Photo-bioconvection: towards light-control of flows in active suspensions
PPhoto-bioconvection: towards light-control of flows in active suspensions
A. Javadi , J. Arrieta , I. Tuval , , and M. Polin , Physics Department, and Centre For Mechanochemical Cell Biology,University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom Instituto Mediterr´aneo de Estudios Avanzados, IMEDEA, UIB-CSIC, 07190, Esporles, Spain Departamento de F´ısica, Universitat de les Illes Balears, 07122 Palma de Mallorca, Spain ∗ The persistent motility of the individual constituents in microbial suspensions represents a primeexample of so-called active matter systems. Cells consume energy, exert forces and move, overallreleasing the constraints of equilibrium statistical mechanics of passive elements and allowing forcomplex spatio-temporal patterns to emerge. Moreover, when subject to physico-chemical stimulitheir collective behaviour often drives large scale instabilities of hydrodynamic nature, with impli-cations for biomixing in natural environments and incipient industrial applications. In turn, ourability for external control of these driving stimuli could be used to govern the emerging patterns.Light, being easily manipulable and, at the same time, an important stimulus for a wide variety ofmicroorganisms, is particularly well suited to this end. In this paper, we will discuss the currentstate, developments, and some of the emerging advances in the fundamentals and applications oflight-induced bioconvection with a focus on recent experimental realisations and modelling efforts.
INTRODUCTION
A quiescent suspension of swimming microorgan-isms often develops spontaneously large scale currents,whirling the microbes around in mesmerisingly complexflows [2]. Changing patterns of tight cell-rich down-welling plumes interspersed with large upwells of lowcell concentration can be observed in intertidal pools [3],and are a very familiar sight in the laboratory (Fig. 1).They reveal macroscopically the incessant activity thatis present at the microscale, which distinguishes in afundamental way the behaviour of these so-called activecomplex fluids [4] from their passive counterparts likecolloidal suspensions. The emergent hydrodynamic in-stabilities, called bioconvection, are amongst the oldestreported collective effects in microbial suspensions [2, 5– (a) (b)
FIG. 1. Bioconvective plumes and pattern formation. (a)Long vertical gyrotactic plumes in a culture flask. (b) Pat-tern formation in a shallow Petri dish (depth 0 . cells/cm ); the illumination is white from below,where the lower half is covered with a red filter (660 nm; con-trast enhanced). The two regions display different biocon-vection patterns: in the top half, white illumination leadscells to swim upwards, with phototaxis supporting gravitaxisand suppressing gyrotaxis, initiating an overturning instabil-ity with broad downwelling structures; in the bottom half,cells do not respond to the red illumination and form finelyfocused gyrotactic plumes. Adapted from [1] a r X i v : . [ phy s i c s . b i o - ph ] S e p teraction between microbial phototaxis and bioconvec-tion (for a recent general review on bioconvection , see[35]). After summarising the main mechanism leadingto bioconvection in Sec. , Sec. will discuss the effect ofphototactic perturbations to an underlying bioconvectiveinstability. We will then proceed to review recent studiesthat are pioneering the use of phototaxis to drive -ratherthan perturb- hydrodynamic instabilities (Sec. ), and thecombination of phototaxis and externally imposed shearflows to alter the macroscopic distribution of phototacticcells (Sec. ). We will then conclude in Sec. . BIOCONVECTION OF MICROBIALSUSPENSIONS: A QUICK PRIMER
Bioconvection, the macroscopic recirculation that de-velops spontaneously in concentrated suspensions ofmicro-organisms, was first observed for bottom-heavy mi-croalgae [2]. In these microorganisms, the centre of masslies behind the hydrodynamic centre of resistance. Thisis commonly ascribed to inhomogeneities in the mass dis-tribution within the microbial cell body, although shapeasymmetries have been proposed to play an importantrole [36, 37]. Indeed, these have been shown to be thedominant cause of bottom-heaviness in the model mi-croalga
Chlamydomonas reinhardtii [38]. The gap be-tween the centres of mass and hydrodynamic resistanceresults in a gravitational torque which biases cell motionupwards [39]. This phenomenon, called gravitaxis, ac-cumulates bottom-heavy cells at the upper surface. Ascells are usually denser than the surrounding fluid, the ac-cumulation makes the ensuing stratification gravitation-ally unstable, leading to the development of plumes andconvective rolls in a manner similar to Rayleigh-B´enardconvection. Sinking bioconvective plumes are then rein-forced by gyrotaxis, the tendency of bottom-heavy cellsto move towards downwelling regions resulting from thebalance between gravitational and hydrodynamic torquesoriginally observed in pipe flow by J. Kessler [39]. Acontinuum model for gyrotactic bioconvection of a dilutesuspension of swimming microorganisms was first stud-ied by Pedley, Hill and Kessler [40] and then by Pedleyand Kessler [14, 41] (Fig. a). Here, the suspension of al-gal cells is modelled as a continuum of identical prolateellipsoids of volume v and density ρ c , that swim with aconstant speed of V s along a direction p and do not inter-act directly with each other. The swimming direction isdetermined by a combination of environmental torquesacting on the cell and the randomness inherent in mi-crobial swimming, and is therefore described locally bya probability distribution f ( p ). This is taken to satisfya quasi-steady Fokker-Planck (FP) equation, balancingadvection by currents in p -due to gravitational and hy-drodynamic torques- with an effective angular diffusionof diffusivity D R due to cell swimming. If the concen- tration of cells at position x and time t is n ( x, t ), and u ( x, t ) and p represent the velocity and pressure of theunderlying fluid, the governing equation for the systemare [41] ∇ · u = 0 , (1) ρ (cid:18) ∂u∂t + u · ∇ u (cid:19) = −∇ p + ∇ · (2 µE + Σ) + nv ∆ ρg, (2) ∂n∂t + ∇ · ( n ( u + V )) = ∇ · ( D · ∇ n ) , (3) ∇ p · ( f ˙ p ) = D R ∇ p f, (4)˙ p = ( I − pp ) · Ω · p + Λ ( I − pp ) · E · p + 1 B ( I − pp ) · k. (5)Here we recognise the Navier-Stokes equations for a fluidof density ρ and dynamic viscosity µ . Mass conservation,Eq. (1), enforces fluid incompressibility; and momentumbalance, Eq. (2), is governed by the rate of strain ten-sor, E = ( ∇ u + ∇ u T ) /
2, the buoyancy forces caused bythe excess cell density (∆ ρ = ρ c − ρ ) considered withinthe Bousinnesq approximation (with g being the gravi-tational acceleration), and the excess deviatoric stress Σwhich will be discussed in detail below. Local cell conser-vation, Eq. (3), depends on the cells’ effective diffusiontensor D , which is non-isotropic for gravitactic species[42], and on advection by both the local fluid flow andswimming speed V . The cells’ gravitational sedimenta-tion has been neglected in this formulation, but its effectwas later investigated by Pedley [13, 43]. The swimmervelocity V = V s (cid:104) p (cid:105) , where (cid:104) p (cid:105) is the local average swim-ming direction: (cid:104) p (cid:105) = (cid:90) S pf ( p ) d p. (6)Note that, although p is a unit vector, the magnitude of (cid:104) p (cid:105) can range from 0 (completely random swimming) to 1(all cells swimming along the same direction). Changesin the local distribution of cell orientation, and the ef-fect of local environmental torques on the swimming di-rection p are modelled by Eqs. (4,5). Here, the distri-bution function f ( p ) is the solution of a quasi-steadyadvection-diffusion equation on the unit sphere S witheffective rotational diffusivity D R . This approach slaves f ( p ) to the instantaneous local orientational currents ˙ p ,a valid assumption if temporal changes in the local flowstructure are slow compared with the intrinsic equilibra-tion timescale of the local distribution of cell orienta-tions. Equation (5), in turn, describes how a cell’s swim-ming direction p changes due to viscous and gravitationaltorques. The former is a combination of uniform rotation(from the local vorticity Ω = ( ∇ u − ∇ u T ) / E ). The relative importance of these (a) (b) No-slip lower surface xz Stress-freeor no-slipvertical boundaries H Uniform light source Is No-slip/Stress-free surface n ( x ,t ) FF l / l / F F/2F/2 l / l / p g FIG. 2. Standard setting for photo-bioconvection. (a) The cell suspension, of instantaneous local cell density n ( x, t ), iscontained within a horizontal chamber of thickness H . The bottom surface is no-slip, while the top surface can be either no-slipor no-stress. The suspension is subjected to a uniform vertical illumination of intensity I s . (b) Typical structures of the maintypes of swimming micro-organisms. Left: a biflagellate microalga (puller-like) Right: a bacterium (pusher-like). Both swimalong the direction p and are subjected to gravity, with gravitational acceleration g . terms depends on the shape parameter Λ, which mea-sures the cell’s eccentricity [44]. It is zero for a sphericalorganism, and tends to unity for rod-like swimmers [44].This is responsible for the famous Jeffery orbits of elon-gated ellipsoids in shear flow [45]. The last term on theright hand-side, instead, is the gravitational torque dueto bottom-heaviness, with the gravitational reorientationtimescale B determined by the balance between the max-imal gravitational torque and the viscous resistance torotation [46]: B = µα ⊥ ρ c gh . (7)Here α ⊥ is the resistance coefficient for the cell’s rotationabout an axis perpendicular to p , and h is the centre ofmass offset of the cell. Note that the physical implicationof any term of the form ( I − pp ) · e on the right-hand-sideof Eq. (5) is to rotate and align p along e .We have not yet discussed the term Σ, added to thestress tensor in Eq. (2). This accounts for the stress con-tribution of the cells to the ambient fluid flow, and is thesum of two terms, Σ = Σ p + Σ s . The former is a passiveterm due to the disturbance flow field caused by the rigidcell bodies (Σ p ), which follows the expression derived byBatchelor for a suspension of identical rigid passive par-ticles [47]. The latter is an active stress term from cellmotility which comes from the stresslet contribution tothe forces exerted by a swimming cell on the surroundingfluid. If the magnitude of the drag force from the bodyon the liquid is F , and this is countered -on average- byan equal and opposite thrust from the motile appendagesat a distance l from the hydrodynamic centre of the body,the Σ s is given by [46]:Σ s = ± nF l (cid:18) (cid:104) pp (cid:105) − I (cid:19) , (8)where n -as above- is the local cell concentration, andthe sign choice depends on whether the microswimmers are puller (+) or pusher ( − ) (Fig. b). Pedley and Kessler[41, 46] showed that this is by far the dominant contri-bution, and therefore only the active stress contributions(Σ s ) are taken into account in Eq. (2). Taking Σ (cid:39) Σ s and using Eq. (8) gives a good approximation of the ef-fects of swimmers on the flow in dilute suspensions (withtypical cell densities of the order 10 cells / cm ) and hasbeen used extensively throughout the literature on collec-tive motion. However, it can also be neglected in manysituations with pre-existing background flows that havecharacteristic viscous stresses much larger than (cid:107) Σ s (cid:107) .Besides contributing to the fluid’s stress tensor, cellswimming leads also to an effective diffusivity tensor D .For the constant swimming speed V s assumed here, it canbe derived from the Green-Kubo relations as [41, 48, 49] D = (cid:90) ∞ (cid:104) V r ( t ) V r ( t − t (cid:48) ) (cid:105) dt (cid:48) ; V r ≡ V s ( p − (cid:104) p (cid:105) ) , (9)which, under the assumption of a constant correlationtime τ reduces to [13] D = V s τ (cid:104) ( p − (cid:104) p (cid:105) )( p − (cid:104) p (cid:105) ) (cid:105) . (10)Typical values of τ range between 1 s and 5 s. These ex-pressions assume that the background fluid shear doesnot influence fluctuations and persistence in cell orienta-tion. The validity of this approximation in a real sys-tem is uncertain, especially for large shear rates. Asa result, Bees and Hill (for spherical swimmers [42])and Manela and Frankel (for axisymmetric swimmers[50]) have used instead a generalised Taylor dispersiontheory (GTD) to formulate the diffusivity of a suspen-sion of gyrotactic micro-swimmers. For sufficiently lowshear rates the difference between the two approachesis not significant, and the previous -simpler- approachto D can be used for example to study the initial de-velopment of bioconvection from a quiescent backgroundfluid. The stability of the suspension is generally de-scribed in terms of a Rayleigh number for active parti-cles R = ¯ ngv ∆ ρH / ( µD ), which depends on the ratiobetween buoyancy and viscous forces and the ratio be-tween diffusion of momentum within the fluid and activediffusion. Here ¯ n is the average cell concentration and H is the container thickness. Through linear stability anal-ysis of Eqs. (1-5), it is possible to predict the onset of theinstability, in terms of a critical Rayleigh number; andthe most unstable wavelength, usually called the biocon-vective wavelength, which should then correspond to thecharacteristic separation between plumes observed exper-imentally (Fig. b,c) [35, 51].The approach that we have briefly outlined offers ingeneral a good qualitative description of the experimen-tal observations of bioconvection. Quantitatively, how-ever, the agreement is often less good. For shallow sus-pensions, for example, theoretical predictions overesti-mate the observed wavelengths (observed: ∼ − ∼ − B, τ, V s . . . ) as variables to be fit-ted [54]. Pedley and Kessler have also suggested thatthe actual non-linear nature of the instability might bea cause of the observed discrepancy between the char-acteristic wavelength for bioconvection observed exper-imentally, and the one predicted from linear stabilityanalysis [41]. Much of the quantitative uncertainty mustcome also from our limited knowledge of the swimmingbehaviour of the microorganisms within the suspension,and possibly of their ability to sense and react to me-chanical stresses [55, 56]. Finally, bioconvection experi-ments show clearly three dimensional patters. Therefore,in principle, the full three-dimensional problem should besolved for a true quantitative comparison between theoryand experiments. Recent 3D simulations are starting toaddress this issue [57–59], and in particular Ghorai et al.(2015) [59] found characteristic wavelengths that were ingood agreement with previous experiments[53], providedthat the diffusion due to randomness in cell swimmingbehaviour is small. PHOTOTACTIC MODIFICATIONS TOSTANDARD BIOCONVECTION
The general framework presented in Sec. served asa starting point for the first studies on the effects onbioconvection of phototaxis, the ability of some micro-bial species (e.g. green microalgae [60] and dinoflag-ellates [61]) to swim towards a light source (positivephototaxis) or away from it (negative phototaxis). Thelight needs to be within the range of wavelengths whichthe light-sensitive organelles are sensitive to (between ∼ −
500 nm for
Chlamydomonas [62]), which over-laps with -but does not necessarily exactly correspond to-the photosynthetically active range [63]. The pioneering continuum studies of photo-bioconvection [64], consid-ered purely phototactic cell suspensions that are exposedto uniform illumination from above or below (Fig. a).Swimming cells are still assumed to be slightly heavierthan the ambient fluid, but gyrotaxis and gravitaxis areneglected and the up-swimming is exclusively due to pho-totaxis. The swimmer velocity, however, depends on theintensity of light I , with positive phototaxis for light in-tensity below a critical value I c ( I < I c ), and negativephototaxis otherwise ( I > I c ). This captures the factthat phototactic organisms move away from intense lightto avoid photodamage [27], and results in a tendency toaccumulate cells where I ≈ I c . The swimming velocityin Eq. (3) is then V = V s T ( I ) k, where T ( I ) (cid:40) ≥ , I ≤ I c < , I > I c . (11)Here k is the vertical unit vector and T ( I ) is the phe-nomenological taxis function with values between -1 and+1 (Fig. a). The shape of taxis function is -in principle-species-dependent, and is usually constructed based onthe simplification of controlled experimental observations[65]. At the same time, the diffusion tensor is assumed tobe constant and isotropic. This non-monotonic responseto light is coupled to changes in light intensity due to theabsorption and scattering of light by cells, or “shading”.This phenomenon is particularly important for dense cellsuspensions like those that can be achieved in the labora-tory or within bioreactors. For thick but sufficiently di-lute suspensions, the local scattering of light by the cellsis weak and the light intensity I ( x, t ) is simply given bythe Beer-Lambert law [66]: I ( x, t ) = I s exp (cid:32) − α (cid:90) Hz n ( x, t ) dz (cid:33) , (12)where I s is the uniform light intensity at the upper sur-face of the suspension ( z = H ), and α is the absorptioncoefficient. The rigid bottom with no slip boundary con-dition is located at z = 0 Light intensity is therefore amonotonically decreasing function of depth ( z ). If I c liebetween the maximum and minimum intensities acrossthe suspension, a sublayer of cell concentration is formedin the interior of the chamber (Fig. b). The region abovethis sublayer is gravitationally stable and it is restingon a gravitationally unstable region below. The bottomlayer can then develop circulating flows which penetratethe stable layer, resulting in bulk motion throughout thesuspension [67]. This phenomenon is called penetrativebioconvection. Linear instability analysis showed that,within this model, the onset of the instability and thelength scales of the ensuing patterns depend on the av-erage cell concentration, the position at which I = I c fora uniform suspension, and the depths of the gravitation-ally stable and unstable layers (Fig. b,c). Depending onthe values of the parameters, this model can develop twodifferent types of large scale flows, either steady or peri-odic. The former is the usual type of bioconvective flow.The latter corresponds to oscillating flow and concentra-tion fields that occur when upswimming due to photo-taxis is dominant over convective downwelling. Pandaand Singh [68] extended Ghorai and Hill’s work to showthat these instabilities are opposed by the presence ofrigid side walls, which enhance stability for some param-eter regimes.The models above usually rest on two major assump-tions: they neglect light scattering from the cells, imply-ing that each cell only receives light vertically from theuniformly illuminating source; and they neglect bottom-heaviness and therefore gyrotaxis. Let us consider theseassumptions individually. If light scattering from cellsis important, then each cell will have to respond tomultiple stimuli besides the main one from the illumi-nating source. This was first discussed by Ghorai etal. [66] by introducing a radiative transfer equation forthe direction-dependent light intensity, and a scatter-ing probability that determines the likelihood for lightto be scattered by a given amount. Both isotropic andanisotropic scattering probabilities were studied [26, 69].The inclusion of scattering complicates the model sig-nificantly, and makes it a formidable task to find theintensity profile I ( x ), which needs to be solved for to-gether with the fluid flow and cell concentration. Linearstability analysis [66] and 2D numerical solutions [69] re-veal that for purely scattering suspensions, the intensity I ( x ) may not vary monotonically with depth. A suspen-sion can therefore exhibit more than one location where I = I c , leading to an altered base state from the sin-gle sublayer discussed earlier. Besides this difference,light scattering from cells does not seem to introduce anyqualitatively new instabilities within the suspension [66].However, it should be stressed that these results hingeon a specific simple model of phototaxis. Light scatter-ing by cells generates light fields with multiple sources,and our knowledge of phototaxis in this situation is cur-rently poor. Although it has been investigated for glidingmotility in cyanobacteria [70], it is not a well-understoodprocess for swimming cells.Thus far, we have only considered models for purephototaxis, where upswimming was just a consequenceof cells’ motion towards light. Many microorganisms,however, are also gravitactic and gyrotactic and should-in principle- be taken into account to describe the sus-pension’s dynamics. The first model of this kind, whichmodified Pedley and Kessler’s model (Sec. ) to accountfor phototaxis, was proposed by Williams and Bees [17].They considered three different models to incorporatethe effects of phototaxis: a photokinesis model wherecell’s swimming speed is a function of light intensity (A);a coupled photo-gyrotactic model where light controlsthe position of a cell’s centre of mass, and therefore its bottom-heaviness (B); and an extra “phototactic torque”in Eq. (5) to reorient cells along gradients of light inten-sity (C). We note that it has recently been demonstratedthat some microbial species can modify their fore-aftmass asymmetry in response to mechanical stimuli [71],and it would not be far fetched to expect that, in somecases, this might also be true for light stimuli. Over-all, the models proposed by Williams and Bees are anexcellent starting point to investigate the coupling be-tween photo- and gyro-taxis. They lead to changes inEqs. (3,5), which become ∂n∂t + ∇ · n (cid:18) u + V s (cid:18) − II c (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) (A) (cid:104) p (cid:105) (cid:19) = ∇ · ( D · ∇ n ) (13)˙ p = 1 B ( I ) (cid:124) (cid:123)(cid:122) (cid:125) (B) ( I − pp ) · k ( I ) (cid:124)(cid:123)(cid:122)(cid:125) (C) +( I − pp ) · (Ω · p + Λ E · p ) , (14)where V s is now the swimming speed in the absence ofany illumination, and I c is the critical intensity fromEq. (11). Model (A) includes an intensity-dependentspeed in Eq. (13) which encodes both a photokinetic ef-fect (i.e. swimming speed that depends on light intensity)as well as the ability of the cells to reverse their swimmingdirection in regions where I > I c . The other two mod-els, instead, modify the equation for the cell orientation(Eq. (5)), and therefore the value of (cid:104) p (cid:105) . The new equa-tion, Eq. (14), includes either an intensity-dependentgravitational reorientation timescale B ( I ) (model B) oran intensity dependent preferred cell orientation k ( I )(model C). The former is an ad hoc manifestation ofthe cell’s ability to change its centre of mass offset asa function of I , which is taken to follow B ( I ) = µα ⊥ ρ c gh (1 − I/I c ) , (15)where h is the centre of mass offset in the dark (see alsoEq. (7)). This assumes that bottom-heavy cells becometop-heavy for I > I c . The latter is the effect of thephototactic torque. The preferred cell orientation k ( I ) isobtained by a balancing the gravitational torque L g =( ρ c vgh ) p × k , and the phototactic torque L p = − f m I c I ( I − I c ) p × ( β π + β ∇ I ) . (16)Here f m is a parameter defining the overall strength ofthe phototactic torque, while β and β quantify the cell’sphototactic response to light incident from an arbitrarydirection π , and to local gradients in the light intensity.Then k ( I ) is simply given by k ( I ) = k + 4 f m mgh ( I/I c )( I/I c − β π + β ∇ I ) . (17) Ghorai 2005 Growth rate: Ghorai 2010(c)(a) (d)(b)
FIG. 3. Penetrative bioconvection. (a) Taxis functions corresponding to different values of I c (non-dimensionalised), and (b)corresponding equilibrium profiles of cell concentration (for I s = 0 . α ¯ nH = 1). (c) Neutral curvesseparating stable perturbations (below) from unstable ones (above). (d) The perturbations’ growth rate depends strongly onthe Rayleigh number. Adapted from [65, 66]. Williams and Bees studied these three models separately,with two cases considered for model C: ( β , β ) = (1 , ,
1) (C II). In general, phototaxis will affectthe probability distribution f ( p ) (Eq. (4)), and throughthis the quantities (cid:104) p (cid:105) (Eq. (6)) and D (Eq. (9)) in a non-trivial way [17]. The coupled photo-gyrotaxis models areable to predict both steady and oscillating flow patterns.Interestingly models A, B and C I produce very similarresults, when illuminated from above, which also agreequalitatively with experiments carried out by Williamsand Bees [1]. The basis for this comparison with ex-periments, however, was deemed questionable for severalreasons. Firstly, the experimental methods did not allowfor the theoretical base state to be reached before in-stabilities appeared; secondly there was a lack of reliabledirect estimates of the critical intensity I c for the suspen-sion; and finally it was unclear whether the wavelengthsbeing measured in the experiment corresponded to thecritical state obtained using the theory. Nevertheless, the qualitative agreement observed between models andexperiment is promising. At the same time, model C IIdiffers significantly from the others due to the existenceof non-hydrodynamic instabilities. In this case, photo-tactic torques due to ∇ I lead to horizontal modulationsof the cell concentration profile, with equally spaced clus-ters of cells just above the depth at which I = I c for the(unstable) base state. This modulation should happenwithout generating a background fluid flow. Finally, wenote that a formal extension of generalised Taylor dis-persion (GTD) to a suspension of phototactic microor-ganisms, in the spirit of the work of Hill and Bees [42],and Manela and Frankel (for gyrotaxis) [50], and Bearon(for chemotaxis) [72–74], has not been attempted yet. Itwill be interesting to see how GTD alters the stabilitydynamics of phototactic suspensions. BIOCONVECTION DRIVEN BY PHOTOTAXIS
A recent experimental endeavour is to use light asthe main factor leading to bioconvection. In these stud-ies, no bioconvection patterns are observed in absence oflight. Convective instabilities only appear in presence ofspecific illumination patterns, used to generate the cell-density inhomogeneity that drives bioconvection.The first instabilities of this type were investigated insuspensions of the microalga
Euglena gracilis [16, 75].Initially homogenous cultures, filled the gap within a2 mm-thick horizontal Hele-Shaw cell, and were then ex-posed to strong uniform illumination from the bottomto induce negative phototaxis. Within a few minutes,cells accumulated in randomly distributed high-densityclusters [16], which then gathered towards the centre ofthe chamber. These localised patterns disappeared inabsence of light. The number of clusters and the meanseparation between neighbouring clusters can be used tocharacterise the dynamics of the system as a functionof depth of the chamber and average cell concentration.Both quantities showed ageing, with the former increas-ing and the latter decreasing as a function of time. Aneffective model based on transition probabilities betweencell layers was able to recapitulate the qualitative dy-namics of cell concentration observed experimentally, in-cluding the process of pattern formation, cell circulationaround a cluster, and dependence of number of clustersand their distance to nearest neighbours on depth andcell density. This agreement provided support for thehypothesis that pattern formation is a consequence ofcell motion transversal to the direction of incoming light,possibly a result of phototaxis to light scattered by themicroorganisms themselves. Although this model lacks aconnection to the underlying hydrodynamics of the sus-pension, it is still a first step towards understanding theemergence of photo-bioconvective patterns. A later studyof localized photo-bioconvection patterns in the same sys-tem [76], showed that the emergence of these high-densitycell clusters, or “bioconvection units”, was dependent onthe spatial distribution of the cells before light exposure.Below a critical cell concentration, no bioconvection unitswere observed for a uniform initial distribution, whereasa single cluster formed for sufficiently spatially hetero-geneous suspensions. This observation suggests that thesystem is multi-stable, possibly as a result of the topologyof the containing chamber.In an interesting study by Dervaux et al. [77], in-triguing bioconvective instabilities were observed insteadwhen a thin suspension of the model unicellular green al-gae
Chlamydomonas reinhardtii was placed in a Petri dishand illuminated from above by a localised circular laserbeam. For intensities within the range of positive pho-totaxis, the cells accumulated beneath the light beam,and radially symmetric convective flows developed as a result of light-induced collective motion. Although thetimescale for the establishment of this instability waslong ( ∼ µ mglass bead) and trap it just above the laser spot (Fig. a-c). Above a critical Rayleigh number, corresponding tohigh cell density and layer depth, a novel instability inthe form of travelling concentration waves was discovered(Fig d-f). Its development was successfully describedthrough a model for cell reorientation that included turn-ing due to the fluid’s vorticity, and a phototactic reori-entation similar to the first term in Eq. (14), but with k replaced by the direction of ∇ I from the laser. Theeffective timescale for phototactic turning, equivalent to B ( I ) in Eq. (14), was left as a fitting parameter. Best fitswere obtained for a timescale (cid:39) . ∼ − Chlamydomonas augustae , a species known to be stronglygyrotactic [1]). This work is the first experimental val-idation of the phototactic torque model in a simplifiedcase.More recently, Arrieta et al. [18] reported a new exam-ple of bioconvection driven by phototaxis, which developsquickly ( ∼
30 s), and can be easily reconfigured. In thiscase, a suspension of
C. reinhardtii is loaded within athin and wide square chamber held vertically, a config-uration notably distinct from similar studies in biocon-vection, where the widest sides of the chamber are hori-zontal. Localised illumination was provided by a 200 µ m-diameter horizontal optical fibre. With no light from thefibre, cells were uniformly distributed within the suspen-sion. However, as the light was switched on, the algaeaccumulated phototactically around the fibre, leading toa gravitational instability and the formation of a singlelocalised sinking bioconvective plume. The system wasdescribed with a purely phototactic model of 2D biocon-vection, based on the Eqs. (1)-(3), with no slip at theboundary and no cell flux through the boundary. Fol-lowing previous studies, [34] the phototactic velocity wastaken to be proportional to the gradient in light inten-sity. The proportionality factor depends on the phototac-tic sensitivity parameter, β , previously shown to exhibitan interesting adaptation possibly as a result of the pho-tosynthetic activity of the cells [34]. Despite neglectinggyrotaxis and gravitaxis, as well as cell-cell interactionsreinforcing alignment at high concentrations [78], themodel showed an excellent agreement with experimen-tal observations and was able to capture the structure ofthe bioconvective flow of cells (Fig. 5). It also predictedthat for average cell densities below a critical value, n c ( ∼ cells / cm for β = 0 . (d) (e) (f)(a) (b) (c)Dervaux2017 FIG. 4. Photo-bioconvection induced by localised laser illumination in [77]. (a-c) An 800 µ m-diameter floating glass bead istransported by the flow field generated by the concentrated cells. (d-f) Time evolution of phototactic cell accumulation at thebottom of the container, and formation of radial waves of concentration. Adapted from [77]FIG. 5. Dynamics of plume formation in localised photo-bioconvection. Cells accumulate around the optical fibre (white disk)and form a single bioconvective plume as the reduced time τ progresses. Panels show the cell density ( n ) from the continuummodel (left side), and the experiments. Plume formation takes ∼
30 s. Adapted from [18] bifurcation was confirmed experimentally, and the criti-cal cell density observed was in good quantitative agree-ment with the one predicted by the model. Overall, themodel predicted a bifurcation boundary compatible withthe simple relation βn c = const. Leveraging the speed atwhich plumes form in this system, the authors then pro-vided a proof of principle that photo-bioconvection canbe used to mix the suspension. This was achieved hereby alternatively blinking a pair of optical fibres to createtwo sets of convective flows with crossing streamlines. PHOTO-FOCUSSING
Light can also be used together with externally im-posed flows to control the motion of phototactic cellsthrough a mechanism based on balance between hydro- dynamic and phototactic torques, analogous to gyrotaxis.This has been proposed to play a role in the cell accu-mulation observed by Dervaux et al. [77], and can beemployed to accumulate cells in specific regions by im-posing background flows externally. This type of cellaccumulation, called photo-focussing, has been investi-gated in a number of recent studies [33, 79–81]. Thefirst direct proof of photo-focussing used a suspension of
C. reinhardtii flowed through a long square capillary ofside 1 mm [79]. The cells’ distribution across the channel,which was uniform without light stimuli, could be mod-ulated dramatically by turning on a light source orientedalong the length of the channel. When cells tried to pho-totax downstream, they self-focussed along the axis of thechannel; whereas for upstream phototaxis, cells accumu-lated at the boundaries (Fig. 6). An initial model forphoto-focussing used a deterministic description of mi- (c)(d)Garcia 2013 range, the cell’s rotation is too fast, and it has no time toorient itself toward the light, a phenomenon similar to thegyrotactic trap observed in phytoplankton [13], and,consequently, the cell cannot migrate. The concentrationof cells in the center results in a reinforcement of thehydrodynamic interactions between cells and, thus, in abroadening of the distribution of CR around the centerwhere hydrodynamic interactions prevail. As shown inFig. 3, the band width depends on the flow rate. Theband width is measured at the half maximum value ofthe cell distribution. The minimum band width obtainedat a flow rate of :
09 ml = min ( _ ! ! : " ) represents22% of the channel width [see the inset in Fig. 3].Self-focusing is associated with hydrodynamic forcingwhich orients the cells toward the center of the flow wherethey migrate at their own velocity perturbed by their mu-tual interactions. When the light is switched off from thefocusing state, hydrodynamic interactions between ori-ented CR [24] reinforced by the concentration of algae atthe flow center cause the cells to mix back in the fluid. Weanalyze the dynamics of such a phenomenon (see Fig. 5) by varying light exposure time: The light is switched on andoff alternately with a time period T ¼ . Fast cameravisualization clearly shows that the half band width varieslinearly and reversibly with time. We found that the cor-responding average velocity for both self-focusing andremixing is " m s " for a flow rate of :
06 ml = min (i.e., _ ! ! : " ). This velocity value emerges from thedynamics of the collective motion of the cells. It is close tothe swimming speed of a single CR which plays an impor-tant role in the phenomenon (see the model below).However, the hydrodynamic interactions between the cellsare known to generate hydrodynamic diffusion (especiallyin concentrated suspensions) that strongly perturb themotion of a single cell and can thus modify the collectivedynamics and its velocity.We then developed a very simple nonlinear model thatdescribes the motion of a swimmer within a Poiseuille flowin a cylinder of radius R with regular orientation toward alight source situated on the right-or left-hand side of thechannel. We do not describe the remixing obtained experi-mentally which is due to a strong repulsion between ori-ented CR. This model is inspired by Ref. [22] and has beenmodified in order to take into account reorientation of themicroswimmer upstream or downstream toward a lightsource. We define e and k the unitary radial and longitu-dinal vectors, respectively. The microswimmer situatedin r ¼ e þ z k is simply seen as moving at velocity V ¼ V þ u ð Þ , where V is the intrinsic velocity of themicroswimmer ( k V k ¼ V ) moving in a fluid flowing atvelocity u ð Þ ¼ u max ð " R Þ k (Poiseuille flow), u max being the maximum value of u ð Þ at the center of theflow ( ¼ ). The direction of V (i.e., the directionof the swimmer) is determined by local flow vorticity ! ð Þ ¼ = r ’ u ¼ " u max =R e $ . The trajectory ofthe microswimmer is thus given by r ð t Þ ¼ Z t V ð Þ dt ; (1)with V ¼ V sin $ ð Þ e þ ½ V cos $ ð Þ þ u ð Þ) k and $ ð Þ ¼ Z t ! ð Þ dt : (2)By solving this system of equations numerically, weobtain trajectories which experience oscillations but nonet migration [see Fig. 6]. Note that the oscillations havea wavelength much larger than the channel radius R [22]and are not visible in Fig. 2. The oscillations result from thecombination of flow vorticity and swimmer velocity.In order to take into account the effect of light, the micro-swimmer is regularly reoriented upstream. This can bedone by adding a torque; for example, Williams and Bees[25] use a phototactic torque balancing a gravitactic torqueand resulting in a mechanical equilibrium (the swimmeris an isolated body and must be torque-free). Instead herewe simply reorient the swimmer by changing the timevarying angle $ ð t Þ to 0 (upstream) or % (downstream) at Time (s) H a l f B a nd w i d t h ( m ) µ FIG. 5 (color online). Band width of CR as a function of timeunder light exposures of 2.5 s separated by dark periods of 2.5 s.A linear variation is observed with a transverse velocity equal onaverage to * " m s " (indicated by the red lines). Light(a)(b) * Light * II I
FIG. 4 (color online). Schematic view of the phenomenon.Cells orient themselves toward light (I) and are rotated by thevorticity (II). (a) Light source upstream, resulting in a self-focusing. (b) Light source downstream resulting in a migrationtoward the wall.
PRL week ending29 MARCH 2013 range, the cell’s rotation is too fast, and it has no time toorient itself toward the light, a phenomenon similar to thegyrotactic trap observed in phytoplankton [13], and,consequently, the cell cannot migrate. The concentrationof cells in the center results in a reinforcement of thehydrodynamic interactions between cells and, thus, in abroadening of the distribution of CR around the centerwhere hydrodynamic interactions prevail. As shown inFig. 3, the band width depends on the flow rate. Theband width is measured at the half maximum value ofthe cell distribution. The minimum band width obtainedat a flow rate of :
09 ml = min ( _ ! ! : " ) represents22% of the channel width [see the inset in Fig. 3].Self-focusing is associated with hydrodynamic forcingwhich orients the cells toward the center of the flow wherethey migrate at their own velocity perturbed by their mu-tual interactions. When the light is switched off from thefocusing state, hydrodynamic interactions between ori-ented CR [24] reinforced by the concentration of algae atthe flow center cause the cells to mix back in the fluid. Weanalyze the dynamics of such a phenomenon (see Fig. 5) by varying light exposure time: The light is switched on andoff alternately with a time period T ¼ . Fast cameravisualization clearly shows that the half band width varieslinearly and reversibly with time. We found that the cor-responding average velocity for both self-focusing andremixing is " m s " for a flow rate of :
06 ml = min (i.e., _ ! ! : " ). This velocity value emerges from thedynamics of the collective motion of the cells. It is close tothe swimming speed of a single CR which plays an impor-tant role in the phenomenon (see the model below).However, the hydrodynamic interactions between the cellsare known to generate hydrodynamic diffusion (especiallyin concentrated suspensions) that strongly perturb themotion of a single cell and can thus modify the collectivedynamics and its velocity.We then developed a very simple nonlinear model thatdescribes the motion of a swimmer within a Poiseuille flowin a cylinder of radius R with regular orientation toward alight source situated on the right-or left-hand side of thechannel. We do not describe the remixing obtained experi-mentally which is due to a strong repulsion between ori-ented CR. This model is inspired by Ref. [22] and has beenmodified in order to take into account reorientation of themicroswimmer upstream or downstream toward a lightsource. We define e and k the unitary radial and longitu-dinal vectors, respectively. The microswimmer situatedin r ¼ e þ z k is simply seen as moving at velocity V ¼ V þ u ð Þ , where V is the intrinsic velocity of themicroswimmer ( k V k ¼ V ) moving in a fluid flowing atvelocity u ð Þ ¼ u max ð " R Þ k (Poiseuille flow), u max being the maximum value of u ð Þ at the center of theflow ( ¼ ). The direction of V (i.e., the directionof the swimmer) is determined by local flow vorticity ! ð Þ ¼ = r ’ u ¼ " u max =R e $ . The trajectory ofthe microswimmer is thus given by r ð t Þ ¼ Z t V ð Þ dt ; (1)with V ¼ V sin $ ð Þ e þ ½ V cos $ ð Þ þ u ð Þ) k and $ ð Þ ¼ Z t ! ð Þ dt : (2)By solving this system of equations numerically, weobtain trajectories which experience oscillations but nonet migration [see Fig. 6]. Note that the oscillations havea wavelength much larger than the channel radius R [22]and are not visible in Fig. 2. The oscillations result from thecombination of flow vorticity and swimmer velocity.In order to take into account the effect of light, the micro-swimmer is regularly reoriented upstream. This can bedone by adding a torque; for example, Williams and Bees[25] use a phototactic torque balancing a gravitactic torqueand resulting in a mechanical equilibrium (the swimmeris an isolated body and must be torque-free). Instead herewe simply reorient the swimmer by changing the timevarying angle $ ð t Þ to 0 (upstream) or % (downstream) at Time (s) H a l f B a nd w i d t h ( m ) µ FIG. 5 (color online). Band width of CR as a function of timeunder light exposures of 2.5 s separated by dark periods of 2.5 s.A linear variation is observed with a transverse velocity equal onaverage to * " m s " (indicated by the red lines). Light(a)(b) * Light * II I
FIG. 4 (color online). Schematic view of the phenomenon.Cells orient themselves toward light (I) and are rotated by thevorticity (II). (a) Light source upstream, resulting in a self-focusing. (b) Light source downstream resulting in a migrationtoward the wall.
PRL week ending29 MARCH 2013 :
09 ml = min ( _ ! ! : " ) represents22% of the channel width [see the inset in Fig. 3].Self-focusing is associated with hydrodynamic forcingwhich orients the cells toward the center of the flow wherethey migrate at their own velocity perturbed by their mu-tual interactions. When the light is switched off from thefocusing state, hydrodynamic interactions between ori-ented CR [24] reinforced by the concentration of algae atthe flow center cause the cells to mix back in the fluid. Weanalyze the dynamics of such a phenomenon (see Fig. 5) by varying light exposure time: The light is switched on andoff alternately with a time period T ¼ . Fast cameravisualization clearly shows that the half band width varieslinearly and reversibly with time. We found that the cor-responding average velocity for both self-focusing andremixing is " m s " for a flow rate of :
06 ml = min (i.e., _ ! ! : " ). This velocity value emerges from thedynamics of the collective motion of the cells. It is close tothe swimming speed of a single CR which plays an impor-tant role in the phenomenon (see the model below).However, the hydrodynamic interactions between the cellsare known to generate hydrodynamic diffusion (especiallyin concentrated suspensions) that strongly perturb themotion of a single cell and can thus modify the collectivedynamics and its velocity.We then developed a very simple nonlinear model thatdescribes the motion of a swimmer within a Poiseuille flowin a cylinder of radius R with regular orientation toward alight source situated on the right-or left-hand side of thechannel. We do not describe the remixing obtained experi-mentally which is due to a strong repulsion between ori-ented CR. This model is inspired by Ref. [22] and has beenmodified in order to take into account reorientation of themicroswimmer upstream or downstream toward a lightsource. We define e and k the unitary radial and longitu-dinal vectors, respectively. The microswimmer situatedin r ¼ e þ z k is simply seen as moving at velocity V ¼ V þ u ð Þ , where V is the intrinsic velocity of themicroswimmer ( k V k ¼ V ) moving in a fluid flowing atvelocity u ð Þ ¼ u max ð " R Þ k (Poiseuille flow), u max being the maximum value of u ð Þ at the center of theflow ( ¼ ). The direction of V (i.e., the directionof the swimmer) is determined by local flow vorticity ! ð Þ ¼ = r ’ u ¼ " u max =R e $ . The trajectory ofthe microswimmer is thus given by r ð t Þ ¼ Z t V ð Þ dt ; (1)with V ¼ V sin $ ð Þ e þ ½ V cos $ ð Þ þ u ð Þ) k and $ ð Þ ¼ Z t ! ð Þ dt : (2)By solving this system of equations numerically, weobtain trajectories which experience oscillations but nonet migration [see Fig. 6]. Note that the oscillations havea wavelength much larger than the channel radius R [22]and are not visible in Fig. 2. The oscillations result from thecombination of flow vorticity and swimmer velocity.In order to take into account the effect of light, the micro-swimmer is regularly reoriented upstream. This can bedone by adding a torque; for example, Williams and Bees[25] use a phototactic torque balancing a gravitactic torqueand resulting in a mechanical equilibrium (the swimmeris an isolated body and must be torque-free). Instead herewe simply reorient the swimmer by changing the timevarying angle $ ð t Þ to 0 (upstream) or % (downstream) at Time (s) H a l f B a nd w i d t h ( m ) µ FIG. 5 (color online). Band width of CR as a function of timeunder light exposures of 2.5 s separated by dark periods of 2.5 s.A linear variation is observed with a transverse velocity equal onaverage to * " m s " (indicated by the red lines). Light(a)(b) * Light * II I
FIG. 4 (color online). Schematic view of the phenomenon.Cells orient themselves toward light (I) and are rotated by thevorticity (II). (a) Light source upstream, resulting in a self-focusing. (b) Light source downstream resulting in a migrationtoward the wall.
PRL week ending29 MARCH 2013 :
09 ml = min ( _ ! ! : " ) represents22% of the channel width [see the inset in Fig. 3].Self-focusing is associated with hydrodynamic forcingwhich orients the cells toward the center of the flow wherethey migrate at their own velocity perturbed by their mu-tual interactions. When the light is switched off from thefocusing state, hydrodynamic interactions between ori-ented CR [24] reinforced by the concentration of algae atthe flow center cause the cells to mix back in the fluid. Weanalyze the dynamics of such a phenomenon (see Fig. 5) by varying light exposure time: The light is switched on andoff alternately with a time period T ¼ . Fast cameravisualization clearly shows that the half band width varieslinearly and reversibly with time. We found that the cor-responding average velocity for both self-focusing andremixing is " m s " for a flow rate of :
06 ml = min (i.e., _ ! ! : " ). This velocity value emerges from thedynamics of the collective motion of the cells. It is close tothe swimming speed of a single CR which plays an impor-tant role in the phenomenon (see the model below).However, the hydrodynamic interactions between the cellsare known to generate hydrodynamic diffusion (especiallyin concentrated suspensions) that strongly perturb themotion of a single cell and can thus modify the collectivedynamics and its velocity.We then developed a very simple nonlinear model thatdescribes the motion of a swimmer within a Poiseuille flowin a cylinder of radius R with regular orientation toward alight source situated on the right-or left-hand side of thechannel. We do not describe the remixing obtained experi-mentally which is due to a strong repulsion between ori-ented CR. This model is inspired by Ref. [22] and has beenmodified in order to take into account reorientation of themicroswimmer upstream or downstream toward a lightsource. We define e and k the unitary radial and longitu-dinal vectors, respectively. The microswimmer situatedin r ¼ e þ z k is simply seen as moving at velocity V ¼ V þ u ð Þ , where V is the intrinsic velocity of themicroswimmer ( k V k ¼ V ) moving in a fluid flowing atvelocity u ð Þ ¼ u max ð " R Þ k (Poiseuille flow), u max being the maximum value of u ð Þ at the center of theflow ( ¼ ). The direction of V (i.e., the directionof the swimmer) is determined by local flow vorticity ! ð Þ ¼ = r ’ u ¼ " u max =R e $ . The trajectory ofthe microswimmer is thus given by r ð t Þ ¼ Z t V ð Þ dt ; (1)with V ¼ V sin $ ð Þ e þ ½ V cos $ ð Þ þ u ð Þ) k and $ ð Þ ¼ Z t ! ð Þ dt : (2)By solving this system of equations numerically, weobtain trajectories which experience oscillations but nonet migration [see Fig. 6]. Note that the oscillations havea wavelength much larger than the channel radius R [22]and are not visible in Fig. 2. The oscillations result from thecombination of flow vorticity and swimmer velocity.In order to take into account the effect of light, the micro-swimmer is regularly reoriented upstream. This can bedone by adding a torque; for example, Williams and Bees[25] use a phototactic torque balancing a gravitactic torqueand resulting in a mechanical equilibrium (the swimmeris an isolated body and must be torque-free). Instead herewe simply reorient the swimmer by changing the timevarying angle $ ð t Þ to 0 (upstream) or % (downstream) at Time (s) H a l f B a nd w i d t h ( m ) µ FIG. 5 (color online). Band width of CR as a function of timeunder light exposures of 2.5 s separated by dark periods of 2.5 s.A linear variation is observed with a transverse velocity equal onaverage to * " m s " (indicated by the red lines). Light(a)(b) * Light * II I
FIG. 4 (color online). Schematic view of the phenomenon.Cells orient themselves toward light (I) and are rotated by thevorticity (II). (a) Light source upstream, resulting in a self-focusing. (b) Light source downstream resulting in a migrationtoward the wall.
PRL week ending29 MARCH 2013 made of polydimethylsiloxane by using soft lithographytechniques [20]. The experimental setup is shown in Fig. 1.The Reynolds number Re ! V R= ! associated withswimming alga in water is very weak (about : " where kinematic viscosity is ! $ m s ). It is wellknown that for vanishing Reynolds numbers, because ofStokes flow reversibility, passive spherical particles rotatein a Poiseuille flow (except at the center) but do notexperience any migration across the flow lines [21]. Inthe case of microswimmers, the situation is quite different:Their motion in a Poiseuille flow follows an oscillatingtrajectory [22] due to flow vorticity combined with swim-ming velocity. Therefore, each cell undergoes time peri-odic cross-stream migration but with no net migrationaveraged over the cell’s period of rotation. Figure 2(a)shows 40 ms microswimmer tracks in such a Poiseuilleflow: The cells are homogeneously distributed over thewidth of the channel. Although the trajectories experience some oscillations, they are not visible at this scale, as thespatial wavelength of the oscillations is expected to beapproximately a few channel widths (see below).When a light source is switched on on the right-handside of the flow—i.e., oriented upstream since the flowgoes from right to left—the situation is noticeably differ-ent: The microswimmers migrate to the center of the flow,resulting in strong self-focusing. Figure 2(b) shows 40 msmicroswimmer tracks in the presence of light. It representsthe final stage where the cells have migrated to the centerof the channel. Figure 3 shows the corresponding distribu-tions of cells though the channel (without and with light)for different values of the imposed flow rate. When thelight source is on the left-hand side—oriented down-stream—with the same flow direction, cells migrate towardthe walls of the channel.Let us now simply sketch the phenomenon. In the pres-ence of light, the swimmer rotates due to flow vorticity, butit also regularly reorients itself toward the light sourcewithin a typical time of about 1 s. Because of local vor-ticity, the swimmer will be more frequently oriented by theflow toward the center where it moves to by swimming[Fig. 4(a)]. Note that if the light source is on the oppositeside, the cells are more frequently oriented toward thewalls to which they migrate [Fig. 4(b)]. In the model below,we have solved numerically a simple nonlinear modelbased on Ref. [22] but including regular swimmer reor-ientations toward the light. It clearly shows motion towardthe center or the walls depending on the direction ofreorientation, i.e., toward the position of the light source.Experimentally, the self-focusing is observed in therange of flow rate :
03 ml = min 09 ml = min , whichin terms of shear rate averaged between the center of thesquared channel and the walls gives : & _ " & : . Below this range, the flow is too weak to forcethe cells to rotate, since a CR can resist the flow rotation[23], and the algae are not oriented by the flow. Above this (a)(b) * FIG. 2. CR trajectories (obtained by superposing 10 images).(a) With no light. CR are transported by the Poiseuille flow fromright to left. (b) With light on the right-hand side. The CR movetoward the center. FIG. 3. Probability distribution of CR in the Poiseuille flow.(a) Without light, (b) with the light source on the right. Inset: Thefull band width at half maximum as a function of the flow rate. MicroscopeFast Camera1 mm1 mm Red light (Filter) x yzFlow FIG. 1 (color online). The setup of the experiment. PRL week ending29 MARCH 2013 (b) made of polydimethylsiloxane by using soft lithographytechniques [20]. The experimental setup is shown in Fig. 1.The Reynolds number Re ! V R= ! associated withswimming alga in water is very weak (about : " where kinematic viscosity is ! $ m s ). It is wellknown that for vanishing Reynolds numbers, because ofStokes flow reversibility, passive spherical particles rotatein a Poiseuille flow (except at the center) but do notexperience any migration across the flow lines [21]. Inthe case of microswimmers, the situation is quite different:Their motion in a Poiseuille flow follows an oscillatingtrajectory [22] due to flow vorticity combined with swim-ming velocity. Therefore, each cell undergoes time peri-odic cross-stream migration but with no net migrationaveraged over the cell’s period of rotation. Figure 2(a)shows 40 ms microswimmer tracks in such a Poiseuilleflow: The cells are homogeneously distributed over thewidth of the channel. Although the trajectories experience some oscillations, they are not visible at this scale, as thespatial wavelength of the oscillations is expected to beapproximately a few channel widths (see below).When a light source is switched on on the right-handside of the flow—i.e., oriented upstream since the flowgoes from right to left—the situation is noticeably differ-ent: The microswimmers migrate to the center of the flow,resulting in strong self-focusing. Figure 2(b) shows 40 msmicroswimmer tracks in the presence of light. It representsthe final stage where the cells have migrated to the centerof the channel. Figure 3 shows the corresponding distribu-tions of cells though the channel (without and with light)for different values of the imposed flow rate. When thelight source is on the left-hand side—oriented down-stream—with the same flow direction, cells migrate towardthe walls of the channel.Let us now simply sketch the phenomenon. In the pres-ence of light, the swimmer rotates due to flow vorticity, butit also regularly reorients itself toward the light sourcewithin a typical time of about 1 s. Because of local vor-ticity, the swimmer will be more frequently oriented by theflow toward the center where it moves to by swimming[Fig. 4(a)]. Note that if the light source is on the oppositeside, the cells are more frequently oriented toward thewalls to which they migrate [Fig. 4(b)]. In the model below,we have solved numerically a simple nonlinear modelbased on Ref. [22] but including regular swimmer reor-ientations toward the light. It clearly shows motion towardthe center or the walls depending on the direction ofreorientation, i.e., toward the position of the light source.Experimentally, the self-focusing is observed in therange of flow rate : 03 ml = min 09 ml = min , whichin terms of shear rate averaged between the center of thesquared channel and the walls gives : & _ " & : . Below this range, the flow is too weak to forcethe cells to rotate, since a CR can resist the flow rotation[23], and the algae are not oriented by the flow. Above this (a)(b) * FIG. 2. CR trajectories (obtained by superposing 10 images).(a) With no light. CR are transported by the Poiseuille flow fromright to left. (b) With light on the right-hand side. The CR movetoward the center. FIG. 3. Probability distribution of CR in the Poiseuille flow.(a) Without light, (b) with the light source on the right. Inset: Thefull band width at half maximum as a function of the flow rate. MicroscopeFast Camera1 mm1 mm Red light (Filter) x yzFlow FIG. 1 (color online). The setup of the experiment. PRL week ending29 MARCH 2013 (a) made of polydimethylsiloxane by using soft lithographytechniques [20]. The experimental setup is shown in Fig. 1.The Reynolds number Re ! V R= ! associated withswimming alga in water is very weak (about : " where kinematic viscosity is ! $ m s ). It is wellknown that for vanishing Reynolds numbers, because ofStokes flow reversibility, passive spherical particles rotatein a Poiseuille flow (except at the center) but do notexperience any migration across the flow lines [21]. Inthe case of microswimmers, the situation is quite different:Their motion in a Poiseuille flow follows an oscillatingtrajectory [22] due to flow vorticity combined with swim-ming velocity. Therefore, each cell undergoes time peri-odic cross-stream migration but with no net migrationaveraged over the cell’s period of rotation. Figure 2(a)shows 40 ms microswimmer tracks in such a Poiseuilleflow: The cells are homogeneously distributed over thewidth of the channel. Although the trajectories experience some oscillations, they are not visible at this scale, as thespatial wavelength of the oscillations is expected to beapproximately a few channel widths (see below).When a light source is switched on on the right-handside of the flow—i.e., oriented upstream since the flowgoes from right to left—the situation is noticeably differ-ent: The microswimmers migrate to the center of the flow,resulting in strong self-focusing. Figure 2(b) shows 40 msmicroswimmer tracks in the presence of light. It representsthe final stage where the cells have migrated to the centerof the channel. Figure 3 shows the corresponding distribu-tions of cells though the channel (without and with light)for different values of the imposed flow rate. When thelight source is on the left-hand side—oriented down-stream—with the same flow direction, cells migrate towardthe walls of the channel.Let us now simply sketch the phenomenon. In the pres-ence of light, the swimmer rotates due to flow vorticity, butit also regularly reorients itself toward the light sourcewithin a typical time of about 1 s. Because of local vor-ticity, the swimmer will be more frequently oriented by theflow toward the center where it moves to by swimming[Fig. 4(a)]. Note that if the light source is on the oppositeside, the cells are more frequently oriented toward thewalls to which they migrate [Fig. 4(b)]. In the model below,we have solved numerically a simple nonlinear modelbased on Ref. [22] but including regular swimmer reor-ientations toward the light. It clearly shows motion towardthe center or the walls depending on the direction ofreorientation, i.e., toward the position of the light source.Experimentally, the self-focusing is observed in therange of flow rate : 03 ml = min 09 ml = min , whichin terms of shear rate averaged between the center of thesquared channel and the walls gives : & _ " & : . Below this range, the flow is too weak to forcethe cells to rotate, since a CR can resist the flow rotation[23], and the algae are not oriented by the flow. Above this (a)(b) * FIG. 2. CR trajectories (obtained by superposing 10 images).(a) With no light. CR are transported by the Poiseuille flow fromright to left. (b) With light on the right-hand side. The CR movetoward the center. FIG. 3. Probability distribution of CR in the Poiseuille flow.(a) Without light, (b) with the light source on the right. Inset: Thefull band width at half maximum as a function of the flow rate. MicroscopeFast Camera1 mm1 mm Red light (Filter) x yzFlow FIG. 1 (color online). The setup of the experiment. PRL week ending29 MARCH 2013 made of polydimethylsiloxane by using soft lithographytechniques [20]. The experimental setup is shown in Fig. 1.The Reynolds number Re ! V R= ! associated withswimming alga in water is very weak (about : " where kinematic viscosity is ! $ m s ). It is wellknown that for vanishing Reynolds numbers, because ofStokes flow reversibility, passive spherical particles rotatein a Poiseuille flow (except at the center) but do notexperience any migration across the flow lines [21]. Inthe case of microswimmers, the situation is quite different:Their motion in a Poiseuille flow follows an oscillatingtrajectory [22] due to flow vorticity combined with swim-ming velocity. Therefore, each cell undergoes time peri-odic cross-stream migration but with no net migrationaveraged over the cell’s period of rotation. Figure 2(a)shows 40 ms microswimmer tracks in such a Poiseuilleflow: The cells are homogeneously distributed over thewidth of the channel. Although the trajectories experience some oscillations, they are not visible at this scale, as thespatial wavelength of the oscillations is expected to beapproximately a few channel widths (see below).When a light source is switched on on the right-handside of the flow—i.e., oriented upstream since the flowgoes from right to left—the situation is noticeably differ-ent: The microswimmers migrate to the center of the flow,resulting in strong self-focusing. Figure 2(b) shows 40 msmicroswimmer tracks in the presence of light. It representsthe final stage where the cells have migrated to the centerof the channel. Figure 3 shows the corresponding distribu-tions of cells though the channel (without and with light)for different values of the imposed flow rate. When thelight source is on the left-hand side—oriented down-stream—with the same flow direction, cells migrate towardthe walls of the channel.Let us now simply sketch the phenomenon. In the pres-ence of light, the swimmer rotates due to flow vorticity, butit also regularly reorients itself toward the light sourcewithin a typical time of about 1 s. Because of local vor-ticity, the swimmer will be more frequently oriented by theflow toward the center where it moves to by swimming[Fig. 4(a)]. Note that if the light source is on the oppositeside, the cells are more frequently oriented toward thewalls to which they migrate [Fig. 4(b)]. In the model below,we have solved numerically a simple nonlinear modelbased on Ref. [22] but including regular swimmer reor-ientations toward the light. It clearly shows motion towardthe center or the walls depending on the direction ofreorientation, i.e., toward the position of the light source.Experimentally, the self-focusing is observed in therange of flow rate : 03 ml = min 09 ml = min , whichin terms of shear rate averaged between the center of thesquared channel and the walls gives : & _ " & : . Below this range, the flow is too weak to forcethe cells to rotate, since a CR can resist the flow rotation[23], and the algae are not oriented by the flow. Above this (a)(b) * FIG. 2. CR trajectories (obtained by superposing 10 images).(a) With no light. CR are transported by the Poiseuille flow fromright to left. (b) With light on the right-hand side. The CR movetoward the center. FIG. 3. Probability distribution of CR in the Poiseuille flow.(a) Without light, (b) with the light source on the right. Inset: Thefull band width at half maximum as a function of the flow rate. MicroscopeFast Camera1 mm1 mm Red light (Filter) x yzFlow FIG. 1 (color online). The setup of the experiment. PRL week ending29 MARCH 2013 made of polydimethylsiloxane by using soft lithographytechniques [20]. The experimental setup is shown in Fig. 1.The Reynolds number Re ! V R= ! associated withswimming alga in water is very weak (about : " where kinematic viscosity is ! $ m s ). It is wellknown that for vanishing Reynolds numbers, because ofStokes flow reversibility, passive spherical particles rotatein a Poiseuille flow (except at the center) but do notexperience any migration across the flow lines [21]. Inthe case of microswimmers, the situation is quite different:Their motion in a Poiseuille flow follows an oscillatingtrajectory [22] due to flow vorticity combined with swim-ming velocity. Therefore, each cell undergoes time peri-odic cross-stream migration but with no net migrationaveraged over the cell’s period of rotation. Figure 2(a)shows 40 ms microswimmer tracks in such a Poiseuilleflow: The cells are homogeneously distributed over thewidth of the channel. Although the trajectories experience some oscillations, they are not visible at this scale, as thespatial wavelength of the oscillations is expected to beapproximately a few channel widths (see below).When a light source is switched on on the right-handside of the flow—i.e., oriented upstream since the flowgoes from right to left—the situation is noticeably differ-ent: The microswimmers migrate to the center of the flow,resulting in strong self-focusing. Figure 2(b) shows 40 msmicroswimmer tracks in the presence of light. It representsthe final stage where the cells have migrated to the centerof the channel. Figure 3 shows the corresponding distribu-tions of cells though the channel (without and with light)for different values of the imposed flow rate. When thelight source is on the left-hand side—oriented down-stream—with the same flow direction, cells migrate towardthe walls of the channel.Let us now simply sketch the phenomenon. In the pres-ence of light, the swimmer rotates due to flow vorticity, butit also regularly reorients itself toward the light sourcewithin a typical time of about 1 s. Because of local vor-ticity, the swimmer will be more frequently oriented by theflow toward the center where it moves to by swimming[Fig. 4(a)]. Note that if the light source is on the oppositeside, the cells are more frequently oriented toward thewalls to which they migrate [Fig. 4(b)]. In the model below,we have solved numerically a simple nonlinear modelbased on Ref. [22] but including regular swimmer reor-ientations toward the light. It clearly shows motion towardthe center or the walls depending on the direction ofreorientation, i.e., toward the position of the light source.Experimentally, the self-focusing is observed in therange of flow rate : 03 ml = min 09 ml = min , whichin terms of shear rate averaged between the center of thesquared channel and the walls gives : & _ " & : . Below this range, the flow is too weak to forcethe cells to rotate, since a CR can resist the flow rotation[23], and the algae are not oriented by the flow. Above this (a)(b) * FIG. 2. CR trajectories (obtained by superposing 10 images).(a) With no light. CR are transported by the Poiseuille flow fromright to left. (b) With light on the right-hand side. The CR movetoward the center. FIG. 3. Probability distribution of CR in the Poiseuille flow.(a) Without light, (b) with the light source on the right. Inset: Thefull band width at half maximum as a function of the flow rate. MicroscopeFast Camera1 mm1 mm Red light (Filter) x yzFlow FIG. 1 (color online). The setup of the experiment. PRL week ending29 MARCH 2013 m a d e o f po l yd i m e t hy l s il ox a n e byu s i ng s o f tlit hog r a phy t ec hn i qu e s [ ] . T h ee xp e r i m e n t a l s e t up i ss ho w n i n F i g . . T h e R e yno l d s nu m b e r R e ! V R = ! a ss o c i a t e d w it h s w i mm i ng a l g a i n w a t e r i s v e r y w ea k ( a bou t : " w h e r e k i n e m a ti c v i s c o s it y i s ! $ m s ) . I ti s w e ll kno w n t h a t f o r v a n i s h i ng R e yno l d s nu m b e r s , b eca u s e o f S t ok e s fl o w r e v e r s i b ilit y , p a ss i v e s ph e r i ca l p a r ti c l e s r o t a t e i n a P o i s e u ill e fl o w ( e x ce p t a tt h ece n t e r) bu t dono t e xp e r i e n cea ny m i g r a ti on ac r o ss t h e fl o w li n e s [ ] . I n t h eca s e o f m i c r o s w i mm e r s , t h e s it u a ti on i s qu it e d i ff e r e n t: T h e i r m o ti on i n a P o i s e u ill e fl o w f o ll o w s a no s c ill a ti ng t r a j ec t o r y [ ] du e t o fl o w vo r ti c it y c o m b i n e d w it h s w i m - m i ngv e l o c it y . T h e r e f o r e , eac h ce ll und e r go e s ti m e p e r i - od i cc r o ss - s t r ea mm i g r a ti onbu t w it hnon e t m i g r a ti on a v e r a g e dov e r t h ece ll ’ s p e r i odo fr o t a ti on . F i gu r e ( a ) s ho w s m s m i c r o s w i mm e r t r ac k s i n s u c h a P o i s e u ill e fl o w : T h ece ll s a r e ho m og e n e ou s l yd i s t r i bu t e dov e r t h e w i d t ho f t h ec h a nn e l . A lt hough t h e t r a j ec t o r i e s e xp e r i e n ce s o m e o s c ill a ti on s , t h e y a r e no t v i s i b l ea tt h i ss ca l e , a s t h e s p a ti a l w a v e l e ng t ho f t h e o s c ill a ti on s i s e xp ec t e d t ob e a pp r ox i m a t e l y a f e w c h a nn e l w i d t h s ( s ee b e l o w ) . W h e n a li gh t s ou r ce i ss w it c h e donon t h e r i gh t - h a nd s i d e o f t h e fl o w — i . e ., o r i e n t e dup s t r ea m s i n ce t h e fl o w go e s fr o m r i gh tt o l e f t — t h e s it u a ti on i s no ti cea b l yd i ff e r- e n t: T h e m i c r o s w i mm e r s m i g r a t e t o t h ece n t e r o f t h e fl o w , r e s u lti ng i n s t r ong s e l f-f o c u s i ng . F i gu r e ( b ) s ho w s m s m i c r o s w i mm e r t r ac k s i n t h e p r e s e n ce o f li gh t . I t r e p r e s e n t s t h e fi n a l s t a g e w h e r e t h ece ll s h a v e m i g r a t e d t o t h ece n t e r o f t h ec h a nn e l . F i gu r e s ho w s t h ec o rr e s pond i ngd i s t r i bu - ti on s o f ce ll s t hough t h ec h a nn e l ( w it hou t a nd w it h li gh t ) f o r d i ff e r e n t v a l u e s o f t h e i m po s e d fl o w r a t e . W h e n t h e li gh t s ou r ce i s on t h e l e f t - h a nd s i d e — o r i e n t e ddo w n - s t r ea m — w it h t h e s a m e fl o w d i r ec ti on , ce ll s m i g r a t e t o w a r d t h e w a ll s o f t h ec h a nn e l . L e t u s no w s i m p l y s k e t c h t h e ph e no m e non . I n t h e p r e s - e n ce o f li gh t , t h e s w i mm e rr o t a t e s du e t o fl o w vo r ti c it y , bu t it a l s o r e gu l a r l y r e o r i e n t s it s e l f t o w a r d t h e li gh t s ou r ce w it h i n a t yp i ca lti m e o f a bou t s . B eca u s e o f l o ca l vo r- ti c it y , t h e s w i mm e r w ill b e m o r e fr e qu e n tl yo r i e n t e dby t h e fl o w t o w a r d t h ece n t e r w h e r e it m ov e s t oby s w i mm i ng [ F i g . ( a )] . N o t e t h a ti f t h e li gh t s ou r ce i s on t h e oppo s it e s i d e , t h ece ll s a r e m o r e fr e qu e n tl yo r i e n t e d t o w a r d t h e w a ll s t o w h i c h t h e y m i g r a t e [ F i g . ( b )] . I n t h e m od e l b e l o w , w e h a v e s o l v e dnu m e r i ca ll y a s i m p l e non li n ea r m od e l b a s e don R e f . [ ] bu ti n c l ud i ng r e gu l a r s w i mm e rr e o r- i e n t a ti on s t o w a r d t h e li gh t . I t c l ea r l y s ho w s m o ti on t o w a r d t h ece n t e r o r t h e w a ll s d e p e nd i ngon t h e d i r ec ti ono f r e o r i e n t a ti on , i . e ., t o w a r d t h e po s iti ono f t h e li gh t s ou r ce . E xp e r i m e n t a ll y , t h e s e l f-f o c u s i ng i s ob s e r v e d i n t h e r a ng e o f fl o w r a t e : m l = m i n < Q < : m l = m i n , w h i c h i n t e r m s o f s h ea rr a t ea v e r a g e db e t w ee n t h ece n t e r o f t h e s qu a r e d c h a nn e l a nd t h e w a ll s g i v e s : s & _ " & : s . B e l o w t h i s r a ng e , t h e fl o w i s t oo w ea k t o f o r ce t h ece ll s t o r o t a t e , s i n cea CR ca n r e s i s tt h e fl o w r o t a ti on [ ] , a nd t h ea l g aea r e no t o r i e n t e dby t h e fl o w . A bov e t h i s ( a ) ( b ) mm mm . mm F l o w d i r ec ti on L i gh t s ou r ce * F I G . . CR t r a j ec t o r i e s ( ob t a i n e dby s up e r po s i ng10 i m a g e s ) . ( a ) W it hno li gh t . CR a r e t r a n s po r t e dby t h e P o i s e u ill e fl o w fr o m r i gh tt o l e f t . ( b ) W it h li gh t on t h e r i gh t - h a nd s i d e . T h e CR m ov e t o w a r d t h ece n t e r . F I G . . P r ob a b ilit yd i s t r i bu ti ono f CR i n t h e P o i s e u ill e fl o w . ( a ) W it hou tli gh t , ( b ) w it h t h e li gh t s ou r ce on t h e r i gh t . I n s e t: T h e f u ll b a nd w i d t h a t h a l f m a x i m u m a s a f un c ti ono f t h e fl o w r a t e . M i c r o s c op e F a s t C a m e r a mm mm R e d li gh t ( F ilt e r) xy z F l o w F I G . ( c o l o r on li n e ) . T h e s e t upo f t h ee xp e r i m e n t . P R L , ( ) P HY S I C A L R E V I E W LETTE R S w ee k e nd i ng29 M A RC H - FIG. 6. Photo-focussing. Individual trajectories of C. reinhardtii flowed within a rectangular tube (a) without and (b) withphototactic illumination (light source from the right). (c,d) Schematic of the combined effect of phototaxis and fluid vorticityon the orientation of cells, leading to photo-focussing in the centre (c) or on the sides (d) of the channel. Adapted from [79]. crobial swimming within a Poiseuille flow [82], modifiedby an effective phototactic torque on the cell, and pro-vided a good qualitative description of the phenomenon,including the existence of a threshold fluid vorticity be-yond which the suspension does not self-focus. The samesystem was later studied numerically for a whole sus-pension of interacting microswimmers [80], where photo-taxis was modelled by the ad hoc reorientation of eachswimmer along the light direction every ∼ η < 1. Best fits were obtained for η = 0 . 25. Although the details of the phototaxis modelused here are not correct, this approach has the advan- tage that the steady state concentration profile can bederived analytically.A systematic theoretical study of photo-focussing ofmicroswimmers in a Poiseuille flow was done by Clarke[81]. He considered a suspension of phototactic cells con-fined between two no-slip horizontal surfaces located at z = ± H/ 2, and subjected to a background fluid veloc-ity profile u = U (1 − z /H ) e x . Unlike the previousstudies, the suspension was uniformly illuminated frombelow. The local distribution function of cell directions, f ( x, p ), satisfied a generalised form of Eq. (4):( u + V ) · ∇ x f + ∇ p · ( f ˙ p ) = D R ∇ p f + D ∇ x f, (18)where D is the translational diffusivity, and p satisfiesEq. (5) with 1 /B = 0. Phototaxis was implementedthrough a light-dependence of the swimming speed givenby model A in [17] (Eq. (13)), which should be appro-priate mostly when the light intensity I ≈ I c . Themodel considered also self-shading within the suspen-sion by modifying the light intensity according to theBeer-Lambert law (Eq. (12)). Within this model, themicroorganisms can be concentrated at specific depthswithin the suspension by adjusting key parameters likethe strength of the flow (defined through the rotationalP´eclet number P e = 2 U/ ( HD R )), the cells’ absorptioncoefficient ( α in Eq. (12)), and the reduced intensity ofthe light source I s /I c (Fig. 7). Unfortunately these stud-ies did not consider the role of bioconvective instabilitiesdue to gravitationally unstable layers induced by photo-0 (a) (b) (c)Clarke2018 FIG. 7. Predicted concentration profiles for phototactic microoganisms within a horizontal Hele-Shaw cell, in presence of ahorizontal background flow and vertical illumination. (a) Weak flow ( P e = 1), strong absorption and I s /I c = 2. (b) Strongflow ( P e = 5), strong absorption, and I s /I c = 2. (c) Strong flow ( P e = 5), strong absorption, and I s /I c = 3. Adapted from[81]. focussing in shear flows. We believe that this will providefertile ground for future investigations. CONCLUSION Vigorous bioconvective flows can easily emerge withina suspension of swimming microorganisms, and play afundamental role in nutrient transport, mixing, and celldistribution within the suspension. This in turn can im-pact on microbial growth and reproduction, and presentsa technological tool to improve the efficiency of bioreac-tors [84]. We believe that a promising avenue for therapid and accurate control of bioconvection is to lever-age the natural response to light of microbial species.However, the possibility to engineer photo-bioconvectionrelies on the ability to predict the phototactic behaviourof individual microorganisms. Unfortunately, our under-standing of this process is still largely incomplete. Light-induced steering in microalgae has been investigated atboth the physiological and biophysical level [34, 60, 85–87], and much is known on the mechanism of light de-tection [88], stimulus relay to the flagella [89] and thebiomechanics that translates the ensuing changes in flag-ellar beating into directional changes of the cell [86, 90].Upon illumination, however, cells often switch dynami-cally between positive and negative taxis, even for a fixedlight source. This seemingly unpredictable behaviour ispossibly linked to the dual role of light as both environ-mental stimulus and energy source, and it highlights theneed to investigate in depth the link between cell motilityand cell metabolism within a holistic approach to photo-taxis. We believe this to be an area prime for substantialdevelopments in the future. Standard microbial photo-bioconvection is currentlyrestricted to motile eukaryotic microalgae, and al-though several species are technologically important (e.g. Dunaliella , Chlamydomonas and Chlorella spp. [9]), thisstill represents a constraint to its general applicabilityin bioreactors. One possible solution is to engineer alight-response within a given species of interest by geneticmodification. For example, light-regulation of swimmingspeed, or photokinesis, has recently been introduced in E. coli , and used to arrange a bacterial population incomplex dynamical patterns of cell concentration in twodimensions [30, 31]. In turn, this should be able to gener-ate bioconvection within bulk cultures illuminated fromabove, providing a first example of genetically engineeredphoto-bioconvection. Yet another possibility could be of-fered in the future by light-driven active colloids [91–93].Currently, they can be made to move microscopic cargo[94, 95] and perform phototaxis [91] in two dimensions.If their density is sufficiently reduced, they could swimin bulk towards a light source and drive bioconvectiveinstabilities even in suspensions of non-motile microor-ganisms.Overall, the possibilities offered by the combination ofphototaxis and bioconvection to control the macroscopicbehaviour of active matter suspensions are only startingto be explored. We look forward to the new and excitingdevelopments that will undoubtedly emerge in the future.AJ, MP and IT gratefully acknowledge support fromThe Leverhulme Trust through grant RPG-2018-345. JAand IT acknowledge the support from the Spanish Min-istry of Economy and Competitiveness (AEI, FEDEREU) grant nos. FIS2016-77692-C2-1-P and CTM-2017-83774-D. 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