Dilute bacterial suspensions 18.S995 - L04 dunkel@mit.edu E.coli (non-tumbling HCB 437) Drescher, Dunkel, Ganguly, Cisneros, Goldstein (2011) PNAS dunkel@math.mit.edu labeled, non-tumbling E. lines coli due as they through a suspenexistence of closed stream to the swam presence of the wall. (H) The flow field of an E. coli “pu presence ofthe the wall.tracer (H) The flow field of an E.measured coli “pusher” decays much when ab dthe by subtracting best-fit dipole from field. The presence of faster, the flagella However, the force sionsince of fluorescent particles. far from it is partially cancelled by the the experimentally flowFor fieldmeasurements of its “puller” image. Results w of itswe “puller” image. d. field Atwalls, distances rfocused <6µ mon theadipole model overestimates the bacterial flow field. (E) Experimentally measured flow to t plane 50 µm from the top and bottom tancesurfaces 2 µm parallel to sample the wall. chamber, (F) Best fitand force-dipole model, (G) residual field. Notewhere the Bacterial fl cell body, th of the recorded ∼ 2 and terabytes of flow 4 (non-tumbling HCB e flowmovie fieldResults ofdata. an E. In coli “pusher” decays much faster, a bacterium swims close thecule surface, fortothe length of th theflow meas fi this data we identified ∼437) 10when rare events when achieved by fitting theByTo measured andminisbest-fit force d decays of cellsBacterial swam in the for > surfaces. 1.5 s. tracking labeled, no flowfocal fieldplane far from resolve the the atspeed variable location fluid tracers in each of the rare events, relating their position of the ced decays of the flow u with cule flow field created by individual bacteria, we tracked gfp- sion of fluor m surfaces. To resolve the minisfield (r >field 8 µm). and labeled, velocity to the position and orientation of the bacterium, disp non-tumbling E. coli as they swam through a suspenof the cell body (Fig. 1D) illustra walls, we fo dividual bacteria, we tracked gfpthe measured and best-fit force dipole field (Fig. 1C). The the specific fitting r and performing an ensemble average over all tracers, we reHowever, sion of fluorescent tracer particles. For measurements far from field displays the characteristic 1/1 dipole length ℓ =of decays of the flow speed u with distance r from thesurfaces center i asministhey the swam through aflow suspensolved time-averaged field in the E. coli swimming he measured walls, we focused on a plane 50 µm from the top and bottom value of F is consis movie data However, the force dipole flow sign oftothe cell (Fig. 1D) illustrate that flow down 0.1% of body the mean swimming speed V0 = 22 ±the 5 measured ticles. For measurements far from ckedplane gfpcell body surfaces of the sample chamber, and recorded terabytes ofresistive 2 ∼ 2their and force t µm/s. As E. coli rotate about their swimming direction, cells swam field displays the characteristic 1/r decay of a force dipole. measured flow to the side of the 4 suspenea 50 µmmovie fromdata. the top and bottom for the be le In this data we identified ∼ 10 rare events when note that in the flow field in three dimensions isbody, cylindrically fluid tracer However, the force dipole flow significantly overestimates the cell where the flow magnitu sber, fartime-averaged from achieved and recorded ∼ 2 terabytes of cells swam in the focal plane forall>components 1.5 s. Byoftracking thebehind the cent µm symmetric. Our measurements capture this measured flow to when the side of the cell body,ofand behind the 4 and velocit didentified bottom for the length the flagellar bund at variab fluid drag on the fla ∼ 10 rare events fluid tracers in each of the rare events, relating their position cylindrically symmetric flow, except the azimuthal flow due to cellof body, where the floworientation magnitude u(r) isof nearly constant and perform abytes of field (r fo > achieved by fitting two opposite and velocity to the position and of the bacterium, the rotation the cell about its body axis. The topology ne for > 1.5fors.theBy tracking the length of the flagellar bundle. The force dipole fitthe was solved the nts when speci the measured flow field (Fig. 1A) is the same as that of a and performing an ensemble average over all tracers, we reat variable locations along the swi are events, relating their position Bacterial flowdown field achieved by fitting two opposite force monopoles (Stokeslets) dipole len plane ckingforce the dipole flow (Fig. 1B), defined by solved the time-averaged flow field in the E. coli swimming field (r > 8 µm). From the best and orientation of the bacterium, dipole flow describ at variable locations along the swimming direction to the far value of E F position µm/s. As plane down to 0.1% of the mean swimming speed V = 22 ± 5 0 the specific fitting routines fit with good and accuracy h i e average over all tracers, we refield (r > 8 µm). From the best fit, which is insensitive to and resist A E. coli r direction, their time-averag ℓF swimming acterium, µm/s. As 2rotate about their ˆ thisµm approximation u(r)in= the 3(r̂.coli d) − 1 r̂, routines A= , r̂fitting = length , regions, [1 ]= we dipole ℓ 1.9 and dipo ow field E. swimming 2 the specific fitting and obtain the note that |r| 8πηdimensions |r| is cylindrically s, we retime-averaged flow field in three symmetric. a wall. Focusing 2 value of F is consistent with optic dipole length 1.9±µm and dipole force F = of 0.42 This µm behin mean swimming speed V0 ℓ==22 5 capture symmetric. Our measurements all components thispN. wimming and applying the s cylindricall and resistive force theory calculatio fluid drag of Fforce, is consistent with optical trapforce measurements [45] where F isvalue the dipole ℓ the distance separating the ut direction, their symmetric flow, except the azimuthal flow due to the = their 22 ±cylindrically 5swimming resulted in a slight rotatio note thatThe in the best fit, the cell d and resistive force theory calculations [46]. It is interesting to pair, η the viscosity the fluid, dˆ the orientation vector theof flow field struct the rotation ofisof the cell about itsunit body axis. topology ion, their three dimensions cylindrically the measur (swimming direction) the best bacterium, and rbehind the distance surfaces, thebfi note thatflow inofthe fit, 1A) theµm cell drag Stokeslet isfrom 0.1 the measured field (Fig. is the same as that oflocated aof the center the cell ndrically nts capture all components of this force dipole Drescher, Dunkel, Ganguly, Cisneros, Goldstein (2011) PNAS Bacterial vector relative to the center of the dipole. Yet there are some ity of a no-slip µm behind the center of thefluid cell body, reflectingbundle. the surf force dipole flow (Fig. 1B), defined by dunkel@math.mit.edu drag possibly on the flagellar nts of this E.coli , except flow bacterium. due to Fig. 1. Averagethe flow fieldazimuthal created by a single freely-swimming (A) Experimentally measured flow field far from a surface. Stream lines indicate local direction of flo dipole Bacterial run & tumble motion movie: V. Kantsler ~20 parts 20 nm Berg (1999) Physics Today Chen et al (2011) EMBO Journal source: wiki movie: V. Kantsler Bacterial run & tumble motion Rep. Prog. Phys. 72 (2009) 096601 E Lauga and T R Powers (a) (b) (c) (d) Figure 15. Bundling of bacterial flagella. During swimming, the bacterial flagella are gathered in a tight bundle behind the cell as it moves through the fluid ((a) and (d)). During a tumbling event, the flagella come out the bundle (b), resulting in a random reorientation of the cell before the next swimming event. At the conclusion of the tumbling event, hydrodynamic interactions lead to the relative attraction of the flagella (c), and their synchronization to form a perfect bundle (d). necessary conditions for phase locking [243–246]. An early are bundled together tightly behind the cell (figure 15(a) for more movies, see also http://www.rowland.harvard.edu/labs/bacteria/movies/index.php dunkel@math.mit.edu n(y) [2N cells] n (y) [N cells] combined with Percoll (2:3 ratio) to match the medium and ion 200 !m). ls by the closest wall. Our model allows us to predict cell buoyant densities [18]. A droplet of the cell mixture is th a constant ulting steady-state cell distribution and is exploited week ending P H Y S I C A L R E V I E W L E T T E R S deposited between two glass cover slips, previously ear the walls 18 JULY 2008 PRL 038102 (2008) ain an estimate for the101, flagellar propulsive force in cleaned in a mixture of ethanol saturated with potassium umanE.spermaming coli. experimental is illustrated Fig. 1. hydroxide, rinsed in with ultrapure water, and allowedby Surfaces nimal sperma-procedure Hydrodynamic Attraction offiltered Swimming Microorganisms (smooth-swimming strain is grown to airHCB-437 dry. The[17]) cover slips are separated by a distance H, 4,* nce of three1 2 2,3 Allison P. Berke, Linda Turner, Howard C. Berg, and Eric Lauga exponential phase in T broth (1% tryptone, 0.5% controlled by layers ofS Iother cover slips (No. 1.5), and Massachusetts 02139, ions of model 1Department week ending of Mathematics, Massachusetts Institute ofETechnology, 77 Mass. Avenue, Cambridge, USA P H Y C A L R V I E W L E T T E R S washed 3PRL times by centrifugation (2200 g for 2 18 JULY 2008 101, 038102 (2008) Rowland Institute at Harvard, 100 Edwin H. Land Boulevard, Cambridge, Massachusetts 02142, USA ] supported an 3 ), and then resuspended a motility medium Department ofinMolecular and Cellular Biology, Harvard University, 16 Divinity Avenue, Cambridge, Massachusetts 02138, USA 4 ynamic interverified by caliperpH measurement. Aofphase-contrast 60 University of California Department Mechanical Aerospace Engineering, 120 H = 100 µm San Diego, M potassium phosphate, 7.0, 0.1 mM EDTA).andmicro9500 Gillman Drive, La Jolla, California 92093-0411, USA scope Optiphot-2) using 600! magnification exp. (N cells) mputations for (Nikon (a) 0 (polyvinylpyrrolidone) is added (0.005%) to pre11 December 2007; published 17 July 2008) theory (depth of field 4:3 !m) and equipped (Received with a shuttered exp. (2N cells) ber swimmers dsorption ofCCD cellsvideo to glass, and the swimming finalElectronics suspension isenvironments Cells in confined the steady-state 80 camera (Marshall V1070) set for are attracted 40 by surfaces. We measure theory distribution of bacteria (Escherichia coli) between two glass plates. In agreement with ned (2:3 ratio) to match thesmooth-swimming medium andpopulation an exposure of 1 ms=frame is used to image the ngewith in Percoll swim- earlier studies,signal we find strong of the cell concentration at the boundaries. We demonstrate of swimming Theof video is asent toisincrease aHMacG4 oyant densities [18]. A cells. droplet the cell mixture theoretically that hydrodynamic interactions of the swimming cells with solid surfaces lead to their 40 equipped with an LG-3 frame grabber (Scion Image) and 20 (b) ted between two glass cover slips, reorientation in the previously direction parallel to the surfaces, as well as their attraction by the closest wall. A of swimming IMAGEJ software [National Institutes Health distribution (NIH), of swimming cells, which compares favorably with our model is derived forpotassium theof steady-state d in a mixture of ethanol saturated with Bethesda, MD]. Wemeasurements. capture 2-second movies 20toframes We exploit our at data estimate the flagellar propulsive force in swimming E. coli. distribution of 0 xide, rinsed per with ultrapure filteredthewater, and allowed cells by 0 second and measure number of swimming 0 PACS20numbers:4047.63.Gd,6087.17.Jj,80 between twoslipscells DOI: 10.1103/PhysRevLett.101.038102 87.18.Ed100 dry. The cover areswimming separated by afaster distance counting at speed than 1H, body length y y (µm) obtain results per second. We start 5 !m above the lower glass surface; lled by layers of other coverofslips (No.microorganisms 1.5), and involved in the cells The(c) majority swimming (d)by the closest wall. Our model allows us to predict n (y) we thenhuman bring the plane and of focus up 10 andinrepeat the ]. We demonthe60resulting steady-state cellµmdistribution and is exploited functions diseases are!m found geometrically H = 200 measurement until we reach within 5 !m of the upper to obtain an estimate for the flagellar propulsive force in confined environments. Spermatozoa in the female reprocell profile is surface. exp. glass Experiments are then repeated with other swimming E. coli. ductive tract1 swim in online). constricted domains [1]. Bacteriaof the FIG. (color Representation experimental protheory cell samples cover slips. uments based Our experimental procedure is illustrated in Fig. 1. make and theirsets wayof through host cells and tissues [2] and 40are mixed with a cedure. (a) Smooth-swimming E. coli cells In our protocol,in two parameters canbiofilms be varied: the dis-[3]. E. coli (smooth-swimming strain HCB-437 [17]) is grown aggregate antibiotic-resistant on surfaces etween swimdensity-matched fluid. (b) The cell mixture is midexponential deposited between tance H between the cover slips (we H "motility 100 or to phase in T broth (1% tryptone, 0.5% Despite the two ubiquitous nature of chose biological near e interactions Htwo 200 !m) and thenot cell density of (separation the finalthemixture, i.e., the H). (c) NaCl), 3 times of by centrifugation (2200 g for surfaces, much is known about physical consequenglass plates distance Thewashed distribution 20 8 min), and then resuspended in a motility medium ces overall of locomotion in a confined environment size of the cell population (when H " 100[4,5]. !m, Perhaps we ection parallel swimming cells is imaged as a function of the distance y from (10 mM potassium phosphate, pH 7.0, 0.1 mM EDTA). the simplest observed effect of locomotion wallsofis the performed additional experiments doubling the near number tial condition the lower surface. (d) Example imagePVP-40 obtained from data is added (0.005%) to pre(polyvinylpyrrolidone) accumulation of swimming oninsurfaces. In an 1963, cells). The experimental results are cells shown Fig. 2; of verti0 adsorption glass, and150 the final 200 suspension is Rothschild measured distribution of bull spermatozoa 0 50of cells to 100 attractioncal of acquisition inthethe first above the glassvent surface. errors bars represent statistics on tenlayer different experiy combined with Percoll y(2:3 ratio) to match the medium and swimming between glass 200in!m). (µm) ments and horizontal errortwo bars theplates depth(separation of field. As cell buoyant densities [18]. A droplet of the cell mixture is cell that distribution was withnear a constant (d)nonuniform, Ref. [6],The we find the cell profile peaks strongly the between Experimental two glass cover slips, previously density in the center strongly increasing near the wallsFIG.deposited 2 (color online). data: number of swimwalls, with a nearly constant cell density about 20 !m 038102-1 2008 The spermaAmerican Physical Society ofofethanol saturated potassium [6]. Similar results were later© obtained for human mingcleaned cells ninasaamixture function the distance to thewith bottom cover Goals • minimal SDE model for microbial swimming • wall accumulation & density profile dunkel@math.mit.edu 1.3 Dilute microbial suspensions A minimalist model for the locomotion of an isolated microorganism (e.g., alga or bacterium) with position X(t) and orientation unit vector N (t) is given by the coupled system of Ito SDEs 1.3 Dilute microbial suspensions p dX = V N dt + 2D ⇤ dB(t), (1.45a) A minimalist model for the locomotion ofT anpisolated microorganism (e.g., alga or bacdNX(t) = and (1 orientation d)DR N dt unit + vector 2DR (IN (t) NN ) ⇤ dWby(t). terium) with position is given the coupled (1.45b) system of Ito SDEs Here, V is the self-swimming speed of the organism, DT the translational di↵usion coefp ficient, and DR dX the rotational di↵usion (I N N ) is an orthogonal projector = V N dt + 2Dcoefficient, (1.45a) T ⇤ dB(t), p with d-dimensional unit matrix I, and B and W are two independent d-dimensional BrowdN =Eq. (1(1.45a) d)DRdescribes N dt + locomotion 2DR (I N Nto ) ⇤translational dW (t). (1.45b) nian motion processes. due di↵usion and self-swimming in the direction of the N , Dandthe Eq.translational (1.45b) models changes in Here, V is the self-swimming speed of orientation the organism, di↵usion coefT orientation on thedi↵usion d-dimensional unit sphere. ficient, and DasR di↵usion the rotational coefficient, (I N N ) is an orthogonal projector To confirm that Eq. (1.45b) conserves the unit length of the orientation vector, |N |2 = 1 with d-dimensional unit matrix I, and B and W are two independent d-dimensional Browfor all t, it is convenient to rewrite Eqs. (1.45) in component form: nian motion processes. Eq. (1.45a) describes locomotion due to translational di↵usion and p orientation N , and Eq. (1.45b) models changes in self-swimming in dX the direction of the 2DT ⇤ dBi (t), (1.46a) i = V Ni dt + p orientation as di↵usion on the d-dimensional unit sphere. dNEq. (1 conserves d)DR Nj dtthe + unit 2Dlength (1.46b) j = R ( jk of N j Norientation k ) ⇤ dWk (t).vector, |N To confirm that (1.45b) the |2 = 1 forFor allthe t, itconstraint is convenient (1.45) we in component |N |2to=rewrite 1 to beEqs. satisfied, must have form: d|N |2 = 0. Applying the dp see Eq. (A.12), to F (N ) = |N |2 , one finds indeed dimensional version of Ito’s formula, dXi = V Ni dt + 2DT ⇤ dBi (t), (1.46a) that p dNj = (1 d)DR Nj dt + 2DR ( jk Nj Nk ) ⇤ dWk (t). (1.46b) d|N |2 = 2Nj ⇤ dNj + @Ni @Nj Nk Nk DR ( ij Ni Nj ) dt i For the constraint |N |2 = 1h to be satisfied, wepmust have d|N |2 = 0. Applying the d(1 d)Dsee dt +(A.12), 2DR to ( jkF (NN)j N ⇤ dW (t) finds + j ⇤ formula, R NjEq. dimensional version= of2N Ito’s =k )|N |2 , kone indeed that @Ni ( jk Nk + Nk jk ) DR ( ij Ni Nj ) dt = 2(1 d)DR dt + d|N |2 = 2N Ni Nj ) dt j ⇤ dNj + @Ni @Nj Nk Nk DR ( ij http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.0020044 h + ik jk ) DR ( ij p i ( jk ik Ni Nj ) dt elf-swimming in the direction , and Eq. (1.45b) models Browchanges in with d-dimensional unit matrixofI,the andorientation B and W areN two independent d-dimensional rientation as di↵usion onEq. the(1.45a) d-dimensional unit sphere. nian motion processes. describes locomotion due to translational di↵usion and Toself-swimming confirm thatinEq. conserves the unit length the(1.45b) orientation the(1.45b) direction of the orientation N , andofEq. modelsvector, changes|N in|2 = 1 BM on the unit sphere as di↵usion the d-dimensional or allorientation t, it is convenient toonrewrite Eqs. (1.45)unit in sphere. component form: 2 To confirm that Eq. (1.45b) conserves the unit length of the orientation vector, |N | =1 p for all t, it isdX convenient to rewrite Eqs. component form: 2DT(1.45) ⇤ dBiin(t), (1.46a) i = V Ni dt + p p dNjdX= (1 d)DR N2D ( jk Nj Nk ) ⇤ dWk (t). (1.46b) j dtT+ ⇤ dB2D (1.46a) i = V Ni dt + i (t),R dNj 2 = (1 p d)DR Nj dt + 2DR ( jk Nj Nk ) ⇤ dW2k (t). (1.46b) or the constraint |N | = 1 to be satisfied, we must have d|N | = 0. Applying the dimensional Ito’s see Eq. we (A.12), to F (N |N Applying |2 , one finds For the version constraintof|N |2 =formula, 1 to be satisfied, must have d|N)|2 = = 0. the d-indeed hat dimensional version of Ito’s formula, see Eq. (A.12), to F (N ) = |N |2 , one finds indeed that d|N |2 =2 2Nj ⇤ dNj + @Ni @Nj Nk Nk DR ( ij Ni Nj ) dt i d|N | = 2Njh⇤ dNj + @Ni @Nj Nk Nk pDR ( ij Ni Nj ) dt = 2Nj ⇤ (1h d)DR Nj dt + p 2DR ( jk Nj Nk ) ⇤ dWik (t) + = 2Nj ⇤ (1 d)DR Nj dt + 2DR ( jk Nj Nk ) ⇤ dWk (t) + @Ni@(N jk( N Ni N k + Nk jk ) DR ( ij N iN j ) dt jk Nk + Nk jk ) DR ( ij j ) dt i = =2(12(1 d)D dtdt++ RR d)D ( jk( jk ) )DDRR(( ijij N NiiNNj )j )dtdt ik +ik ikjkjk ik + = =0. 0. (1.47)(1.47) To understand the dynamics (1.46), it is useful to compute the orientation correlation, To understand the dynamics (1.46), it is useful to compute the orientation correlation hN (t) · N (0)i = E[N (t) · N (0)] = E[Nz (t)], hN (t) · N (0)i = E[N (t) · N (0)] = E[Nz (t)], (1.48) where we have assumed (w.l.o.g.) that N (0) = ez . Averaging Eq. (1.46b), we find that (1.48) = 2Nj ⇤ (1 d)DR Nj dt + 2DR ( jk Nj Nk ) ⇤ dWk (t) + @Ni ( jk Nk + Nk jk ) DR ( ij Ni Nj ) dt = 2(1 d)DR dt + ( jk ik + ik jk ) DR ( ij Ni Nj ) dt = 0. Orientation correlations (1.47) To understand the dynamics (1.46), it is useful to compute the orientation correlation, hN (t) · N (0)i = E[N (t) · N (0)] = E[Nz (t)], (1.48) where we have assumed (w.l.o.g.) that N (0) = ez . Averaging Eq. (1.46b), we find that d E[Nz (t)] = (1 dt d)DR E[Nz (t)], (1.49) implying that, in this model, the memory loss about the orientation is exponential hN (t) · N (0)i = e(1 12 d)DR t , (1.50) Mean square displacement which is approximately true for many microorganisms. Another relevant quantity is the mean square displacement E[X(t)2 ], assuming that X(0) = 0. Using Ito’s formula, d|X|2 = 2Xj ⇤ dXj + @Xi @Xj Xk Xk DT ij dt = 2Xj ⇤ dXj + ( jk ik + ik jk ) DT ij dt = 2Xj ⇤ dXj + 2d DT dt p = 2Xj [V Nj dt + 2DT ⇤ dBj (t)] + 2d DT dt, (1.51) averaging and dividing by dt, gives d E[X 2 ] = 2V E[X(t)N (t)] + 2d DT . dt (1.52) The expectation value on the rhs. can be evaluated by making use of Eq. (1.50): Z t E[X(t) · N (t)] = E dX(s) · N (t) 0 Z t = VE ds N (s) · N (t) 0 Z t = V ds hN (t) · N (s)i Z0 t = V ds e(1 d)DR (t s) 0 ⇥ V = 1 (d 1)DR e (1 d)DR t ⇤ . By inserting this expression into Eq. (1.52) and integrating over t, we find = V 0 Z t ds hN (t) · N (s)i Mean square displacement = V ds e Z0 t (1 d)DR (t s) 0 ⇥ V = 1 (d 1)DR e (1 d)DR t ⇤ . By inserting this expression into Eq. (1.52) and integrating over t, we find 2 ⇥ 2V 2 E[X ] = (d 2 2 (d 1) DR 1)DR t + e (1 d)DR t ⇤ 1 + 2dDT t. (1.53) If DT is small, then at short times t ⌧ DR 1 the motion is ballistic E[X 2 ] ' V 2 t2 + 2dDT t, (1.54) At large times, the motion becomes di↵usive, with asymptotic di↵usion constant E[X 2 ] 2V 2 lim = + 2dDT . t!1 t (d 1)DR (1.55) Inserting typical values for bacteria, V ⇠ 10µm/s and DR ⇠ 0.1/s, and comparing with DT ⇠ 0.2µm2 /s for a micron-sized colloids at room temperature, we see that active swimming and orientational di↵usion dominate the di↵usive dynamics of microorganisms at long times. 13 n(y) [2N cells] n (y) [N cells] combined with Percoll (2:3 ratio) to match the medium and ion 200 !m). ls by the closest wall. Our model allows us to predict cell buoyant densities [18]. A droplet of the cell mixture is th a constant ulting steady-state cell distribution and is exploited week ending P H Y S I C A L R E V I E W L E T T E R S deposited between two glass cover slips, previously ear the walls 18 JULY 2008 PRL 038102 (2008) ain an estimate for the101, flagellar propulsive force in cleaned in a mixture of ethanol saturated with potassium umanE.spermaming coli. experimental is illustrated Fig. 1. hydroxide, rinsed in with ultrapure water, and allowedby Surfaces nimal sperma-procedure Hydrodynamic Attraction offiltered Swimming Microorganisms (smooth-swimming strain is grown to airHCB-437 dry. The[17]) cover slips are separated by a distance H, 4,* nce of three1 2 2,3 Allison P. Berke, Linda Turner, Howard C. Berg, and Eric Lauga exponential phase in T broth (1% tryptone, 0.5% controlled by layers ofS Iother cover slips (No. 1.5), and Massachusetts 02139, ions of model 1Department week ending of Mathematics, Massachusetts Institute ofETechnology, 77 Mass. Avenue, Cambridge, USA P H Y C A L R V I E W L E T T E R S washed 3PRL times by centrifugation (2200 g for 2 18 JULY 2008 101, 038102 (2008) Rowland Institute at Harvard, 100 Edwin H. Land Boulevard, Cambridge, Massachusetts 02142, USA ] supported an 3 ), and then resuspended a motility medium Department ofinMolecular and Cellular Biology, Harvard University, 16 Divinity Avenue, Cambridge, Massachusetts 02138, USA 4 ynamic interverified by caliperpH measurement. Aofphase-contrast 60 University of California Department Mechanical Aerospace Engineering, 120 H = 100 µm San Diego, M potassium phosphate, 7.0, 0.1 mM EDTA).andmicro9500 Gillman Drive, La Jolla, California 92093-0411, USA scope Optiphot-2) using 600! magnification exp. (N cells) mputations for (Nikon (a) 0 (polyvinylpyrrolidone) is added (0.005%) to pre11 December 2007; published 17 July 2008) theory (depth of field 4:3 !m) and equipped (Received with a shuttered exp. (2N cells) ber swimmers dsorption ofCCD cellsvideo to glass, and the swimming finalElectronics suspension isenvironments Cells in confined the steady-state 80 camera (Marshall V1070) set for are attracted 40 by surfaces. We measure theory distribution of bacteria (Escherichia coli) between two glass plates. In agreement with ned (2:3 ratio) to match thesmooth-swimming medium andpopulation an exposure of 1 ms=frame is used to image the ngewith in Percoll swim- earlier studies,signal we find strong of the cell concentration at the boundaries. We demonstrate of swimming Theof video is asent toisincrease aHMacG4 oyant densities [18]. A cells. droplet the cell mixture theoretically that hydrodynamic interactions of the swimming cells with solid surfaces lead to their 40 equipped with an LG-3 frame grabber (Scion Image) and 20 (b) ted between two glass cover slips, reorientation in the previously direction parallel to the surfaces, as well as their attraction by the closest wall. A of swimming IMAGEJ software [National Institutes Health distribution (NIH), of swimming cells, which compares favorably with our model is derived forpotassium theof steady-state d in a mixture of ethanol saturated with Bethesda, MD]. Wemeasurements. capture 2-second movies 20toframes We exploit our at data estimate the flagellar propulsive force in swimming E. coli. distribution of 0 xide, rinsed per with ultrapure filteredthewater, and allowed cells by 0 second and measure number of swimming 0 PACS20numbers:4047.63.Gd,6087.17.Jj,80 between twoslipscells DOI: 10.1103/PhysRevLett.101.038102 87.18.Ed100 dry. The cover areswimming separated by afaster distance counting at speed than 1H, body length y y (µm) obtain results per second. We start 5 !m above the lower glass surface; lled by layers of other coverofslips (No.microorganisms 1.5), and involved in the cells The(c) majority swimming (d)by the closest wall. Our model allows us to predict n (y) we thenhuman bring the plane and of focus up 10 andinrepeat the ]. We demonthe60resulting steady-state cellµmdistribution and is exploited functions diseases are!m found geometrically H = 200 measurement until we reach within 5 !m of the upper to obtain an estimate for the flagellar propulsive force in confined environments. Spermatozoa in the female reprocell profile is surface. exp. glass Experiments are then repeated with other swimming E. coli. ductive tract1 swim in online). constricted domains [1]. Bacteriaof the FIG. (color Representation experimental protheory cell samples cover slips. uments based Our experimental procedure is illustrated in Fig. 1. make and theirsets wayof through host cells and tissues [2] and 40are mixed with a cedure. (a) Smooth-swimming E. coli cells In our protocol,in two parameters canbiofilms be varied: the dis-[3]. E. coli (smooth-swimming strain HCB-437 [17]) is grown aggregate antibiotic-resistant on surfaces etween swimdensity-matched fluid. (b) The cell mixture is midexponential deposited between tance H between the cover slips (we H "motility 100 or to phase in T broth (1% tryptone, 0.5% Despite the two ubiquitous nature of chose biological near e interactions Htwo 200 !m) and thenot cell density of (separation the finalthemixture, i.e., the H). (c) NaCl), 3 times of by centrifugation (2200 g for surfaces, much is known about physical consequenglass plates distance Thewashed distribution 20 8 min), and then resuspended in a motility medium ces overall of locomotion in a confined environment size of the cell population (when H " 100[4,5]. !m, Perhaps we ection parallel swimming cells is imaged as a function of the distance y from (10 mM potassium phosphate, pH 7.0, 0.1 mM EDTA). the simplest observed effect of locomotion wallsofis the performed additional experiments doubling the near number tial condition the lower surface. (d) Example imagePVP-40 obtained from data is added (0.005%) to pre(polyvinylpyrrolidone) accumulation of swimming oninsurfaces. In an 1963, cells). The experimental results are cells shown Fig. 2; of verti0 adsorption glass, and150 the final 200 suspension is Rothschild measured distribution of bull spermatozoa 0 50of cells to 100 attractioncal of acquisition inthethe first above the glassvent surface. errors bars represent statistics on tenlayer different experiy combined with Percoll y(2:3 ratio) to match the medium and swimming between glass 200in!m). (µm) ments and horizontal errortwo bars theplates depth(separation of field. As cell buoyant densities [18]. A droplet of the cell mixture is cell that distribution was withnear a constant (d)nonuniform, Ref. [6],The we find the cell profile peaks strongly the between Experimental two glass cover slips, previously density in the center strongly increasing near the wallsFIG.deposited 2 (color online). data: number of swimwalls, with a nearly constant cell density about 20 !m 038102-1 2008 The spermaAmerican Physical Society ofofethanol saturated potassium [6]. Similar results were later© obtained for human mingcleaned cells ninasaamixture function the distance to thewith bottom cover ‘Hydrodynamic’ fields Concentration profile between two walls An interesting question that is relevant from a medical perspective concerns the spatial distribution of bacteria and other swimming microbes in the presence of confinement. Restricting ourselves to dilute suspensions10 , we may obtain a simple prediction from the model (1.45) by considering the FPE for the associated PDF p(t, x, n). Given p and the total number of bacteria Nb in the solutions, we obtain the spatial concentration profile by integrating over all possible orientations Z c(t, x) = Nb dn p(t, n, x). (1.56a) Sd The associated mean orientation field reads Z u(t, x) = Nb dn p(t, n, x) n. (1.56b) Sd The FPE for the Ito-SDE (1.45) can be written as a conservation law @t p = (@xi Ji + @ni ⌦i ), (1.57a) where Ji = (V ni DT @xi )p ⌦i = DR (1 d)ni p @nj [( ij ni nj )p] . (1.57b) (1.57c) Focusing on the three-dimensional case, d = 3, we are interested in deriving from Eq. (1.57) the stationary concentration profile c of a suspension that is confined by two quasi-infinite Sd The associated mean orientation field reads Z u(t, x) = Nb dn p(t, n, x) n. Concentration field (1.56b) Sd The FPE for the Ito-SDE (1.45) can be written as a conservation law @t p = (@xi Ji + @ni ⌦i ), (1.57a) where Ji = (V ni DT @xi )p ⌦i = DR (1 d)ni p @nj [( ij ni nj )p] . (1.57b) (1.57c) Focusing on the three-dimensional case, d = 3, we are interested in deriving from Eq. (1.57) the stationary concentration profile c of a suspension that is confined by two quasi-infinite parallel walls, which are located z = ±H. That is, we assume that the distance between the walls is much smaller then their spatial extent in the (x, y)-directions, 2H ⌧ Lx , Ly . To obtain an evolution equation for c, we multiply Eq. (1.57a) by Nb and integrate over n with Z dn @ni ⌦i = 0. (1.58) Sd This yields the mass conservation law @t c = r · (V u DT rc). (1.59) To obtain also an evolution equation for u, we multiply Eq. (1.57a) by nk , @t (nk p) = @xi (nk Ji ) n k @ n i ⌦i . (1.60) b where Sd The associated meanJorientation field DTreads @xi )p (1.57b) i = (V ni Z ⌦i = DR (1 d)ni p @nj [( ij ni nj )p] . (1.57c) u(t, x) = Nb dn p(t, n, x) n. (1.56b) Sd Focusing on the three-dimensional case, d = 3, we are interested in deriving from Eq. (1.57) the stationary profilecan c ofbe a suspension is confinedlaw by two quasi-infinite The FPE for concentration the Ito-SDE (1.45) written as athat conservation parallel walls, which are located z = ±H. That is, we assume that the distance between @t p spatial = (@ ⌦i ),(x, y)-directions, 2H ⌧ Lx(1.57a) xi Ji +in@nthe the walls is much smaller then their extent , Ly . i To obtain an evolution equation for c, we multiply Eq. (1.57a) by Nb and integrate over n where with Z Ji = (V ni DT @xi )p (1.57b) dn @ni ⌦i = 0. (1.58) ⌦i = DR Sd(1 d)ni p @nj [( ij ni nj )p] . (1.57c) Orientation (velocity) field This yields on thethe mass conservation law Focusing three-dimensional case, d = 3, we are interested in deriving from Eq. (1.57) the stationary concentration profile c of a suspension that is confined by two quasi-infinite @t c z== ±H. r · (V u is, DTwe rc). (1.59) parallel walls, which are located That assume that the distance between the is much thenequation their spatial in the Eq. (x, y)-directions, Towalls obtain also ansmaller evolution for u,extent we multiply (1.57a) by nk ,2H ⌧ Lx , Ly . To obtain an evolution equation for c, we multiply Eq. (1.57a) by Nb and integrate over n with @t (nk p) = @xi (nk Ji ) nk @ni ⌦i . (1.60) Z and note that dn @ni ⌦i = 0. (1.58) Sd nk @ni ⌦i = @ni (nk ⌦i ) (@ni nk )⌦i = @ni (nk ⌦i ) This yields the mass conservation law ik ⌦i . 10 (1.61) The simplifying assumption that microbes can be considered as non-interacting is usually justified when their volume filling fraction is less @t cthan = 1%. r · (V u DT rc). (1.59) To obtain also an evolution equation for 14 u, we multiply Eq. (1.57a) by nk , This yields the mass conservation law Orientation @c = r(velocity) · (V u D rc). field t T (1.59) To obtain also an evolution equation for u, we multiply Eq. (1.57a) by nk , @t (nk p) = @xi (nk Ji ) n k @ n i ⌦i . (1.60) and note that nk @ni ⌦i = @ni (nk ⌦i ) 10 (@ni nk )⌦i = @ni (nk ⌦i ) ik ⌦i . (1.61) The simplifying assumption that microbes can be considered as non-interacting is usually justified This their allows us tofilling rewrite (1.60) as than 1%. when volume fraction is less @t (nk p) = = @xi (nk Ji ) + ⌦k @ni (nk ⌦i ) 14 @xi [V nk ni p DT @xi (nk p)] + DR 2nk p @nj [( kj nk nj )p] @ni (nk ⌦i ) = @xi [V nk ni p DT @xi (nk p)] 2DR nk p @nj (nk ⌦j + ( kj nk nj )p). (1.62) Multiplying by Nb and integrating over n with appropriate boundary conditions gives @ t uk = @xi [V Nb hnk ni ip DT @xi uk ] 2DR uk , where we have abbreviated hni nk · · ·i = Z Sd dn p(t, n, x) ni nk · · · . (1.63) To obtain a closed linear system of equations for the fields (c, u), we neglect11 the higherorder moments Nb hnk ni i in (1.63) and find @t u ' 2DR u + DT r2 u. (1.64) This yields the mass conservation law Orientation @c = r(velocity) · (V u D rc). field t T (1.59) To obtain also an evolution equation for u, we multiply Eq. (1.57a) by nk , @t (nk p) = @xi (nk Ji ) n k @ n i ⌦i . (1.60) and note that nk @ni ⌦i = @ni (nk ⌦i ) 10 (@ni nk )⌦i = @ni (nk ⌦i ) ik ⌦i . (1.61) The simplifying assumption that microbes can be considered as non-interacting is usually justified This their allows us tofilling rewrite (1.60) as than 1%. when volume fraction is less @t (nk p) = = @xi (nk Ji ) + ⌦k @ni (nk ⌦i ) 14 @xi [V nk ni p DT @xi (nk p)] + DR 2nk p @nj [( kj nk nj )p] @ni (nk ⌦i ) = @xi [V nk ni p DT @xi (nk p)] 2DR nk p @nj (nk ⌦j + ( kj nk nj )p). (1.62) Multiplying by Nb and integrating over n with appropriate boundary conditions gives @ t uk = @xi [V Nb hnk ni ip DT @xi uk ] 2DR uk , where we have abbreviated hni nk · · ·i = Z Sd dn p(t, n, x) ni nk · · · . (1.63) To obtain a closed linear system of equations for the fields (c, u), we neglect11 the higherorder moments Nb hnk ni i in (1.63) and find @t u ' 2DR u + DT r2 u. (1.64) Z Stationary profiles hni nk · · ·i = Sd dn p(t, n, x) ni nk · · · . (1.63) To obtain a closed linear system of equations for the fields (c, u), we neglect11 the higherorder moments Nb hnk ni i in (1.63) and find @t u ' 2DR u + DT r2 u. (1.64) To find the stationary density and orientation profiles, we look for solutions of the form c = ⇢(z) and ux = uy = 0, uz = u(z). According to Eqs. (1.59) to (1.63), the functions ⇢ and uz must satisfy 0 = V u DT c 0 , 0 = 2DR u + DT u00 , (1.65) (1.66) and it is physically plausible that they also fulfill the symmetry12 requirements ⇢(z) = ⇢( z) and u(z) = u( z). Hence, solution takes the form u(z) = A sinh(z/⇤), V⇤ ⇢(z) = A [cosh(z/⇤) DT (1.67a) 1] + ⇢0 , (1.67b) p where ⇤ = D? /(2DR ). The cosh-profile (1.67b) agrees qualitatively with experimental measurements for dilute bacterial suspensions [BTBL08, LT09]. 11 Ad hoc simplifications of this type are usually referred to as ‘truncation (of the moment hierarchy)’ or ‘closure conditions’ - they are (almost) always unavoidable when one tries to derive continuum equations from ODEs, SDEs or FPEs. Closure conditions are not unique, for example, we could also have adopted the mean field approximation Nb2 hnk ni i ' uk ui , which leads to a nonlinear set of equations for (c, u). 12 Neglecting gravity is valid approximation, provided the density of the microbes roughly matches that But … E. coli RT (χ2 = 0.018) E. ecoli (smooth) P. aeruginosa RT (χ2 = 0.016) B. subtilis RT (χ2 = 0.092) BD (χ2 = 0.021) BD (χ2 = 0.026) BD (χ2 = 0.024) BD (χ2 = 0.13) RT (χ2 = 0.074) ρ 1 0.5 ρ± /ρ 0 1 Density profiles seem ok … what about fluxes? 0.5 0 0 5 10 0 (z − zm )(µm) 5 (z − zm )(µm) 10 0 5 (z − zm )(µm) 10 0 5 (z − zm )(µm) joint work with Peter Lu, Rik Wensink & Jeff Guasto 10 But … E. coli RT (χ2 = 0.018) E. ecoli (smooth) P. aeruginosa RT (χ2 = 0.016) B. subtilis RT (χ2 = 0.092) BD (χ2 = 0.021) BD (χ2 = 0.026) BD (χ2 = 0.024) BD (χ2 = 0.13) 5 5 5 5 RT (χ2 = 0.074) ρ 1 0.5 ρ± /ρ 0 1 0.5 0 0 (z − zm )(µm) 10 0 (z − zm )(µm) 10 0 (z − zm )(µm) 10 0 (z − zm )(µm) 10 rotation translation PT 2 tD rotation PR translation PT time ⌧ = tv0 /` û =PR DR (T t + û (1) bw ⇥ û) R G Laboratoire de Physique des Solides, Université Paris-Sud & CNRS, UMR 8502, 91405 Orsay, France Let us start with the coupled Langevin equations for September 4, 2014) 0.04 ± 0.005 0.1 ± 0.01 0.11 ± 0.005 0.31 ± 0.0051 = k(Dated: with thermal energy. The bacteriummensionless form B T the the center-of-mass position r and orientation vector û of We propose two simple statistical models to describe the density and orientation profiles of E. Coli wall forces emerge from a simple segment potential, 0.02 ± 0.005 collisions 0.07 ± are 0.005 0.07 and ± 0.001 0.34 0.001 and other ± swimming bacteria near aP flat hard wall. The models focus on purely Brownian dynamics Mutual bacteria neglected contribu`/2 a swimming bacterium, linearized in time: versus explicit ‘run and tumble’ (RT) dynamics of non-interacting bacteria near a flat structure2 F = r + ↵û|) with Yukawa form 1 0.14 ± 0.05 0.02 ± 0.005 0.10 ± 0.001 0.09 ± 0.001 bw r ons of O( t ) are negligible for small enough time disless substrate. Using best fit values for ↵= the e↵ective translational and rotational di↵usivities both `/2 u(|r ` r =P models generate similar density profiles – all in quantitative agreement with the experimental ones retizations t. Fbw (Tbw denotes 0.04 the force (torque) u(x) =di↵erent u0 exp( x/ )/x. amplitude u0 isInimmaterial 0.02 ± 0.005 0.08 ± )0.005 ± 0.001 0.005 –0.49 but yield± distinctly stationary density fluxesThe with respect to the wall normal. global p 1 Need to include run & tumbling comparison, the RT dynamics provides a much more appropriate description of the non-equilibrium xerted by the flat, repulsive wall r on=the bacterium, v whereas the screening range can be tuned to match the 0 v of bacteria close to2 confining substrates. Compared to purely Brownian dynamics it turns DT · Fbw t structure + û t + tD r 0 T G =P out that run-and-tumble dynamics generates qualitatively di↵erent stationary orientational wall or-û100). the swim velocity and DR and DT = DT I are the efpeak position near the wall (z ⇡ 1µm, `/ ⇡ 80 m p der with distinct anti-nematic features and a narrow boundary layer of bacteria swimming towards the wall. Eq. (2) then permits a two-parameter (PT , PR ) best fit ctive rotational and translationalûbacterial di↵usivities = DR (T 2 tDR ûG (1) bw ⇥ û) t + 1 range gnoring the intrinsic k / ? contributions for simplicof the bacterial density profiles overwith the spatial = Pécle kB DYNAMICS rotation: I. BROWNIAN DYNAMICS (typical) bacterium length ` to define the inverse 1in the fitted 1 density proy). rG represents a random Gaussian displacement zm < z < 10µm. Error-bars numbers PT = DT /v0 ` and PR = DR `/v0 and rescale wall forces em timecomparing ⌧ = tv0 /` to recast the equations of motion in d n 3D Cartesian space whereas a Gaussian are estimated from results for the bare Let us start with thefiles coupled Langevin equations for Mutual 1ûbacteria collisions are neglected and contribuG denotes P Fbw the center-of-mass position r and orientation vector û of mensionless form: 2 r =tvector ⌧a negligible isplacement of the orientational unit the surbacterium aspect of the bacterial body Fto =q of an r bwthose refswimming bacterium, linearized in time: )Faareon for small enough time disangle ✓ between thetions `of O( 1 1 1 ` r = PT `Fbw ⌧ + û ⌧ + 2 ⌧ PT rG ce of the unit sphere. Both stochastic variables have fective hydrodynamic aspect ratio ae↵ based on p q the total ✓ ◆ ✓ ◆ he bacterium duringcretizations t. Fbw (T ) denotes the force (torque) u(x) = u0I). exp( 1 1 =bw DT · Fbw t + v0 û t + 2 tDT rG û = PR ( Tbw ⇥ û) Table ⌧ + 2 ⌧P ûG (2 Tbw û and the flagellum R ero mean and unit variance. It is expedient to use rthe lengthpof the body (see BD: (SF-SDE) RT: we assume this angleexerted by û = ⇠ flat, repulsive ⇥ û ⌧ +⇥wall ⇥ ⌧v0 û = D (T û) t +ûon 2 tD the û (1) ⇥ û the bacterium, whereas the scre with = k T the thermal energy. The bacterium Fa ` ⌧tumble wall forces emerge from a simple segment potentia bacteria neglected is the swim velocityMutual and DRcollisions andareD Dcontribuefpeak position ne P T =and T I areF the (6) = r u(|r + ↵û|) with Yukawa form tions of O( t ) are negligible for small enough time disR bw R G 1 2 B r bw `/2 ↵= `/2 cretizations t. F (T ) denotes the force (torque) u(x) = u exp( x/ )/x.Eq. The amplitude u is immateria fective rotational and translational bacterial di↵usivities (2) then pe by the flat, repulsive wall on used the bacterium, v fit. whereas the screening range can be tuned to match th TABLE I: Mainexerted bacterial parameters for the ) (3)(ignoring is the swim D and(✓ D =, D I are thefor efpeakAway position near the wall ⇡ 1µm, `/ ⇡ 80 100 the intrinsic k velocity /û?=andcontributions simplicof (zthe bacterial with tumbling vector û ' ) û(0). new T R fective rotational and translational bacterial di↵usivities Eq. (2) then permits a two-parameter (P , P ) best fi culture ` (µm) a (ae↵ ) v0 (µm/s) ✓T ttumble (s) trun (s) 2 (ignoring the intrinsic k / ?exhibits contributions zero for simplicof the bacterial densityzprofiles overzthe< spatial rang ity). r represents a random Gaussian displacement < 10µm from the wall, each bacterium net translaG m E. coli ⇠ 3 ⇠ity). 2(5.6) ⇠ a20 68 displacement ⇠ 0.1 z < z < ⇠ 1 Error-bars in the fitted density pro r represents random Gaussian 10µm. 3D Cartesian space whereas a Gaussian files are estimated fromfiles comparing results for the bar tion and ⇠û3reorients linearly a forward-directed in 3D Cartesian space whereas ûGû denotes a0 Gaussian are estima tified from observaE. coli (smooth) ⇠in 2(5.6) ⇠ 20towards 0denotes 1 displacement of the orientational unit vector on the surbacterium aspect of the bacterial body to those of an e cone parameterized byunit(✓sphere. Near the wall bacof the stochastic variables have fective hydrodynamic ratio a based on the tota T⇠, ' R ).unit displacement orientational vector on surbacterium aspec B. subtilis ⇠of 5 the ⇠face 6(11.5) 50Both ⇠ 40 ⇠ 0.1the ⇠ 0.5 aspect of tumbling bactezero mean and unit variance. It is expedient to use the length of the body and the flagellum (see Table I). terial dynamics is governed by a nontrivial interplay beTABLE II: Best fit parameters for the intrinsic bacterial di↵usivities. Errors are estimated from di↵erent fit results P. aeruginosa ⇠ 2 ⇠ 4(9.8) ⇠ 40 ⇠ 110 ⇠ 0.1 ⇠ 0.5 gle appears stronglyface of the unit sphere. Both stochastic variables have fective hydrodyn or and ae↵ we draw byistheexpedient wall and ato ‘tumbling’ bw exerted It In aaddition zero tween mean the andtorque unit Tvariance. use the length of the bo TABLE I: Main bacterial parameters used for the fit. bw bw 0 0 0 T R T m T G R m G e↵ torque Ttumble ⇠ ⇠RRT ✓T / ttumble . ` (µm) a (a ) vBD distribution on the culture (µm/s) ✓ t (s) t (s) 1 1 1 ⇠ 20 1 E. coli ⇠ 3 ⇠ 2(5.6) 68 ⇠ 0.1 ⇠ 1 The above can be easily implemented for N in- PT rotationscheme PR translation PT rotation PR translation acterium unit vector E. coli (smooth) ⇠ 3 ⇠ 2(5.6) ⇠ 20 0 0 1 E. coli 0.04 ± 0.005 0.1 ± 0.01 0.11 ± 0.005 0.31 ± 0.005 dependent bacteria with random initial B. subtilis ⇠ 5 conditions. ⇠ 6(11.5) ⇠ 50 Time ⇠ 40 ⇠ 0.1 ⇠ 0.5 e orthonormal frame aeruginosa ⇠ 2 ⇠ 4(9.8) 40 ⇠ 110 ⇠ 0.5 E. coli (smooth) 0.02 0.005‘run’ 0.07stage ± P. 0.005 0.07 ± 0.001 ± 0.001⇠ 0.1 intervals for± the are drawn from a⇠0.34 Poisson TABLE I: 0.001 Main 0.09 bacterial parameters used fo 1 0.10 ± B. subtilis 0.14 ± 0.05 0.02 ± 0.005 ± 0.001 distribution with ⌧run = ⇤ ln(1 U ) with U a ranP. aeruginosa 0.02 ± from 0.005 a 0.08 ± 0.005 0.04 ± 0.001 0.49 ± 0.005 culture ` (µm) a (a ✓T dom number uniform distribution in the interval e↵ ) v0 (µm/s) 'R e2 + cos ✓T û (4) [0, 1] and ⇤ = f `/v0 a E. dimensionless tumbling coli ⇠ 3 ⇠frequency. 2(5.6) ⇠ 20 68 wensink@lps.u-psud.fr e↵ 0 T tumble run Need to include run & tumbling E. coli RT (χ2 = 0.018) E. ecoli (smooth) P. aeruginosa RT (χ2 = 0.016) B. subtilis RT (χ2 = 0.092) BD (χ2 = 0.021) BD (χ2 = 0.026) BD (χ2 = 0.024) BD (χ2 = 0.13) 5 5 5 5 RT (χ2 = 0.074) ρ 1 0.5 ρ± /ρ 0 1 0.5 0 0 (z − zm )(µm) 10 0 (z − zm )(µm) 10 0 (z − zm )(µm) 10 0 (z − zm )(µm) 10