Cell Motility and the Cytoskeleton 53:103–124 (2002)
Computer Simulation of Flagellar Movement
VIII: Coordination of Dynein by Local
Curvature Control Can Generate Helical
Bending Waves
Charles J. Brokaw*
Division of Biology, California Institute of Technology, Pasadena
Computer simulations have been carried out with a model flagellum that can bend
in three dimensions. A pattern of dynein activation in which regions of dynein
activity propagate along each doublet, with a phase shift of approximately 1/9
wavelength between adjacent doublets, will produce a helical bending wave. This
pattern can be termed “doublet metachronism.” The simulations show that doublet
metachronism can arise spontaneously in a model axoneme in which activation of
dyneins is controlled locally by the curvature of each outer doublet microtubule.
In this model, dyneins operate both as sensors of curvature and as motors. Doublet
metachronism and the chirality of the resulting helical bending pattern are regulated by the angular difference between the direction of the moment and sliding
produced by dyneins on a doublet and the direction of the controlling curvature for
that doublet. A flagellum that is generating a helical bending wave experiences
twisting moments when it moves against external viscous resistance. At high
viscosities, helical bending will be significantly modified by twist unless the twist
resistance is greater than previously estimated. Spontaneous doublet metachronism must be modified or overridden in order for a flagellum to generate the planar
bending waves that are required for efficient propulsion of spermatozoa. Planar
bending can be achieved with the three-dimensional flagellar model by appropriate specification of the direction of the controlling curvature for each doublet.
However, experimental observations indicate that this “hard-wired” solution is not
appropriate for real flagella. Cell Motil. Cytoskeleton 53:103–124, 2002.
© 2002 Wiley-Liss, Inc.
Key words: cilia; flagella; helix; motility; spermatozoa; writhe
INTRODUCTION
The various patterns of bending produced by eukaryotic flagella and cilia require the coordinated operation of tens of thousands of individual motor enzymes
(dyneins) in each flagellum or cilium. Mathematical
modelling has shown that the coordination required for
oscillation and propagation of planar bending waves by a
flagellum could result from a simple local control of
dynein activity by the curvature of the flagellum, if
bending is restricted to a single plane [Brokaw, 1971,
1972a]. Many flagella normally generate non-planar, often nearly helical, bending patterns, and helical bending
can be induced in some sperm flagella that normally
generate planar bending waves [Brokaw, 1966; Woolley
© 2002 Wiley-Liss, Inc.
and Vernon, 2001]. In a helical bending wave, regions of
dynein activity are phase shifted around the circumference of the axoneme, in a pattern that will be referred to
as doublet metachronism. Doublet metachronism suggests that regions of dynein activity propagate around the
circumference, as well as along the length. This article is
*Correspondence to: Dr. Charles J. Brokaw, Kerckhoff Marine Laboratory, 101 Dahlia, Corona del Mar, CA 92625.
E-mail: [email protected]
Received 11 March 2002; Accepted 3 May 2002
Published online 12 August 2002 in Wiley InterScience (www.
interscience.wiley.com). DOI: 10.1002/cm.10067
104
Brokaw
a first attempt to examine whether local control of dynein
activity by curvature can provide the internal coordination required in a flagellum capable of bending in any
direction.
Previous study of the mechanisms of flagellar
bending has been concentrated on two-dimensional
bending because planar bending waves can be photographed easily to obtain precise quantitative description,
and because the mathematical analysis is easier in two
dimensions. Initial analysis was influenced by the idea
that flagellar bending waves were similar to the bending
waves that can be propagated on an elastic filament in a
viscous medium, if one end is driven by an external
oscillation [Machin, 1958; Rikmenspoel, 1965; Brokaw,
1966]. It was recognized that flagella must be different in
having active elements distributed along the length, to
generate active bending moment to overcome the energy
dissipated against viscous resistance of the surrounding
fluid. A different view was introduced in 1985, by computer simulations that demonstrated generation of bending waves by models containing only active moments
and elastic resistances [Brokaw, 1985]. These models
were able to generate stable movements in the absence of
viscous resistances because the model for generating
active moments incorporated the realistic feature of decreasing force at increased velocity. The effect of adding
an external viscous resistance could then be examined
over a full range of viscosities, starting from 0.
The present paper, which examines a model flagellum that is able to bend in any direction, also begins by
considering a model that operates in the absence of
external viscous resistance. The extension to three-dimensional bending is further simplified by assuming that
the dyneins generate shear forces that are parallel to the
outer doublets, and do not generate moments that cause
the axoneme to twist. Experiments have shown that flagella have relatively low twist resistance when exposed
to external forces [Gibbons and Gibbons, 1974]. However, previous analyses have found that the amount of
twist generated by internal forces is likely to be small
[Hines and Blum, 1983, 1984, 1985]. These simple models, without forces that might cause the axoneme to twist,
are sufficient to demonstrate that the coordination of
dynein activity required for generation of a helical bending pattern can be provided by a simple local control of
dynein activity by curvature of the flagellum. Two versions of the models are examined, one that uses a mathematical formalism for calculating active shear forces, as
in Brokaw [1985], and another that uses stochastic modelling of individual dyneins, as in Brokaw [1999].
The effects of external viscous resistances are then
introduced by a straightforward extension of the methods
used with earlier two-dimensional modelling. This approach turns out to be less appropriate for three-dimen-
sional modelling than for earlier two-dimensional modelling. New methods have been developed elsewhere for
analyzing three-dimensional bending of cilia attached to
a surface, using a more accurate treatment of external
viscous resistances [Gueron and Levit-Gurevich,
2001a,b]. Some of these methods may need to be adapted
for future studies of helical bending, particularly with
helices having high pitch angles.
METHODS
For numerical analysis, the flagellum is modelled
as a series of N straight segments of equal length, ⌬s,
which can bend at the joints between each segment. In
previous modelling studies [Brokaw, 1972a, 1985, 1999]
the bending at all of the joints was restricted to a single
plane. The shape of the flagellum was defined by an array
of scalar values of curvature ␬[j] at each joint j from 1 to
N⫺1. As the modelling computations proceeded through
time, the time rate of change of curvature ␬⬘[j] within a
time step was calculated, and then the curvature values
were updated to the end of the time step. For a flagellum
that is allowed to bend in three dimensions, the curvature
at each joint is a vector quantity, and both its magnitude
and direction must be calculated in order to determine the
shape of the flagellum. There are 7 steps in extending the
previous modelling methods to three-dimensional bending, but it is primarily just step 4 that introduces novel
mathematics. Much of the groundwork for this analysis
has been developed in an important paper by Hines and
Blum [1983], which is a valuable introduction to these
methods.
Step 1: Local Coordinates and Vectors
In each segment along the length of the flagellum,
there is a local x,y,z coordinate system, the body coordinate system. The ⫹z axis points towards the tip of the
flagellum, with the segment on the ⫹z axis from 0 to ⌬s.
Coordinates in the plane perpendicular to the segment are
shown in Figure 1. The space curve that is the centerline
of the flagellum has a curvature that is always perpendicular to the tangent to the curve, so this curvature must
lie in the x,y plane, with components ␬x and ␬y. When
there is no internal twist, the curvature component ␬z ⫽
0, and the shape of the flagellum is then completely
defined by two arrays of values of local curvature components, ␬x[j] and ␬y[j]. The outer doublet microtubules
of the axoneme are located on a circle in the x,y plane, as
shown in Figure 1. A center-to-center spacing between
the doublets of 60 nm is used, corresponding to placement of the outer doublets on a circle with a diameter of
175 nm. The activity of dynein arms on doublet 1 produces an active shear moment per unit length with magnitude m, which causes sliding of doublet 2 towards the
Simulation of Helical Bending
Fig. 1. The body coordinate system for the cross-section of an
axoneme at any point along the length. This view is from the base
towards the tip of the axoneme, and the positive z axis, which is
tangent to the centerline of the axoneme, points into the plane of the
paper. The positions of doublets 1 through 4 around the circumference
of the axoneme are indicated by small numbered circles, and the
directions of vectors relevant to doublet 1 are shown. Dynein arms on
doublet 1 push doublet 2 towards the tip of the flagellum, equivalent
to rotation around the axis labelled m, for the active shear moment
vector produced by doublet 1. This is also the direction used for the
sliding between doublets 1 and 2. ␬c represents the direction of the
curvature that controls dyneins on doublet 1. It may be different from
the direction of m. It could be different for inner and outer arms, but
that possibility is not used here. The angle between the m and ␬c
vectors is referred to as the m␬ divergence angle, or ␪m␬.
tip of the flagellum, equivalent to rotation around the axis
shown by the m vector in Figure 1. In the present study,
mz is assumed to be 0, and the m vector is in the plane of
Figure 1. This vector is drawn perpendicular to a line
connecting the locations of two adjacent doublets, at an
angle ␪m measured from the ⫹x axis in the direction of
the ⫹y axis. It is convenient to maintain ␪m[3] ⫽ ⫹␲,
with m[3] pointing in the ⫺x direction. In a normal
axoneme with 9 outer doublets, this locates doublet 1
near the ⫹y axis. For all cases
␪ m关i兴 ⫽ ␲ ⫹ 共2i ⫺ 6兲␲/n,
(1)
where i is the index for a particular doublet and n is the
number of axonemal doublets, usually 9. The m[i] vector
also specifies the direction of rotation for the shear and
shear rate that influence active shear moment generated
by dyneins on doublet i.
Sliding in segment j will produce curvature at joints
j and j⫺1, if there are resistances to sliding in other
segments. At joint j, the sliding will produce curvature
with the same direction as the m vector shown in Figure
1, and at joint j–1 the curvature will be in the opposite
sense. The conventions used here are consistent with the
105
direction of action of dynein motor enzymes determined
by Sale and Satir [1977], and are not the same as in
previous modelling work, beginning with Brokaw
[1971].
The modelling in this and previous papers is designed to examine the consequences of the hypothesis
that the active shear moment generated by dyneins is
regulated by the local curvature of the axoneme. In this
paper, the dyneins on each doublet can be regulated
independently, by the curvature of that doublet. For each
doublet, the direction of the curvature controlling dyneins on that doublet must be specified by a vector ␬c,
shown for doublet 1 in Figure 1. The direction of ␬c need
not be in the same direction as m, but will be assumed to
be determined by ␬x and ␬y, neglecting any possible ␬z
component. The angle between the m and ␬c vectors of
a doublet is an important parameter that will be referred
to as the m␬ divergence angle, and symbolized by ␪m␬. In
the simplest cases, ␪m␬ will be the same for each doublet,
and this will be the default assumption unless differences
are specified. Although each dynein motor enzyme might
be independently regulated by curvature, the present
programming assumes that all of the dyneins in one
length segment along one doublet are regulated as a unit.
Local control means that m in a segment j along the
length is regulated by the curvature at that segment.
Since curvature is defined at the joints between segments,
the controlling curvature for segment j is obtained from
0.5(␬[j] ⫹ ␬[j⫺1]). To estimate more accurately the
shear moment generated in a segment during the next
time step, an estimate of curvature in the middle of the
next time step is used for calculation of ␬c:
␬关t ⫹ 0.5⌬t兴 ⫽ ␬关t兴 ⫹ 0.5␬⬘关t兴⌬t,
(2)
where ␬⬘ represents the time derivative of ␬.
Step 2: A Simple Mathematical Formalism for
Active Shear Moment
A mathematical formalism that produces active
shear moment that decreases linearly with increasing
shear velocity (sliding velocity) has been useful for twodimensional flagellar modelling [Brokaw, 1985]. The
decrease in shear moment towards 0 as the velocity
increases allows stable results to be computed without
introducing the complications of external viscous resistances. Although this formalism allows very rapid computations of the behavior of the flagellar model, it is
limited to producing a linear decrease, and cannot produce other, more realistic, relationships between shear
moment and shear velocity. The same mathematical formalism is used here for three-dimensional flagellar modelling, with the difference that active shear moment is
computed independently for each outer doublet. At
steady state in the absence of sliding, the active shear
106
Brokaw
moment per unit length has a constant magnitude, mA. In
response to a rapid change in shear ⌬␴, the momentgenerating system acts like an elastic shear resistance,
and its moment m becomes mA(1 ⫺ ⌬␴ ESCB), where
ESCB is an elastic shear resistance parameter. Whenever
m ⫽ mA, there is a first order recovery process by which
m approaches mA with a rate constant k1. This prescription leads to a differential equation [Brokaw, 1985] that
has a steady-state solution
m ⫽ m A共1 ⫺ ESCB ␴⬘/k1),
(3)
where ␴⬘ ⫽ d␴/dt is the sliding rate. For a short time
interval ⌬t in which ␴⬘ is taken to be constant, the
non-steady-state solution [Brokaw, 1985] is
m共t ⫹ ⌬t兲 ⫽ m共t兲 ⫹ 共m A ⫺ m共t兲兲共1 ⫺ e⫺k1⌬t兲
⫺ mAESCB␴⬘ 共1 ⫺ e⫺k1⌬t兲/k1.
(4)
For each doublet, ␴⬘x and ␴⬘y are obtained by summation
of ␬x⬘ ⌬s and ␬y⬘ ⌬s from the base of the axoneme, with
no sliding allowed in segment 1. The ␴⬘ in the appropriate direction for Eq. 3 is in the direction of m[i], and is
obtained by adding the components of ␴⬘x and ␴⬘y in the
m[i] direction for a particular doublet. In each segment,
the x and y components of the total active shear moment,
mx and my, are obtained by summing the x and y components of m[i] for all n doublets.
Control by curvature is effected for each doublet i
by the local curvature magnitude ␬c[i] in the direction
shown for doublet 1 by ␬c in Figure 1, at ␪␬[i] ⫽ ␪m[i]
⫹ ␪m␬. The value of ␬c[i] is given by
␬ c 关i兴 ⫽ ␬ xcos共␪␬关i兴兲 ⫹ ␬ysin共␪␬关i兴兲.
(5)
Active shear moment is turned on and off by comparing
␬c[i] with a curvature control parameter, ␬0. When ␬c[i]
falls below ⫺␬0, mA[i] ⫽ 0. When ␬c[i] rises above ⫹␬0,
mA[i] ⫽ a constant m0 that is the same for each doublet,
and is always positive. Between ⫺␬0 and ⫹␬0, mA[i]
retains its current value. The result is a negative feedback
control by which active shear moment in a segment is
turned off when the magnitude of the curvature that it is
producing in the basal direction reaches the control magnitude. This causes activity to propagate from base to tip
of the flagellum. The hysteresis ensures a phase lag
between ␬ and m along the length of each doublet, which
is essential for balancing elastic bending resistances
[Brokaw, 1971, 1985].
Step 3: Moment Balance Equations
At each of the N⫺1 joints along the length of the
model, there is an unknown rate of change of curvature
with time, ␬⬘[j]. These values of ␬⬘ are found by solving
a system of N⫺1 vector equations for the balance between active moments and moments resulting from elas-
tic and viscous resistances [Brokaw, 1972a]. These moments can be functions of ␬, ␬⬘, ␴, and/or ␴⬘. The shear
rate, ␴⬘, can be obtained by integration of ␬⬘ using the
specification that no sliding is allowed at the base of the
flagellum. In order to allow all of the integrations to
proceed from the base of the flagellum, an Nth equation
is used for the unknown shear moment MS[1], which has
a value that causes the shear moment MS[N] at the end of
the flagellum to be 0. In the absence of twist and external
forces, the moment balance problem reduces to two
independent moment balance equations, for the x and y
components of the moments [Crowley et al., 1981; Hines
and Blum, 1983]. The rationale for this statement, as
explained by Hines and Blum [1983], can be seen by
considering a moment M that must be constant throughout a length of the flagellum that is not generating shear
moments or experiencing external forces. Since this moment corresponds to shear forces transmitted along the
outer doublets, the moment M is constant in body coordinates, rather than in global coordinates. Therefore, the
x and y components of M also remain constant in body
coordinates.
Each equation is set up and solved as in previous
work, using implicit formulations to obtain stability for
active moments, as in Eq. (4), and elastic bending moments. For example, for the x component of moment ME
resulting from elastic bending resistance EB
M Ex关j, t ⫹ ⌬t兴 ⫽ ⫺ EB␬x关j, t兴 ⫺ EB␬x⬘关j兴⌬t.
(6)
This is added to the MAx[j, t ⫹ ⌬t] obtained by integrating mx[t ⫹ ⌬t] from 1 to j, to form the simplest moment
balance equation. Other details are given in Brokaw
[1985]. When shear resistance contributed by elastic
(nexin) linkages between the doublets was included, it
was usually calculated using the non-linear formulation
of Hines and Blum [1978]:
m S ⫽ ES␴共1 ⫺ 共1 ⫹ 0.75␴2兲 ⫺ 1/2兲.
(7)
Since this shear resistance is non-linear, it must be calculated separately for the shear force generated between
each doublet, as is done for the active shear forces. The
elastic shear resistance constant, ES, is usually low
enough that an implicit integration term is not required
for stability. Linear elastic resistances can be calculated
more simply just by using x and y components, as has
been done for the elastic bending resistance [Hines and
Blum, 1983]. Each of the two systems of equations, for
the x and y components of ␬⬘[j], is solved by Gaussian
elimination and back substitution. The values of curvature are then updated by
␬ x关j, t ⫹ ⌬t兴 ⫽ ␬x关j, t兴 ⫹ ␬x⬘ 关j兴⌬t,
with an analogous equation for the y components.
(8)
Simulation of Helical Bending
Step 4: Computing the Shape of the Flagellum
To visualize the solutions, the shape of the flagellum must be described in a fixed coordinate system,
rather than the body coordinate system that reorients as
the flagellum bends at each joint. In the absence of
external viscosity, motion of the flagellum in space is not
defined. Without loss of generality, the base of the flagellum can remain at the origin of a base X,Y,Z coordinate system, with the first segment aligned along the Z
axis. Let A be a 3 ⫻ 3 transformation matrix that transforms a vector in the body x,y,z coordinate system of a
segment to the base X,Y,Z coordinate system. For the
first segment, A[1] is the identity matrix, and the position
of the segment end at z ⫽ ⌬s is also at Z ⫽ ⌬s. At joint
1, bending is represented by a curvature vector ␬ specified in the local coordinate system of segment 1. In the
initial case, without twist, the curvature is obtained from
the components ␬x[1] and ␬y[1] obtained after solving
the moment balance equation and updating ␬x and ␬y.
The magnitude of the bending at joint 1 is a rotation
angle ␾ ⫽ ␬ ⌬s, where ␬ is the magnitude of ␬. Let a be
a unit vector in the direction of this curvature vector ␬.
The transformation matrix is used to convert a in body
coordinates to a* in base coordinates:
a* ⴝ Aa.
(9)
A[1] is then rotated to obtain A[2] for segment 2, by
applying the rotation formula (Equation 4 –22 of Goldstein [1980]) for rotation of a vector v
v* ⫽ vcos(⫺ ␾) ⫹ a*(a* 䡠 v)共1 ⫺ cos(⫺␾兲)
⫹ 共v X a*)sin(⫺␾兲
(10)
to each of the unit vectors of A[1]. Note that ⫺␾ is used
for rotation of a transformation matrix, rather than ⫹ ␾
that would be used for rotation of a vector. Equivalently,
a rotation matrix R can be computed by applying the
rotation formula to each of the vectors of an identity
matrix, and then A[2] ⫽ RA[1], etc. Explicit formulation
of the rotation matrix allows it to be examined during the
computations, which reveals that it is not precisely antisymmetric for segments as long as 1 ␮m. Consequently,
in such cases, the finite transformations method described here may be more exact than the infinitesimal
transformations method used by Hines and Blum [1983].
After A[2] is obtained, it is then used to add a distance ⌬s
along the body z direction to the position of the end of
segment 1, to locate the end of segment 2 in the base
coordinate system. This sequence is then repeated for
each segment along the flagellum. The result is an array
of position vectors, S[0. . .N], giving the positions of the
ends of each segment, in the base coordinate system. An
array of tangent vectors, T[1. . .N], representing the
length and orientation of each segment, is also generated.
107
Steps 1 to 4, above, are sufficient to convert previous programming so that three-dimensional bending patterns can be generated (steps 1 to 3) and visualized (step
4) using a simple model for dynein motor activity, as
long as there are no external or internal forces that twist
the axoneme. This model, referred to as a Level 1 model,
is useful as an intermediate step in developing more
complete models and is also useful to obtain results with
less computing time when no twist can occur.
Step 5. Complete Expansion to Handle ThreeDimensional Moments and Twist
External forces, such as those resulting from external viscous resistance, can cause the flagellum to twist,
even if mz ⫽ 0 [Hines and Blum, 1983]. Before adding
external forces from movement against viscous resistances, the moment balance must be changed from two
independent sets of equations that balance moments in
the x and y directions, to a single set of equations that
balances moments in x, y, and z directions. This generates a 3N ⫻ 3N matrix, which may include, for example,
terms for the dependence of the x component of moment
at joint j on ␬y⬘ and ␬z⬘ as well as ␬x⬘ . An important
addition is the twist resistance of the axoneme, EBz.
Hines and Blum [1983] calculated that the twist resistance resulting from the sum of the twist resistances of
the microtubular components would be about 2.2 times
the bending resistance, so this value has been used as a
starting point. When movement against external viscous
resistances is included, a further expansion is required, to
include X,Y,Z components of the unknown linear (V)
and angular (W) velocities of the base of the flagellum.
This generates a (3N⫹6) ⫻ (3N⫹6) matrix. This expanded model will be referred to as the Level 2 model.
With mz ⫽ 0 and no external viscosity, it produces results
that are identical to those produced by the Level 1 model.
In such cases, the Level 2 model requires computing
times that are about 8 times as long as the Level 1 model.
The Level 2 model has the advantage of allowing the
internal twist, ␬z, to be computed and examined, to verify
that it is 0.
Step 6. Addition of External Viscous Resistance
To facilitate comparison with earlier work with
two-dimensional models, external viscous resistances are
again obtained from the resistance coefficient method
pioneered by Gray and Hancock [1955]. The integrations
required to obtain the viscous bending moments are
based on the methods used in two dimensions [Brokaw,
1972a]. In three dimensions, if the velocity of a segment
is represented by a vector v, the force on that segment
resulting from viscous resistance to movement is
⌬F ⫽ ⫺C Nv N⌬s ⫺ CLvL⌬s,
(11)
108
Brokaw
where the velocity is resolved into normal and tangential
components, vN and vL, and multiplied by normal and
tangential drag coefficients, CN and CL. For consistency
with earlier work [Brokaw, 1985, etc.], a value of CL ⫽
2.16 ⫻ 10⫺9 pN s nm⫺2 has been used for normal
viscosity. This value was originally calculated for experimental conditions used with sperm flagella. It changes
only slowly with wavelength and is also appropriate for
a flagellum with a wavelength of 10 ␮m at standard
viscosity of 1 cp [Lighthill, 1976]. The drag coefficient
ratio, CN/CL, has been maintained at 1.8, as in previous
work. If the segment is represented by its tangent vector,
T, which has magnitude ⌬s, then vL ⫽ (v 䡠 T)T/(⌬s)2,
vN ⫽ v ⫺ vL, and
⌬F ⫽ (C N⫺C L)(v 䡠 T)T/⌬s ⫺ C Nv⌬s.
(12)
Eq. 12 can be used to derive a 3 ⫻ 3 matrix that
multiplies the three unknown components of v to obtain
the three components of ⌬F. The force F at joint j is the
sum of all of the ⌬F for segments from the base to joint
j. The sum must also include the force at the base of the
flagellum, which may be 0 or a larger value resulting
from the viscous resistance of a cell body or attachment
at the base. This sum must ⫽ 0 at the free distal end of
the flagellum, which provides three of the equations in
the expanded matrix. To carry out this summation, all
vectors must be expressed in the base coordinate system,
which is automatic if v and T are expressed in this
coordinate system.
The primary component of the viscous bending
moment, MV[j], is then obtained in the conventional
manner as the sum of ⌬MV[j] ⫽ T[j] X F[j⫺1] [e.g.,
Hines and Blum, 1983]. The sum of the viscous bending
moments must also equal 0 at the free distal end of the
flagellum, resulting in the final three equations of the
expanded matrix, and this sum may include moments at
the basal end resulting from viscous rotational resistance
of a cell body. For completeness, three additional terms
may be added to ⌬MV[j]:
0.5T关j兴 X ⌬F关j兴 ⫺ ⌺k共CN⌬s/12兲T关j兴 X (w关k兴 X T关j兴)
⫺ ⌺kCw(w[k] 䡠 T关j兴) T关j兴/⌬s. (13)
The first term adds the moment resulting from force on
segment j. The second term adds the moments resulting
from rotation of segment j around an axis through the
midpoint of the segment and normal to the segment,
caused by angular velocity w[k] resulting from ␬ⴕ at joint
k. These first two terms improve accuracy when ⌬s is
large. They become negligible for small ⌬s. The last term
results from rotation of segment j about its z axis, with
viscous drag given by the coefficient CW. For a flagellum
with a diameter of 200 nm, a reasonable value for CW is
72,000 CL [Chwang and Wu, 1971; Lighthill, 1976]. This
term does not vanish when ⌬s is small, but it is nevertheless too small to have any influence on the results
described here; an increase in CW by about 100-fold is
required to have a noticeable effect on the results.
These steps are straightforward, and can be applied
directly to the velocities resulting from the unknown
velocity V at the base of the flagellum. The complications arise in obtaining the required velocity and angular
velocity vectors in the basal system from the unknown ␬⬘
vectors in the body system. This is done in the same
manner as the two-dimensional case, by using the shape
of the flagellum at the beginning of the time step, computed as in step 4, above. This shape provides the transformation matrix A[j] that converts the unknown ␬⬘[j]
vectors from body coordinates to ␬⬘[j]* in base coordinates. Then, in base coordinates,
v关j兴 ⫽ V ⴙ W X D关0, j兴 ⫹ ⌺ 共␬⬘关k兴* X (D关k, j兴兲) ⌬s.
(14)
D[k,j] is the distance from joint k to the midpoint of
segment j, obtained from the shape computation described in step 4. The summation is from k ⫽ 1 to k ⫽
j⫺1. Each of the ␬⬘[k] vectors from 1 to j⫺1 contributes
a term for v[j], and Eq. 12 is applied to each term, giving
terms for ⌬F[j] depending upon V, W, and each of these
unknown ␬⬘[k] vectors.
This procedure yields viscous bending moment
terms at joint j representing vector components in the
base coordinate system. Before they are added to the
other terms in the moment balance equations matrix,
which are represented in the body coordinate system,
they are converted to the body system using the inverse of the transformation matrix A[j]. They are also
multiplied by ⫺1 to convert them to the sign conventions used for m and ␬ in the body coordinate system
(step 1).
This is an approximate method that assumes that
changes in shape of the flagellum during one time step
are too small to invalidate the use of a constant shape for
converting between body and basal coordinate systems.
The same assumption was used for two-dimensional
modelling, but there it is assisted by the fact that errors
tend to cancel out in two-dimensional oscillations. It is
less safe with helical wave shapes, but it is essential for
maintaining a system of linear equations. An additional
error is introduced by the fact that in generating the
helical shape by rotation of the curvature vector, as
represented by successive rotations of the transformation
matrix A[j], there is a rotation of the body coordinate
system. This rotation produces the writhe of the helical
curve. In generating the shape of the flagellar curve, the
writhe is introduced by successive additions of ␬[j] to the
results of bending at previous joints. On the other hand,
Simulation of Helical Bending
in obtaining velocities for the viscous resistances,
each ␬ⴕ[j] is assumed to act independently. Independent rotation by ␬⬘[j] is not equivalent to addition of
␬⬘[j] to ␬[j] before rotation. The result is that velocities calculated by solving the equations and putting the
␬⬘[j] values back into the equations do not match the
velocities obtained from the differences between S[t]
and S[t⫺⌬t]. One effect is to add a spurious VZ component to the velocity of the base of the flagellum.
These problems were approached in a different manner
in the method for simulation of three-dimensional
ciliary movement introduced by Gueron and Liron
[1993]. In their method, the velocities (v) were the
unknowns. After solving the moment balance equations to obtain these velocities, the curvature, torsion,
and the shape of the cilium were obtained by differentiation of the velocities. In contrast, the method used
here explores the limits of the method used previously
for two-dimensional modelling, which obtains velocities by integration of the unknown curvatures and the
previous shape. It cannot be improved directly by
incorporating the methods of Gueron and Liron
[1993].
Accurate results for the movement of the base of
the flagellum, described by the V and W vectors, were
obtained by a “half-iterative” method. After one step
of computation, as described, the updated values of
␬[j] were used to compute the new shape. Velocities of
each segment, in the base coordinate system, were
computed from the difference between the new and old
shapes of the model. These known velocities were then
used to compute forces and moments, as described
above, and used with a reduced 6 ⫻ 6 matrix to
compute values of V and W. The movement of the
flagellum can then be defined in a third, global, coordinate system, where the position, P, of the base of the
flagellum is obtained from P[t ⫹ ⌬t] ⫽ P[t] ⫹ V[t] ⌬t.
Initially, a similar iteration was used to obtain the
orientation angles of the base of the flagellum in the
global system as a function of time. However, in some
cases this would work for several cycles, and then
become unstable. This problem was solved by using a
transformation matrix for the orientation of the basal
coordinate system, rotating it by W as described in
step 4. The question then arises whether updating the
values of curvature by ␬⬘[j] should also be done with
a rotation matrix. This method was tried out, but the
results were identical to those obtained with the simpler update procedure of Eq. 8. Values of VZ ⌬t and
WZ ⌬t were summed over the final cycle of computation to approximate forward velocity and body rotation
values for one cycle of bending.
109
Step 7: Generating Active Shear Moment by
Stochastic Modelling of Dyneins
Steps 1 to 6, above, are sufficient to convert previous programming so that three-dimensional bending patterns can be generated using a simple mathematical formulation for dynein motor activity, with or without
external viscous resistance. A potentially more realistic
model can be obtained by replacing Eq. 4 with computation of active shear moment by a stochastic treatment of
each individual dynein in the axoneme. The method is
referred to as stochastic, because each individual dynein
is followed through time. In each time step, the state of
each dynein is determined by comparing a random variable with transition probabilities determined by the kinetic constants in the mechanochemical cycle of the
dynein motor enzymes [Brokaw, 1976]. By varying these
kinetic constants, dynein models with a variety of relationships between shear moment and shear velocity can
be obtained, such as those used in Brokaw [1999]. These
relationships cannot be obtained easily with the simple
mathematical formulation for dynein motor activity.
The methods are the same as those used in the
previous paper in this series [Brokaw, 1999], and, for
simplicity, the dynein models are identical. The only
significant changes are the use of up to 9 doublets,
instead of 8, and a more exact treatment of the sliding of
each doublet relative to the centerline of the axoneme, to
determine which dyneins remain in each segment when
the flagellum is bent. The dyneins on each segment of
each doublet can be either in an active state or in an
inactive state in which the rate for formation of stronglybound force-producing crossbridges is set to 0. After
switching to the inactive state, existing crossbridges must
complete their normal cycle in order to detach. This is an
additional difference between stochastic modelling and
modelling with the simple mathematical formulation for
dynein activity. In the latter case, switching to the inactive state immediately turns off force production, as if all
existing crossbridges are immediately detached. Switching between these active and inactive states is controlled
by local curvature just as for the model with a mathematical formulation for dynein activity, as described in
the last paragraph of step 2. In all of the examples shown
in this study, inner and outer arm dyneins are controlled
synchronously. Accurate computations of the dynein kinetics require very short time steps. As a result, the
computation time for the stochastic models is much
greater than for the simple models that use a mathematical formulation for dynein motor activity, so the simple
models remain useful for at least preliminary explorations of changes in parameters. Stochastic modelling of
dyneins can be used with both Level 1 and Level 2
models.
110
Brokaw
Validation
The method for calculating the three-dimensional
shape of the flagellum, given the values of ␬[j], was
checked by putting in appropriate sinusoidally varying
␬x[j] and ␬y[j] values for a helix, and confirming that the
result was a helix with correct pitch and wavenumber.
Models were operated over a large range of values
of ⌬t and ⌬s. With 400 or more time steps per flagellar
bending cycle, changing ⌬t has no effect on the results.
Slight differences were detectable with 200 time steps
per bending cycle. Because the non-linear control of
active moment by curvature switches activity on a segmental basis, changing the size of ⌬s produces changes
in the frequency, and sometimes the shape, of the resulting bending patterns. This sensitivity is probably in part
a reflection of the inability of local control by curvature
to determine completely the solution to the moment
balance equations [Brokaw, 1985]. With the two-dimensional models, this dependency upon ⌬s could be reduced by using a more complicated and possibly more
realistic control of bend initiation at the basal end of the
flagellum [Brokaw, 1985, 1999], but extension of this
type of control to three-dimensional bending models has
not been examined. Most of the computations shown
here used 100 length segments (⌬s ⫽ 0.4 ␮m), as a
compromise between small switching units and reasonable computing times.
The swimming velocities that can be calculated
when external viscosity is included in the moment balance should be 0 if CN/CL ⫽ 1.0. For helical bending
waves, the swimming velocity for normal CN/CL ⫽ 1.8
should also be 0 when there is no rotational resistance of
a cell body or sperm head [Chwang and Wu, 1971].
Values of Vz computed by the half-iterative method described in step 6 for these two test cases were close to 0,
as predicted. When bending is restricted to a plane, the
values of swimming velocity are not significantly modified by using the half-iterative method, and these planar
results are consistent with results obtained with earlier
two-dimensional models [Brokaw, 1985, 1999]. This
paragraph is the only place in this paper where results of
computations of swimming velocity are presented or
used.
These computer programs, as Macintosh applications, are available at www.its.caltech.edu/⬃brokawc/
software.html
RESULTS
Helical Bending in the Absence of External
Viscosity
Figure 2 shows a typical result, from computations
with a model that obtains active shear moments by sto-
Fig. 2. Example of bending produced by a flagellar model in the
absence of external viscous resistance, with stochastic modelling of
dynein force production using the same models for inner and outer arm
dynein used in Brokaw [1999]. Identical parameters are used for each
of the 9 axonemal doublets. A: Active states of dynein arms along each
doublet at the end of the computation. Black bars represent segments
in which dyneins are turned on, allowing them to form force-producing
cross-bridges. Lighter lines represent segments where dyneins are
turned off, and not allowed to form cross-bridges. B: Curvature at
0.25-cycle intervals covering the last beat cycle. C: Three views of the
shape of the flagellum at the end of the computation, in each of the
coordinate planes of the basal X, Y, Z coordinate system. Computed
for 8 beat cycles with 100 length steps of 0.4 ␮m and time steps of 10
␮s. In B, there are 2,215 time steps between plots. The elastic bending
resistance parameter for the axoneme, EB, is 2.0 ⫻ 108 pN nm2. The
non-linear elastic shear resistance parameter for each doublet, ES, is
5.0 pN. The curvature control parameter, ␬0, is 0.18 rad ␮m⫺1 and ␪m␬
is ⫹0.20 rad. Computed results include frequency, 11.3 cycles s⫺1,
and an average ATP turnover per dynein of 48 s⫺1, equivalent to a
total energy input of 0.12 pJ s⫺1.
Simulation of Helical Bending
chastic modelling of the chemical kinetics of each dynein. The dynein models are the same as those used in a
previous article modelling two-dimensional bending
[Brokaw, 1999]. The arrangement of dyneins along the
outer doublets is also the same, with outer arm dynein
motors at 24-nm intervals and inner arm dynein motors at
32-nm intervals along the length of each outer doublet
(40 ␮m). The elastic bending resistance (EB ⫽ 2 ⫻ 10⫺8
pN nm2) is also the same. Standard procedure was to start
the model by activating all of the dyneins on doublets 2,
3, and 4 throughout the length of the flagellum and to
continue the computation for a time period equivalent to
4 or more bend cycles. With the model used for Figure 2,
the model could be started by activating dyneins on two
adjacent doublets, but not by activating dyneins on only
one doublet. A full examination of starting conditions
and transients has not yet been performed. Figure 2
shows results in the last of 8 bend cycles. The final result
was illustrative of a consistent and stable pattern of
bending in the previous bend cycles. Figure 2A shows
the pattern of activation of dyneins on each outer doublet
at the end of this computation. Regions of dynein activity, indicated by dark bars on the lines for each doublet
(Fig. 2A) propagate along the length of the flagellum.
Observation of the pattern of activation during the computation shows that this longitudinal propagation occurs
towards the tip of the flagellum. Activation of dyneins on
doublet i precedes activation of dyneins on doublet i⫹1
by approximately 1/9 of the longitudinal wavelength, as
expected for a helical bending wave. This could be
interpreted as propagation of dynein activity around the
circumferential ring of outer doublets, with a phase shift
of 2␲/9 rad between doublets. As in this example, circumferential propagation is in positive numerical order
(clockwise when viewed as in Fig. 1) when the angle
between the directions of produced and controlling curvatures for each doublet, referred to as the m␬ divergence
angle, or ␪m␬, is ⫹0.2 rad. This produces a left-handed
helix. Circumferential propagation is in negative (counterclockwise) order when ␪m␬ is ⫺0.2 rad. This produces
a right-handed helix. This pattern of activation becomes
well established within the second bending cycle, and
continues in a stable manner throughout the computation.
This model stalls in a bent configuration with ␪m␬ values
in the range of ⫺0.1 to ⫹0.1 rad.
Figure 2B shows plots of curvature as a function of
length, in the body coordinate system. Five plots, at equal
time intervals, are shown in each panel. The number of
time steps between each plot was adjusted to 2,215 by
trial and error so that this set of plots encompasses one
bending cycle, with the first and last curvature plots
approximately coinciding. This establishes the repeat frequency for the curvature vs. length plots, which is also
the frequency of rotation of the curvature vector in the
111
x,y plane of the body coordinate system. Unless otherwise specified, all references to frequency in this study
refer to this bend cycle frequency in the body coordinate
system. These plots show that the x and y components of
curvature vary similarly, but are out of phase by 1/4
cycle. The y component ␬y (bending in the x,z plane) lags
behind ␬x, which is appropriate for a left-handed helix.
The longitudinal wavelength, L, for the waves of dynein
activation and curvature can be determined from Figure
2A or B. This wavelength increases gradually as the
bends propagate. Values of L were obtained by measuring the half-wave closest to the midpoint of the flagellar
length. By definition, a circular helix has constant values
of curvature and torsion. If the bending pattern is sufficiently regular to be approximated by a circular helix, its
curvature, ␬, can be determined from the peaks of the
plots of the x and y components of curvature, in Figure
2B. Values of L and ␬ then provide sufficient information
to calculate the pitch angle, ␾H, torsion, ␶, and radius, r,
of the helix [Coxeter, 1969; Gueron and Liron, 1993;
Lighthill, 1996]
tan( ␾H) ⫽ ␬L/2␲,
(15)
␶ ⫽ 2 ␲ /L ⫹ ␬ z,
(16)
r ⫽ ␬ /共 ␬ 2 ⫹ ␶ 2 兲 ⫽ 共L/2 ␲ 兲 sin( ␾H) cos(␾H)
⫽ sin2(␾H)/␬.
(17)
The torsion is the rate of rotation of the curvature vector
in the x,y plane of the body coordinate system as it
moves along the length of the flagellum. In this case, the
wavelength of 36 ␮m and curvature of 0.22 rad ␮m⫺1
predict a helix with a pitch angle, ␾H, of 0.9 rad, a radius
of 2.8 ␮m, and a pitch of 14 ␮m.
Figure 2C shows 3 views of the approximately
helical shape of the flagellum at the end of the final cycle,
projected onto the X,Y plane, the X,Z plane, and the Y,Z
plane of the basal coordinate system. The orientation of
the axis of the helix varies, depending upon the details of
bend initiation at the base of the flagellum, and typically
is not aligned with the Z axis. The wavenumber is greater
than the wavenumber of the curvature plots in Figure 2B.
In each turn of the helix, the curvature vector must rotate
by 2␲ radians around the axis of the helix, in the basal
coordinate system. Rotation around the helix axis resulting from the torsion is 2␲ cos(␾H). The additional rotation is rotation of the body x,y,z coordinate system,
referred to as the writhe of the curve, which accumulates
as the flagellum is bent by the curvature [Fuller, 1971;
Maggs, 2001]. This additional rotation increases the
wavenumber by a factor of 1/cos(␾H).
The flagellar model used for Figure 2 includes
elastic shear resistance, using the non-linear formulation
of Hines and Blum [1978]. This formulation is designed
112
Brokaw
to model the effect of elastic nexin linkages between the
outer doublets. With this non-linear formulation, there is
a low resistance to small amounts of shear, and the
resistance increases up to its parameter value, ES, as the
shear increases. At a shear angle of 2 rad, the resisting
shear moment has half of its parameter value. In this
example, ES ⫽ 5 pN means a shear moment of 5 pN nm
per nm of doublet length, equivalent to a shear force of
2.0 pN for each 24-nm outer arm dynein repeat along the
length when the shear angle is 2.0 rad. If this shear
resistance is omitted from the model, the results show a
greater increase in wavelength and helix radius as the
bends propagate along the length, so that the bending
pattern is less accurately described as a circular helix.
Computing the model of Figure 2 with 60 length steps
instead of 100 length steps produces a result with almost
the same shape, but a significantly lower frequency (7.9
s⫺1).
Examples of results with smaller and larger wavelengths were obtained by varying the elastic bending
resistance, and are illustrated in Figures 3 and 4. In these
figures, the B panels show plots of shear angle along the
length, rather than curvature. Earlier work with planar
models [Brokaw, 1985] indicated that the wavelength of
the bending waves is regulated primarily by the ratio
between active moment and elastic bending resistance,
when external viscous resistance is absent. In Figure 3,
using a lower elastic bending resistance, EB ⫽ 1.0 ⫻ 108
pN nm2, a higher value of the curvature control parameter, ␬0, can be used, to obtain higher curvature. Since the
wavelength is shorter, the pitch angle (1.0 rad) is not
much greater, but the radius of the helix is reduced. The
stiffer flagellar model in Figure 4, with EB ⫽ 4.0 ⫻ 108
pN nm2, requires a lower value of ␬0. These examples
have used negative values of ␪m␬, to demonstrate the
right-handed chirality of the helix.
Because these models involving stochastic computation of individual dyneins require extended computing
times, a more extensive exploration of the effects of
varying the curvature control parameter, ␬0, and the m␬
divergence angle, ␪m␬, has been performed with the
models in which active shear moment is obtained from
the mathematical formulation introduced in Brokaw
[1985], as described in Methods, step 2. Results are
summarized in Figure 5, from computations using the
same elastic bending resistance (EB ⫽ 2.0 ⫻ 108 pN
nm2) used for the model in Figure 2, and a lower elastic
shear resistance parameter (ES ⫽ 2.0 pN). Active moment parameters m0 and k1 were chosen for similarity to
the standard result shown in figure 2 of Brokaw [1985],
as discussed in Planar Bending Patterns, and similarity to
the result shown in Figure 2 of this study. With this set of
specifications, and a low value of 0.08 rad ␮m⫺1 for the
curvature control parameter ␬0, helical bending can be
Fig. 3. Example of helical bending in the absence of external viscous
resistance, using a flagellar model with a lower elastic bending resistance, EB ⫽ 1.0 ⫻ 108 pN nm2. The presentation follows the same
format as Figure 2, except that shear angles, obtained by integrating
curvature along the length, are presented in B instead of curvatures.
The curvature control parameter, ␬0 is 0.28 rad ␮m⫺1 and ␪m␬ is
⫺0.30 rad. All other parameter specifications are the same as for
Figure 2. Computed frequency is 5.7 cycles s⫺1.
obtained over a wide range of values of ␪m␬, from ⫺1.3
to ⫹1.3 rad. The helix pitch angle increases with the
magnitude of ␪m␬, but the frequency decreases dramatically from 62 s⫺1 at ␪m␬ ⫽ 0 to 1.25 s⫺1 at ␪m␬ ⫽ 1.3.
The increased helix pitch angle reflects increases in both
curvature and wavelength. With ␪m␬ ⫽ 1.4, a helically
bent flagellum is produced, but the movement stalls, so
the frequency is 0. The chirality of the helix changes
Simulation of Helical Bending
Fig. 4. Example of helical bending in the absence of external viscous
resistance, using a flagellar model with a higher elastic bending resistance, EB ⫽ 4.0 ⫻ 108 pN nm2. The presentation follows the same
format as Figure 3, with shear angles presented in B. The curvature
control parameter, ␬0, is 0.10 rad ␮m⫺1 and ␪m␬ is ⫺0.30 rad. All
other parameter specifications are the same as for Figure 2. Computed
bend cycle frequency is 16.0 cycles s⫺1.
from right-handed to left-handed between ␪m␬ values of
0.005 and 0.010 rad, indicating a small programming
bias. With values of ␬0 of 0.12 rad ␮m⫺1 and higher,
there is a region of ␪m␬ values near 0 that does not give
helical bending. The usual result is an erratic bending
pattern that is attempting to become helical, but does not
stabilize, and may stall completely. At the highest values
113
Fig. 5. Computations of helical bending in the absence of external
viscous resistance, using the mathematical formulation for active shear
force [Brokaw, 1985]. Results are shown for various values of curvature control parameter, ␬0, and m␬ divergence angle, ␪m␬. The parameters for this active shear model are m0 ⫽ 7.0 pN, ESCB ⫽ 4.0, and
k1 ⫽ 870 s⫺1. Elastic resistance parameters were EB ⫽ 2.0 ⫻ 108 pN
nm2 and ES ⫽ 2.0 pN. Values of wavelength and curvature were
measured manually on the curvature vs. length plots as in Figure 2B,
and Eq. 15 was used to calculate the helix pitch angle, shown in A.
Results obtained with a particular value of curvature control parameter, ␬0, and various values of ␪m␬ are connected by lines. Values of ␬0
of 0.08, 0.12, 0.16, 0.18, and 0.20 rad ␮m⫺1 were used and are
identified in A. B: Computed bend cycle frequency plotted against the
helix pitch angle, with results at each value of ␬0 connected by lines.
Computations were performed with 60 length segments and 400 time
steps per bend cycle, for at least 12 bend cycles.
of ␬0, the region of ␪m␬ values that give stable helical
bending is reduced to the region around 0.3 to 0.4 rad, or
close to ␲/9 rad. In this region, models with higher values
of ␪m␬ or ␬0 often start generating a helical pattern, but
the movement subsequently deteriorates as regions appear at the distal end where the active force is too low to
bring the curvature up to the control value, ␬0. Bending
dies out, starting from the distal end, until the model
stalls completely. Similar behavior was found previously
with two dimensional models [Brokaw, 1985]. With pla-
114
Brokaw
nar models, improved behavior at the distal end can
usually be obtained by controlling the last 2 ␮m by the
curvature 2 ␮m from the tip, but this prescription was not
very useful with helical bending.
These results suggest that there is a maximum helix
pitch angle of about 1.15 rad that can be obtained by
varying the parameters of this model. Increasing the
shear moment parameter, m0, of the active shear model
allows higher values of curvature control parameter to be
used, but does not produce significantly greater helix
pitch angles, because the wavelength is decreased. Irrespective of the choices of ␬0 and ␪m␬, the plot in Figure
5B shows that there is a decrease in frequency as the
helix pitch angle is increased. The reduction in frequency
as ␾H is increased reflects the shape of the force vs.
velocity curve of the dynein model. When very high
forces are needed to reach a high ␬0, the velocity must be
close to 0, and the frequency must be low.
When ␪m␬ ⫽ 0, the models using stochastic computation of individual dyneins can produce helical patterns of activation, as in Figure 2A, with some parameter
specifications. However, these patterns are irregular, and
do not give clearly periodic bending patterns. Values of
␪m␬ with a magnitude of 0.2 rad or greater are required to
obtain stable bending patterns such as shown in Figure 2.
With ␪m␬ ⫽ 0.35 rad, a value of ␬0 as high as 0.28 rad
␮m⫺1 can be used to obtain a stable helical pattern with
a helix pitch angle of 1.1 rad using the same dynein
model that was used for Figure 2. This result is shown in
Figure 6. The ability to use a higher value of ␬0 with the
stochastic dynein model may be explained by the difference between the non-linear force vs. velocity behavior
of the dynein models [Brokaw, 1999] used in the stochastic dynein model and the linear force vs. velocity
curve obtained with the models using a mathematical
formulation for dynein force. However, with the stochastic dynein models, no combination of parameters was
found that gave significantly higher pitch angles than the
result shown in Figure 6.
Planar Bending Patterns
Planar bending can be obtained with these models,
by abandoning the specification that ␪m␬ is the same for
all doublets, and using variable values of ␪m␬ such that
the controlling curvature direction, ␬c, for doublets 1
through 5 is at ␪ ⫽ ␲ radians, and ␬c for doublets 6
through 9 is at ␪ ⫽ 0 radians. This specification produces
a preferred bending plane parallel to the y,z coordinate
plane. Figure 7 shows an example with a model using the
mathematical formulation for dynein force, with parameters adjusted such that the result matches the result
shown for the two-dimensional model in figure 2 of
Brokaw [1985]. These parameters were also used to
obtain the results shown in Figure 5. The parameter
Fig. 6. Example of helical bending in the absence of external viscous
resistance using a model that obtains active shear force by stochastic
modelling of dyneins, as in Figures 2– 4. This result shows the maximum pitch angle of 1.1 rad that was obtained with this model using
EB ⫽ 2.0 ⫻ 108 pN nm2 and ES ⫽ 2.0 pN, as used for the modelling
in Figure 5. The curvature control parameter, ␬0 ⫽ 0.28 rad ␮m⫺1, and
␪m␬␬ ⫽ 0.35 rad. The computation used 60 length steps and time steps
of 40 ms. Computed bend cycle frequency is 1.9 s⫺1.
adjustments are small, involving only adjustments in m0
and k1 to compensate for the change from a model with
two doublets to a model with nine doublets. Even though
the model is free to bend in the x,z plane, no bending
occurs in this plane because, for both doublet set 1
through 5 and doublet set 6 through 9, the sum of the
moments produced by dyneins in each doublet set is
parallel to the x axis if all of the doublets are equally
active. Figure 8 shows that this specification of ␬c directions can also generate planar bending when the dynein
force is obtained by stochastic modelling of the dyneins.
The parameters of this example are close to those used
for the model in Figure 3, which generated helical bending. The plots of shear angle in the x,z plane, in Figure
8B, show that the movement is not as planar as with the
model in Figure 7, which used a mathematical formulation for active shear moment. The stochastic computations of dynein force introduce random fluctuations that
are probably responsible for the bending in the x,z plane.
As in these examples, the wavelength of the planar
bending pattern is typically longer than that obtained
when helical bending is generated, and the frequency of
the planar bending pattern is lower.
Using the models with a mathematical formulation
for dynein force, stable planar bending can also be obtained under some conditions when ␪m␬ is close to 0 for
all doublets. Figure 9 shows an example, using the model
of Figure 7, with ␪m␬ ⫽ 0 and ␬0 ⫽ 0.12 rad ␮m⫺1.
When the simulation is initiated with dyneins on doublets
Simulation of Helical Bending
Fig. 7. Example of planar bending by a three-dimensional model in
the absence of external viscous resistance. This model used the mathematical formulation for active shear force [Brokaw, 1985]. To obtain
planar bending, dyneins on doublets 1 through 5 are turned on by
curvature in the ⫺x direction (␪ ⫽ ␲ rad), and dyneins on doublets 6
through 9 are turned on by curvature in the ⫹x direction. Parameters
are matched as closely as possible to reproduce the result obtained
from a two-dimensional model, in figure 2 of Brokaw [1985]. Elastic
bending resistance, EB ⫽ 1.0 ⫻ 10⫺8 pN nm2. Elastic shear resistance
is 0. The curvature control parameter, ␬0, is 0.20 rad ␮m⫺1. The
parameters for the active shear model are m0 ⫽ 7.0 pN, ESCB ⫽ 4.0,
and k1 ⫽ 870 s⫺1. The computed bend cycle frequency is 31 s⫺1.
Computed with 60 length steps and 160 time steps per beat cycle. The
presentation follows the same format as Figure 3.
2, 3, and 4 active, bending is sufficient to turn off the
dyneins on these doublets and turn on dyneins on doublets 7 and 8. These dyneins can be turned on and off
115
Fig. 8. Example of planar bending in the absence of external viscous
resistance, using the model that obtains active shear force by stochastic
modelling of dynein, as in Figures 2– 4. Dyneins on doublets 1 through
5 are turned on by curvature in the ⫺x direction (␪ ⫽ ␲ rad), and
dyneins on doublets 6 through 9 are turned on by curvature in the ⫹x
direction. Elastic bending resistance, EB ⫽ 1 ⫻ 108 pN nm2. The
non-linear elastic shear resistance parameter, ES, is 3.0 pN. The
curvature control parameter, ␬0, is 0.34 rad ␮m⫺1. The computed bend
cycle frequency is 28.4 s⫺1. Computed with 100 length steps and
10-␮s time steps. The presentation follows the same format as Figure
3.
regularly to produce a stable planar pattern, but the
curvature of doublets 1, 5, 6, and 9 never becomes high
enough to activate the remaining dyneins on these doublets. This planar bending pattern is noticeably asymmetric. These doublet sets are not evenly matched because
⌺cos␪[i] for i ⫽ 2,3,4 is ⫺2.53 rad while this sum for i ⫽
7,8 is ⫹ 1.89 rad. In contrast, when dyneins on all of the
116
Brokaw
from ⫺0.006 to ⫹0.009 rad. Outside of this range, the
movement switches to a stable helical pattern. This range
can be increased modestly by increasing the elastic bending resistance for bending in the x,z plane, or by increasing the elastic shear resistance between doublets 5 and 6.
With values of ␬0 of 0.16 rad ␮m⫺1 or greater and ␪m␬ ⫽
0, activation of dyneins on doublets 2 and 4 becomes
erratic. The stable bending pattern breaks down and is
followed by erratic helical bending, which usually stalls
in a bent configuration.
The type of planar bending described in the preceding paragraph has never been obtained with the models that obtain active shear force by stochastic modelling
of the dyneins. The stochastic models have a hyperbolic
force/velocity curve, making it more likely that the curvature control value can be exceeded, and also have more
variability in force, so the stable planar bending that
depends on failure to activate dyneins on some doublets
is not obtainable. Using values of ␪m␬ close to 0 in these
models, it was not possible to induce stable planar bending by increasing the elastic bending resistance for bending in the x,z plane.
Movement in the Presence of External Viscous
Resistance
Fig. 9. Example of bending by a three-dimensional model in the
absence of external viscous resistance, where planar bending results
from failure to activate dyneins on some doublets. This model uses the
mathematical formulation for active shear force [Brokaw, 1985]. This
figure is like Figure 7, except that dyneins on each doublet are
controlled independently by the curvature of that doublet, as in the
models such as Figure 2. For each doublet, ␪m␬ is 0.0 rad. Parameters
are as in Figure 7 except that there is an elastic shear resistance
parameter of 2.0 pN and the curvature control parameter, ␬0, is 0.12
rad ␮m⫺1. Computed with 60 length steps and 160 time steps per beat
cycle. The computed bend cycle frequency is 32.8 s⫺1.
doublets are activated, as in the preceding paragraph, the
sum for i ⫽ 1,2,3,4,5 is ⫺2.88 rad and the sum for i ⫽
6,7,8,9 is ⫹2.89 rad, so that an essentially symmetric
planar bending wave is produced. With the parameters
used for the example shown in Figure 9, planar bending
is stable for at least 20 beat cycles when ␪m␬␬ is varied
Planar bending. Effects of external viscous resistance on planar bending patterns will be discussed first,
since the programs are least modified from earlier twodimensional modelling. In figure 7 of Brokaw [1985], the
effect of external viscous resistance on a two-dimensional model, with dynein force obtained from a mathematical formulation, was demonstrated. A set of parameters was chosen such that the same wavelength mode
was maintained at viscosities of 0, 1, 8, and 64 times
normal viscosity, while the frequency decreased from 39
s⫺1 at 0 viscosity to 2.6 s⫺1 at 64 times normal viscosity.
Using the three-dimensional model and the mathematical
formulation for dynein shear moments, with planar bending imposed by appropriate specification of doublet ␬c
directions, as in Figure 7 of this study, the results obtained with the two-dimensional model [Brokaw, 1985]
were replicated, with just one significant difference. At
viscosities of 8 or 64 times normal, the movement became non-planar and irregular, unless the bending resistance EBy in the x,z plane was increased. At 64 times
normal viscosity, increasing EBy by a factor of 20 restored stable planar bending, but increasing this resistance by a factor of 10, or increasing the twist resistance,
EBz, by a factor of 100, was not sufficient to obtain planar
bending. The model shown in Figure 9, which generated
planar bending even when all of the ␪m␬ values are 0,
switches to a helical bending pattern when external viscosity (4 times normal) is included.
Simulation of Helical Bending
A model using stochastic modelling of dyneins,
which produced the planar bending results shown in
Figure 8, with a frequency of 28.4 s⫺1 at 0 viscosity, was
examined in the presence of external viscosity. At a
viscosity of 4 times normal, the frequency decreased to
13.9 s⫺1, while the shape of the bending waves remained
approximately constant. At higher viscosities, the movement switches to a shorter wavelength mode, as previously observed with 2-dimensional models [Brokaw,
1972b; Hines and Blum, 1978], and becomes less stable.
Bending can be stabilized by increasing the elastic bending resistance in the x,z plane, EBy, and/or increasing the
twist resistance. Figure 10 shows results obtained with 16
times normal viscosity, with a 10-fold increase in twist
resistance, EBz, and a 4-fold increase in x,z bending
resistance, EBy. Similar results were obtained with no
increase in twist resistance and a 40-fold increase in EBy.
Results were slightly less stable with a 100-fold increase
in twist resistance and no increase in EBy. Note that the
movement of the flagellar model in the global coordinate
system, which can be computed when the model operates
in the presence of external viscous resistance, is not
shown in Figures 10 through 14. The bottom panels of
these figures show the shape of the flagellum in the base
coordinate system, so that the bending can be compared
directly with the results in earlier figures.
Helical bending. Computing the model of Figure 2
at normal viscosity gives results very similar to Figure 2.
There is only a slight decrease in frequency, from 11.3 to
10.8 s⫺1, in contrast to the large reduction in frequency
observed with planar bending when external viscosity is
added. When there is no rotational resistance at the basal
end of the flagellum, provided by a cell body or otherwise, there is very little movement of the helix relative to
the viscous environment [see Chwang and Wu, 1971], so
viscosity has little effect on the movement. However, the
results show that a twist of about 0.15 rad develops in the
midregion of the flagellum. At higher viscosities, the
twist becomes greater and significantly alters the shape
of the flagellum. Figure 11 shows results at 8 times
normal viscosity, with normal twist resistance, where a
twist of about 1.2 rad develops in the midregion of the
flagellum. Figure 12 shows results with an increased
twist resistance, to demonstrate that when twist is eliminated, the shape of the bending wave is very similar to
that obtained at 0 viscosity. The reduction in frequency,
to 8.5 s⫺1, is still small compared to effects of viscous
resistance on planar bending.
Twist can be modified and sometimes reversed by
adding rotational resistance equivalent to a head or cell
body at the basal end of the flagellum. A full examination of
the effects of a cell body at the basal end is beyond the
scope of this study, but one example is shown in Figure 13.
117
Fig. 10. Planar bending obtained with the model presented in Figure
8, with external viscosity included at 16 times normal. Elastic twist
resistance, EBz, has been increased by a 10-fold factor to 44 ⫻ 108 pN
nm2, and elastic bending resistance in the x,z plane, EBy, has been
increased by a 4-fold factor to 8 ⫻ 108 pN nm2. Computed with 100
length steps and 40-␮s time steps. The computed bend cycle frequency
is 10.8 s⫺1. Bending of the flagellum, in C, is shown in the basal
coordinate system, as in Figure 2; movement in the viscous environment is not shown.
Axonemes With Fewer Than Nine Doublets
Axonemes with only 3 doublets have been described from the parasitic protozoan Diplauxis hatti.
These flagella, known as “3⫹0” flagella because they
also lack central pair microtubules, generate approximately helical bending with frequencies of about 1.5 s⫺1
[Prensier et al., 1980]. Helical bending obtained with a
118
Brokaw
Fig. 11. Helical bending with the model presented in Figure 2. This
result was obtained with external viscosity included, at 8 times normal.
All other specifications are the same as for Figure 2. Note that A (right)
shows the twist angle. Bending of the flagellum, in B, is shown in the
basal coordinate system. Computed with 100 length steps and 40-␮s
time steps. Computed bend cycle frequency is 8.2 s⫺1.
3⫹0 version of the model using stochastic modelling of
individual dyneins is shown in Figure 14. To model the
simplest possible flagellum, only the inner arm dyneins
are included, using the model from Brokaw [1999]. The
elastic bending resistance, EB, has been reduced to 0.4 ⫻
108 pN nm2. Additional computations have been performed with 3⫹0 models, with active shear moment
derived from a mathematical formulation, similar to the
computations performed to obtain the results in Figure 5.
These computations indicate that slightly higher values
of ␪m␬ are required for the 3⫹0 flagellum, compared to
the 9 doublet flagellum. For example, with m0 ⫽ 4 pN,
EB ⫽ 0.5 ⫻ 108 pN nm2, ES ⫽ 2 pN, and ␬0 ⫽ 0.18 rad
␮m⫺1, stable helical movement was obtained with values
of ␪m␬ from 0.4 to 0.7 rad. The required increase in ␪m␬
is much less than the factor of 3 increase in the angular
separation between outer doublet microtubules. A few
computations have been performed with model axonemes containing 6, 7, or 8 doublets, using the mathematical formulation for active shear moment. No notable
differences were observed between these results and re-
Fig. 12. Results obtained from a model with the same parameters and
conditions as Figure 11, except that the elastic twist resistance, EBz,
has been increased by a 10-fold factor to 44 ⫻ 108 pN nm2. Computed
bend cycle frequency is 8.5 s⫺1.
sults obtained with model axonemes containing 9 doublets.
DISCUSSION
Local Curvature Control Can Easily Generate
Doublet Metachronism and Helical Bending
“Helical undulation . . . makes no specially complex demands on the organization of patterns of relative
sliding of adjacent tubules in an axoneme” [Lighthill,
1996, p 51]. In the simulations shown in this study,
regions of dynein activation propagate along each outer
doublet just as seen in previous analyses of two-dimensional bending waves. From doublet to adjacent doublet,
the pattern of activation is phase-shifted by 2␲/n radians,
where n is the number of doublets. This pattern is illustrated, for example, in Figure 2A. I propose to refer to
this pattern of activation as “doublet metachronism,” in
order to emphasize its similarity to ciliary metachronism,
in which a spatial succession of phase differences between the beat cycles of adjacent cilia arrayed on a cell
Simulation of Helical Bending
119
Fig. 13. Results obtained from a model with the same parameters and
conditions as Figure 12, except that resistance coefficients equivalent
to the drag of a spherical head with a radius of 2 ␮m have been added
at the base of the flagellum. Computed bend cycle frequency is 10.4
s⫺1.
surface allows the generation of a smooth metachronal
wave that is effective for fluid propulsion. In both cases,
the phase differences allow independent oscillators to be
coordinated so that they can operate in a productive
manner, rather than interfering with each other. Doublet
metachronism, as described here, is a refinement of the
idea of unidirectional transfer of active sliding, with a
transmission delay, around the ring of outer doublets
[Machemer, 1977]. Patterns of dynein activity showing
doublet metachronism were illustrated in an earlier computer simulation study by Sugino and Naitoh [1982],
which related specified patterns of dynein-driven sliding
to the bending pattern of a cilium. That work did not
address the mechanism for control of dynein activity, and
it is difficult to evaluate since the mathematical details
were never published.
Metachronism is a purely descriptive term that does
not imply any particular mechanism for its production.
Modelling studies [Gueron and Levit-Gurevich, 1998]
have confirmed earlier suggestions that ciliary metachronism could result from fluid dynamical interactions between the cilia. No other signalling mechanism between
cilia is required, consistent with earlier experimental tests
Fig. 14. Results obtained from a model with 3 doublets, using
stochastic modelling of inner arm dyneins [Brokaw, 1999]. Flagellar
length is reduced to 20 ␮m. Elastic bending resistance is reduced to
0.4 ⫻ 108 pN nm2, and elastic shear resistance is 2.0 pN. The curvature
control parameter ␬0 is 0.18 rad ␮m⫺1 and ␪m␬ is 0.5 rad. A normal
level of external viscous resistance is included, and viscous resistance
coefficients equivalent to the drag of a spherical head with a radius of
2 ␮m have been added at the base of the flagellum. Computed bend
cycle frequency is 3 s⫺1.
that provided evidence against signalling through the
interior of the cells. Similarly, the simulations reported
here show that doublet metachronism can be obtained
without any novel signalling mechanism for propagation
of activation from doublet to doublet around the circumference of the axoneme. The phase lags that establish a
120
Brokaw
helical bending pattern require only the mechanical coupling that is an inescapable feature of the structure of the
axoneme, and external viscosity is not involved or required. Consequently, local control of dynein activity by
curvature of the flagellum, suggested previously for control of planar bending waves [Brokaw, 1971], can easily
generate helical bending if a flagellum is free to bend in
three dimensions. The novel feature of these three-dimensional models is that dyneins on an outer doublet are
independently controlled by the curvature of that doublet.
In the simplest case, where the control relationship is the
same for each doublet, helical bending waves can be
generated.
The range of helical bending patterns attainable
from the simulations using local curvature control encompasses many known examples. The bending patterns
of spermatozoa of the eel, Anguilla anguilla, have been
studied by Gibbons et al. [1985] and by Woolley [1998].
The flagellum lacks outer arm dyneins and central pair/
radial spoke structures, so its movement is generated
entirely by inner arm dyneins. It generates a left-handed
helix with a radius that has a maximum of about 3 ␮m
near the middle of the flagellar length. The overall movement is complicated by the presence of an asymmetric
head, which appears to increase the efficiency of forward
swimming by a design that maximizes the rotational drag
to reduce roll of the spermatozoon about its longitudinal
axis. The helical pitch is about 20 ␮m, giving a pitch
angle of about 0.75 rad. Spermatozoa of the Asian horseshoe crab (two species of Tachypleus) generate righthanded helical bending, with a pitch angle of 0.82 rad
[Ishijima et al., 1988]. Limited data from other observations of helical bending by spermatozoa are summarized
by Brennan and Winet [1977].
Helical bending waves with larger pitch angles
have been confirmed in two studies. Dinoflagellates have
two flagella, one of which, known as the transverse
flagellum, produces a bending wave propagating from
base to tip and typically contained in a groove or cingulum running transversely around the cell body. Gaines
and Taylor [1985] examined many species, finding many
examples of helical bending, with small radii and pitch
lengths, and left-handed chirality. The example illustrated in their paper has a pitch of about 4 ␮m and pitch
angle of about 1.1 rad. In sea water solutions containing
methyl cellulose, to produce a high viscosity, sea urchin
sperm flagella have been observed to switch from planar
bending patterns and generate right-handed helical
waves, with a pitch of 3 ␮m or less and a pitch angle of
about 1.25 rad [Woolley and Vernon, 2001]. With such
small pitch distance, fluid dynamical interactions between adjacent gyres of the helix are likely to be significant, invalidating the simple assumptions of resistance
coefficient analysis of effects of viscous resistance on
flagellar bending and propulsion.
The current work has been limited to just one
particular paradigm for switching dynein activity on and
off as a function of the local curvature. It is also limited
to the case where the active shear moment is directed
along the length of the doublet, and has no mz component
that would tend to twist the flagellum. There is extensive
evidence that inner arm dyneins can cause microtubule
rotation during in vitro motility assays [Vale and Toyoshima, 1988; Kagami and Kamiya, 1992]. These observations suggest that moment produced by inner arm
dyneins might have an mz component, but this is only a
suggestion because the conditions within an axoneme
might differ significantly from those obtained during in
vitro motility assays. The existence of an internal mechanism that can generate twist of the axoneme is also
indicated by observations on quail sperm flagella [Woolley and Vernon, 1999]. The present results demonstrate
that a non-zero mz component is not required for generation of helical bending. However, active moments with
an mz component may be essential for generation of
helices with pitch angles larger than the 1.15 rad values
found with the current model, such as those observed by
Woolley and Vernon [2001]. Additional modelling work
is needed to explore this possibility.
These modelling results demonstrate that local control of dynein activity by curvature is a possible mechanism for generation of helical bending by flagella. There
is still almost no evidence that real flagella actually use a
curvature control mechanism, even though this hypothesis has been in existence for more than 30 years. Possibly the most important conclusion is that the early
evolution of flagellar movement might have been possible by using a very simple control of dynein activity by
curvature. Results with the simple model containing only
inner arm dyneins on 3 doublets (Fig. 14) support this
idea.
m␬ Divergence Angle, ␪m␬, Is an Important New
Parameter
This work has identified a new parameter that is
important for describing the control of dynein activation
by curvature. This parameter, called the m␬ divergence
angle, or ␪m␬, is the angle between the direction of active
shear moment and curvature generated by dyneins and
the direction of the curvature that regulates these dyneins. These directions do not need to be the same. The
modelling results show that larger and more stable helical bending patterns can be obtained with a non-zero m␬
divergence angle. The m␬ divergence angle also determines the sense (chirality) of the helix.
The relationship between the m␬ divergence angle
and the chirality of helical bending can be understood in
Simulation of Helical Bending
the following manner. Consider a situation where only
the dyneins on doublet 3 are active. The moment vector
for doublet 3 will be in the ⫺x direction (Fig. 1). If
sliding is restricted at the basal end of the axoneme, these
dyneins will produce positive curvature (a curvature vector in the ⫹x direction) in the basal region of the axoneme. In a normal axoneme with 9 outer doublets, dyneins
on doublets 7 and 8 will have moment vectors with
angles of ⫺␲/9 and ⫹␲9 rad, respectively. If ␪m␬ ⫽ 0,
the controlling curvature (␬
␬c) vectors for doublets 7 and
8 will also point in these directions, and these dyneins
will be equally regulated by the curvature produced by
dyneins on doublet 3. If ␪m␬ is ⬎0, the ␬ c vector for
doublet 7 will be closer to the ⫹x direction of the
curvature produced by dyneins on doublet 3, and dyneins
on doublet 7 will be activated before the curvature rises
to the level that will activate dyneins on doublet 8.
Earlier activation of dyneins on doublet 7 than on doublet
8 corresponds to propagation of activity in a clockwise
direction around the axoneme, which leads to a helical
bending wave with left-handed chirality. The direction of
␬c for dyneins on doublet 7 will coincide with the ⫹x
direction of the curvature produced by dyneins on doublet 8 when ␪m␬␬ ⫽ ␲/9 rad. This should be the optimal
direction for activation of dyneins on doublet 7; this may
explain the apparent optimum ␪m␬␬ values indicated by
the results in Figure 5 at high values of ␬0.
The increased curvature and helix pitch angle that
is obtained with higher values of ␪m␬ may be explained
by the following arguments: In the absence of viscous
resistances, the peak active moment is needed between
bends of a planar bending wave [Brokaw, 1971, 1994],
so active moment should be switched on symmetrically
around the inflection points between bends. This can be
obtained by switching active moment on when the curvature peaks in one direction, and switching it off when
the curvature peaks in the opposite direction. This relationship between curvature and active moment will be
achieved by switching dyneins on and off at ⫾␬0, but
only if ␬0 is close to the peak curvature. If ␪m␬ ⬎ 0,
switching will occur later in the clockwise propagation of
curvature and moment, allowing the curvature to continue to rise to a value greater than ␬0, with peak curvature still coinciding with switching of the active moment.
If ␪m␬ is an important parameter for generation of
helical bending waves by real flagella, it would be unlikely to find flagella that switch helical chirality, and
unlikely to find both chiralities in a population. Most
recent studies of helical bending waves cited here did not
report variations in helical chirality within species, and
are therefore consistent with the idea that a fixed value of
␪m␬ is an important part of the control of dynein activity
by curvature. However, spermatozoa that swim with
nearly planar bending waves typically swim in helical
121
paths and show other evidence that they roll around their
longitudinal axis. This roll may result from a helical
component of the bending wave, corresponding to the
interpretation that the near-planar bending wave is actually a helical bending wave with high eccentricity. Variations within a sperm population in roll direction and the
chirality of the helical swimming path of spermatozoa
have been described by Ishijima et al. [1992] and Ishijima and Hamaguchi [1993]. Occasional reversals of
swimming path chirality of individual spermatozoa were
also reported. Development of a model for planar bending that admits a small helical component will be required to evaluate these observations.
Generation of Planar Bending Becomes a
Problem When a Flagellum Can Bend in Three
Dimensions
For spermatozoa with minimal heads, planar flagellar bending waves are more effective for propulsion than
helical bending waves [Gray, 1953], because helical
bending waves can only provide forward propulsion if
the head is large enough to restrict rotation of the flagellum by the helical bending waves. Because of the ease of
generating helical bending waves, and their appearance
in very simple flagella [Prensier et al., 1980], planar
bending waves may be a later evolutionary development
that has modified the control mechanisms that can generate helical bending waves.
A distinction should be made between mechanisms
that prevent doublet metachronism and impose planar
bending, and mechanisms that determine the plane of
bending. These two mechanisms are combined when
planar bending patterns are obtained with three-dimensional models by specifying appropriate ␪m␬ values for
each doublet, such that dyneins on doublets 1 to 5 are
controlled as a single unit, and dyneins on doublets 6 to
9 are controlled as a single unit (Fig. 8). This is an
important result, because it shows that the previous restriction to a two-dimensional world is not an absolute
necessity for obtaining planar bending patterns with curvature-controlled models. However, obtaining planar
bending patterns by this combined specification does not
take into account accumulated experimental evidence
about planar bending waves. In the first place, there is
extensive evidence that the central pair microtubules are
important for determining the bending plane [reviewed
by Omoto et al., 1999]. The central pair microtubules
may also be important for establishing planar bending,
but planar bending has been observed in mutant flagella
with central pair microtubule deficiencies [Brokaw and
Luck, 1985]. A particularly compelling example is provided by two species of horseshoe crab spermatozoa, one
of which has central pair microtubules and generates
planar bending waves, the other of which lacks central
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Brokaw
pair microtubules and generates helical bending waves
[Ishijima et al., 1988]. Whatever mechanism is responsible for imposing planar bending, there must be a way
that it can be overridden, allowing helical bending, since
planar bending can be converted to helical bending when
sea urchin sperm flagella are exposed to high viscosities
[Woolley and Vernon, 2001]. Some computer models
with ␪m␬ ⫽ 0 can generate planar bending patterns at 0
viscosity (Fig. 9), but switch to helical bending patterns
at higher viscosity. This result is probably irrelevant
since it has not been obtainable with the models that
obtain active shear moment from stochastic modelling of
individual dyneins.
Observations of Gibbons et al. [1987] demonstrated
that the bending plane of a sea urchin sperm flagellum
could be rotated by rotating the plane of an imposed
vibration of the sperm head. This rotation may be accompanied by rotation of the central pair microtubules,
but it appears to occur without rotation of the entire
axoneme [Shingyoji et al., 1991]. This result, therefore,
provides evidence that the mechanisms for imposing
planar bending and for selecting the bending plane are
separable, and are not like the combined method used to
obtain planar bending with the models presented here
(Figs. 7, 8). It suggests that once planar bending has been
established, the bending plane can be determined by
mechanical conditions, such as the plane of minimal
bending resistance. In the experiments with imposed
vibration, the central pair microtubules may rotate to
match the imposed bending plane. In other cases, the
orientation of the central pair may determine the bending
plane simply because it establishes an increased resistance for bending in the plane of the central pair. A
detailed calculation by Hines and Blum [1983] showed
that the bending resistance of the central pair could have
only a small effect on the total bending resistance of the
axoneme, giving about a 10% increase in bending resistance for out of plane bending compared to in plane
bending. Crowley et al. [1981] calculated that out of
plane bend resistance might be increased by a factor of
up to 3 by shear resistance between doublets 5 and 6,
associated with the “5– 6 bridge” that appears instead of
dynein arms in some axonemes. Brokaw [1988] proposed, without detailed justification, that even small differences in bending resistance could establish a preferred
bending plane, if the oscillatory mechanism were nonlinear. Computations with the models presented here
indicate that even much larger ratios between EBy and
EBx are insufficient to overcome the natural tendency of
the curvature-controlled model to establish doublet metachronism and helical bending. These results do not disprove the idea that the bending plane can be determined
by the plane of minimum bending resistance, but they do
show that overcoming doublet metachronism and impos-
ing planar bending requires more than just a plane of
minimum bending resistance. Other mechanisms for constraining the flagellum to beat in a plane, without requiring a particular plane of bending, are required. These
might require that dyneins not only sense local curvature,
but also receive information about the status of dyneins
on adjacent doublets.
Some nearly planar bending patterns have been
considered to be elliptical helices with high eccentricity
[Hiramoto and Baba, 1978], and elliptical helices with
eccentricities of 0.2 to 0.5 have been described for human
and bull spermatozoa [Rikmenspoel, 1965; Ishijima et
al., 1992]. Exploration of the changes to the models that
are required to obtain elliptical helices may be a fruitful
approach to solving the problem of finding a mechanism
for planar bending wave generation that is consistent
with the experimental observations.
An additional problem became apparent with the
version of the three-dimensional model that was modified to produce planar bending (Figs. 7, 8, 10). At high
viscosity, the bending pattern was unstable unless a relatively large additional bending resistance was added in
the plane perpendicular to the bending plane. When the
bending pattern becomes unstable, external viscous
forces can induce twisting, so the effects of instability
can be minimized by increasing twist resistance, but the
out of plane bending resistance is the primary requirement for stability. The stability of planar flagellar bending at high viscosities is currently unexplained. Solving
this problem will require a better understanding of the
mechanism that imposes planar bending.
Twist Resistance of the Axoneme Becomes
Important When a Flagellum Can Bend in Three
Dimensions
At higher than normal viscosities, flagellar models
generating helical bending waves are subjected to significant forces that tend to twist the flagellum and distort the
helical bending pattern (Fig. 11). The magnitude of twist
and distortion depends upon the elastic twist resistance
parameter, EBz. A value of 2.2 times the elastic bending
resistance, EB, was used, based on the detailed analysis
of Hines and Blum [1983]. The basic twist resistance of
the axoneme is the sum of the twist resistances of its
microtubular components, so the main part of the problem is to calculate the twist resistance of an outer doublet
microtubule, relative to its bending resistance. Hines and
Blum [1983] used the simplest assumptions. The need for
a higher twist resistance may suggest that, instead, axonemal outer doublet microtubules have a specialized
design that significantly increases the ratio between their
twist resistance and their bending resistance. This solution appears essential, because very little increase in twist
resistance can be provided by linkages between outer
Simulation of Helical Bending
doublet microtubules, even if the linkages are attached in
a manner that allows them to resist twisting distortion
[Hines and Blum, 1984, 1985].
Limitations and Future Directions
The stochastic modelling of dynein kinetics in
these models is based on our understanding of myosin
operation in skeletal muscle, where large numbers of
myosin heads are assumed to operate independently under relatively uniform conditions. Actual information
about dynein function is very limited. No models take
full consideration of the diversity of dynein within an
axoneme, which is likely to be functionally important
[Asai, 1995]. The assumption that a dynein operates
independently of its neighbors is challenged by observations of a non-random grouping of distinct dynein conformations [Burgess, 1995] and by observations that the
size of dynein arms is such that interactions between
adjacent dyneins may be inevitable [Goodenough and
Heuser, 1982]. Real flagella may have additional mechanisms that limit the independent, stochastic operation of
individual dyneins in order to generate more stable and
robust bending patterns. Dyneins also possess capabilities for oscillatory and/or processive sliding [Sakakibara
et al., 2000; Shingyoji et al., 1998] that are not recognized in these flagellar models. Since, as already mentioned, some dyneins have been shown to be capable of
producing rotation of microtubules in motility assays,
models in which dyneins can produce shear force that is
not parallel to the microtubules, causing an mz component of active shear moment, need to be examined.
In the current models, dyneins in one segment
along the length, on each doublet, are regulated as a unit.
With segments of 0.4 ␮m, there are then about 16 outer
arm dyneins and 12 inner arm dyneins in each unit. It
should be relatively easy to dispense with this grouping
and regulate each dynein individually by curvature, assuming a linear change in curvature within each length
segment. This improvement may eliminate much of the
effect of segment length on the performance of the models, but the effect of this change on the stability of the
models is difficult to predict.
The methods used in this paper for introducing
external viscous resistances are a crude approximation,
which may only be justifiable as a means to explore the
limitations of the methods. The resistance coefficients
method is not appropriate for modelling the helical bending waves seen by Woolley and Vernon [2001], in which
successive gyres of the helix are separated in space by
only a few microns. The hydrodynamic methods developed by Gueron and Liron [1992] for handling situations
where viscous interactions between adjacent cilia are
important will probably be needed for accurate modelling of helical bending waves that have high pitch angles,
123
and trying to understand the transition between planar
and helical bending. Newer methods for analysis of
three-dimensional bending in the presence of external
viscosity, developed by Gueron and Levit-Gurevich
[2001a,b], will need to be evaluated for applicability to
the types of helical bending waves examined in this
work.
Many cilia perform three-dimensional bending that
is not characterizable as helical, because their lengths are
short compared to the wavelength characteristics of the
bending. At best, these bending patterns are comparable
to the bending in the basal regions of flagella and flagellar models that generate approximately helical bending
waves. The basal bending of the flagellar models has not
been characterized in detail. In addition, the asymmetry
of most ciliary bending patterns implies differences in
the control of dyneins on different doublets, in contrast to
the uniform control that has been emphasized here. As an
example, a detailed analysis of curvature and torsion in a
three-dimensional ciliary bending pattern is given by
Teunis and Machemer [1994]. Modelling of such patterns has been discussed by Sugino and Naitoh [1982]
and by Gueron and Levit-Gurevich [2001a]. There is no
assurance that local curvature control is an appropriate
paradigm for these bending patterns, or even for the
extremely asymmetric two-dimensional bending patterns
that can be seen, for example, in sperm flagella at high
calcium ion concentration [Brokaw, 1979]. Modelling
these bending patterns with the methods developed in
this study remains a challenge for the future.
ACKNOWLEDGMENTS
I thank Dr. C.K. Omoto for valuable comments on
an early version of this manuscript.
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