A Study of Bacterial Flagellar Bundling

Heather FloresUniversity of Nebraska-Lincoln

hflores@math.unl.edu

Edgar LobatonSeattle University

lobatoe@seattleu.edu

Stefan Mendez-DiezUniversity of Chicago

stefan@uchicago.edu

Svetlana TlupovaTulane University

stlupova@tulane.edu

Ricardo CortezTulane University

cortez@math.tulane.edu

November 3, 2003

Abstract

Certain bacteria, such as Escherichia coli (E. coli) and Salmonella ty-phimurium (S. typhimurium), use multiple flagella often concentrated at oneend of their bodies to induce locomotion. Each flagellum is formed in a left-handed helix and has a motor at the base that rotates the flagellum in acorkscrew motion. We present a computational model of the flagellar motionand their hydrodynamic interaction. The model is based on the equations ofStokes flow to describe the fluid motion. The elasticity of the flagella is modeledwith a network of elastic springs while the motor is represented by a torqueat the base of each flagellum. The fluid velocity due to the forces is describedby regularized Stokeslets and the velocity due to the torques by the associatedregularized rotlets. Their expressions are derived. The model is used to ana-lyze the swimming motion of a single flagellum and of a group of three flagellain close proximity to one another. When all flagellar motors rotate counter-clockwise, the hydrodynamic interaction can lead to bundling. We present ananalysis of the flow surrounding the flagella. When one of the motors changesits direction of rotation, the same initial conditions lead to a tumbling behaviorcharacterized by the separation of the flagella, changes in their orientation, andno net swimming motion. The analysis of the flow provides some intuition forthese processes.

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1 Introduction

Single-celled bacteria, such as Escherichia coli (E. coli) induce locomotion throughthe use of multiple flagella often concentrated at one end of the organism. Eachflagellum is flexible and maintains roughly a helical shape. At the base of eachflagellum is a small rotary motor that can turn clockwise or counterclockwise. It isknown that when the motors all turn counterclockwise, the flagella tend to gathertogether to form a single helix (see for example [14]). This movement, known asbundling, results in forward motion of the cell. When some or all of the motorsturn clockwise, the flagella go through a sequence of shape transformations changingamplitude, pitch, and sometimes handedness. The consequence is that the bundleunravels and the flagella separate. This motion is known as tumbling and changesthe movement of the organism into one without a preferred direction, resulting in noappreciable net motion [1]. Bundling and tumbling together enable the organism tomove in one direction and reorient itself to move in another direction where theremay be more favorable conditions for survival. This process may be in response tochemical stimuli (chemotaxis) or other factors such as temperature and light intensity.

Previous work on the modeling of helical swimming motions has combined ana-lytical and numerical methods and has focused mainly on organisms with a singleflagellum. Lighthill [9, 10] provided mathematical analysis of the motion of a thintube with helical shape using slender-body theory. In this theory, the flagellum is re-placed with a distribution of Stokeslets and dipoles along its centerline. The analysisprovided refined resistance coefficients for this motion and modifications to accountapproximately for the drag force on the cell body. Higdon [6, 5] combined mathemat-ical and numerical analyses applied to the motion of an organism with a sphericalbody and a single flagellum. The flagellum was also modeled using slender-bodyapproximations. The body was assumed to be spherical which allowed the use ofimages to impose approximate boundary conditions on its surface. A rotlet at thecenter of the cell body was included in order to balance the rotation induced by theflagellum. Ramia et al. [13] used a computational approach based on the boundaryelement method to model the motion of an organism with spherical body and a singlehelical flagellum. Their study included motions near walls and near another organismof the same shape. The description of the organism is similar to that in [6]; however,the Green’s function approach is traded for a boundary element method in order toaddress cases with multiple walls (of finite extent) and multiple organisms.

The works mentioned above assumed from the outset the shape of the flagellumfor all time. For helical waves, the centerline was assumed to be given by a helix ofthe form

(x, y, z) = (x,E(x) cos(kx− ωt), E(x) sin(kx− ωt))

which implies that throughout the motion, the flagellum rotates and translates as asolid body without deformation. This assumption simplifies the mathematical anal-ysis but is not realistic for bacteria like E. coli and S. typhimurium, whose flagellashould not be considered rigid screws since they assume a variety of distinct helicesdepending on their environment (see for example [7]).

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Powers [11] considered a single straight but flexible filament which is rotated atone end in a circular fashion around an axis parallel but not coincident with the fil-ament. The rotation simulates the cell body rotation and the filament represents asingle flagellum. Based on steady states of the filament driven by various rotation fre-quencies, conclusions were drawn regarding the possibility of bundling. However, onlya single filament was considered and the hydrodynamic interactions among neighbor-ing flagella were not taken into account. The Stokes flow was included only throughthe use of a transverse friction coefficient following local resistive-force theory (for arod) and slender-body approximations, and the conclusions were based on the helicalshape of the rotating isolated filament.

The goal of the present work is to determine conditions that tend to producebundling of various flagella in close proximity to one another. The emphasis here ison the role of the hydrodynamic interaction of the flagella in the processes of bundlingand tumbling. No restrictions on the wave amplitudes or flagellum dimensions areimposed. In this way, the methodology used here can also be applied to other organ-isms that may not be slender or that display large-amplitude waves in their motion.It is important not to assume a priori the helical shape of the flagella for all timebut allow for deformations during the interactions. Therefore, our model includesa mechanism designed to provide a certain amount of elasticity to the flagellum sothat a helical shape is preferred but deviations from it are allowed. The simulationsare based on solutions of Stokes equations in the presence of external forces given byshape functions that smoothly approximate delta distributions. This is the basis ofthe method of regularized Stokeslet [2, 3] used here. In addition, the rotation inducedby the motors at the base of each flagellum is modeled with a regularized rotlet whichis derived as the antisymmetric part of the derivative of the regularized Stokeslet.The rotlet represents a localized torque.

2 Equations

The fluid dynamics in problems of microorganism motion, where length and velocityscales are very small, is well-modeled by the Stokes equations for incompressible flows

0 = −∇p+ µ∆u + f

0 = ∇ · u

where p is the fluid pressure, u is the fluid velocity, µ is the viscosity and f is theexternal force density. If we define L and U to be a characteristic length and acharacteristic speed in the problem, we can define the dimensionless variables

~x =1

Lx, ~u =

1

Uu, p =

L

µUp, ~f =

L2

µUf .

Then, after some simplification, the Stokes equations in dimensionless form become

0 = −∇p+ ∆~u +~f (1)

0 = ∇ · ~u. (2)

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These are the equations we use in our model. We mention that a typical length ofan E. coli flagellum is L = 10 − 20µm and the wave speed of the flagellum is aboutU = 250µm/sec.

2.1 Solutions of the Stokes Equations

When a force ~f is exerted on the fluid, the resulting velocity field ~u and pressure pare the solution of Eq. (1)-(2). The particular case of a single point force ~f0 exertedat ~x0 results in a velocity field called a Stokeslet and is given by

~Us(~x;~x0,~f0) =~f0

8πr+

[~f0 · (~x− ~x0)](~x− ~x0)

8πr3

where r = ‖~x−~x0‖. Note that this flow is undefined at ~x = ~x0 although a distributionof forces on a surface yields a flow that is defined everywhere.

In our computations, the motion is generated by forces along the surface of theflagellum and by a torque at the base of it. The torque represents the one transferredto the flagellum by the motor at the junction with the body of the bacterium (which isnot part of the model). The forces are designed to keep the flagellum in approximatelythe same helical shape while providing some flexibility to it.

The velocity field that satisfies Stokes equations when a torque ~L0 is applied at asingle point ~x0 is called a rotlet and is given by

~Ur(~x;~x0, ~L0) =~L0 × (~x− ~x0)

8πr3. (3)

This flow is more singular than the Stokeslet and is no longer integrable even if it weredistributed over a surface. This implies that the fluid velocity becomes arbitrarilylarge as the evaluation point approaches the point where the torque is applied. Thenumerical method that we use here is based on the systematic regularization of theflows described above by considering forces and torques that are applied not at singlepoints but within small spheres centered at those points. In this way, the forcesand torques are highly concentrated but are spread over a small neighborhood of theapplication points. The precise form of the force is given by a cutoff function φδ(~x)which we will take to be radially symmetric and to satisfy∫∫∫

R3

φδ(~x)d~x = 1,

where δ is a numerical parameter that controls the spread of the function (see Fig-ure 1). Throughout this article, we will use the cutoff function

φδ(~x) =15δ4

8π(r2 + δ2)7/2(4)

where r = ‖~x‖.

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Figure 1: The cutoff function in Eq. (4) for three values of δ.

When the force in Eq. (1) is given by a cutoff centered at ~x0, ~f(~x) = ~f0φδ(~x−~x0),one can derive the exact solution of the Stokes equation to get the regularized Stokeslet(see Appendix).

~Uδ,s(~x;~x0,~f0) =~f0(r2 + 2δ2)

8π(r2 + δ2)3/2+

[~f0 · (~x− ~x0)](~x− ~x0)

8π(r2 + δ2)3/2. (5)

Notice that as δ approaches zero, we recover the Stokeslet expression. However, theregularized Stokeslet represents a flow that is bounded for all ~x as long as δ > 0.

The regularized rotlet is derived from the antisymmetric part of the derivative ofthe regularized Stokeslet (see Appendix). The result for a torque ~L0 centered at ~x0 is

~Uδ,r(~x;~x0, ~L0) =(2r2 + 5δ2)

16π(r2 + δ2)5/2[~L0 × (~x− ~x0)]. (6)

As δ → 0 we recover the original rotlet expression. It is important to emphasize thatthe regularized rotlet yields finite velocities everywhere simply because of the assumedform of the torque. At points far from the torque, the regularized expression is nearlyindistinguishable from the singular counterpart. Near the torque, the regularizedexpression provides a model for the fluid motion that can be used in computations.

3 The numerical method

In order to build the model, we start by creating the structure of the flagellum. Eachflagellum is a tubular structure made of discrete particles connected by springs. Someof the springs connect particles around the cross-sections and others connect particles

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between neighboring cross-sections. The forces between two particles, ~xj and ~xk, arecomputed using Hooke’s Law:

~fjk = k0

(Ljk

Ljk0− 1

)~xk − ~xjLjk

, ~fkj = −~fjk, Ljk =‖ ~xk − ~xj ‖

where k0 is the stiffness constant, Ljk is defined to be the distance between ~xk and~xj at time t, and the spring resting length, Ljk0 , is defined to be the initial distancebetween ~xk and ~xj, so at time t = 0 the force between the particles is zero. Asthe particles that describe the flagellum move, the distance between them changescausing spring forces to be applied on both particles in equal and opposite pairs.The forces are designed to maintain approximately the initial resting length betweenthe particles by preventing them from moving too far apart or getting too close.The helical shape is achieved by having springs of varying resting lengths along theflagellum. The stiffness constants may also be different for each spring, and theirvalues control the flexibility (elasticity) of the flagellum. Higher stiffness results inmore rigid structures.

In our model, a particle may be connected by springs to several other particles.We use the total force applied at each point in the computation of the velocity. Thetorque is applied to only one point at the base of each flagellum to simulate therotation due to the motors.

The motion of the flagellum is computed as follows. Given the positions of allparticles at time t, all forces are computed based on the geometry of the flagellum.The torque and its location are also computed from the particle positions. Once theforces and torque are known, the velocity at any location ~xk is computed using theregularized Stokeslet and rotlet formulas:

d~xkdt

= ~u(~xk) =Nr∑i=1

~Uδ,r(~xk; ~yi, ~Li) +Ns∑j=1

~Uδ,s(~xk;~zj,~fj)

where Nr is the number of rotlets of strengths ~Li located at ~yi and Ns is the numberof Stokeslets of strengths ~fj located at ~zj. The expressions in this formula are givenby Eq. (5)–(6).

The position of each particle changes according to the fluid velocity so that eachparticle position can be updated after a small time interval. At that time, the newparticle positions define new forces and a new torque which are used for the next timestep. In this way, the forces impose a time scale in the problem where the velocityof the particles is the superposition of the regularized Stokeslets and the rotlet. Thetime evolution of the particle positions is computed using a fourth-order Runge-Kuttamethod.

4 A Single Flagellum

In E. coli, each flagellum is shaped into a left-handed helix that extends from the cellbody. Our goal is to create a solid, yet flexible, representation of the flagellum. We

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define the flagellum to be a helical tube of total length ` with cross-sections perpen-dicular to the tangent vector (see Figure 2). Each cross-section is an n-sided polygon.The helix has a varying radius, R(s), which is implemented using an arctangent enve-lope that allows the radius to start from zero and increase to some fixed value. Thisis similar to the envelope used in [6, 13]. The initial conditions for the particles areas follows:

x(s) = α(s)

y(s) = −R(s) cos(

2πnp

(s`

))z(s) = R(s) sin

(2πnp

(s`

))where the amplitude R(s) is given by

R(s) = Rh

(arctan

(β(s`− γ)

+π

2

)),

np represents the number of turns in a helix, and α(s) is found so the tangent vector(x′(s), y′(s), z′(s)) has unit length. We emphasize that the shape of the flagellumfor t > 0 is not specified but found as part of the computation. The benefit of thearctangent envelope is that it defines an axis of rotation in the front of the flagellumwhile creating a helix of constant radius in the rear as can be seen in Figure 2. Foreconomy of computation, we choose n = 3 for the cross-sections, the smallest numberpossible, so that each flagellum is a structure made of 3 helices defining triangularcross-sections. This choice, however, is not a restriction.

Figure 2: Side view of one helical flagellum.

In Figure 3, we show the spring connections that are defined between differentparticles. First, each particle on a helix is connected to the corresponding particleson the other two helices (Figure 3a). This defines the cross-sections of the flagellumand will be referred to as cross-sectional forces with stiffness constants of ka. Next,each point on a helix is connected to adjacent points on the same helix using springforces, this is shown in Figure 3b. These will be referred to as consecutive forceswith stiffness constants kb. We also define diagonal forces around the surface of theflagellum. The bold lines are the diagonal forces with stiffness constants kc (Figure 3c).

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This structure has the benefit of being composed entirely of triangles. Triangles,unlike other polygons, have the property that they preserve angles when preservingthe length of the sides.

Figure 3: Spring Connections (a) Cross-sectional springs (b) Consecutive Springs (c)Diagonal Springs.

The torque is positioned at the center of the first cross-section of the flagellum.Because of the varying helix amplitude, the first cross-section is perpendicular to thecentral axis of the helical flagellum. This makes it appropriate to define the torqueorthogonal to the first cross-section.

Figure 4 shows the initial conditions for this problem. Since the flow due tothe rotlet decays as r−2 for large values of r, nearby cross-sections experience morerotation than ones far from the rotlet. The distances between contiguous cross-sections deviate from their resting-lengths due to the rotation. This is particularlypronounced near the front of the flagellum. In response to the stretching, the springsexert forces that pull the rest of the flagellum, making the entire structure rotate.The diagonal forces are essential in this process since they have a significant effect onpropagating rotation along the entire flagellum.

4.1 Parameter Dependence of a Single Flagellum

The motion of a single flagellum is induced by a torque applied at its base. Theparameters were chosen so that the flagellum held together in its helical form whilestill allowing it to rotate with some elasticity. The rotation of the entire structure ina viscous fluid necessarily results in forward swimming motion of the flagellum. Inthis section, we discuss the effect of some parameters in the model on the swimmingspeed and angular speed of the flagellum. In most of the numerical experiments, the

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Figure 4: Initial Conditions for a single flagellum.

flagellum length was fixed at ` = 1.3 (dimensionless units) which means that thelength scale used to non-dimensionalize Eq. (1)-(2) is approximately the flagellumlength. Based on numerical experiments, most of the reported results use N = 51cross-sections to discretize the flagellum since larger numbers of cross-sections hadno significant effect on forward motion or stretching. In all numerical experiments,the maximum helix radius was fixed to Rh = 0.06, which is about 5% of the length.The radius of the flagellum was fixed to a value of 0.012, which is about 1% of theflagellum length (consistent with a somewhat short flagellum of 10 nm in radius and1µm in length. Other computational experiments use longer flagella as indicated inTable 4).

In a computational flagellum, there are many different parameters that affect itsmotion. To analyze the effect of one parameter, we monitored the angular velocity,forward motion, and structural stability. Here we discuss the changes caused byvarying the stiffness constant values (ka,kb,kc), the regularization parameters for boththe Stokeslet (δs) and the rotlet (δr), the magnitude of torque applied (L), number ofperiods in a flagellum (np), pitch (p), and arclength (`).

The spring constants must be set to large enough values in order to keep thehelical shape of the flagellum. Once the spring constants are sufficiently large, theyhave little effect on the structure of the flagellum and forward motion. As can be seenin Table 1, for the range of spring constant values chosen for the parameter analysis,variations have little effect on forward motion. Although at smaller constant values,there is slightly more stretching of the flagellum. For these reasons, the stiffnessconstants were set equal to 12 for all computations in this section.

The regularization parameters for the Stokeslet and the rotlet are independent ofone another. These were set to a multiple of the distance between the cross-sections

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ka, kb, kc Speed(×10−4)

4,4,4 1.22728,8,8 1.2245

12,12,12 1.221916,16,16 1.2204

Table 1: Comparison of spring constant values. The parameters used were δs = δr =0.052, L = 0.001, np = 3, p = 0.2399, ` = 1.3.

in the flagellum. For N cross-sections along the helix of length `, this distance is`/(N − 1). For the tests in Tables 1, 2 and 4, this cross-sectional distance was 0.026.Changing the Stokeslet’s parameter δs has contrasting effects on forward motionand the stretching of the flagellum. The value of δs should be comparable to theseparation between cross-sections along the flagellum so that the cutoff functions ofnearby forces can overlap. Our results show that larger values of δs yielded fasterforward motion but also produced more stretching. This is to be expected sincelarger values of δs produce forces that are spread over larger regions and reduce themaximum value of the cutoff function being used. This produces smaller reactionforces by the springs and allows more stretching. The convergence of the method ofregularized Stokeslets as the discretization is refined and the regularization parameteris reduced is discussed in [3].

δs Speed(×10−4) Stretching

1*0.026 0.7692 0.0032*0.026 1.2219 0.0103*0.026 1.4522 0.030

Table 2: Effect of the Stokeslet regularization parameter δs. The parameters usedwere ` = 1.3, δr = 0.0526, L = 0.001, np = 3, p = 0.2399

The torque magnitude L and the rotlet regularization parameter δr have alarge effect on forward motion and the structure of the flagellum. We varied theamount of torque applied and looked at its effect on forward motion and angularvelocity. Physical experimentations performed by Purcell [12] assumed linear rela-tionships between the torque magnitude and the angular speed, and between theswimming speed and angular speed, consistent with Stokes flow. He expressed theserelationships with the equations

F = Av +Bω, (7)

L = Cv +Dω, (8)

where F is the net applied external force magnitude, v is swimming speed, ω is theangular speed, L is the net torque magnitude, and A, B, C, and D are constants that

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depend on the geometry of the flagellum. There is no net external force in our system,so F = 0 in Eq. (7) and the correspondence between v and ω is a line through theorigin. This, together with Eq. (8), results in a linear relation between ω and the nettorque, and also between v and the net torque. We computed the flagellar motion forvarious torque magnitudes L and verified that our model produces a linear relationbetween the swimming speed v and ω, between the torque magnitude L and ω, andtherefore, between the torque magnitude and the swimming speed (see Figure 5).The linear relationships hold for different values of δr. Eq. (6) indicates that the

rotlet velocity can be written as ~Uδ,r(~x;~x0, ~L0) = δ−2r~Uδ,r(~x/δ;~x0/δ, ~L0) so that as δr

is reduced, the torque is concentrated in a smaller region and its maximum valueincreases, resulting in faster rotation.

Figure 5: Torque magnitude L vs. angular speed ω (left); Torque magnitude L vs.swimming speed v (middle); angular speed ω vs. swimming speed v (right).

The number of periods, np, in the helix comprising each flagellum also hasan effect on both its swimming motion and its ability to hold its shape. We reportresults using four, three and two helical periods while keeping the pitch p constant.Since flagella with more periods of a given pitch are longer than those with fewerperiods, one expects that for a given driving torque, the viscous drag would have alarger effect on a flagellum with more periods causing a smaller angular velocity and,therefore, slower swimming motion. Our results, shown in Table 3, show that havingfewer periods causes less stretching and resulted in faster swimming, as expected. Forthis experiment, the number of cross-sections along the flagellum was increased as thenumber of periods increased in order to maintain a constant cross-sectional spacing.

The pitch of the flagellum has almost no effect on ω for a fixed rotlet. Thepitch was varied by fixing the arclength and changing the number of turns in thehelix. Having more or fewer turns per arclength did not have an effect on the angularvelocity ω. However, the model shows that the maximum swimming speed is achievedfor a given pitch which does not seem to depend on arclength. Intuitively, if the pitchis very large, the flagellum is nearly a straight tube and does not swim efficiently.On the other hand, if the pitch is very small, the helix is tightly wound and the

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np ` N Speed(×10−4) ω

2 0.867 34 1.70 2.03 1.30 51 1.22 1.54 1.73 68 0.964 1.0

Table 3: Comparison of number of periods with constant pitch. The parameters usedwere δs = δr = 0.052, torque magnitude L = 0.001 and pitch p = 0.24.

flagellum does not swim efficiently either. Thus, one expects a selected pitch to yielda maximum swimming speed. Table 4 shows the results for flagella of fixed arclengthand variable number of turns (different pitch). This was done with two differentarclengths while maintaining the same cross-sectional distance. Based on the resultswe estimate the optimal pitch to be about 0.45 dimensionless units, which gives aratio of pitch-to-length of p/` = 0.35. This ratio compares well with the ratio foundexperimentally in [14, 8] of p/` ≈ 2.3/7 = 0.33 for normal flagellar filaments.

` (N) np Pitch ω Speed(×10−4)

1.7 (51) 2 0.761 0.8 0.7071.7 (51) 3 0.436 0.8 0.7901.7 (51) 4 0.228 0.8 0.483

2.5 (75) 2 1.168 0.6 0.3752.5 (75) 3 0.747 0.6 0.5482.5 (75) 4 0.512 0.6 0.5942.5 (75) 4.5 0.422 0.6 0.566

Table 4: Comparison of parameters defining pitch. The parameters used were δs =δr = 0.052, and L = 0.001.

4.2 Flow Generated by a Flagellum

The motion of the flagellum generates fluid flow around it. Figure 6 shows the par-ticular flow at planes parallel to the cross-sections of the flagellum. The flow near thefront is largely influenced by the torque applied at that location, while the flow fur-ther back along the flagellum is influenced by the spring forces that cause the entireflagellum to rotate. Figure 7 shows the flow projected onto the xy-plane. The flowindicates the forward motion of the flagellum since there is flow coincident with thehelix tangent. The figure also shows regions of fluid rotation in alternating directionsthat approximately coincide with the helix shape.

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Figure 6: Flow fields at first, 4th, 10th, and 51st (last) cross-sections of a flagellum.

13

Figure 7: Initial position and flow field through a middle plane in the flagellum att = 350.

Figure 8: View of Initial Conditions for the three flagella case.

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5 Three Flagellum Model

In this section, we discuss the interactions among three flagella through the fluid flowthey generate. In particular, we are interested in the role of the fluid motion in theprocesses of bundling and tumbling. Our model includes three flagella constructed inthe same way as described in the previous sections. Each one has a motor modeled bya torque in the center of the first cross-section. While we do not construct explicitlythe body of the organism, the model includes features related to the effects of thebacterial body on the motion. The three flagella were placed equally-spaced arounda circle whose radius represents the radius of the bacterial body. To simulate thefront of these flagella being connected to a rigid body and not being able to freelychange their distances and orientation relative to one another, the front sections ofthe flagella were connected by springs. These springs connect the center points ofthe first three cross-sections of each flagellum with the corresponding cross-sectionsof the other two flagella. Figure 8 shows these connections. Under this constructionthe center points of the first cross-sections of the three flagella formed an equilateraltriangle and remained at approximately the same distance throughout the simulation.

Figure 9: Torque setup for the three flagella case.

Since the representation of the flagella in our model is by discrete points, thereis a possibility that in the computation they may get closer than physically possible.We prevent two flagella from crossing each other by defining repulsive forces whichare turned off when the flagella are apart. We define the minimum distance beforethe flagella come into physical contact to one another and set this as the thresholdfor activation of the repulsive forces. In the remaining computations, the activationthreshold was set to 2.3 times the flagellar radius.

The torques located at the base of each flagellum generate rotation throughout thefluid. Since we are interested in the flagellar interaction due to the fluid motion andnot due to the rotation of the flagella around one another, we introduce a counter-torque to represent the counter rotation effect of the bacterial body. This counter-torque is in the opposite direction of the three motor torques and is designed to keepthe bases of the flagella from rotating around each other. The counter-torque is placed

15

at the center of the triangle defined by the center points of the first cross sections ofthe three flagella, so that it is equidistant from the other three torques located at thevertices of the triangle. Its direction is perpendicular to the plane of the triangle andpointing in the opposite direction of the other three torques, see Figure 9.

We compute the magnitude of the counter-torque dynamically at every time stepto cancel the rotation of the front flagellar cross-sections (where the motors are lo-cated) around each other. This was accomplished by computing first the velocitiesof these cross-sections due to all the forces and motor torques and projecting thesevelocities onto the circle defined by the location of the motors. The magnitude of thecounter-torque was computed to cancel the average projected velocity.

This definition of the counter-torque leads to simulations in a reference frame inwhich the bacterial body does not rotate. This case is similar to a laboratory setup inwhich each flagellum is connected at the base to an stationary motor which does notallow it to spin around the other flagella [8]. We expect to learn the flow interactionsamong the flagella from the analysis of this experiment. This is not unrealistic on atime scale of a few flagellar rotations since the bacterial body of E. coli rotates witha frequency of 10 rev

sec(see [11]) whereas the flagella rotate with a frequency of 100 rev

sec

(see [1]). In this frame, the angular momentum will be conserved only approximatelyand the net torque will not be exactly zero. It is possible to adjust the location ofthe counter-torque in order to minimize the net torque, however this was not donehere since the masses of the body and the flagella are not considered.

5.1 Bundling of Three Flagella

One important feature of bundling is how close the flagella get to one another. Inorder to characterize the distance between the flagella, we computed and monitoredthe distances between the centers of the cross-sections. We first defined the distancebetween one cross-section of a flagellum to another flagellum to be the minimum dis-tance between the center of the cross-section and the center of all the cross-sectionsof the other flagellum. By computing the minimum, maximum and average of thesedistances, one can monitor the separation between flagella. Figure 10 shows the aver-age distance between flagella for the case analyzed in the next section (see Table 5).This average distance drops substantially in time as the flagella start to bundle.

Another property that we examined was the forward swimming motion of theflagella. This was measured by comparing the position of the center of the firstcross-section of the three flagella and their corresponding initial positions. Similarly,we computed the corresponding quantity at the last cross-section. The two curvesare shown in Figure 10. The gap between these curves represents stretching which iscaused by the elasticity of the flagella. Notice that most of the stretching occurs earlyin the motion. The slope of the curve gives the swimming speed of the flagella. As theflagella get closer together, the computation of the motion becomes more demandingin order to prevent the numerical crossing of any two flagella. Consequently, the timestep in the computation must be reduced, making a long-time simulation challenging.Figure 11 shows the final snapshot of the simulation using the parameters given inTable 5.

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Parameter Value

number of cross-sections 51Radius of flagellum 0.01Helix radius (Rh) 0.06

Arc Length of flagellum 1.3Number of periods 2

Radius of flagella separation 0.05ka, kb, kc 10

δs 0.039δr 0.052

Magnitude of Torque 0.002

Table 5: Table of parameter values for three-flagellum case.

Figure 10: Plot of average distance between flagella (left) and forward motion ofthe front and last cross-sections (right). The difference between the latter representsstretching of the flagellum.

Figure 11: Side view of simulation of three flagella.

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5.2 Flow Generated by Three Flagella

Our numerical experiments indicate that the process of bundling can be acceleratedby placing at least the tail ends of the flagella slightly closer. In the experimentsof this section, we use initial conditions in which the helix axes are not parallel buttilted slightly so that they are closer at the tails than at the front. The helix axeswere tilted typically only by 1◦–2◦. The only purpose of this is to reach a bundledstate earlier in the simulation. We believe that even without tilting the helix axesthe bundling would continue further in time from that shown in Figure 11 but thecomputation time would have to be longer. We present results using the parametersgiven in Table 5 and an angle of incline of 1.2◦. The parameter values fall within therange computed for the single flagellum for favorable swimming speed.

Figure 12: Cross-sectional view of fluid flow near the three flagella (t = 400). Theplots show the flow on planes at the front, back and two intermediate locations alongthe flagella.

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We focus on the fluid flow around the flagella. The flow generated by the threeflagella has a different character depending on the distance from the torques. Theflow near the first cross-sections of the flagella is mainly dominated by the effect ofthe torques, see top-left plot in Figure 12. At this location, which is the plane wherethe torques are located, the forces act to keep the cross-sections from deforming andto maintain their relative distances approximately constant. One can see the effectof the counter-torque at the center of the graph.

Further away, the torques generate a more uniform single rotation. The effect ofthe torques also decreases as the distance from the torques increases, and the effectof the forces along nearby cross-sections becomes more significant. For the particularresults displayed in Figure 12, the flow beyond a quarter length of the flagella isdominated by the forces (Stokeslets) while the torques (rotlets) have little effect. Thetop-right plot shows the flow near the front but far enough away from the torquesthat there is a single center of rotation (where the flow is zero in the plot) betweenthe three cross-sections. The bottom-left plot in Figure 12 shows how the rotationis altered by the effect of the forces in the regions close to the flagella. The centerof rotation is no longer on the center axis of the three flagella and the resulting flowpulls the flagella located furthest away from the center of rotation toward the otherflagella. We point out that the flagella can be seen to be very close to one another inthe bottom-left plot of Figure 12. The bottom-right plot shows the flow at the tail.For all the observed time, the cross-section located furthest away from the center ofrotation was always pulled in the its direction. This type of flow, in turn, results inbundling.

Figure 13 shows the location of the center of rotation about a quarter length ofthe flagella from the front. The center of rotation moves in a spiral pattern causingthe flagella to come together during bundling.

By analyzing the flow on the xy-plane (see Figure 14) we observe the appearanceof regions of circulation. This flow is an expected result from the motion of theflagella [4, 9]. The direction of the flow rotation changes when the concavity of thehelix changes as viewed from the side. These side views of the flow also make evidentthe end-effect of the flagella. The force balance at the end of the flagella is differentfrom the balance in the middle and this affects the flow. The changes in the flowrotation pattern can be observed more clearly on the bottom plot of Figure 14. Onecan also compare this plot with Figure 11 to notice that the small angle of inclineused in this section does speed up the bundling process. With the small tilt of thehelix axes, the flagella wrap around one another for about half the total length afterthe same final simulation time. We point out that the same phenomenon where partsof the flagella come close to one another while other sections remain apart has alsobeen observed experimentally [8].

For comparison with the numerical simulations of the next section, we show inFigure 15 four snapshots of the flagella entering into the bundle. In the relatively shortsimulation time shown in this figure, the average distance between flagella decreasesfrom 0.0866 to 0.0653 and continues to decrease in time. It is also clear from the figurethat the orientation of the flagella remains the same throughout the simulation.

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Figure 13: Position of the center of rotation at the 15th cross-section of the flagella asa function of time. The center of rotation spirals inward in time promoting bundling.

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Figure 14: Side view of flow for three flagella case.

Figure 15: The flagella entering into a bundle.

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5.3 Results on Tumbling

Parameter Value

No. cross-sections 51Radius of Flagellum 0.008Helix Radius (Rh) 0.05

Arc Length of Flagellum 1.3Number of Periods 3

Radius of Flagella Separation 0.05ka, kb 10kc 12δs 0.039δr 0.052

Magnitude of Torque 0.008Angle of Incline 0.0

Table 6: Table of parameter values for three flagella case.

In this section we test whether the same construction that produced bundling canproduce tumbling. Tumbling occurs when one or more of the motors change theirdirection to a clockwise rotation. When this occurs in laboratory experiments, theflagella are observed to behave erratically and the organism changes its orientation inapparently random fashion [14]. We look at the case in which all the parameters andinitial conditions are as in the previous section except that we change the directionof rotation of only one of the motors. We use the parameters shown in Table 6throughout this section.

The resulting motion is depicted in Figure 16 and should be compared with Fig-ure 15. Notice that although only the direction of rotation of one motor was changed,the motion is dramatically different from that in Figure 15. Instead of getting closerand wrapping around one another, the flagella move away from one another and theychange orientation. In the span of one hundred units of time, the maximum distancebetween the flagella increases from 0.0867 to 0.3305, and the average distance be-tween flagella increases from 0.0866 to 0.1285. This also gives an indication that thetumbling occurs faster than bundling. In E. coli the bundling time is about ten timeslonger than the tumbling time [14].

The changes in orientation are also clear from Figure 16 and it becomes more andmore drastic as time progresses. The x-position of the tail does not change. Thisindicates that there is no net forward motion, but only a rotation of the flagella thatchanges their orientation.

When analyzing the fluid flow generated by the flagella, one can get an idea ofhow the change in the direction of rotation of some motors may cause tumbling. Onthe left side of Figure 17, the flagellum in the lower right rotates clockwise creatinga dipole effect with the flagella that rotate counterclockwise. Since the cross-sectionsnear the front of the flagella are at a fixed distance from one another, as they would

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Figure 16: A tumble at different times.

Figure 17: The 1st cross section during the beginning of the tumble.

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Figure 18: The 20th cross section during the tumble.

be if they were connected to the bacterium body, the flow forces the front to changeits orientation significantly rather than to separate the flagella.

Except for the front few sections of the flagella, which are tied together, therest of the flagella are free to move independently as they are carried by the flow.The combination of the dipole flows created by the counter-rotations of the motorsgenerates a flow that separates the flagella along their lengths. Figure 18 showssnapshots of a cross-section about halfway along the flagella length and clearly showsthe process of separation.

6 Conclusions

A computational model of the interaction of multiple bacterial flagella was presented.The method of regularized Stokeslets is based on the superposition of exact solutionsof the Stokes equations with external force given by a linear combination of cutofffunctions. These functions concentrate a given force in a small sphere rather than ata point, as is the case with delta distributions. The resulting velocity expression iscalled a regularized Stokeslet and the corresponding regularized rotlet was derived.These two types of elements were used to introduce the torque generated by themotors that rotate the flagella and the forces that provide elasticity to the flagella.

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The forces in the model presented here are based on a network of springs of a givenresting length and stiffness; however, other elasticity models can easily be used.

The tests performed for a single flagellum document the effect of different param-eters on its motion. In general, the applied torque at a single location at the frontof the flagellum introduces the rotation and the elastic forces maintain the structuralintegrity of the flagellum. The helical shape and the rotation necessarily generateforward swimming motion in a Stokes flow. The values for some of the parametersthat lead to a faster swimming speed were estimated. In particular, the results showthat the maximum swimming speed is achieved when the pitch of the helix has a givenvalue regardless of the length of the flagellum. In addition, the linear dependence ofthe swimming and angular speeds on the torque was verified. The proportionalityconstants in these relations depend on numerical parameters such as the regulariza-tion parameters δs and δr. Since these represent the spreading distance of the forcesand torques, they should be comparable in size to the flagellar radius. More specificdetails of the dependence of errors on the numerical parameters can be found in [3].

The far-field velocity is not significantly affected by the regularization. Therefore,away from the front of the flagellum where the torques are located, the effect of theforces is more substantial than the effect of the torque since the rotlet expressiondecays faster than the Stokeslet expression as a function of distance. Of course,the forces depend implicitly on the torque since the entire system is coupled. Thefluid flow surrounding the flagellum show the rotation of the cross-sections and alsoshow circulation regions of alternating sign coincident with the helix wavelength (seeFigure 7).

The model was then applied to three interacting flagella. The front sections ofthe flagella were kept approximately at a fixed distance in order to simulate theirconnection to the bacterial body. In order to analyze the hydrodynamic interactionof the flagella without artificially twisting them around one another, a counter-torquewas added to cancel the rotation of each flagellum around the others that the motorsgenerate. The counter-torque can also be thought of as the effect of the counter-rotating bacterial body.

When the three flagellar motors rotate counterclockwise, the hydrodynamic inter-action of the flagella can lead to bundling. Our results show that the flow near thefront of the flagella is mainly a combination of all the torques, which have compara-ble magnitude. Further away from the front, the flow on planes perpendicular to theaxes of the flagella appears to become a single rotation whose center is outside thetriangle defined by the three flagellar cross-sections (see Figure 12). This flow causesthe flagella to rotate in circles as a unit but also draws them closer together, resultingin bundling. In this case the structure maintains its orientation.

When one of the motors turns in the opposite direction, the flow patterns changedramatically. The generated flow near the cross-sections has a dipole character due tothe counter-rotation of two nearby flagella. This causes them to separate and forcesthe front sections to change their orientation erratically consistent with tumbling.There is no significant net forward motion in this case and the tumbling occurs in afaster time scale than bundling.

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7 Appendix

7.1 Derivation of the Regularized Stokeslet

The Stokes equations are

0 = −∇p+ ∆~u +~f (9)

0 = ∇ · ~u. (10)

We assume that the force is given by ~f(~x) = ~f0φδ(~x−~x0). For simplicity we write~X = ~x− ~x0 and r = | ~X|.

We take the divergence of Eq. (9) and use Eq. (10) to eliminate the velocity andget an equation for the pressure

∆p = ∇ ·~f = ~f0 · ∇φδ

whose solution is

p(~x) = ~f0 · ∇Gδ( ~X), where ∆Gδ = φδ.

Substituting this expression for p(~x) into Eq. (9) results in the following equationfor the fluid velocity

∆~u = (~f0 · ∇)∇Gδ −~f0φδ

whose solution is

~u(~x) = (~f0 · ∇)∇Bδ( ~X)−~f0Gδ( ~X), where ∆Bδ = Gδ.

For the particular cutoff function used throughout this article

φδ(~x) =15δ4

8π(r2 + δ2)7/2

we have that

p(~x) = (~f0 · ~X)2r2 + 5δ2

8π(r2 + δ2)5/2(11)

and

~u(~x) = ~f0r2 + 2δ2

8π(r2 + δ2)3/2+

(~f0 · ~X) ~X

8π(r2 + δ2)3/2. (12)

Eq. (12) is also displayed as Eq. (5).

7.2 Derivation of the Regularized Rotlet

We begin with the regularized Stokeslet found in the previous section, Eq. (11)–(12).

P0 =(5δ2 + 2r2)(~f0 · ~X)

8π(r2 + δ2)5/2, ~U0 =

~f0(r2 + 2δ2)

8π(r2 + δ2)3/2+

(~f0 · ~X) ~X

8π(r2 + δ2)3/2

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where ~f0 is an arbitrary vector constant. Now, (P0, U0) satisfy ∇P0 = ∆ ~U0 + ~f0φδ.Other solutions can be found by differentiation. For example, let ~g be an arbitraryconstant vector, and define

P1 = ~g · ∇P0 and ~U1 = (~g · ∇) ~U0.

One can check that (P1, ~U1) are solutions of the Stokes equations with forcing term

(~g · ∇)~f0φδ. If we expand ~U1 using Eq. (12), we obtain

U1 =−(r2 + 4δ2)

8π(r2 + δ2)5/2(~g · ~X)~f0 +

1

8π(r2 + δ2)3/2(~f0 · ~X)~g

+(~g ·~f0) ~X

8π(r2 + δ2)3/2− 3(~f0 · ~X)(~g · ~X) ~X

8π(r2 + δ2)5/2. (13)

We look for the antisymmetric part of this expression with respect to ~f0 and ~g.Notice that the third and fourth terms on the right side of Eq. (13) are symmetric

with respect to ~f0 and ~g (switching ~f0 and ~g makes no difference). To determine thesymmetry in terms one and two, we may write them as

(a+ b)

2[(~g · ~X)~f0 + (~f0 · ~X) ~g ] +

(a− b)2

[(~g · ~X)~f0 − (~f0 · ~X) ~g ]

where

a =−(r2 + 4δ2)

8π(r2 + δ2)5/2and b =

1

8π(r2 + δ2)3/2.

Therefore, the antisymmetric part of Eq. (13) is

~Uδ,r(~x) =(a− b)

2[(~g · ~X)~f0 − (~f0 · ~X)~g]

=−(2r2 + 5δ2)

8π(r2 + δ2)5/2[(~g · ~X)~f0 − (~f0 · ~X)~g].

Defining ~L = ~f0 × ~g, our equation for ~Uδ,r becomes

~Uδ,r(~x) =(2r2 + 5δ2)

16π(r2 + δ2)5/2(~L× ~X)

which is the equation given in Eq. (6). ~Uδ,r represents the flow due to a regularized

rotlet of strength ~L. We mention that as δ approaches zero, we recover the singularform of the rotlet, Eq. (3). However, the regularized rotlet is a bounded function thatcan be evaluated everywhere.

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8 Acknowledgments

This work was funded by NSF grant #DMS-0094179. The work was conducted inthe Center for Computational Science at Tulane and Xavier Universities which issupported in part by DOE grant #DE-FG02-01ER63119.

References

[1] H. C. Berg, The rotary motor of bacterial flagella, Annu. Rev. Biochem. 72(2003), 19–54.

[2] R. Cortez, The method of regularized Stokeslets, SIAM J. Sci. Comput. 23 (2001),1204–1225.

[3] R. Cortez, L. Fauci, and A. Medovikov, The method of regularized Stokesletsin three dimensions: Analysis, validation, and application to helical swimming,Submitted, 2003.

[4] J. Gray, Animal locomotion, Weidenfeld & Nicolson, London, 1968.

[5] J. J. L. Higdon, A hydrodynamic analysis of flagellar propulsion, J. Fluid Mech.90 (1979), 685–711.

[6] , The hydrodynamics of flagellar propulsion: Helical waves, J. Fluid Mech.94 (1979), 331–351.

[7] R. Kamiya and S. Asakura, Helical transformations of salmonella flagella invitro, J. Mol. Biol. 106 (1976), 167–186.

[8] M. Kim, J. C. Bird, A. J. Van Parys, K. S. Breuer, and T. R. Powers, A macro-scopic scale model of bacterial flagellar bundling, Submitted to PNAS, 2003.

[9] M. J. Lighthill, Flagellar hydrodynamics, SIAM Rev. 18 (1976), 161–230.

[10] , Helical distributions of Stokeslets, J. Eng. Math. 30 (1996), 35–78.

[11] T. R. Powers, Role of body rotation in bacterial flagellar motion, Phys. Rev. E65 (2002), 040903(R).

[12] E. M. Purcell, The efficiency of propulsion by a rotating flagellum, Proc. Natl.Acad. Sci. USA 94 (1997), 24–29.

[13] M. Ramia, D. L. Tullock, and N. Phan-Thien, The role of hydrodynamic inter-action in the locomotion of mircroorganisms, Biophys. J. 65 (1993), 755–778.

[14] L. Turner, W. S. Ryu, and H. C. Berg, Real-time imaging of fluorescent flagellarfilaments, J. Bacteriol. 182 (2000), no. 10, 2793–2801.

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