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Flexibility of bacterial flagella in external shear results in complex
swimming trajectories
M. Tournus1, A. Kirshtein1, L. V. Berlyand1, and I. S. Aranson2
1Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802, USA2Materials Science Division, Argonne National Laboratory, 9700 S.Cass Avenue, Argonne, IL 60439, USA
Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, IL 60202
Email : [email protected]
Abstract
Many bacteria use rotating helical flagella in swimmingmotility. In search for food or migration towards a newhabitat, bacteria occasionally unbundle their flagellar fil-aments and tumble, leading to an abrupt change in di-rection. Flexible flagella can also be easily deformed byexternal shear flow, leading to complex bacterial trajecto-ries. Here we examine the effects of flagella flexibility onthe navigation of bacteria in two fundamental shear flows:planar shear and Poiseuille flow realized in long channels.On the basis of slender body elastodynamics and numer-ical analysis, we discovered a variety of nontrivial effectsstemming from the interplay of self-propulsion, elasticity,and shear-induced flagellar bending. We show that in pla-nar shear flow the bacteria execute periodic motion, whilein Poiseuille flow they migrate towards the center of thechannel or converge toward a limit cycle. We also findthat even a small amount of random reorientation can in-duce a strong response of bacteria leading to the overallnon-periodic trajectories. Our findings exemplify the sen-sitive role of flagellar flexibility and shed new light on thenavigation of bacteria in complex shear flows.
1 Introduction
Bacteria are among the oldest and simplest living organ-isms on Earth. Bacterial activity influences the planetsenvironmental dynamics in multiple ways, from maintain-ing soil structure to controlling the biochemistry and pho-tosynthetic productivity of the oceans [1]. To search forfood or populate new territories, bacteria often migrateen mass over large distances. This collective behavior isknown as swarming motility. Besides many obvious evolu-tionary advantages, collective behavior appears to be alsoan effective strategy to prevail against antibiotics [2].
The flow produced in dense bacterial colonies in thecourse of swarming can be very complex because of the in-teraction between the bacteria and the fluid [3, 4, 5, 6, 7].While the flow might visually resemble the turbulent mo-
tion emerging in rapidly stirred fluids, there is a funda-mental difference: in hydrodynamic turbulence the me-chanical energy is injected at the macroscopic scale, e.g.,by stirring the liquid. In contrast, in “bacterial turbu-lence”, the energy is injected at the microscopic scale bythe rotation of helical bacterial flagella, which makes itsphysical properties deeply distinguished from the turbu-lent flow of liquid. In particular, the scale of large vorticesgenerated by the collective bacterial locomotion does notdepend on the energy injection rate [5, 6]. This complexphenomena, arising due to the intricate interplay betweenfluid motion and bacterial motility, are difficult to charac-terize experimentally. Despite significant recent progressin modeling of bacterial collective behavior [8, 9, 10, 11],a predictive model that describes multiple aspects of bac-terial turbulence (see for example [12]) has not emergedto date.
Many bacteria use rotating helical flagella for motilityand for periodic reorientation, e.g., in response to chemi-cal gradients (chemotaxis) [13]. Peritrichously-flagellatedbacteria, such as common B subtilis or E coli have multi-ple flagellar filaments distributed over its entire bacterialbody. In the course of swimming the filament bundlesare spun together. However, bacteria unbundle the fila-ments and tumble, leading to abrupt reorientation. Thetransition between the two phases (coiled and unbundle)is triggered by the reversal of a driving motor torque[14].The flagellar filaments are typically at least twice longerthat the bacterial body, and are very flexible, suggestingthat they can be easily deformed either by the externalshear flow or by the flows of other bacteria. In combi-nation with self-propulsion, the bending of flagella couldresult in a significant effect on bacterial rheotaxis, i.e. apropensity of microorganism to turn into an oncomingcurrent. It was shown, e.g., that uni-flagellated bacteriaexploit flagella buckling to change direction [15, 16]. Inaddition, flagellum flexibility possibly affects the rheotac-tic behavior of sperm cells [17, 18, 19].
Recently, the trajectories of self-propelled ellipsoidalrigid swimmers in a 2D Poiseuille flow have been explored.Since the proximity of the boundary can significantly af-fect bacterial rheotaxis [20, 21], the swimmer is assumedto stay far enough from the walls to exclude those interac-
1
tions. The propulsion force enables the swimmer to moveacross the streamlines. The main result is that a self-propelled swimmer in a two-dimensional channel exhibitsperiodic stable oscillations around the centerline [22, 23].These periodic trajectories are strongly influenced by ran-dom reorientations (tumbling) [24] and the proximity ofrigid walls [25, 26]. Bacterial transport appears to be sup-pressed in shear Poiseuille flow, leading to the depletion ofswimmers at the center of the channel [27]. However, theresponse of bacterial flagella to an applied shear remainspoorly understood.
A variety of specific models for helical flagellum havebeen derived and studied numerically. The pioneeringwork in that direction is due to [28], who derived fromfirst principles the flow around a moving helix, calculatedthe associated force and torque, and determined therebythe motion of the helix attached to a large body. In orderto understand the propulsion, [29] examined the linear re-lationships between forces and torques and between trans-lational and angular velocities of helical objects. Specificmodels were developed for microswimmers. For example,the rotational dynamics of a superhelix towed in a Stokesfluid was studied in [30], and a recent review can be foundin [31].
It is also important to distinguish between differentmechanisms leading to the alignment and reorientationof microswimmers. Some are due to biological mecha-nisms, such as chemotaxis [13], whereas others originatedue to pure hydrodynamic effects, as a combined effectof the surrounding fluid flow and the specific shape ofthe body considered. Here we focus on the latter mech-anisms. There is also a significant body of works dealingwith the response of microswimners to an external shearflow (rheotaxis), we will mention just a few for review.The study was pioneered in Ref. [17] which pointed outthat spermatozoa tend to align in the flow direction, andthat this phenomenon is purely due to passive hydrody-namic effects. In [32] the gyrotaxis effect in biflagellatedswimming algae was described. Ref. [33] demonstratedthat the differences in drag forces between body and flag-ella lead to preferential alignment with the flow. Recentexperiments suggest that rheotaxis may significantly af-fect the navigation of mamalian sperm cells [18, 19]. Thenontrivial rheotactic response of B. subtilis bacteria dueto chirality of flagella was predicted in [31].
Here, in order to examine the complex response of bac-teria in an external shear, we model the swimmer as a flex-ible entity consisting of a rigid ellipsoidal body attached toa flexible flagellum. The trajectory of the swimmer is rep-resented by the center of mass of the body. To simplify theanalysis and exclude side wall effects, we consider bacte-ria swimming in wide channels. Using elastic slender bodydynamics in Stokes flow, we have shown that bacteria canexhibit complex trajectories due to the combined effectsof flagellar flexibility and self-propulsion. This behavioris fundamentally different from that predicted for passiveellipsoidal particles [34] that move along streamlines andwhose orientation is described by Jeffery orbits [22]. Wehave examined the behavior of bacteria in two external
shear flows: planar shear between two flat walls movingin opposite directions and two-dimensional Poiseuille flowrealized in long flat channels. We have shown that in thecase of planar shear flow the bacteria execute periodic or-bits, with both the period and the amplitude determinedby the shear rate, the shape of the swimmer, and theflexibility of the flagella. For the case of Poiseuille flowwe have found that for a realistic range of the parame-ter values, the swimmer migrates towards the centerlineof the channel and swims against the flow. Moreover,for softer flagella, our analysis indicates a different trend:the swimmer executes a periodic motion – a limit cycle.The amplitude of the oscillations around the centerlinedepends on the flexibility of the flagellum. We have alsofound that the occasional tumbling of bacteria may havea profound effect on the bacterial swimming trajectoriesdue to the complex interplay between flagellar flexibilityand self-propulsion. In particular, we have found thattumbling can result in long non-periodic excursions of thebacteria, mediated by periods of steady swimming alongthe centerline.
Throughout this paper, we will highlight the differencepassive ellipsoid particles [34], active fixed shape ellip-soidal swimmers [22], flagellated non-motile objects (e.g.,dead bacteria), and flagellated self-propelled swimmers.
2 Model
Physical framework
Our main hypothesis is that, via its nontrivial couplingbetween the body orientation and self-propulsion, a flex-ible flagellum can significantly affect bacterial swimmingtrajectories and possibly the rheological property of thesuspension [36, 37, 38]. We consider a microswimmer ina viscous fluid constituted of an ellipsoidal body linkedto a flexible thin rod (flagellum) to explore its behaviorin a Poiseuille flow and planar shear flow. For the sakeof simplicity we replace the helicoidal flagellum by a thinrod. Also, we neglect effects associated with the counter-rotation of the bacterial body [35]. The self-propulsionis implemented via a tangential force density (traction)distributed uniformly along the rod.
In our model, the geometrical and physical propertieswe take into account are the length of the flagellum, itspropulsion-force and its elasticity.
The bacterial body undergoes rotation described byclassical Jeffery equations [34], rigidly attached to a flag-ellum of length L (see Figure 1) and placed in shear flow.We derive the equation of motion of the flagellum in theframework of slender body theory in the Stokes flow ap-proximation [39]. A somewhat similar derivation is per-formed in [40] for a free one-armed-swimmer modeled asa slender elastica (not attached to another body), whichdeforms under the effect of external magnetic torques in aStokes flow. The novelty of our model is in the interplaybetween the flexibility of the flagellum described by theslender elastica, self-propulsion, and external shear flow.
2
For simplicity we consider that the attachment of flagellafilaments to the bacterial body is rigid1. We also assumethat the microswimmer does not affect the surroundingfluid. In order to exclude wall effects, we consider swim-ming in wide channels where hydrodynamic and steric in-teractions with the walls are negligible.
y
x
N0
θ0u
θ (s)
τn
d
l
Ly
x
Figure 1: Schematic representation of the swimmer (body+ flagellum). Inset: Limiting configuration of the swim-mer in Poiseuille flow (on the centerline, resisting theflow).
Mathematical model
The key ingredient of our model is the classical Jefferyequation (for the body) modified by an additional torqueterm (coming from the flagellum). This term is deter-mined via coupling the Jeffery’s orbits with a correspond-ing nonlinear 4th order equation for the flagellum derivedfrom basic physical principles. The flow is assumed to beparallel to the x axis and its velocity vector is u(y)ex.Here we summarize the model in dimensionless form (seeMethods 6.1).
Since the bacterial body is modeled by a rigid ellipsoid,we describe its motion by a Jeffery equation [34] withparameter β, the Bretherton constant of the body (β = 0for needles and 1/2 for spheres), where the shear rateis taken to be the flow rate u′(yh) at the center of theellipsoid (yh is the y−coordinate of the center of mass ofthe ellipsoid, and the fluid velocity does not depend onx). We add an additional term N0 corresponding to thenormal internal stress coming from the presence of theflagellum. The modified Jeffery equation is written as
1Individual flagellar filament is attached to body via a softhook. As it was shown in [16], the flexibility and buckling of thehook plays a profound role in the tumbling of uni-flagellatedbacteria. However, in the case of bacteria with multiple flagel-lar filaments distributed over the bacterial body, such as Bacil-
lus subtilis, flexibility due to soft hook coupling is not impor-tant since the flagella form tight bundles.
dθ0dt
= −u′(yh)(
(1− β) sin2 θ0 + β cos2 θ0)
+3L
lkrN0,
θ0(0) = θ0,in,(1)
where θ0 is the orientation of the swimmer body (see Fig-ure 1), l and L are respectively the length of the bodyand of the flagellum, and kr is a ratio between the dragcoefficients of the body and the flagellum (end of Section6.1).
The flagellum is represented by a 1D elastica of bend-ing regidity Kb parametrized by the non-dimensional ar-clength 0 ≤ s ≤ 1. We denote by θ(s, t) the angle betweenthe flagellum and the x-axis (see Figure 1). We denote byN the integrated internal stress in the flagellum and byN0 the force exerted by the flagellum on the body (nottangential due to different torques in the body and theflagellum), decomposed on the Frenet basis (τ ,n) of theflagellum as (Λ is the tangential stress (tension))
N = Λτ +Nn, N0 = Λ0τ +N0n. (2)
The equations describing the shape and motion of theflagellum are derived from the balances of forces and in-ternal torques (see Methods, equations (20) and (22)).The forces taken into account are the fluid friction force(proportional to the relative velocity between the flagel-lum and the fluid according to resistive force theory) andthe propulsion force density Fp generated by an internalmotor.
The balance of forces provides a vectorial relation whichwe decompose on the Frenet basis. Substituting balanceequations into geometrical identities (see Methods, equa-tions (27)) we obtain two scalar equations for angle θ andtension Λ
∂2Λ
∂s2=
1
αΛ
(
∂θ
∂s
)2
−Kb
(
∂2θ
∂s2
)2
− u′(y)
2sin(2θ)
− (α+ 1)
αKb
∂3θ
∂s3∂θ
∂s, t ≥ 0,
(3)
and
∂θ
∂t=− Kb
α
∂4θ
∂s4+
(
1
αΛ+Kb
(
∂θ
∂s
)2)
∂2θ
∂s2
+
(
α+ 1
α
∂Λ
∂s+ Fp
)
∂θ
∂s− u′(y)
(
sin2 (θ))
,
t > 0, θ (s, 0) = θin (s)
(4)
where α is a drag anisotropy factor taking into accountthe shape of the flagellum.
The physical hypotheses included in equations (3) and(4) are the inextensibility and the elasticity of the flagel-lum (see Methods, equations (21) and (26)).
Equations (3) and (4) come with a set of boundaryconditions, encoding the fact that the end of the flagellumis free at s = 1,
∂θ
∂s(1, t) =
∂2θ
∂s2(1, t) = Λ (1, t) = 0, (5)
3
and the interface between the body and the flagellum isrigid
θ (0, t) = θ0 (t) , Λ (0, t) = Λ0 (t) , (6)
The balance of internal torques combined with therigidity of the attachment between the body and the flag-ellum also gives us the expression of N0
N0(t) = −Kb
∂2θ
∂s2(0, t) . (7)
Another consequence of the rigidity of the attach-ment is the equality of the velocity at the interfacebody/flagellum. This vectorial equality provides directlytwo scalar equations, corresponding to the tangential part
krΛ0 =α
L
(
u(y(0))− u(yh))
cos(θ0) +∂Λ
∂s+ Fp − ∂θ
∂sN0,
(8)and to the normal part of the velocity equality.
( α
αh
+3α
2
)
krN0 = −α
Lsin(θ0)
(
u(y(0))− u(yh))
+αl
2Lu′(y)
[
(1− β)sin2(θ0) + βcos2(θ0)]
+[
−Kb
∂3θ
∂s+
∂θ
∂sΛ]
,
(9)
These two last equations are algebraic equations and onlyhold at s = 0.
To close the system, since the fluid velocity, and thusthe shear rate (u′(y)), may depend on swimmer coordi-nate y, we need to know at each time where the swimmeris located. The y-coordinate of the body satisfies the fol-lowing ordinary differential equation, similarly to [23]
dyh(t)
dt= krΛ0(t) sin(θ0(t)) +
krαh
N0(t) cos(θ0(t)),
yh(0) = yh0 .
(10)and the position along the flagellum follows directly fromthe geometrical considerations
∂y(s, t)
∂s= sin(θ(s, t)),
y(0, t) = yh(t) +l
2Lsin(θ0(t)).
(11)
The x-coordinates are not needed, however, their expres-sion is provided in the Methods section, since they areused to plot the trajectories.
3 Results
In this section, the model is studied numerically fortwo types of shear flows: a planar shear (linear) and aPoiseuille flow (nonlinear). The equations for the flagel-lum, (3) and (4), are solved using a centered finite differ-ence scheme on a uniform grid (∽ 100 point per micron onthe flagellum and 10−2s timestep). In (4), the fourth andsecond order derivatives are taken implicitly, which still
allows for the resolution of the scheme without iterations.We compute N0 from the boundary condition and substi-tute its value into (1), which we solve with the forwardEuler scheme. The numerical scheme is implemented inC++. See Supplementary Material S3 for more details onthe numerical scheme.
3.1 Planar shear flow
In a planar shear flow, we compare the outcome of the nu-merical study to a theoretical result derived in the asymp-totic limit of large bending stiffness Kb ≫ 1 of the (rigid)flagellum. The fluid speed in planar shear flow is given by
u(y) = γy. (12)
Our numerical procedure yields that the trajectories areperiodic (see Figure 2). When starting close to the center-line, the trajectory is closed (see Figure 2(a)). However,when starting on the upper part of the channel, there is adrift toward the right after one period, because the swim-mer spends more time in the upper part of the channelthan in the lower part. The value of the drift linearly de-pends on the y-initial position. It vanishes when startingat y = 0. For a visualisation of the motion, see the elec-tronic Supplementary Material, video S2. To obtain ananalytical result, we linearize Eqs. (1),(3)-(9) around thestate given by Kb → ∞, and derive a solution analyticallyby a multiscale perturbation method.
From (3)-(9), in the limit of large Kb the evolution ofthe bacterial body angle is given asymptotically by
θ00(t) = arctan
[
√
b
1− btan
(
t√
b(1− b)
(
1 +c
Kbb
)
)]
.
(13)Here b and c are geometrical constants, which only dependon L, l, α, αh, kr and β (see Supplementary Material S1).
Equation (13) describes how the period of the bodyangle θ0 depends on the bending stiffness Kb. For largebending stiffnesses, the body rotates according to the Jef-fery equation with parameter b < β. As expected, theeffective aspect ratio of the flagellated swimmer is largerthan that of the non-flagellated one. The flagellum ampli-fies the contrast between slow rotation (swimmer paral-lel to the flow) and fast rotation (swimmer perpendicularto the flow). The relation given by Eq. (13) as well asthe corresponding numerical solution are plotted in Fig-ure 2(b), using the values established in the literature forB. Subtilis. The result shows that the period of the bodyrotation, determined by the geometrical parameters of theswimmer with rigid flagellum, decreases when the flagel-lum becomes softer (smaller bending stiffness). As ex-pected, in the limit of large Kb there is a good agreementbetween the asymptotic analysis and numerics.
The decrease of the rotation period with the decreasein bending stiffness can be understood as follows. A softerflagellum on the average bends more than a rigid one. As aresult, a swimmer with a bent flagellum has an effectivelysmaller aspect ratio than with a more rigid one. In turn,
4
−200 −100 0 100 200
−10
0
10
20
30
40
Drift
x/L
y/L
0 0.05 0.1 0.15 0.2 0.25 0.3 0.3515
20
25
30
35
40
45
50
Asymptotic analysis
Numerics
1/Kb
Period
(a) (b)
Figure 2: (a): Trajectories of the self-propelled swimmer in a planar shear flow for two different initial positions.
Green arrows indicate the orientation θ0(t) of the swimmer body. (b): Dependence of the rotation period on1
Kb
,
where Kb is the bending stiffness (dimensionless, see Methods 6.1, rescaling) The red line is the period extracted byformula (13). The dots are corresponding numerical results. Other parameters are taken as written in Table 1. Forcomparison, the rotation period of similar non-flagellated swimmer is about 23 s.
the rotation period decreases with the increase in aspectratio.
Thus, in a planar shear flow, as for non-flagellatedswimmers [34], the trajectories of flagellated swimmersare periodic, and the period is mainly determined by theelasticity of the flagellum: the softer the flagellum, thesmaller the period.
Dependence of the period on the length of the flagellumcan be thought as a competition of two effects. On onehand, longer flagellum increases effective aspect ratio ofthe flagellated swimmer. On the other hand, the longerthe flagellum is, the easier it is to bend. Numerical resultsshow that the period has local maximum when L ≈ 9·10−6
(with other parameters as written in Table 1).
3.2 Poiseuille flow
The majority of experiments [19, 20, 27] are performed ina rectangular channel of width w (or in circular channelof radius r0). In the planar geometry, assuming no-slipboundary conditions, the x-component of fluid velocityhas the following parabolic profile,
u(y) = p(
y2 − w2
4
)
, (14)
where p is the applied pressure normalized on the dynamicviscosity of suspending liquid.
First, we demonstrate the stability of steady-stateswimming along the centerline. In [23], the trajectoriesof a self-propelled ellipsoidal swimmer (with point-forcepropulsion) in a Poiseuille flow are shown to be periodic.The presence of the flagellum makes our system more dif-ficult to deal with, nevertheless, in the large Kb limit the
properties of the trajectories can be studied analytically.In particular, it is clear that the states [y = 0, θ = 0] and[y = 0, θ = π] (i.e. the swimmer with a rigid (straight)flagellum swims along the centerline) are the only station-ary states of the system. In the large Kb limit, the state[y = 0, θ = π] (i.e. swimming against the flow) is theonly linearly stable equilibrium.
The linearized system around the stationary state [y =0, θ = π] is of the form:
dθ0(t)
dt= −2py(t) + 3N0(t),
dy(t)
dt=
Fp cos(π)
3θ0(t) +
cos(π)
4N0(t),
(15)
where N0 is determined by
∂θ
∂t(s, t) = −Kb
2
∂4θ
∂s4+
Fp
3(1− s)
∂2θ
∂s2,
N0 = −Kb
∂2θ
∂s2(s = 0)
N0 = −βp
4y − Kb
2
∂3θ
∂s3(s = 0) +
Fp
3
∂θ
∂s(s = 0),
∂θ
∂s(1, t) =
∂2θ
∂s2(1, t) = 0, θ(0, t) = θ0(t).
(16)We recall that Fp < 0, which implies that the second orderterm in the first line of (16) causes a buckling instability.If there is no flagellum, i.e. N0 = 0, the matrix of thelinear system d
dt(θ0, y)
T = A (θ0, y)T is
A =
(
0 −2pFp cos(π)
30
)
.
5
Tr(A) = 0 and Det(A) > 0, which means that the eigen-values are imaginary. From (16), N0 is computed as
N0 = −1
3
(dθ0dt
+ βpy)
. (17)
Substitution of (17) into the linear system (15) gives us,after straightforward computations, that [y = 0, θ = π] isstable whereas [y = 0, θ = 0] is not. This is in agreementwith the numerical observation that in Poiseuille flow, forlarge enough values of the bending stiffness, the bacteriaswim opposite to the flow direction for large times.
Next, we performed a numerical study of a bacteriumswimming in the Poiseuille flow. Select results are shownin Figures 3. For certainty, the bacterium was launched aty = 5, with an angle θ0 = 0 (i.e. parallel to the walls) anda straight flagellum. The swimmer performs large loops,first in the upper part of the channel, then in both halves(see Figure 3(a)). The amplitude of the loops is decreasingwith time, and at the end, the swimmer converges to thecenterline of the channel and orients against the flow (seeinset to Figure 1).
The flow strength was taken sufficiently high to avoidany collision between the swimmer and the wall. Thus theswimmer always drifts downstream. Due to the vorticityof the flow, the swimmer first (short time, x ≈ −104 onFigure 3 (a)) performs a “tumbling” motion (as definedin [22]), i.e. the body is undergoing complete rotation,in one part of the channel and has a slight drift towardthe center of the channel 2. Due to this drift, after sometime, the swimmer crosses the centerline. It then begins(at an intermediate time) to visit both sides of the chan-nel, performing a “swinging” motion (as defined in [22]),i.e. the body oscillates around the angle π, because thedifference of vorticity between the upper- and lower-sideof the channel prevents it from undergoing a complete ro-tation. However, due to flagella flexibility, the amplitudeof the excursions becomes smaller and smaller, and forlarge time, the swimmer reaches a stable stationary state,in stark contrast to the behavior of a swimmer modeledby a rigid body without a flagellum. The equations de-scribing the trajectory of a self-propelled fixed shape el-lipsoidal swimmers have been studied in [23] where theauthors observed two different periodic behaviors, “tum-bling” and “swinging”, depending on the y-coordinate ofthe initial position. In our case, there is a drift towardsthe center of the channel due to presence of the flagel-lum, which breaks the periodicity of the trajectories andallows the swimmer to switch from “tumbling” motion to“swinging” motion (see Figure 3(b), non-closed and closedcurves respectively).
2Here we distinguish “tumbling” motion due to vorticity ofthe flow from “run-and-tumble” behavior of bacteria due tounbundling of flagella
3.2.1 Heuristics for the convergence towardthe centerline for large bending stiff-ness
Here we present a simple explanation of why the swim-mer converges toward the centerline due to its flagellumflexibility in the case where the flagellum is rigid enough.Two mechanisms are responsible for the convergence to-ward the center: the elastic response of the flagellum tendsto displace the center of mass of the swimmer, and thepropulsion force amplifies this phenomenon.
The propulsion force amplifies the drift toward
the center. First, we point out that the convergencetoward the center is not a sole property of self-propelledswimmers. Numerical results show that our flagellatedobject (e.g., dead bacterium) with no propulsion force (i.efor Fp = 0) undergoes a slight drift toward the center ofthe channel (see Figure 3 (d), Kb = 3 · 10−23). Underassumptions used in [34] passive ellipsoid is expected todrift along the streamlines. This small drift toward thecenter is negligible comparing to the lift force that oc-curs to the nonlinear profile of the Poiseuille flow [42],which in contrast, makes the swimmer migrate away fromthe center (see Section 6.2.1). However, for self-propelledswimmers, the migration toward the center is amplifiedby the propulsion force, whereas the lift force is not.
The elastic response of the flagellum tends to
displace the center of mass of the swimmer. Forlarge enough bending stiffness (e.g., larger than 3.10−23)there is an overall tendency of the flagellum to straightendue to the bending rigidity described by the fourth orderterm in (4). This phenomenon results in a net displace-ment of the body, for both self and non-self propelledswimmers, and the direction of the body displacement isthe orientation of the body. The y−component of thisdisplacement determines if the swimmer drifts toward oraway from the center. The higher the shear rate experi-enced by the flagellum, the more important the verticaldisplacement. Because the shear rate experienced by theflagellum is higher when further from the centerline, thisdisplacement is more important when the swimmer is ori-ented toward the center.
Why convergence toward the center does not
occur in planar shear flow? The migration towardthe center is not observed in planar shear flow. The rea-son is the following: In a planar shear flow, whatever theorientation of the swimmer is, the shear experienced byeach point of the flagellum only depends on the positionof the body (since the shear rate is constant everywhere).Displacement away from the center occur then as often asdisplacement toward the center.
3.2.2 Non-monotone dependence of the lim-iting behavior on the bending stiffness
The heuristic explanation we provided above is no longervalid for small values of the bending stiffness. For the cho-sen value of dimensionless pressure gradient p = 104 andsmaller bending stiffness Kb = 10−24 (representative value
6
|x| large x ≈ −106 x ≈ −5 · 104 x ≈ −104-20
-10
0
10
20
flow
x = 0
y
x
(a)
y
θ00 π 2π
-20
-10
0
10
20
θ0
y
0 π 2π-15
-10
-5
0
5
10
15
(b) (c)
−7 −6 −5 −4 −3 −2 −1 0
4.85
4.9
4.95
5
5.05
5.1
5.15
×104
x
y
Kb = 1 · 10−24
Kb = 3 · 10−23
Kb = 2 · 10−25
4.5
5
5.5
flow
x
y
10−26 10−25 10−24 10−23 10−22
−6
−4
−2
0
2
4
6
Vy,R
Bending stiffness Kb
Vy
R
(d) (e)
Figure 3: (a): Representative trajectory (x(t), y(t)) of the swimmer in the channel for different intervals of time,for the physical value of the bending stiffness (Kb = 3 · 10−23). The arrows indicate the orientation θ0(t) of theswimmer body. The blue circles indicate the sign of the vorticity. (b) and (c): Phase portraits of the trajectoriesof the self-propelled swimmer. For large values of the bending stiffness, the swimmer converges toward the centerof the channel swimming against the flow (b), for a smaller value of the bending stiffness, there is a limit cycle (c).(d): Averaged trajectory over a period of the passive swimmer for three different bending stiffness. Depending of thebending stiffness value, there is an averaged drift toward or away from the center. The inset corresponds to the realshape of the trajectory. The convergence is slow, so we need to consider large values on the x− axis (downstream).(e): Averaged y velocity of the passive swimmer 〈Vy〉 vs bending stiffness compared to the size of the limit cycle R vsbending stiffness for the self-propelled swimmer. Dots correspond to values given by the numerical scheme, the dottedlines are an interpolation of the dots. R is the maximum distance from the center of the channel that the swimmerreaches in large time (i.e R := lim supt→∞
|y0(t)|). Except for Kb, parameter values were taken as written in Table1 and p = 104 1/ms.
7
for Bacillus subtilis is Kb = 3 · 10−23) the self-propelledswimmer no longer drifts toward the center, but insteadconverges towards a limit cycle (see Figure 3(c)), wherethe swimmer swings around the centerline. When fur-ther decreasing the bending stiffness (Kb = 2 · 10−25), thebehavior is again similar to Kb = 3 · 10−23. This sug-gests that the behavior of the system is highly nontrivial.It has been observed numerically that depending on thebending stiffness of the flagellum, the non self-propelledflagellated object (Fp = 0) can either drift toward thewalls or toward the center (see Figure 3(d),Kb = 10−24
and Kb = 2 · 10−25 respectively). Also, we observed theself propelled swimmer reaching either a stable stationarystate, or a limit cycle. For different bending stiffness val-ues the y-averaged velocity of a non self-propelled object isplotted on Figure 3(e). If the averaged velocity is negative(e.g. Kb ≥ 3 · 10−23), the non self-propelled object tendsto reach the center of the channel. If the averaged veloc-ity is positive (e.g. Kb = 10−24), the swimmer migratesaway from the center. Clearly, there is a strong correlationbetween the size of limit cycle R for self-propelled swim-mer and average y velocity 〈Vy〉 of non-motile flagellatedbody, see Figure 3(e). Ranges of the bending stiffnesswhere the averaged velocity of the non self-propelled ob-ject is negative correspond approximately to ranges wherethe self-propelled swimmer converges toward the center(size of the limit cycle = zero). Conversely, ranges ofthe bending stiffness where the averaged velocity of thenon-motile body is positive correspond to ranges wherethe self-propelled swimmer converges toward a limit cy-cle. It means that the bending stiffness value determinesthe qualitative behavior of the swimmer. When decreas-ing Fp, but staying close to the physical value (50%), thecurve R is slightly shifted on the left and has a smalleramplitude. The pressure gradient of the flow p on theother hand does not affect qualitatively the large time be-havior of the swimmer, but it has a quantitative effect:the ranges of values of Kb where the swimmer convergestoward the stationary state or the limit cycle are the same,whatever p. However, increasing p accelerates the conver-gence toward the steady state, and reduces the size R ofthe limit cycle.
3.2.3 Random reorientation (tumblingevents)
For large values of the bending stiffness, the swimmerasymptotically converges toward the centerline. However,tumbling, that is in that context an abrupt reorientationof a swimmer, may destabilize the steady state. To ver-ify this assumption, we incorporated random reorientationin our model. A good approximation to the probabilitydistribution of tumbling events in time is a Poisson distri-bution, [41]. It implies that the total number of tumblesoccurring between time s and t is a Poisson random vari-able with parameter λr(t−s), where λr is the intensity ofthe Poisson law. The Poisson process determines the in-stant of reorientation, and the angle changes according tothe uniform distribution on [0,2π]. The flagellum reorients
as well with the same angle. With random reorientationpresent, the relaxation toward the steady state no longeroccurs. We display on Figure 4 two selected time intervalsat which tumbling occurs. Recall that the fluid is movingis the negative direction, so that each plot has to be readfrom right to left. On plots (a) and (b), the swimmer un-dergoes a “swing” when the tumble event occurs. If norandom reorientation, the angle of the body would haverelaxed toward π and the y− amplitude of the trajectorywould have decreased to y = 0. The tumble event reori-ents the swimmer and makes it switch to another point ofthe phase portrait Figure 3, panel (b). On Figure 4, panel(a), after the tumble event the swimmer is still ”swing-ing”, but the amplitude of the trajectory has increased.On plots (c) and (d), the swimmer was also undergoinga “swinging” motion when the tumble event occurs. Thenew orientation of the swimmer is 3π/2, and the motionswitches to a “tumbling” motion (i.e. the swimmer is un-dergoing a complete rotation) on the lower part of thechannel. Thus, the qualitative behavior of the swimmermay drastically change due to a relatively rare randomreorientation, eliminating the overall convergence to thecenterline.
4 Conclusions
We introduced a non-linear model based on partial differ-ential equations that couples body motion of a swimmerwith flexible flagellum attached to it. Linear asymptoticanalysis of this model for planar shear flow shows how clas-sical Jeffery orbits change due to the presence of the flag-ellum. In particular, we found the dependence of the bodyrotation period on the bending stiffness of the flagellum.Next, we performed numerical analysis of this model inPoiseuille flow and made several important observations.First, the model exhibits non-periodic trajectories due tothe presence of the flagellum. This stands in contrast withclassical periodic trajectories for Jeffery equation for pas-sive particles, and even more striking, when compared toperiodic trajectories for fixed shape active swimmers [23]that are self-propelled but have no flagellum. Second, weobserve that the large time behavior of our system is non-monotonic with respect to the bending stiffness. Namely,for large bending stiffness, the system reaches a steadystate. When bending stiffness is decreased, it convergestowards limit cycle. Further decrease in the bending stiff-ness results in the system reaching the steady state again.Finally, we observe that even a small amount of randomtumbling events drastically affect the above described be-havior.
5 Acknowledgments
The theoretical work was supported by the NIH grant1R01GM104978-01. The simulations at Argonne andwork of I.S.A. were supported by the U.S. DOE BES,Division of Materials Science and Engineering (IA)
8
-1.43 -1.42 -1.41 -1.4 -1.39 -1.38
×105
-15
-10
-5
0
5
10
15
flow
x
y
-1.43 -1.42 -1.41 -1.4 -1.39 -1.38
×105
π
2
π
3π
2
x
θ0
(a) (b)
-2.64 -2.63 -2.62 -2.61 -2.6 -2.59
×105
-20
-15
-10
-5
0
5
10
15
flow
x
y
-2.64 -2.63 -2.62 -2.61 -2.6 -2.59
×105
0
π
2
π
3π
2
2π
x
θ0
(c) (d)
Figure 4: (a) and (c): Trajectories of the self-propelled swimmer in the (x, y) space for two selected time intervals.The fluid flows from the right to the left and the blue circles indicate the sign of the vorticity. Green arrows indicatethe orientation θ0(t) of the swimmer body. The symbol ’+’ shows the tumbling events. (b) and (d): Orientation θ0of the swimmer as a function of x. The intensity of the Poisson process is taken to be 0.1/min, which means that atumbling event occurs in average every 10 minutes. Other parameters are taken as written in Table 1.
6 Methods
6.1 Derivation of the equations of mo-tion
The flexible flagellum
To derive the equations of motion we employ resistiveforce theory and theory of the elasticity of a slender body.We use the arclength 0 ≤ s ≤ L to parametrize the flagel-lum. We denote by θ(s, t) the angle between the flagellumand the x-axis (see Figure 1). We also introduce the localtangent, normal, and binormal vectors,
τ =
cos (θ)sin (θ)
0
,n =
− sin (θ)cos (θ)
0
,b =
001
,
and write the velocity of the point of the flagellum as
v = vττ + vnn. (18)
Denote byN the integrated internal stress in the flagellumand by N0 the force exerted by the flagellum on the body,
N = Λτ +Nn, N0 = Λ0τ +N0n. (19)
In the framework of the resistive force theory, the forceexerted by the fluid on the flagellum is proportional tothe relative velocity between the flagellum and the fluid.The force balance states that the divergence of the in-ternal stress is equal to the external force applied to the
flagellum, yielding the following relations
∂
∂s
(
Λτ +Nn)
= ζf(
vτ − u(y) cos θ)
τ
+ αζf(
vn + u(y) sin θ)
n− Fpτ .(20)
Here, Fp is the propulsion force density generated bythe flagellum. The parameter ζf is the drag coefficientof the flagellum, and α is a drag anisotropy factor dueto the shape of the flagellum. Following [40], we setζf = 2πη0/ log(L/D), where η0 is the viscosity of thefluid, and α = 2 (slender body). The elasticity of theflagellum is represented by the constitutive relation
M = Kbκb, (21)
where κ is the local flagellum curvature, defined by κ =∂θ
∂s. The balance of internal torques gives
∂M
∂s+ τ ×N = 0, (22)
which combined with (21) gives an expression of the nor-mal component of the internal stress
N = −Kb
∂2θ
∂s2. (23)
The force balance vector relation (20) can be separatedinto the tangential and normal parts
ζfvτ =ζfu (y) cos θ +∂Λ
∂s+ Fp − κN,
αζfvn =− αζfu (y) sin θ +∂N
∂s+ κΛ.
(24)
9
Now we derive the equations for Λ and θ, using thegeometrical Frenet relations
∂τ
∂s= κn,
∂n
∂s= −κτ,
yielding
dτ
dt=
dv
ds=(∂vτ
∂s− κvn
)
τ +(∂vn
∂s+ κvτ
)
n. (25)
The inextensibility of the flagellum can be cast in theform
dτ
dt=(∂θ
∂t
)
n (26)
By identifying the terms in (25) and (26) we obtain
∂vτ∂s
= κvn,∂θ
∂t=
∂vn∂s
+ κvτ , (27)
which gives us the following equations on Λ and θ
α∂2Λ
∂s2=
(
∂θ
∂s
)2
Λ− αKb
(
∂2θ
∂s2
)2
− α
2u′(y)ζf sin (2θ)
− ∂θ
∂s(α+ 1)Kb
∂3θ
∂s3,
(28)
αζf∂θ
∂t=−Kb
∂4θ
∂s4+
(
Λ+ αKb
(
∂θ
∂s
)2)
∂2θ
∂s2
+(
(α+ 1)∂Λ
∂s+ αFp
)∂θ
∂s− u′(y)αζf sin
2(θ).
(29)Boundary conditions. The bacterium flagellum bound-
ary is free at s = L and the connection between the bodyand the flagellum is rigid. It gives the set of boundaryconditions:
∂θ
∂s(L, t) =
∂2θ
∂s2(L, t) = Λ (L, t) = 0, (30)
θ (0, t) = θ0 (t) , Λ (0, t) = Λ0 (t) , (31)
where θ0(t) and Λ0(t) are respectively the angle that therigid bacterial body makes with the x-axis and its tangen-tial internal stress at the interface between the body andflagellum. They are both determined by conditions on theinterface, so the system is closed. We add the condition
N0(t) = −Kb
∂2θ
∂s2(0, t) , (32)
which is also a consequence of the rigid connection be-tween the body, the flagellum, and (23).
Motion of the bacterial body
Motion of passive rigid ellipsoid is described by the Jef-fery’s equation. In the first order approximation, the flowaround the ellipsoid can be treated as a planar shear flow.We describe the bacterial body motion by the Jefferyequations, where the shear rate is taken to be the flow
rate at the center of the ellipsoid. We add an additionalterm corresponding to the normal internal stress comingfrom the presence of the flagellum. The modified Jefferyequation is written as
dθ0dt
= −u′(yh)( l2
l2 + d2sin2 θ0(t) +
d2
l2 + d2cos2 θ0(t)
)
+l
2ζrN0(t),
(33)where yh is the y−coordinate of the center of mass of theellipsoid. The velocity of the body center of mass hastwo components, the first comes from the velocity of thefluid taken at its center, and the second comes from thestress applied to the flagellum. In the modeling we neglectperturbations produced by the body on the flow and willdescribe its motion by the motion of the center of mass.
Body-flagellum interface conditions
The body makes contact with the flagellum at s = 0. Therigidity of the junction enables the local velocities at thecontact of body and flagellum to be equal.
We have an expression of the local velocity of the flag-ellum using (24)
vτ (0) =u (y(0)) cos θ +1
ζf
[∂Λ
∂s(0) + Fp − κN0,
]
vn(0) =− u (y(0)) sin θ +1
ζf
[∂N
∂s(0) + κΛ0
]
.
(34)
The velocities of the body are given by
vhτ = u(yh) cos θ0 +1
ζhΛ0
vhn = −u(yh) sin θ0 +1
ζhαh
N0,
(35)
where ζh is the drag coefficient of the body, ζh =2πη0l/ log(l/d), and αh is an anisotropy drag factor, takento be 2 (slender body approximation, [40]). The tangen-tial components of the velocities are the same, but thereis an extra term for the normal velocities given that vn(0)represents the normal velocity at the flagellum-body con-tact whereas vn,0 is the normal velocity of the center ofthe body. The angular velocities are the same, but thecenter of the body and the flagellum-body contact are at
a distancel
2of each other.
vτ (0) = vτ,0, vn(0) = vn,0 +l
2
dθ0dt
. (36)
The equalities in (36) holding at s = 0 are expressed asfollows, using (34) and (35).
10
1
ζhΛ0 = cos(θ0)
(
u(y(0))− u(yh))
+1
ζf
[∂Λ
∂s+ Fp +Kb
∂θ
∂s
∂2θ
∂s2
]
,
1
ζhαh
N0 +l
2
dθ0dt
= − sin(θ0)(
u(y(0))− u(yh))
+1
αζf
[
−Kb
∂3θ
∂s+
∂θ
∂sΛ]
.
(37)
Expression of the coordinates
Since swimmer’s velocity depends on the local velocity ofambient fluid, which itself depends on the y−coordinate,we express yh(t) and y(s, t), the vertical position of thecenter body and of the flagellum respectively. We alsostudy the evolution of the x-coordinate, as it is useful todescribe the trajectories. Due to geometrical consider-ations, the body position (xh, yh) satisfies the followingequations:
dxh(t)
dt= vτ,0 cos(θ0(t))− vn,0(t) sin(θ0(t)),
dyh(t)
dt= vτ,0 sin(θ0(t)) + vn,0 cos(θ0(t)),
xh(0) = xh0 , yh(0) = yh
0 ,
(38)
where (xh0 , y
h0 ) is the initial position of center of mass of
the body. We substitute the expressions of the tangentialand normal velocities (35) and obtain
dxh(t)
dt= u(xh(t), yh(t)) +
1
ζhΛ0(t) cos(θ0(t))
− 1
αhζhN0(t) sin(θ0(t)),
dyh(t)
dt=
1
ζhΛ0(t) sin(θ0(t)) +
1
αhζhN0(t) cos(θ0(t)),
xh(0) = xh0 , yh(0) = yh
0 .(39)
The positions of the flagellum may then be written inthe form
∂x(s, t)
∂s= cos(θ(s, t))
∂y(s, t)
∂s= sin(θ(s, t))
x(0, t) = xh(t) +l
2cos(θ0(t)),
y(0, t) = yh(t) +l
2sin(θ0(t)).
(40)
Rescaling
We first define the order of magnitude of the shear rateas γ0. The variables are rescaled as
s =s
L, u =
u
Lγ0, t = γ0t, Λ =
Λ
ζfγ0L2,
Λ0 =Λ0
ζfγ0L2, N0 =
N0
ζfγ0L2, ζr =
l2ζh6
,
Kb =Kb
ζfγ0L4, Fp =
Fp
ζfγ0L, kr =
Lζfζh
,
β =d2
l2 + d2,
In Section 2, we use these rescaled variables but we omitthe tildes, to simplify the notation.
6.2 Random reorientation (tumbling)
Define Trun as the time between two tumbles. If we denoteby Nt the number of tumbles occurring before time t, wehave
P[(Nt −Ns) = k] = e−λr(t−s)
(
λr(t− s))k
k!, (41)
where P is probability of the event.This causes the distribution of run duration to be expo-
nential. Indeed, if Trun1 is the time at which first tumbleoccurs, we have
P(Trun1 ≥ t) = P(Nt = 0) = exp (−λrt), (42)
which means that the time of the first run has an expo-nential distribution. Each other time of the run has also
exponential distribution. The parameter1
λr
is the expec-
tation of Trun and is taken to be 10 min.To simulate the exponential distribution, we gener-
ate a uniform random variable U using a random num-ber generator in C++. We denote by G the inverse ofthe repartition function of the exponential law, namely
G(u) = − 1
λr
ln(1 − u), then the random variable G ◦ U
has an exponential distribution, since
P(G ◦ U ≤ t) = P(U ≤ F (t)) = F (t). (43)
6.2.1 Effects of the nonlinear flow profile
In the analysis above we neglected the effect due to thenonlinear profile of the flow. In the context of a sphericalparticle in an unbounded flow, the lift force exerted bythe fluid on a body whose center is in yh is of the form
Lift = 6.46 η0r∆u
√R
4, R = ρu′(yh)
r2
η0, (44)
where ρ is the ratio between the solid and the fluid density,∆u is the slip velocity of the particle (the particle velocityminus the undisturbed velocity at the particle center), andr is the radius of the body (see [42, 43]). This expression
11
is valid only for spherical swimmers. In our case, theswimmer is close to a rod of length l. Thus, in (44) we
replace the radius r by the effective radius,l
2| sin(θ0)|,
which corresponds to the radius seen by the fluid. This
results in ∆u = |u(yh +l
2sin(θ0))− u(yh)| ≈ |4l sin(θ0)|.
In the Poiseuille flow, assuming that ρ = 1, the formulabecomes
Lift =6.46
4
√η0√
u′(yh)pl3| sin3(θ0)|yey, (45)
With the order of magnitude of the parameters, Lift ≈10−14N , which can be neglected, in comparison to thetotal propulsion force L×Fp, which is of the order of fewpN. The model including this lift effect was implemented,and no noticeable change in the behavior was observed.
Glossary
α Drag anisotropy factorβ Bretherton constant of the ellipsoidal bodyγ Shear rated Thickness of the bodyη0 Viscosity of the surrounding fluidFp Propulsion force densityKb Bending stiffnesskr Non-dimensional parameterΛ Internal stress of the flagellum (tangential part)λr Parameter of the Poisson processL Length of the flagelluml Length of the bodyN Internal stress of the flagellum (normal part)p Pressure gradient of the Poisseuilles flows Current coordinate on the flagellumt Timeθ0 Body angle (swimmer orientation)θ Angle of the flagellum(τ ,n,b) Frenet systemu Fluid velocity(vτ , vn) Flagellum velocity in Frenet coordinates(vτ,0, vn,0) Body velocity in Frenet coordinatesw Radius of the channel
(xh, yh) Coordinate of the body centerζf Friction coefficient for the flagellumζh Friction coefficient for the head
References
[1] R. Stocker, Marine Microbes See a Sea of Gradients.Science 338, 628-633, 2012.
[2] M. T Butler, Q. Wang, and R. M. Harshey, Cell Den-sity and Mobility Protect Swarming Bacteria AgainstAntibiotics. PNAS 107, 3776, 2010.
[3] C. Dombrowski, L. Cisneros, S. Chatkaew, R. E. Gold-stein, and J. O. Kessler, Self-Concentration and Large-Scale Coherence in Bacterial Dynamics. Phys. Rev.
Lett. 93, 098103, 2004.
Parameter Parameter value
α 2
β 0.0192
γ 0.1 s−1
d 7× 10−7 m
η0 10−3 Pa s
Fp 10−7 N m−1
Kb 3× 10−23 N m2
L 10−5 m
l 5× 10−6 m
ζf 10−3 N s m−2
ζh 1.6× 10−8 N s m−1
p 104 m−1 s−1
w 4× 10−4 m
Table 1: Parameter values used in the numerical sim-
ulations
Rescaled variable Representative value
b 5× 10−3
c −0.0045
Fp 100
Kb 30
kr 0.65
E 1
Table 2: Numerical values of the rescaled variables
[4] A. Sokolov, I.S. Aranson, J.O. Kessler, and R.E. Gold-stein, Concentration dependence of the collective dy-namics of swimming bacteria. Phys. Rev. Lett. 98,158102, 2007.
[5] A. Sokolov and I. S. Aranson, Physical Properties ofCollective Motion in Suspensions of Bacteria. Phys.
Rev. Lett. 109, 248109, 2012.
[6] J. Dunkel, S. Heidenreich, K. Drescher, H. H.Wensink, M. Br, and R. E. Goldstein, Fluid Dynamicsof Bacterial Turbulence. Phys. Rev. Lett. 110, 228102,2013.
[7] S. D. Ryan, B. M. Haines, L. Berlyand, F. Ziebert, andI. S. Aranson, Viscosity of bacterial suspensions: Hy-drodynamic interactions and self-induced noise. Phys.Rev. E 83, 050904, 2011.
[8] D. Saintillan and M. Shelley, Instabilities and Pat-tern Formation in Active Particle Suspensions: Ki-netic Theory and Continuum Simulations. Phys. Rev.Lett. 100, 178103, 2008.
[9] I. S. Aranson, A. Sokolov, J. O. Kessler, and R. E.Goldstein, Model for Dynamical Coherence in ThinFilms of Self-Propelled Microorganisms. Phys. Rev. E75, 040901, 2007.
[10] H. H. Wensink, J. Dunkel, S. Heidenreich, K.Drescher, R. E. Goldstein, H. Lwen, and J. M. Yeo-mans, Meso-scale Turbulence in Living Fluids. PNAS
109, 14308, 2012.
12
[11] S. D. Ryan, A. Sokolov, L. Berlyand, and I. S. Aran-son, Correlation properties of collective motion inbacterial suspensions. New J. Phys. 15, 105021, 2013.
[12] L. H. Cisneros, J. O. Kessler, S. Ganguly, and R. E.Goldstein. Dynamics of swimming bacteria: Transi-tion to directional order at high concentration. Phys.
Rev. E 83, 061907, 2011.
[13] J. Adler. Chemotaxis in bacteria. Science 153, 708-716, 1966.
[14] R. Vogel and H. Stark. Motor-driven bacterial flagellaand buckling instabilities. Eur. Phys. J. E 35, 1-15,2012.
[15] R. Vogel and H. Stark. Rotation-induced polymor-phic transitions in bacterial flagella. Phys. Rev. Lett.
110, 158104, 2013.
[16] K. Son, J. S. Guasto, and R. Stocker. Bacteria canexploit a flagellar buckling instability to change direc-tion. Nature Phys. 9, 494-498, 2013.
[17] F. P. Bretherton and L. Rotshchild. Rheotaxis ofspermatozoa. Proc. R. Soc. Lond. B. Biol. Sci. 153,490-502, 1961.
[18] K. Miki and D.E. Clapham, Rheotaxis guides mam-malian sperm. Curr Biol. 23, 443-52, 2013.
[19] V. Kantsler, J. Dunkel, M. Blayney, R. E. Gold-stein. Rheotaxis facilitates upstream navigation ofmammalian sperm cells. eLife 3, e02403, 2014.
[20] P. Denissenko, V. Kantsler, D. J. Smith, and J.Kirkman-Brown. Human spermatozoa migration inmicrochannels reveals boundary-following navigation.PNAS 109, 8007-8010, 2012.
[21] V. Kantsler, J. Dunkel, M. Polin, and R. E. Gold-stein. Ciliary contact interactions dominate sur-face scattering of swimming eukaryotes. PNAS 110,11871192, 2012.
[22] A. Zottl and H. Stark. Non linear dynamics of aMicroswimmer in Poiseuille flow. Phys. Rev. Lett. 108,21804, 2012.
[23] A. Zottl and H. Stark. Periodic and quasiperiodicmotion of an elongated microswimmer in Poiseuilleflow. Eur. Phys. J. E 36, 1-10, 2012.
[24] S. Chilukuri, C. H. Collins and P. T. Underhill. Im-pact of external flow on the dynamics of swimming mi-croorganisms near surfaces. J. Phys. Condens. Matter.
26, 115101, 2014.
[25] A. P. Berke, L. Turner, H. C. Berg and E. Lauga, Hy-drodynamic Attraction of Swimming Microorganismsby Surfaces. Phys. Rev. Lett. 101, 038102, 2008.
[26] K. Drescher, J. Dunkel, L. H. Cisneros, S. Ganguly,and R. E. Goldstein, Fluid dynamics and noise in bac-terial cellcell and cellsurface scattering. PNAS 108,10940-10945, 2011.
[27] R. Rusconi, J. S. Guasto and R. Stocker. Bacterialtransport suppressed by fluid shear. Nature physics
10, 212-217, 2014.
[28] G. I. Taylor. The action of waving cylindrical tails inpropelling microscopic organisms. Proc. R. Soc. Lond.221, A211-225, 1952.
[29] E. M. Purcell. The efficiency of propulsion by a ro-tating flagellum. PNAS 94, 11307-11311, 1997.
[30] S. Jung, K. Mareck, L. Fauci and J. M. Shelley. Ro-tational dynamics of a superhelix towed in a Stokesfluid. Physics of Fluids 19, 103-105, 2007.
[31] Marcos, H. C. Fu, T. R. Powers, and R. Stocker.Bacterial rheotaxis. PNAS 109, 4780-4785, 2012.
[32] J. O. Kessler. Hydrodynamic focusing of motile algalcell. Nature 313, 218-220, 1985.
[33] L. Karp-Boss, L.E. Boss, and P. A. Jumars. Mo-tion of dinoflagellates in a simple shear flow. Limnol.
Oceanogr. 45, 15961602, 2000.
[34] G. B. Jeffery. The motion of ellipsoidal particles in aviscous fluid. Proc. R. Soc. Lond. 179, 102-161, 1922.
[35] W.R. DiLuzio, L. Turner, M. Mayer,P. Garstecki,D.B. Weibel, H.C. Berg, and G. W. Whitesides. Es-cherichia coli swim on the right-hand side. Nature,435, 1271-1274, 2005.
[36] A. Sokolov, I.S. Aranson, Reduction of viscosity insuspension of swimming bacteria. Phys. Rev. Lett.
103, 148101, 2009.
[37] J. Gachelin, G. Mino, H. Berthet, A. Lindner, A.Rousselet, and E. Clement, Non-Newtonian Viscosityof Escherichia coli Suspensions. Phys. Rev. Lett. 110,268103, 2013.
[38] B.M. Haines, A. Sokolov, I.S. Aranson, L. Berlyand,D.A. Karpeev, Three-dimensional model for the effec-tive viscosity of bacterial suspensions. Phys. Rev. E
80, 041922, 2009.
[39] G. K. Batchelor. Slender-body theory for particles ofarbitrary cross-section in Stokes flow. J. Fluid Mech.
44, 419-440, 1970.
[40] M. Roper, R. Dreyfus, J. Baudry, M. Fermigie, J.Bibette, and H.A. Stone. On the dynamics of mag-netically driven elastic filaments. J. Fluid Mech. 554,167-190, 2006.
[41] H. C. Berg. E. coli in motion. Springer-Verlag, 2004.
[42] G. Segre and A. Silverberg. Behaviour of macroscopicrigid spheres in Poiseuille flow. J. Fluid Mech. 14, 136-157, 1962.
[43] P. G. Saffman. The lift on a small sphere in a slowshear flowing. J. Fluid Mech. 22, 385-400, 1965.
13
Supplementary Material
Asymptotic analysis through multiscale method
We use an asymptotic expansion based on multiscale method (as in [1]). We consider that Kb =1
δ, where
δ is a small parameter, and we introduce a so-called “short time” τ . Even though τ = δt, we treat the fastand short time as independent variables. Expand all variables with respect to the small parameter δ to find
θ(s, t, τ) = θ0(s, t, τ) + δθ1(s, t, τ),
θ0(t, τ) = θ00(t, τ) + δθ10(t, τ),
Λ(s, t, τ) = Λ0(s, t, τ) + δΛ1(s, t, τ),
Λ0(t, τ) = Λ0
0(t, τ) + δΛ1
0(t, τ),
N0(t, τ) = N0
0 (t, τ) + δN1
0 (t, τ).
We plug these expansions in the non-linear system, and identify the terms of same order in δ, resultingin a decoupled system.
First order - An effective Jeffery equation. Solving analytically the system corresponding to the first order(O (δ)) gives us an effective Jeffery’s equation for the angle of the body
∂θ00∂t
(t, τ) = −(
(1− b) sin2 θ00(t, τ) + b cos2 θ00(t, τ))
,
θ00(0, τ) to be determined.
(1)
with
E =3Lkrα
2l(α
αh
+3α
2)kr + 2l
, b =β
1 + E
(
1− El
L
)
.
Without a flagellum, the coefficient b is the usual coefficients of the Jeffery equation, b = β. At first order,in the presence of the flagellum the body rotates according to a Jeffery equation, but the coefficients are nolonger given merely by the geometry of the body. The swimmer is rotating because the coefficients of the
effective Jeffery equation have the same sign. Sincel
L< 1, we have 1− b ≥ 1− β and b ≤ β. Since β <
1
2, it
means that the flagellum amplifies the contrast between slow rotation when the ellipsoid is orientated parallelto the flow and fast rotation when the ellipsoid is orientated perpendicular to the flow.
The initial condition θ00(0, τ) is determined in the next step by setting a condition on convergence of theexpansion, preventing any resonance from appearing at the following order.
Correction to the period. We solve the second order system and we obtain the following equation on θ10 :
(
1 + E)(∂θ00
∂τ+∂θ10∂t
)
=[
− 1 + 2β − 2E(
βl
L+
1
2
)]
θ10 sin(2θ0
0)+
+ C1
(
a sin2(θ00) + b cos2(θ00))
sin(2θ00) + C2 sin(2θ0
0) cos(2θ0
0) + c cos(2θ00),
(2)
where C1, C2 and c are constants depending on the geometrical parameters. The only secular term, is theterm in cos(2θ00). Indeed, solving
dθ10dt
= cθ10 sin(2θ0
0(t)) + cos(2θ00(t)) (3)
1
would lead to a resonance phenomenon between order 0 and order 1 in the regular perturbation series. Toprevent this resonance from appearing, we impose
∂θ00∂τ
= c cos(2θ00(t, τ)), (4)
where c = −0.0045. We then are left to solve the following partial differential equation
∂θ00∂t
(t, τ) = (1 − b) sin2 θ00(t, τ) + b cos2 θ00(t, τ),
∂θ00∂τ
(t, τ) =c
1 + Ecos(2θ00(t, τ)),
θ00(0, 0) = 0.
(5)
The solution on {t = 0} is given by
θ00(0, τ) =1
2arcsin
(1− exp(4cτ)
1 + exp(4cτ)
)
. (6)
We now remember that τ = δt and find the solution of the effective Jeffery’s equation
θ00(t) = arctan
[
b√
b(1− b)tan
(
t√
b(1− b) + arctan( 1− b√
b(1− b)tan(
1
2arcsin
(1− exp(4Ct)
1 + exp(4Ct)))
)]
, (7)
which after simplification gives us the result claimed in Section 3.1.
Description of videos
Videos 1-3 are obtained by numerical solution of Eqs. 1-11. The parameters values are described in the tableat the end of the paper.
Video 1
Left panel : Orientation of the body + Shape of the flagellum displayed in the moving frame in planar shearflow. Right panel: Trajectory of the center of mass of the body in x − y plane. For a realistic value of thebending stiffness Kb ≈ 3× 10−23 N m2, the flagella are almost straight.
Video 2
Parameters are the same as for Supplementary Video 1. To illustrate deformation of the flagella, in thisvideo the bending of the flagella is artificially increased (the difference θ(s, t)− θ0(t) is multiplied by factor1.5× 104).
Video 3
Parameters are the same as for Supplementary Video 1, except for the extremly high shear rate. The videoexhibits that the bending of the flagella can be seen to the naked eye when the shear rate is sufficiently large.
2
Video 4
This video illustrates spontaneous buckling of the flagella for an exceedingly small bending stiffness Kb =4×10−25N m2 (almost two orders of magnitude smaller than a representative value ofKb for Bacillus subtilis).The body of bacterium is kept fixed, and no external flow is imposed.
To characterize the buckling instability, we plotted time averaged total variation of the angle TV(θ) :=
1
T2 − T1
∫ T2
T1
∫ L
0
∣
∣
∣
∣
∂θ
∂s(s, t)
∣
∣
∣
∣
ds dt as a function of the bending stiffness Kb, see Figure 1. For our model, Eq. 1-
11, the buckling threshold is about Kb ≈ 8.5×10−25N m2. When the body is kept fixed (as in SupplementaryVideo 4), the threshold is slightly lower (about Kb ≈ 6× 10−25N m2). Both values are about hundred timesbelow a realistic value for the apparent bending stiffness of Bacillus subtilis flagella.
0 0.2 0.4 0.6 0.8 1 1.20
2
4
6
fixed body
free body
×10−24Kb
Figure 1: Time averaged total variation TV(θ) as function of bending stiffness Kb
About the numerical scheme
We discretize the time axis with a time step ∆t and the flagellum with a mesh of size ∆x. We denote byθ0[n], yh[n], Λ0[n], θ[k, n], Λ[k, n] the approximated values of the variables characterizing the swimmer attime n∆t and at point k∆x. We assume that all quantities are known at the time step n. We present herea chart describing how we compute the state of the system at the time step n+ 1. We observed numericallythat the scheme is converging, and we validated it on a few test cases.
References
[1] C. M. Bender and S. A. Orszag. Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, 1978.
3
1 : The flagellum exerts a normal stress on thebody.• Computation of N0[n+1] - Formula (7).
2 : The body orientation changes under the com-bined action of the shear flow stress generated bythe flagellum.• Computation of θ0[n+1] - Solving ODE (1).
3 : Update the shape of the flagellum, usingθ0[n+1] as a boundary condition.• Computation of θ[k, n+1] - Solving the parabolicPDE (4) + (5) + (6).
4 : Prepare the boundary conditions for 5 .• Computation of an algebraic equation on Λ0[n+1]and the discrete derivative ∆Λ
∆x(0, n+1) - Solving a
linear system (8) + (9).
5 : Update the tangential internal stress along theflagellum.• Computation of Λ[k, n+1] and Λ0[n+1] - Solving
a boundary value problem (3) + (5) + 4 .
6 : Update the position of the body and theflagellum (useful if the flow depends on the y −coordinate).• Computation of yh[n+1] - Solving ODE (10) +(11).
(1)
. (2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
4
