iii) and are λ0 = 5, χ = 0.6 for case (iv).See [16] for movies of swimmer concentration dynamics.a bacterial tumble event as a function of the chemoat-tractant temporal gradient DtC = Ct + (p + u) · ∇Calong a swimmer’s path. Experiments [14] show thatwhen DtC > 0 the tumbling frequency is reduced, andis otherwise constant, as captured by the biphasic formλ(DtC) = λ0 max(min(1− χDtC, 1), 0), a linearized ver-sion of an earlier model [8, 12]. The fluxes in (2) areẋ = p + u, ṗ = (I− pp) (γE + W) p. (3)The particle velocity ẋ includes swimming at constantspeed (non-dimensionalized to unity) in the axis direc-tion p (|p| = 1), translation by the fluid velocity u. (Inthe KS model [5], swimmer speed is linear in the chem-ical gradient.) The angular velocity ṗ follows Jeffrey’sequation [15] where E = (∇xu + ∇xuT )/2 is the rate-of-strain tensor, W = (∇xu −∇xuT )/2 is the vorticitytensor. For rod-like swimmers, the shape factor is γ ∼ 1.The fluid velocity u(x, t) produced by the suspensionsatisfies the Stokes equations driven by an “active” stressΣa arising from particle locomotion:− ∇2xu + ∇xq = ∇x ·Σa, ∇x · u = 0 (4)Σa(x, t) = α∫dpΨ(x,p, t)pp. (5)The active stress is an orientational average of the forcedipoles αpp the cells exert on the fluid [9], where α is anO(1) constant by our rescaling. A cell that self-propelsby front-actuation (a puller) has stresslet strength α > 0,and a rear-actuated cell (pusher) has α < 0. The caseof “neutral” cells (α = u = 0) is the closest this modelapproaches the KS [5] and RT models [8, 12].We first illustrate the effect hydrodynamics has onaggregation. From nonlinear simulations of Eqs. (1-6),Fig. 1 shows the swimmer concentration Φ(x, t) and meanorientation n =∫dpΨ/Φ at late times, having startednear uniform isotropy, for neutral, puller, and pushersuspensions. All share a dominant self-aggregation in-stability, but differing (or no) hydrodynamic interactions.Neutral swimmers show aggregation and pattern coarsen-ing. Pullers show limited aggregation into circular spotskept apart by non-trivial fluid flows. Pushers create com-plex fluid flows and fragmented aggregation regions.These behaviors can be understood through a stabilityanalysis of uniform isotropic suspensions. For simplicity,consider rod-like (γ = 1) swimmers and a quasi-staticchemoattractant field Pe−1∇2C − β1C = −β2Φ, whichslaves C to Φ. The tumbling frequency is simplified toλ(p) = λ0 (1− χp · ∇C). A steady state is Ψ0 = 1/4π(Φ = 1), u = 0, and C0 = β1/β2. Perturbations of theform ε(Ψ̃(p,k), C̃(k)) exp(ik · x + σt), yield(σ + λ0 + ik · p)Ψ̃ =λ04π(ikχβ2(k̂ · p)β1 + k2Pe−1+ 1)Φ̃− 3α4π(k̂ · p)p · (I− k̂k̂)Σ̃pk , (6)where Φ̃ =∫dp′Ψ̃′ and k = kk̂. Since Σ̃p =∫dp′Ψ̃′p′p′,this is a linear eigenvalue problem for Ψ̃ and σ. The firstterm on the RHS is chemotactic (C) and has unstabledynamics restricted to the zeroeth azimuthal mode on|p| = 1. The second is hydrodynamic (H), with unstabledynamics restricted to the first azimuthal mode. This3FIG. 2: (color online). Linear stability analysis. (a) Branches of the hydrodynamic instability, with and without tumbling,for pushers. (b) Chemotactic branch for χ = 35, λ0 = 0.25, β1 = β2 = 1/4 and Pe = 20. Regimes diagram for (c) neutralswimmers and pullers, and (d) pushers. Solid curves give linear stability boundaries for long waves. Dashed lines show shiftedboundaries in nonlinear simulations at finite box size. Encircled labels (i-iii) denote parameters used in simulations (Fig. 1a-c).yields uncoupled relations for growth rates σC,H ,2λ0= R[2 + aC log(aC − 1aC + 1)]− 1iklog(aC − 1aC + 1)4k3iα= 2a3H −43aH + (a4H − a2H) log(aH − 1aH + 1), (7)where aC,H = (σC,H +λ0)/ik and R = χβ2/(β1+k2/Pe).We refer to these as the chemotactic and hydrodynamicrelations, respectively. The first induces growth in con-centration fluctuations, while the second increases orien-tational order. The two are coupled only through thebasal tumbling rate λ0 which in the hydrodynamic re-lation only shifts the growth rate. Further, the chemo-tactic instability gives rise to normal stresses of the formΣ̃p = k̂k̂ − k̂⊥k̂⊥, while the hydrodynamic instabilitygives shear stresses of the form Σ̃p = k̂k̂⊥ + k̂⊥k̂.For pushers (α < 0) the hydrodynamic instabilityhas a finite bandwidth (Fig. 2a), though with maxi-mal growth rates at k = 0. Tumbling shifts downthe Re(σH(k)) branch by λ0 for all k, further stabi-lizing the system. Long-wave asymptotics of (7) givetwo solution branches: σH1 ' −α/5 − λ0 + 15/7αk2and σH2 ' −λ0 + O(−αk2). There is no hydrody-namic instability for pullers [9]. Fig. 2b shows thechemotactic growth rate. Small k asymptotics yieldsσC ≈ k2/(3λ0)[(χβ2/β1)λ0 − 1]: for (χβ2/β1) > 1/λ0there are wavenumbers with Re(σC(k)) > 0, shown inone case as a finite band of unstable modes whose widthis controlled by chemo-attractant diffusion.From Fig. 2a, we can obtain a range for λ0 for whichthere is a hydrodynamic instability in pusher suspen-sions. Heuristically, λ0 sets an effective rotational dif-fusivity, and λ0 ≥ 0.2 turns off the hydrodynamic insta-bility for any system size. For L = 50 and the diffusionconstants used in simulations, λ0 ≥ 0.09 suffices. Thisinformation is assembled in Fig. 2 c,d as phase diagramsthat relate the parameters to various dynamical regimes.Numerical studies of the full nonlinear dynamics (1–5)were done in 2D, with a box size L = 50 large enough toinclude several unstable linear modes. Swimmer transla-tional and rotational diffusions are added in the model tocontrol the growth of steep gradients over long-times. Aninitial random perturbation of the uniform isotropic stateis used: Ψ(x,p, 0) = 1/2π+Σiεi cos(ki·x+ξi)Qi(pi) withrandom coefficients |εi| < 0.01, ξi an arbitrary phase andQi a low-order polynomial. The initial chemo-attractantconcentration is uniform with C(x, 0) = β1/β2 = 1. Fig-ure 1 shows long-time swimmer concentration Φ for fourillustrative cases. In each case, concentration C closelytracks Φ. Cases (i-iii) share the same chemotactic insta-bility, but differ in swimming actuation: α = 0, 1,−1.The expected regimes of these three cases are shown inFigs. 2c,d. For neutral swimmers, aggregation dominatesand the dynamics is typified by the formation of a fewregions of steadily increasing concentration that slowlycoarsen (Fig. 1a). The maximum swimmer concentra-tion (Fig. 3) shows little sign of the rapid self-focussingassociated with finite-time chemotactic collapse [17] ofthe KS model, which here may be due to the fixed swim-ming speed [18]. While the initial aggregation for pullers(Fig. 1b) is similar to that for neutral swimmers (Fig.3) its long-time behavior is very different. Concentrationgrowth and coarsening cease as the dynamics enters anear steady-state with circular regions of high concentra-tion. Active-stress driven flows suppress further coarsen-ing by pushing nearby peaks apart and apparently main-tain the few remaining high concentration regions.For pushers (Fig. 1c), linear theory gives only a chemo-tactic instability, and the dynamics is indeed initiallydominated by aggregation as is evidenced by the earlyrapid growth of normal stresses relative to shear stresses(not shown). However, aggregation into a regions withhigh swimmer concentration creates a destabilizing ac-tive stress, giving rise to unsteady fluid flows. Theseflows fragment the peaks while pushing them around the4FIG. 3: (color online) Measures of growth: maximum swim-mer concentration for cases (i-iii) in Fig. 1.domain. The dynamics is one of constant aggregationand flow instability, which apparently suppresses furthergrowth in swimmer concentration (Fig. 3).Lastly, we examine in Fig. 1(iv) the dynamics thatarises with parameters close to those measured bySaragosti et al [12] (before our rescaling) in their experi-ments of E. coli chemotaxis. These parameters lie far tothe right of the aggregation regime of Fig. 2d as λ0 is 20times higher than at the predicted threshold for suppress-ing hydrodynamic instabilities. Not surprisingly, thesimulations show chemotactic aggregation into very highpeaks. Once the swimmer concentration in those peaksis large enough, the active stresses give rise to small-scaleand localized fluid flows (cf. Fig. 1(iii)). These local flowsdo appear to be implicated in the slow “wriggling” we ob-serve of the saturated aggregates (see Supplementary Ma-terial [16]). The experiments of Saragosti et al [12], whichare performed in confined micro-channels and capillaries,show instead the development of traveling concentrationwaves of chemotactic bacteria. These traveling waveswere initiated in the experiments through an initial con-centration by centrifugation of the swimmer populationto one end of the channel. We do not observe the spon-taneous formation of such traveling waves here thoughours is an open system (though confined geometricallyby the assumed periodicity length) and the initial swim-mer state is un-oriented and nearly homogeneous. Thecombined effects of a confining geometry and the initialconcentration of swimmers has yet to be examined in ourtheoretical system.We have shown that the intrinsically generated fluidflows arising from collective swimming of microorgan-isms can modify patterns of chemotactic aggregationand, most importantly, can limit aggregate concentra-tion. This is unlike chemotactic models that predictconcentration blow-up or include artificial terms to capgrowth. While we have emphasized hydrodynamic ef-fects in attractive chemotactic dynamics, it is importantto remember that ours is a dilute to semi-dilute theorythat does not capture near-interactions between swim-mers, hydrodynamic or otherwise. In denser suspensionsswimmer size limits local swimmer density through stericinteractions though as yet well-founded models that com-bine these with hydrodynamic interactions do not exist.Nonetheless, we expect similar results when large-scalecoherence is driven by steric effects [10, 19]. On thatnote, steric effects with no hydrodynamics may also limitaggregation of chemotactic random walkers [20].Finally, these auto-chemotactic effects can be seen ascomplementary to the enhanced mixing by swimmers[11] that has also been explored for microfluidic appli-cations [21]. Systematic studies of the interplay betweenchemotaxis and locomotion-generated fluid flow shouldbe possible through controlled introduction of exoge-neous chemoattractants to trigger aggregation, throughthe interplay of quorum sensing and chemotaxis [22], andperhaps by specific genetic engineering of the dynamicsof locomotion and chemosensing [23].This work was supported in part by NSF grants DMS-0652775 and DMS-0652795, and DOE grant DEFG02-00ER25053 (E.L., M.J.S.) and an ERC Advanced Inves-tigator Grant 247333 (R.E.G.).[1] C. Dombrowski, et al., Phys. Rev. Lett. 93, 098103(2004); L.H. Cisneros, et al., Exp. Fluids 43, 737 (2007).[2] S. Ramaswamy, Ann. Rev. Condens. Matter Phys. 1, 323(2010).[3] E.O. Budrene and H.C. Berg, Nature 349, 630 (1991).[4] B.L. Bassler, Cell 109, 421 (2002).[5] E.F. Keller, L.A. Segel, J. Theor. Biol. 30, 225 (1971).[6] H.C. Berg, Random Walks in Biology Princeton Univer-sity Press, 1993[7] R.N. Bearon, T.J. Pedley, Bull. Math. Bio. 62, 775(2000).[8] K.C. Chen, et al., J. Math. Bio. 47, 518 (2003).[9] D. Saintillan and M.J. Shelley, Phys. Rev. Lett. 100,178103 (2008); also Phys. Fluids 20, 123304 (2008).[10] L.H. Cisneros, et al., Phys. Rev. E 83, 061907 (2011).[11] E. Lushi and M.J. Shelley, preprint (2012).[12] J. Saragosti, et al., Proc. Natl. Acad. Sci. USA 108,16235 (2011).[13] R. Stocker, et al., Proc. Natl. Acad. Sci. USA 105, 4209(2008).[14] R.M. Macnab and D.E. Koshland, Proc. Natl. Acad. Sci.USA 69, 2509 (1972); N. Mittal, et al., Proc. Natl. Acad.Sci. USA 100, 13259 (2003).[15] G.B. Jeffery, Proc. R. Soc. London, Ser. A 102, 161(1922).[16] See Supplementary Material at [URL to be inserted bypublisher] for movies of the swimmer concentration.[17] S. Childress and J.K. Percus, Math. BioSci. 56, 217(1981).[18] M.J. Schnitzer, et al., Symp. Soc. Gen. Microbiol 46, 15(1990).[19] K. Drescher, et al., Proc. Natl. Acad. Sci. USA 108,10940 (2011).[20] J. Taktikos, et al., Phys. Rev. E 85, 051901 (2012).[21] M.J. Kim and K.S. Breuer, Anal. Chem. 79, 955 (2007).[22] S. Park, et al.. Science 301, 188 (2003).[23] C. Liu, et al.. Science 334, 238 (2011).

Collective chemotactic dynamics in the presence of self-generated fluid flows

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