A study of bacterial flagellar bundling.

Bulletin of Mathematical Biology 67 (2005) 137–168
www.elsevier.com/locate/ybulm
A study of bacterial flagellar bundling
Heather Floresa, Edgar Lobatonb, Stefan Méndez-Diezc,
Svetlana Tlupovad, Ricardo Cortezd,∗
aDepartment of Mathematics, University of Nebraska, 203 Avery Hall, P.O. Box 880130, Lincoln,
NE 68588, USA
bDepartment of Mathematics, University of California at Berkeley, 970 Evans Hall #3840, Berkeley,
CA 94720, USA
cDepartment of Applied Mathematics and Scientific Computing, University of Maryland,
3103 Mathematics Building, College Park, MD 20742, USA
dDepartment of Mathematics, Tulane University, 6823 St. Charles Avenue, #424, New Orleans,
LA 70118, USA
Received 3 November 2003; accepted 30 June 2004
Abstract
Certain bacteria, such as Escherichia coli (E. coli) and Salmonella typhimurium (S. typhimurium),
use multiple flagella often concentrated at one end 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 a
corkscrew motion. We present a computational model of the flagellar motion and their hydrodynamic
interaction. The model is based on the equations of Stokes flow to describe the fluid motion. The
elasticity of the flagella is modeled with a network of elastic springs while the motor is represented
by a torque at the base of each flagellum. The fluid velocity due to the forces is described by
regularized Stokeslets and the velocity due to the torques by the associated regularized rotlets. Their
expressions are derived. The model is used to analyze the swimming motion of a single flagellum
and of a group of three flagella in close proximity to one another. When all flagellar motors rotate
counterclockwise, the hydrodynamic interaction can lead to bundling. We present an analysis of the
flow surrounding the flagella. When at least one of the motors changes its direction of rotation, the
same initial conditions lead to a tumbling behavior characterized by the separation of the flagella,
∗ Corresponding author.
E-mail addresses: hflores@math.unl.edu (H. Flores), edgar_loba@yahoo.com (E. Lobaton),
stefan@alumni.uchicago.edu (S. Méndez-Diez), tlupova@tulane.edu (S. Tlupova), cortez@math.tulane.edu
(R. Cortez).
0092-8240/$30 © 2004 Society for Mathem atical Biology. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.bulm.2004.06.006

138 H. Flores et al. / Bulletin of Mathematical Biology 67 (2005) 137–168
changes in their orientation, and no net swimming motion. The analysis of the flow provides some
intuition for these processes.
© 2004 Society for Mathematical Biology. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Single-celled bacteria, such as Escherichia coli (E. coli) induce locomotion through the
use of multiple flagella often concentrated at one end of the organism. Each flagellum is
flexible and maintains roughly a left-handed helical shape. At the base of each flagellum
is a small rotary motor that can turn clockwise or counterclockwise. It is known that
when the motors all turn counterclockwise (when viewed from behind the flagella), the
flagella tend to gather together to form a single helix [see for example Turner et al.
(2000)]. This movement, known as bundling, results in forward motion of the cell. When
some or all of the motors turn clockwise, the flagella go through a sequence of shape
transformations changing amplitude, pitch, and sometimes handedness. The consequence
is that the bundle unravels and the flagella separate. This motion is known as tumbling and
changes the movement of the organism into one without a preferred direction, resulting
in no appreciable net motion (Berg, 2003). Bundling and tumbling together enable the
organism to move in one direction and reorient itself to move in another direction
where there may be more favorable conditions for survival. This process may be in
response to chemical stimuli (chemotaxis) or other factors such as temperature and light
intensity.
Previous work on the modeling of helical swimming motions has combined analytical
and numerical methods and has focused mainly on organisms with a single flagellum.
Lighthill (1976, 1996) provided mathematical analysis of the motion of a thin tube with
helical shape using slender-body theory. In this theory, the flagellum is replaced with a
distribution of Stokeslets and dipoles along its centerline. The analysis provided refined
resistance coefficients for this motion and modifications to account approximately for
the drag force on the cell body. Higdon (1979a,b) combined mathematical and numerical
analyses applied to the motion of an organism with a spherical body and a single flagellum.
The flagellum was also modeled using slender-body approximations. The body was
assumed to be spherical which allowed the use of images to impose approximate boundary
conditions on its surface. A rotlet at the center of the cell body was included in order to
balance the rotation induced by the flagellum. Ramia et al. (1993) used a computational
approach based on the boundary element method to model the motion of an organism with
spherical body and a single helical flagellum. Their study included motions near walls and
near another organism of the same shape. The description of the organism is similar to
that in Higdon (1979b); however, the Green’s function approach is traded for a boundary
element method in order to address cases with multiple walls (of finite extent) and multiple
organisms. Goto et al. (2001) also use the boundary element method to compute the
swimming speed and cell body rotation of a singly-flagellated bacterium. Given the angular
velocity of the motor, the geometry of the cell body and flagellum, and assuming both move
as rigid bodies, they are able to compute the six unknowns that represent the swimming

H. Flores et al. / Bulletin of Mathematical Biology 67 (2005) 137–168 139
velocity and angular velocity of the cell body. These computations are not dynamic.
Goto et al. (1999) performed computations using a cell body and three flagella using
the same boundary element method as in Goto et al. (2001); however, the dynamics
of the motion, particularly bundling, were not addressed since only the linear and
angular velocities of the cell body were determined from a static balance of forces and
torques.
The works mentioned above assumed from the outset the shape of the flagellum for all
time. For helical waves, the centerline was assumed to be given by a helix of the 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 a solid
body without deformation. This assumption simplifies the mathematical analysis but is
not realistic for bacteria like E. coli and S. typhimurium, whose flagella should not be
considered rigid screws since they assume a variety of distinct helices depending on their
environment [see for example Kamiya and Asakura (1976)].
Powers (2002) considered a single straight but flexible filament which is rotated
at one end in a circular fashion around an axis parallel but not coincident with the
filament. The rotation simulates the cell body rotation and the filament represents a single
flagellum. Based on steady states of the filament driven by various rotation frequencies,
conclusions were drawn regarding the possibility of bundling. However, only a single
filament was considered and the hydrodynamic interactions among neighboring flagella
were not taken into account. The Stokes flow was included only through the use of a
transverse friction coefficient following local resistive-force theory (for a rod) and slender-
body approximations, and the conclusions were based on the helical shape of the rotating
isolated filament.
While some of the works cited above apply only to eukaryotic flagella and some may
apply also to prokaryotic ones, the goal of the present study is to determine conditions that
tend to produce bundling of various prokaryotic flagella in close proximity to one another.
The emphasis here is on the role of the hydrodynamic interaction of the flagella in the
processes of bundling and tumbling. No restrictions on the wave amplitudes or flagellum
dimensions are imposed. In this way, the methodology used here can also be applied to
other organisms 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 time but
allow for deformations during the interactions. Therefore, our model includes a mechanism
designed to provide a certain amount of elasticity to the flagellum so that a helical shape
is preferred but deviations from it are allowed. The simulations are based on solutions of
Stokes equations in the presence of external forces given by shape functions that smoothly
approximate delta distributions. This is the basis of the method of regularized Stokeslet
(Cortez, 2001; Cortez et al., in press) used here. In addition, the rotation induced by the
motors at the base of each flagellum is modeled with a regularized rotlet which is derived as
the antisymmetric part of the derivative of the regularized Stokeslet. The rotlet represents
a localized torque.

140 H. Flores et al. / Bulletin of Mathematical Biology 67 (2005) 137–168
Table 1
Factors for the conversion from dimensionless units to dimensional variables
Dimensionless variable Multiply by
Length L = 10−5 m
Velocity U = 10−1 m s−1
Time T = 10−4 s
Angular velocity T −1 = 104 s−1
Force density F = µU/L2 = 106 N m−3
Torque density Tq = µU/L = 10 N m−2
We note that the force (torque) density is the force (torque) per unit volume.
2. Equations
The fluid dynamics in problems of microorganism motion, where length and velocity
scales 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 of the fluid and
˜f is the external force density. If we define L and U to be a characteristic length and a
characteristic speed in the problem, we can define the dimensionless variables
x =
1
L
˜x, u =
1
U
˜u, P =
L
µU
˜P, f =
L
2
µU
˜f.
Then, after some simplification, the Stokes equations in dimensionless form become
0 =−∇ P + �u + f (1)
0 =∇ · u. (2)
These are the equations we use in our model. We mention that a typical length of an E. coli
flagellum is 10–20 µm (Turner et al., 2000; Kim et al., 2003). A typical forward swimming
speed of the cells is 10–40 µm s−1 (Turner et al., 2000; McClaine and Ford, 2002) and
the motor rotation is on the order of 100 Hz (revolutions per second) (Turner et al., 2000;
Berry, 2001; Berg, 2003; Kim et al., 2003). All computations will be performed using
dimensionless variables. In all cases, the dimensional values will be computed using the
viscosity of water, µ = 10−3 kg m−1 s−1 and the following parameters: L = 10−5 m
and U = 10−1 m s−1. The last two values provide a time scale of T = L/U = 10−4 s.
Table 1 shows explicitly the conversion factors used throughout this article.
2.1. Solutions of the Stokes equations
When a force f is exerted on the fluid, the resulting velocity field u and pressure P are
the solution of Eqs. (1) and (2). The particular case of a single point force f0 exerted at 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

H. Flores et al. / Bulletin of Mathematical Biology 67 (2005) 137–168 141
where r = ‖x − x0‖. Note that this flow is undefined at x = x0 although a distribution of
forces on a surface yields a flow that is defined everywhere.
Our computations show the motion of a flagellum or a group of flagella without a body
(or head). The motion is generated by forces along the surfaces of the flagella and by a
torque at the base of each. The torque represents the one transferred to the flagellum by
the motor at the junction with the body of the bacterium, which is currently not part of
the model. The forces, based on points and springs, are designed to keep the flagellum
in approximately the same helical shape while providing some flexibility to it. These are
described in the next section.
The velocity field that satisfies Stokes equations when a torque L0 is applied at a single
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 were
distributed over a surface. This implies that the fluid velocity becomes arbitrarily large as
the evaluation point approaches the point where the torque is applied. In our computations,
the rotlets will be placed at specific points in the fluid domain, and therefore, we will have
to compute the fluid velocity at points arbitrarily close to the rotlet location.
The singularities in the velocity expression are due to the assumption of having
point-forces and point-torques. However, the singularities can be eliminated through the
systematic regularization of the flows described above by considering forces and torques
that are applied not at single points but within small spheres centered at those points.
In this way, the forces and torques are highly concentrated but are spread over a small
neighborhood of the application points. The precise form of the force is given by a cutoff
function φ
δ
(x) which we will take to be radially symmetric and to satisfy
∫ ∫ ∫
R
3
φ
δ
(x) dx = 1,
where δ is a numerical parameter that controls the spread of the function (see Fig. 1).
Throughout this article, we will use the cutoff function
φ
δ
(x) =
15δ4
8π(r2 + δ2)7/2
(4)
where r = ‖x‖.
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 A).
U
δ,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, the
regularized Stokeslet represents a flow that is bounded for all x as long as δ > 0. Regardless
of the value of δ, the regularized Stokeslet in Eq. (5) is an exact solution of the Stokes
equation for the given form of the force.

142 H. Flores et al. / Bulletin of Mathematical Biology 67 (2005) 137–168
Fig. 1. The cutoff function in Eq. (4) for three values of δ.
The associated regularized rotlet is derived from the antisymmetric part of a directional
derivative of the regularized Stokeslet (see Appendix A). The result for a torque L0
centered at x0 is
U
δ,r (x; x0, L0) =
(2r2 + 5δ2)
16π(r2 + δ2)5/2
[L0 × (x − x0)]. (6)
Here too, as δ → 0 we recover the original rotlet expression. It is important to emphasize
that the regularized rotlet yields finite velocities everywhere simply because of the assumed
form of the torque. At points far from the torque, the regularized expression is nearly
indistinguishable from the singular counterpart. Near the torque, the regularized expression
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. Each
flagellum is a tubular structure made of discrete particles connected by springs. Some of the
springs connect particles around the cross-sections and others connect particles between
neighboring cross-sections. The forces between two connected particles, x j and xk , are
computed using Hooke’s Law:
f j k =
k0
L jk0
(L jk − L jk0 )
xk − x j
L jk
, fkj = −f j k, L jk = ‖xk − x j‖ (7)
where k0 is the stiffness constant, L jk is defined to be the distance between xk and x j at
time t , and the spring resting length, L jk0 , is defined to be the initial distance between xk
and x j , so at time t = 0 the force between the particles is zero. Since the torque models the
effect of the motor, it is applied only at one point at the base of each flagellum. When the

H. Flores et al. / Bulletin of Mathematical Biology 67 (2005) 137–168 143
torque is applied and the particles that describe the flagellum move, the distance between
them changes, activating spring forces applied to both particles at the spring endpoints, in
equal and opposite pairs. In this way, the total force is always zero. The forces are designed
to maintain approximately the initial resting length between the 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 the flagellum. The stiffness constants may
also be different for each spring, and their values control the flexibility (elasticity) of the
flagellum. Higher stiffness results in more rigid structures.
In our model, a particle may be connected by springs to several other particles.
Therefore, the force at a particle may have several contributions from several springs. We
use the sum of the forces at each point in the computation of the velocity as described
below. A torque of constant magnitude and perpendicular to the base of the flagellum (see
Section 4) is applied only at one point at the base of each flagellum to generate the rotation
due to the motors. As the flagellum moves, the torque direction is adjusted so that it remains
perpendicular to the base of the flagellum.
The motion of the flagella is computed as follows. Given the positions of all particles at
time t , all forces f j are computed based on the geometry of each flagellum. A torque Li of
a fixed magnitude is applied at the base of each flagellum in the direction perpendicular to
the cross-section of the base. Once the forces and torques are known, the velocity at any
location xk is computed using the regularized Stokeslet and rotlet formula:
dxk
dt
= u(xk) =
Nr
∑
i=1
U
δ,r (xk; yi , Li ) +
Ns
∑
j=1
U
δ,s(xk; z j , f j ) (8)
where Nr is the number of rotlets of strengths Li located at yi and Ns is the number
of Stokeslets of strengths f j located at z j . The expressions in this formula are given by
Eqs. (5) and (6).
The position of each particle changes according to the fluid velocity so that each particle
position can be updated after a small time interval. At that time, the new particle positions
define new forces and new torques which are used for the next time step. In this way,
the forces impose a time scale in the problem where the velocity of the particles is the
superposition of the regularized Stokeslets and the rotlet. The time evolution of the particle
positions is computed using a fourth-order Runge–Kutta method.
3.1. Comments on the numerical method
While the use of rotlets is new, the method of regularized Stokeslets, given by
dxk
dt
= u(xk) =
Ns
∑
j=1
U
δ,s(xk; z j , f j ),
has been used in two and three dimensions. The method can be used in two ways. The
forward method consists of computing the velocity field due to given forces that are
calculated from the geometry of the body, for example. The inverse problem consists of
computing the forces (or the Stokeslets strengths) for the body to move with a prescribed
velocity. The latter is necessary when the body’s velocity is known in advance, and it

144 H. Flores et al. / Bulletin of Mathematical Biology 67 (2005) 137–168
requires the inversion of the Stokeslet operator. More details are found in Cortez (2001)
and Cortez et al. (in press). This is the approach used in Lighthill (1976), Goto et al. (1999,
2001) and Ramia et al. (1993) because the geometry was known. In our case, changes in
the geometry are part of the solution and so the motion is not known in advance.
In the current problem, the forces along the flagella are computed at every time step
from Hooke’s law based on their geometry. Then, the velocity field due to those forces (and
the rotlets) is computed directly from Eq. (8). No operator needs to be inverted since the
flagellar velocities are part of the computed solution, not a prescribed boundary condition.
The method does not distinguish between a point on the body surface or a fluid marker
at the same location. Since the body’s velocity is computed using the same formula as
the fluid velocity, the no-slip boundary condition is automatically satisfied. The fluid is
dragged by the body as it moves. This is the same type of approach used by other methods
(Dillon and Fauci, 2000; Peskin, 2002).
The analysis of the convergence of the method of regularized Stokeslets as the
discretization is refined and the regularization parameter is reduced is found in Cortez et al.
(in press). The main result is that the error in the velocity field near the body, as compared
to a boundary integral formulation, is O(δ)+ O(�s2/δ3), where �s2 is a discrete element
of area on the body. The error decreases to O(δ2) + O(�s2/δ3) away from the body. This
allows one to choose the regularization parameter δ relative to the surface discretization in
such a way that the method converges (for example, �s ∼ δ2).
4. A single flagellum
We present the model of a single flagellum without a body. In E. coli, each flagellum
is shaped into a left-handed helix that extends from the cell body. Our goal is to create a
solid, yet flexible, representation of the flagellum. We define the flagellum to be a helical
tube of total length � with cross-sections perpendicular to the tangent vector (see Fig. 2).
Each cross-section is an n-sided polygon. The helix has a varying radius, R(s), which is
implemented using an arctangent envelope that allows the radius to start from zero and
increase to some fixed value. This is similar to the envelope used in Higdon (1979b) and
Ramia et al. (1993). The initial conditions for the particles are as follows:
x(s)=α(s)
y(s)=−R(s) cos
(
2πn p
( s
�
))
z(s) = R(s) sin
(
2πn p
( s
�
))
where the amplitude R(s) satisfies 0 ≤ R(s) ≤ Rh and is given by
R(s) = Rh
[
1
π
arctan
(
β
( s
�
− γ
))
+
1
2
]
,
n p 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 flagellum for t > 0
is not specified but found as part of the computation. The benefit of the arctangent envelope
is that it defines an axis of rotation at the base of the flagellum while creating a helix of

H. Flores et al. / Bulletin of Mathematical Biology 67 (2005) 137–168 145
Fig. 2. Side view of one helical flagellum.
Fig. 3. Spring connections: (a) cross-sectional springs; (b) consecutive springs; (c) diagonal springs.
constant radius in the rear as can be seen in Fig. 2. For economy of computation, we
choose n = 3 for the cross-sections, the smallest number possible, so that each flagellum
is a structure made of 3 helices defining triangular cross-sections. This choice, however, is
not a restriction.
In Fig. 3, we show the spring connections that are defined between different particles.
First, each particle on a helix is connected to the corresponding particles on the other two
helices [Fig. 3(a)]. This defines the cross-sections of the flagellum and 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 spring forces; this is shown in
Fig. 3(b). These will be referred to as consecutive forces with stiffness constants kb. We also
define diagonal forces around the surface of the flagellum. The bold lines are the diagonal
forces with stiffness constants kc [Fig. 3(c)]. We note that these stiffness constants actually
have units of force density as defined in Eq. (7). However, since the resting lengths of the
springs (L jk0 ) will remain constant throughout the simulations, the parameters (ka , kb, kc)
are appropriate.
A biological flagellum is a helical tube composed of flagellin monomers arranged
in a pseudohexagonal lattice (Jones and Aizawa, 1991). Some monomer strands

146 H. Flores et al. / Bulletin of Mathematical Biology 67 (2005) 137–168
Fig. 4. Initial configuration of a single flagellum.
(protofilaments) are nearly parallel to the filament axis while others (so called 5-start and
6-start) form helices along the surface of the flagellum. The latter are represented by the
diagonal springs in our model and the protofilaments are represented by the consecutive
springs. The number of particles used in the construction of our model has been reduced
for computational purposes only, but this is not essential. Our structure has the benefit of
being composed entirely of triangles. Triangles, unlike other polygons, have the property
that they preserve angles when preserving the length of the sides. A slight modification of
our model can be made to represent the flagellin monomers as depicted in Fig. 6 of Jones
and Aizawa (1991). This has been done recently in Lim and Peskin (2004) to address
whirling instabilities of some elastic filaments.
The external torque is positioned at the center of the first cross-section of the flagellum
(see also Fig. 9). Because of the varying helix amplitude, the first cross-section is
perpendicular to the central axis of the helical flagellum. This makes it appropriate to define
the torque orthogonal to the first cross-section. We note that since the torque is imposed
externally to simulate the motor, the net torque will not be zero.
Fig. 4 shows the initial conditions for this problem. Since the flow due to the rotlet
decays as r−2 for large values of r , nearby cross-sections experience more rotation than
ones far from the rotlet. The distances between contiguous cross-sections deviate from
their resting-lengths due to the rotation. This is particularly pronounced near the front
of the flagellum. In response to the stretching, the springs exert 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 on propagating rotation along the entire
flagellum. They also affect the amount of twist developed along the flagellum.
4.1. Parameter dependence of a single flagellum
The motion of a single flagellum is induced by a torque applied at its base. The
parameters were chosen so that the flagellum held together in its helical form while still
allowing it to rotate with some elasticity. The rotation of the entire structure in a viscous
fluid necessarily results in forward swimming motion of the flagellum. In this section,

H. Flores et al. / Bulletin of Mathematical Biology 67 (2005) 137–168 147
Table 2
Comparison of forward swimming speed as a function of the spring stiffness
ka , kb, kc Dimensionless speed, v (×10−4) Speed = Uv, (µm s−1)
4, 4, 4 1.2272 12.272
8, 8, 8 1.2245 12.245
12, 12, 12 1.2219 12.219
16, 16, 16 1.2204 12.204
The dimensionless parameters used were δs = δr = 0.052, L = 0.001, n p = 3, p = 0.2399, � = 1.3.
we discuss the effect of some parameters in the model on the swimming speed and angular
speed of the flagellum. In most of the numerical experiments, the flagellum length was
fixed at � = 1.3 (equivalent to 13 µm). In all numerical experiments, the maximum
helix radius was fixed to Rh = 0.06, which is about 5% of the flagellum length. The
radius of the flagellum was fixed to a value of 0.012, which is about 1% of the flagellum
length. This corresponds to a somewhat thick flagellum of 120 nm in radius, which is
thicker than a typical E. coli flagellum (Turner et al., 2000). Alternatively, one may assume
that the computational flagellum is short for its thickness. Experimental studies also have
been conducted on flagellar models which have a small length-to-radius ratio (Kim et al.,
2003). We present other computational experiments which use longer flagella as indicated
in Table 5. Based on numerical experiments, most of the reported results use N = 51
cross-sections to discretize the flagellum since larger numbers of cross-sections had no
significant effect on forward motion or stretching.
In a computational flagellum, there are many different parameters that affect its motion.
To analyze the effect of one parameter, we monitored the angular velocity, forward
displacement, and structural stability. Here we discuss the changes caused by varying the
stiffness constant values (ka, kb, kc), the regularization parameters for both the Stokeslet
(δs) and the rotlet (δr ), the magnitude of torque applied (L), number of periods in a
flagellum (n p), pitch (p), and arclength (�). All computations in this section were run
up to a final dimensionless time of 400, corresponding to t = 0.04 s. The linear speed of
the flagellum was computed from its final and initial positions.
The spring constants must be set to large enough values in order to maintain the helical
shape of the flagellum. Once the spring constants are sufficiently large, they have little
effect on the structure of the flagellum and forward motion. As can be seen in Table 2,
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 stiffness constants were set equal to 12
for all computations in this section.
The regularization parameters for the Stokeslet and the rotlet are independent of one
another. These were set to a multiple of the distance between the cross-sections in the
flagellum. For N cross-sections along the helix of length �, this distance is �/(N − 1).
For the tests in Tables 2, 3 and 5, this cross-sectional distance was 0.026. Changing the
Stokeslet’s parameter δs has contrasting effects on forward motion and the stretching
of the flagellum. The value of δs should be comparable to the separation between cross-
sections along the flagellum so that the cutoff functions of nearby forces can overlap.

148 H. Flores et al. / Bulletin of Mathematical Biology 67 (2005) 137–168
Table 3
Effect of the Stokeslet regularization parameter δs on the swimming speed and filament stretching
δs Speed (µm s−1) Stretching (% of �)
0.026 7.692 0.23
0.052 12.219 0.77
0.078 14.522 2.31
The dimensionless parameters used were � = 1.3, δr = 0.0526, L = 0.001, n p = 3, p = 0.2399.
This is needed for accuracy purposes (Cortez et al., in press). Our results, shown in
Table 3, show that larger values of δs yielded faster forward motion but also produced
more stretching. This is to be expected since larger values of δs produce forces that are
spread over larger regions and reduce the maximum value of the cutoff function being
used. This produces smaller reaction forces by the springs and allows more stretching.
The torque magnitude L and the rotlet regularization parameter δr have a large
effect on forward motion and the structure of the flagellum. We varied the amount of
torque applied and studied its effect on forward motion and angular velocity. Physical
experiments performed by Purcell (1997) assumed linear relationships between the torque
magnitude and the axial angular speed, and between the swimming speed and angular
speed, consistent with Stokes flow. He expressed these relationships with the scalar
equations
F = Av + Bω, (9)
L = Cv + Dω, (10)
where F is the net applied external force magnitude, v is swimming speed, ω is the angular
speed, L is the net torque magnitude, and A, B , C , and D are constants that depend on
the geometry of the flagellum. These equations reflect the relationships in the direction
of the axis of the flagellum under the assumption that the other components average to
zero as the flagellum corkscrews its way through the fluid. There is no net external force
in our system, so F = 0 in Eq. (9) and the correspondence between v and ω is a line
through the origin. This, together with Eq. (10), results in a linear relation between ω and
the net torque, and also between v and the net torque. We computed the flagellar motion
for a wide range of torque magnitudes L and verified that our model produces a linear
relation between the swimming speed v and ω, between the torque magnitude L and ω,
and therefore, between the torque magnitude and the swimming speed (see Fig. 5). The
linear relationships hold for different values of δr . Eq. (6) indicates that the rotlet velocity
can be written as U
δ,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 value increases, resulting in
faster rotation.
The number of periods, n p , in the helix comprising each flagellum also has an effect
on both its swimming motion and its ability to hold its shape. We report results 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 fewer periods, one expects that
for a given driving torque, the viscous drag would have a larger effect on a flagellum with

H. Flores et al. / Bulletin of Mathematical Biology 67 (2005) 137–168 149
Fig. 5. Magnitude of torque density L vs. angular speed ω (left); magnitude of torque density L vs. linear
swimming speed v (middle); angular speed ω vs. swimming speed v (right).
Table 4
Comparison of the forward swimming speed and angular speed of a flagellum with constant pitch as a function
of the number of helical periods along its length
n p � (µm) N Speed (µm s−1) ω (rev s−1)
2 8.7 34 17.0 20
3 13.0 51 12.2 15
4 17.3 68 9.6 10
The driving torque was fixed with magnitude L = 0.001. The dimensionless parameters used were δs = δr =
0.052, and pitch p = 0.24.
more periods causing a smaller angular velocity and, therefore, slower swimming motion.
Our results, shown in Table 4, show that having fewer periods causes less stretching and
resulted in faster swimming, as expected. For this experiment, the number of cross-sections
along the flagellum was increased as the number 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 rotlet of fixed magnitude.
The pitch was varied by fixing the flagellum length and changing the number of turns in
the helix. Having more or fewer turns per arclength did not have an effect on the angular

150 H. Flores et al. / Bulletin of Mathematical Biology 67 (2005) 137–168
Table 5
Comparison of speed as a function of pitch for flagella of different lengths and different number of helical periods
� (µm) N n p Pitch (µm) ω (rev s−1) Speed (µm s−1)
17 51 2 7.61 100 37.0
17 51 3 4.36 100 40.6
17 51 3.5 3.20 100 34.7
17 51 4 2.28 100 24.6
25 75 2 11.68 70 19.5
25 75 3 7.47 70 27.9
25 75 4 5.12 70 30.7
25 75 4.5 4.22 70 29.4
The dimensionless parameters used were δs = δr = 0.052, and L = 0.005.
velocity ω. However, the model shows that the maximum swimming speed is achieved
for a given pitch which is not particularly sensitive to arclength. Intuitively, if the pitch
is 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 flagellum
does not swim efficiently either. Thus, one expects a selected pitch to yield a maximum
swimming speed [see also Cortez et al. (in press)]. Table 5 shows the results for flagella
of fixed length and variable number of turns (different pitch). This was done with two
different arclengths while maintaining the same cross-sectional distance. We define the
optimal pitch to be the one that yields the largest swimming speed. Based on the results we
estimate the optimal pitch to be about 4.5–5 µm. For comparison, we mention that pitch
measurements of stationary normal flagellar filaments found in Turner et al. (2000) and
Kim et al. (2003) are in the range 1–3 µm.
4.2. Flow generated by a flagellum
The motion of the flagellum generates fluid flow around it. Two-dimensional projections
of the fluid flow on planes perpendicular to the axis of the flagellum are shown in Fig. 6.
The triangle in each plot is the projection of a flagellum cross-section. The top-left
plot shows the first cross-section, where the torque is applied. The flow here is largely
dominated by the torque. Further back along the flagellum, as the effect of the torque
decays, the flow is influenced more substantially by the spring forces that cause the
flagellum cross-sections to rotate in circles. Fig. 7 shows the flow projected onto a plane
that includes the flagellum axis. The top plot is the initial position of the flagellum and
the bottom plot shows the flagellum and the flow around it at t = 0.02 s. The forward
motion of the flagellum is apparent and is also indicated by the fluid motion since there is
flow coincident with the helix tangent. The figure also shows regions of fluid rotation in
alternating directions that approximately coincide with the helix shape.
5. Three flagellum model
In this section, we discuss the interactions among three flagella through the fluid flow
they generate. In particular, we are interested in the role of the fluid motion in the processes

H. Flores et al. / Bulletin of Mathematical Biology 67 (2005) 137–168 151
Fig. 6. Flow fields at the 1st, 3rd, 10th, and 51st (last) cross-sections of a flagellum.
of bundling and tumbling. Our model includes three flagella constructed in the same
way as described in the previous sections. Each one has a motor modeled by a torque
in the center of the first cross-section. While we do not construct explicitly the body of
the organism, the model includes features related to the effects of the bacterial body on
the motion. The three flagella were placed equally-spaced around a circle whose radius
represents the radius of the bacterial body. To simulate the front of these flagella being
connected to a rigid body and not being able to freely change their distances and orientation
relative to one another, the front sections of the flagella were connected by springs. These
springs connect the center points of the first three cross-sections of each flagellum with
the corresponding cross-sections of the other two flagella. Fig. 8 shows these connections.
Under this construction the center points of the first cross-sections of the three flagella
form an equilateral triangle and remain at approximately the same distance throughout the
simulation.