The Geometry of Ciliary Dynamics

The Geometry of Ciliary Dynamics
ED10625
The Geometry of Ciliary Dynamics
Mark A. Peterson
Mount Holyoke College∗
(Dated: May 20, 2009)
Abstract
Cilia are motile biological appendages that are driven to bend by internal shear stresses between
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tubulin filaments. A continuum model of ciliary material is constructed that incorporates the
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essential ciliary constraints: (1) one-dimensional inextensibility of filaments, (2) three-dimensional
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incompressibility, and (3) shear strain only longitudinally along filaments. It is shown that twist of
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filaments about each other is not an independent degree of freedom under ciliary constraints. The
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constraint on twist appears in the equations of motion for cilia as a term not previously recognized.
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PACS numbers: 45.10.Na, 46.70.Hg, 47.63.Gd
Keywords:
∗
Electronic address: mpeterso@mtholyoke.edu
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bacterial flagellae is proposed.
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As another application of the same geometrical idea, a general approach to the polymorphism of
(May 20, 2009)
I.
INTRODUCTION
The beating of cilia, or eukaryotic flagella, has been the subject of research for over fifty
years. The hydrodynamics of propulsion is one aspect of these studies, concerned with Stokes
flows past the cilium at a mesoscale of microns, where the cilium is considered a plane curve
or space curve moving in a prescribed way [1][2][3][4] [5][6][7]. The length scale is set by the
length of the cilium, with the small diameter of the cilium serving only as a cutoff in the
logarithmic singularity of the hydrodynamics.
Another focus of cilia research has been the mechanism by which the cilium generates
motion, concerned with the dynein-mediated sliding of microtubule doublets on each other
at a nanoscale of tens of nanometers [8] [9][10][11][12][13]. The length scale is set by the
diameter of the cilium or even by the size of the motor proteins.
The standard ciliary model, motivated by the structure of the cilium, or axoneme, is
called the sliding filament model. The axoneme structure is shown schematically in Fig. 1.
There are 9 microtubule doublets around the outside, and 2 in the center. Other components
are also visible in electron micrographs: radial spokes connecting the outer microtubules to
the central ones, and dynein arms, the motor protein for the system, positioned to walk
along adjacent microtubule doublets and generate shear stress between them.
In the sliding filament model the microtubule doublets (the ‘filaments’) are made to slide
along each other by the dynein motors. Because shear strain is tightly coupled to bending,
the cilium bends, and thus moves. The tight coupling between shear strain and bend,
essential for this mechanism, is always assumed either explicitly or implicitly in the form of
ciliary constraints: (1) the filaments are inextensible, (2) the material is incompressible in the
3-dimensional sense, and (3) shear strain is longitudinal along filaments. In this paper I make
these constraints precise in a hypothetical continuum model of “ciliary material,” abstracted
from the structure of the axoneme, and I deduce the consequences for the possible states
and motion of this material, independent of all other details. Some of these consequences
have not been recognized before. I summarize the results in this introductory section.
The ciliary constraints couple shear strain and bend so tightly that it is possible to take as
the basic kinematic variables not the shape of the cilium, as is usually done, with equations
of motion for that shape, but rather the two independent components of shear strain. This
choice is well adapted to modeling the molecular mechanism of ciliary motion, since local
2
(May 20, 2009)
3
4
2
5
1
6
9
7
8
FIG. 1: Schematic cross-section of a cilium showing the 9+2 arrangement of microtubules (dots)
and microtubule doublets (double dots), the radial spokes (solid) and the dynein arms (dotted).
strain at the nanoscale knows very little about the largely irrelevant mesoscopic shape of the
cilium. The cilium shape is implicit in the state of the shear strain, and can be found from
the shear strain by integration, but it is irrelevant for the dynamics (unless one includes the
non-local part of the hydrodynamic interaction, which is mesoscopic, and does depend on
shape, but is well understood, a separate issue).
The ciliary constraints constrain the twist degree of freedom, i.e. differential rotation
about a given filament (differential rotation about the central microtubules of the axoneme,
for example). Thus twist is not an independent degree of freedom in ciliary matter. What
the ciliary constraints mean for the 9+2 axoneme is illustrated in Fig. 2. Twist has been
treated somewhat inconsistently in the past. Twist was not an issue as long as the cilium
was modelled as a curve with negligible thickness, so until recently equations of motion
just ignored it. More recently, in work of Gueron and Levit-Gurevich [6] and Hilfinger and
Jülicher [15], twist is a degree of freedom on a par with other ones. If the ciliary constraints
are obeyed (more on this in a moment!), neither of these approaches is quite right. Ciliary
matter does twist, and its equations of motion contain terms reflecting this, but these terms
involve a Lagrange multiplier, not recognized before, enforcing the ciliary constraints. Thus
the ciliary equations of motion incorporating the generally accepted ciliary constraints are
derived here in their complete form for the first time.
3
(May 20, 2009)
5
4
3
6
2
7
1
8
4
9
3
2
5
1
6
9
7
8
FIG. 2: Under the ciliary constraints, the filaments in the neighborhood of a given filament twist
about it in a way that is determined by the geometry of the given filament as a space curve (the
displacement vector between the filaments is parallel transported along the given filament). The
above 9+2 cilium, which is assumed to have taken a helical shape with curvature κ = 1 and torsion
τ = 2, is shown in cross-section as it intersects a plane containing the helix axis. It twists by about
38◦ on each turn.
The ciliary constraints forbid the kind of motion shown in Fig. 3, in which initially
straight filaments forming a cylinder slide along each other (non-zero shear strain) in order
to spiral about the cylinder, but without bending the cylinder. From an evolutionary point
of view it is clear that a motile organism depending on its cilia for propulsion would not
want this decoupling of shear and bend to happen. It would want to control twist in order
to control the orientation of the ciliary beat in space. The ciliary constraints accomplish
that. Hilfinger and Jülicher in [15] nonetheless argue that twist is an independent degree
of freedom in at least some cilia. Now it turns out that these are not 9+2 cilia, but 9+0
cilia, lacking the central microtubules. They are motile, but their purpose is not to propel
an organism but rather to drive a flow with a definite chirality. (Other 9+0 cilia are not
4
(May 20, 2009)
FIG. 3: In a twist transition the microtubules shear along each other, but do not generate bend,
i.e., the cylinder remains straight. Such a motion violates the ciliary constraints. It may be possible
in a 9+0 axoneme, lacking the central 2 microtubules, but not in a 9+2 axoneme, in which the
central microtubules enforce the ciliary constraints. The center of the cylinder is deliberately shown
empty here.
even motile.) Apparently the cilia of [15] had to lose the central microtubules to be able
to adopt a spiral handedness, as if the central microtubules would have enforced the ciliary
constraints that these cilia violate. With a little more insight into the meaning of the ciliary
constraints we can see intuitively why the ciliary constraints forbid the motion of Fig 3. As
I will show, the ciliary constraints require neighboring filaments to be as parallel as possible,
given that they also bend. In the motion of Fig. 3 the outer filaments cannot remain parallel
to the central ones, if central ones exist. But if there is nothing in the center, there is nothing
there to be parallel to. Perhaps the radial spokes are structures in 9+2 cilia whose purpose
is precisely to enforce the ciliary constraints in propulsive cilia for the control of twist.
Because ciliary matter is an abstract model, characterized just by certain constraints
and no other stipulations, it might apply to other dense assemblages of filaments than the
axonemal structure that suggested it. In a final section I consider what ciliary constraints
would mean for the protofilaments of the prokaryotic flagellum and the question of its typically helical shape. Without answering that question I do reduce it to a definite question
about strain in a certain surface, assuming ciliary constraints. Remarkably, there might be
5
(May 20, 2009)
an analogue of the 9+2 versus 9+0 phenomenon in the prokaryotic world.
The plan of the paper is as follows. Section II gives an abstract characterization of the
three dimensional material that I call ‘ciliary material,’ made up of ‘filaments’ subject to the
constraints of the sliding filament model: incompressible, inextensible along the filaments,
and admitting shear strain only longitudinally along the filaments. This is an attempt to
capture, in a continuum model, the material properties of the cilium. The unexpected result
for the states of this material is that one filament can take any shape as a space curve, and
all the rest are determined by that one.
Section III determines how the ciliary material can move, subject to its constraints,
what I call ciliary flow (note: not the flow driven in the surrounding aqueous medium by a
beating cilium, but the motion of the three dimensional ciliary material itself, thought of as
a kind of liquid crystal). Consistent with the result of the preceding section, its motion is
parameterized by the motion of any single one-dimensional filament, and this one filament
can move arbitrarily. Thus the hypothetical ciliary material, motivated by the structure of
cilia, combines, in an unexpected way, three-dimensional and one-dimensional properties.
Section IV derives the equation of motion of ciliary material from standard assumptions
of continuum mechanics. It turns out to be the usual equation of motion of cilia but with
a new term that enforces the constraint on twist. The result is remarkable because the
continuum mechanics assumptions say nothing about the cilia shape, and do not even refer
to it.
Section V investigates what the ciliary constraints would mean for the material of bacterial
flagellae. There are hints that this material may also obey ciliary constraints.
The geometrical methods in Sections II and III are well described in the physics textbooks
by Schutz [16] and Frankel [17] for any reader not familiar with them.
II.
CILIARY MATERIAL
In terms of smooth coordinates (x1 , x2 , x3 ) in space one can describe the deformation of
any material by the trajectories of its constituent particles, solutions of equations of motion
dxi
= V i (x, t)
dt
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(May 20, 2009)
(1)
where
V = V i ∂i
(2)
is the velocity vector field, and t is time. Metric relations among particles are given by
ds2 = gij dxi dxj
(3)
where gij is the Euclidean metric. Coordinatize the material object by Lagrangian coordinates convected by the flow, i.e., let every material point keep the same coordinates that
it had originally. The changing metric relationship of material points is then expressed entirely by the change in the metric components gij , and the rate of change is given by the Lie
derivative
∂gjk
= V gjk + g([∂j , V ], ∂k ) + g(∂j , [∂k , V ])
∂t
(4)
Here [ , ] is the Lie bracket of vector fields.
As an example of such a computation, let V be the shearing flow
V = yS∂x
(5)
in the plane, where S is a constant. Then if x and y are initially Cartesian coordinates,
the metric tensor in convected coordinates at time t is given by the Lie-Taylor series (which
terminates in this case)


tS
t
 1

g + t£V g + £V £V g = 
 ,
2
2
2!
tS 1 + t S
2
(6)
and evaluating at t = 1 one has the Euclidean metric

g=

1

S
S 1+S
2


,
(7)
in skew coordinates corresponding to constant shear strain S as shown in Fig. 4.
The flow V of Eq. (5) is an example of a ciliary flow, and the form of the metric tensor
g in Eq. (7) encodes the result. The filaments, parallel to the x-axis and labeled by the y
coordinate, slide on each other inextensibly, as one sees in the metric component g11 = 1.
The resulting shear strain is visible in the off-diagonal component g12 = S. Finally, det g = 1
means that the flow is incompressible. The three dimensional generalization of this form to
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(May 20, 2009)
y
S
( 0,1)
1
( 0,0)
x
FIG. 4: In the skew coordinates (x,y) the distance between (0,0) and (0,1) is correctly given as
√
1 + S 2 by the metric tensor of Eq. (7) .
a general ciliary configuration is

S
 1

2
g=
S 1+S

T
ST
T
ST
1 + T2






.
(8)
where now S and T , two independent components of shear strain, depend on Lagrangian
coordinates (x, y, z). Here x is arclength along filaments, up to an additive constant reflecting
the arbitrariness of choosing an origin in each filament, and the coordinates (y, z) label the
filaments. It should be noted that this is not the most general outcome of an incompressible
flow of the filaments, because the only motion that has been allowed to them is sliding
longitudinally along each other (and of course bending). These are exactly the constraints
of the sliding filament model, encoded in the form of g. The precise statement of the ciliary
constraints is that the metric tensor should have this form in Lagrangian coordinates, which
is really a stipulation of the form of the strain.
Associated with the metric tensor g is an orthonormal frame field
e1 = ∂x
(9)
e2 = ∂y − S∂x
(10)
e3 = ∂z − T ∂x .
(11)
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(May 20, 2009)
where e1 is everywhere tangent to filaments. This is not, however, a coordinate frame,
because the vector fields do not commute as differential operators, in general.
The metric g determines the shape of each filament up to a rigid motion, because for
each (y, z) it determines the filament’s curvature κ and torsion τ as a function of x (see
Appendix A). These are the Frenet data for the filament as a space curve. One has
q
Sx2 + Tx2
Txx Sx − Sxx Tx
τ =
κ2
κ =
(12)
(13)
where the subscript x denotes the derivative ∂x along the filament. The Frenet equations then
~
determine the filaments as space curves R(x)
along with an orthonormal frame {T̂ , N̂, B̂}
at each point of each curve according to
~ x = T̂
R
(14)
T̂x = κ N̂
(15)
N̂x = −κ T̂ + τ B̂
(16)
B̂x = −τ N̂
(17)
In the context of ciliary geometry it is more natural to describe the filaments in terms of
the shear strains S and T directly than to translate them into the Frenet language. This
is an alternative, equivalent description of space curves. It is even possible to translate the
other way, from the Frenet description to the ciliary description, inverting the relations in
Eqs. (12)-(13),
Sx = κ cos φ
(18)
Tx = κ sin φ
(19)
φx = τ
(20)
~
In this formulation the space curves R(x)
are the solutions of
~ x = e1
R
(21)
∂x e1 = κ cos φ e2 + κ sin φ e3
(22)
∂x e2 = −κ cos φ e1
(23)
∂x e3 = −κ sin φ e1
(24)
φx = τ
9
(25)
(May 20, 2009)
There is an arbitrary constant in the angle φ since, unlike N̂ and B̂ in the Frenet picture,
e2 and e3 are only determined up to a global rotation about e1 . This amounts to a gauge
freedom in the ciliary description. An advantage of the ciliary description is that it remains
well defined where the curvature vanishes, a nuisance in the Frenet description. These
equations imply that the displacement vector from a given filament toward a neighboring
filament, e2 for example, is parallel transported along the given filament. In more physical
language, the neighboring filament is as parallel as it can be to the given filament.
The shear strains S and T cannot be arbitrary functions, because g is the Euclidean
metric, even if it is expressed in peculiar coordinates. Its associated Riemannian curvature
tensor must therefore vanish identically, and hence S and T must obey the following identities
(see Appendix B)
0 = ∂x (e2 S) = ∂x (e3 S) = ∂x (e2 T ) = ∂x (e3 T )
(26)
0 = e3 S − e2 T .
(27)
The second of these equations can be recognized as
[e2 , e3 ] = 0 .
(28)
This says that e2 and e3 together form an integrable distribution, and thus there exist
surfaces normal to the filaments, at least locally. Moreover, since e2 and e3 are now an
orthonormal coordinate frame field on these surfaces, they are isometric to Euclidean planes.
A computation shows that their second fundamental form vanishes ([e2 , e1 ] and [e3 , e1 ] are
orthogonal to e2 and e3 ). Thus they are not just isometric to Euclidean planes, they actually
are Euclidean planes, and I will call them normal planes.
By allowing the filaments to reptate along their length inextensibly, one can bring Eq. (26)
to the simpler form
0 = e2 S = e3 S = e2 T = e3 T
(29)
that is, the strains S and T can be made constant in normal planes, as in Fig. 5. Thus the
configuration of filaments in a neighborhood of a given filament is entirely determined by
that filament. The given filament determines the direction of neighboring filaments, since
it determines the normal planes, and it determines their curvatures and torsions, since it
determines the values of S and T . The geometrical meaning of Eqs. (26)-(27) is that these
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(May 20, 2009)
x
θ
FIG. 5: If filaments are bent into arcs of circles, with arclength x along the filaments and plane
polar coordinates (r, θ), then x = rθ, and hence shear strain S = −∂x/∂r = −θ = −x/r is constant
on lines perpendicular to the filaments, a special case of Eq. (29). Also note Sx = −1/r is the
curvature, a special case of Eq. (18). The minus sign is because the curvature vector is opposite
to e2 = ∂/∂r.
two potentially conflicting descriptions are consistent. Thus the ciliary geometry is almost
rigid. The freedom that it possesses corresponds essentially to a single free space curve.
I turn now to the motions possible in ciliary matter, ciliary flows (note: not flows external
to a cilium, but flows within the cilium). Since one of those motions is reptation of filaments,
I will also justify the assertions of the preceding paragraph.
III.
CILIARY FLOWS
A ciliary flow, by definition, must maintain the form of Eq. (8) even as g evolves according
to Eq. (4). If V is the flow with components (α, β, γ), namely
V = α∂x + β∂y + γ∂z ,
(30)
then for V to be ciliary (α, β, γ) must obey the conditions (see Appendix C)
0 = αx + Sβx + T γx
(31)
0 = e2 β
(32)
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(May 20, 2009)
0 = e3 γ
(33)
0 = e2 γ + e3 β
(34)
0 = (e2 γ)x − βx Tx + γx Sx
(35)
0 = e2 e2 γ
(36)
0 = e3 e3 β
(37)
where the partial derivative ∂x is indicated by the subscript x. Eqs. (36)-(37) are not
independent of the others, since for example e3 e3 β = −e3 e2 γ = −e2 e3 γ = 0. In fact all
higher derivatives of β and γ in the normal planes vanish by this argument, so that β and γ
restricted to a normal plane can only be affine linear functions. I return to this consideration
below.
The simplest nontrivial ciliary flow is reptation, αx = β = γ = 0. By Eq. (4) the shear
strains change under this flow at the rate
St = αy + αSx
(38)
Tt = αz + αTx ,
(39)
where subscripts indicate partial derivatives with respect to the corresponding variable. It is
possible to construct a reptation α that alters S and T in such a way that Eq. (29) holds in
one normal plane, at least locally (see Appendix D). Then by Eq. (26) S and T are constant
in every normal plane along the filaments. This proves the assertions made at the end of
the last section. I will now assume that S and T obey the simpler Eq. (29), since this can
always be arranged by a reptation.
The ciliary conditions require β and γ to be affine linear functions in the normal planes.
To see that there exist non-trivial solutions to these conditions, imagine specifying β(x) and
γ(x) arbitrarily along one filament. Integrating Eq. (35) determines e2 γ = −e3 β along the
filament up to an arbitrary global constant. These are all the data required to extend β
and γ in each normal plane as affine linear functions. This extension continues to satisfy
Eq. (35) on neighboring filaments because of the easily verified identities
e2 [(e2 γ)x − Tx βx + Sx γx ] = 2Sx [(e2 γ)x − Tx βx + Sx γx ]
(40)
e3 [(e2 γ)x − Tx βx + Sx γx ] = 2Tx [(e2 γ)x − Tx βx + Sx γx ]
(41)
Repeated differentiation of Eq. (35) using e2 and e3 shows that if (e2 γ)x − Tx βx + Sx γx
vanishes on a filament, then all its normal derivatives also vanish there. Thus given only
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(May 20, 2009)
that it is represented by its Taylor series, it is constant, and hence zero, and the constructed
solution obeys all the ciliary conditions. V is a non-trivial ciliary flow, determined by its
values on one filament.
The conditions Eqs. (31)-(37) confirm that the ciliary configuration is entirely determined
by one filament. If one tries to move the filaments, one can specify β(x) and γ(x) on only
one given filament. The ciliary conditions then determine α(x) along that filament up to a
constant (a reptation). They further determine β and γ almost uniquely as linear functions
in normal planes. Neighboring filaments intersect the normal planes (each plane labelled by
x, its intersection with the given filament), and in the course of the motion these intersection
points rotate about the given filament at an angular velocity
ω(x) = (e2 γ)(x)
(42)
which is not arbitrary but is determined up to a global constant by β and γ according to
Eq. (35). (The undetermined constant in ω describes a global twisting rotation in which
every normal plane rotates at the same rate, a motion which is possible but probably not
physically relevant.) The remaining freedom in α corresponds to reptations of neighboring
filaments. Thus, to summarize, any filament determines the configurations of its neighbors,
and any motion of that filament determines the motion of its neighbors. It is worth noting
what was not obvious a priori, that non-trivial motions of the filaments are possible, that
is, the ciliary material is not completely rigid. One filament can move arbitrarily (but
inextensibly), and all the others must follow it.
IV.
CILIARY DYNAMICS
I now model the dynamics of the cilium by applying standard ideas of continuum mechanics to a thin cylinder of ciliary matter of length L and radius ρ, all dynamical quantities
depending just on x, the coordinate along one given filament. That filament can of course
have any shape, so the cylinder is not necessarily straight. Let the cilium have elastic moduli,
so that the energy of a configuration is given by the shear energy and bending energy
µ
E=
2
Z
L
0
κc
[(S − F ) + (T − G) ] dx +
2
2
2
Z
0
L
(Sx2 + Tx2 ) dx ,
(43)
where µ is the shear modulus and κc is the bending modulus (with dimensions appropriate
to one dimension, not three). Here F and G are target shear strains, such that the shear
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(May 20, 2009)
energy would be minimal if the shear strains S and T could relax to F and G. The operation
of the ciliary “engine” would be to make F and G functions of time, so that the equilibrium
becomes a moving target. Asymmetries in the nanoscopic structure of the cilium could be
built into this expression, which is taken symmetrical here for illustration.
Conjugate to shear strains S and T are internal stresses
Φ = δE/δS = µ(S − F ) − κc Sxx
(44)
Ψ = δE/δT = µ(T − G) − κc Txx
(45)
which, by the above mentioned one-dimensionality, have the units of force.
Continuing this variational approach to the dynamics, one finds the generalized forces
conjugate to the displacement of a cilium along V = α∂x + β∂y + γ∂z of Eq. (30), together
with the required rotation ω of Eq.(42) in normal planes. In the orthonormal basis, V takes
the form
V = (α + βS + γT )e1 + βe2 + γe3 ,
(46)
a representation that becomes increasingly appropriate as one moves up to the mesoscopic
scale. Note that β and γ are the components of velocity normal to the cilium, but that they
also contribute to the tangential velocity if the shear strains are non-zero. This is a residual
piece of three dimensional information in the one-dimensional description of the cilium as a
curve.
From Eq. (4), imposing Eqs. (31)-(37), the strains under the flow V change at the rate
St = (α + βS + γT )Sx + βx + e2 α + ωT
(47)
Tt = (α + βS + γT )Tx + γx + e3 α − ωS
(48)
The terms e2 α and e3 α represent reptations within the cilium, a possibility that is ignored
from now on. The constraints
0 = αx + Sβx + T γx
(49)
0 = ωx − Tx βx + Sx γx
(50)
are handled with Lagrange multipliers λ and ν in the expression
′
E =E+
Z
0
L
λ(αx + Sβx + T γx ) dx +
14
Z
0
(May 20, 2009)
L
ν(ωx − Tx βx + Sx γx ) dx
(51)
Then the generalized interior forces on the cilium are
Fα = −δE ′ /δα = −ΦSx − ΨTx + λx
(52)
Fβ = −δE ′ /δβ = −ΦSSx − ΨSTx + Φx + (λS)x − (νTx )x
(53)
Fγ = −δE ′ /δγ = −ΦT Sx − ΨT Tx + Ψx + (λT )x + (νSx )x
(54)
Fω = −δE ′ /δω = νx − ΦT + ΨS
(55)
Generalized external forces on the cilium, due to the fluid medium of viscosity η in which
it is immersed, can be found from the dissipation function D for the corresponding Stokes
flow. A thorough analysis of this problem has been done by Gueron and Liron [4]. For
simplicity I take just the leading term indicated by their analysis,
D=
Z
L
0
CT
CN 2
Cω 2
(α + βS + γT )2 +
(β + γ 2 ) +
ω dx
2
2
2
(56)
where the C’s are constant resistance coefficients. The viscous force conjugate to displacement by α is then −δD/δα, etc. Requiring that elastic forces and viscous forces balance,
i.e.,
δE ′
δD
=
,
δα
δα
(57)
etc., leads to the surprisingly simple dynamical laws connecting the motion of ciliary matter
to its internal stresses,
CT (α + βS + γT ) = −ΦSx − ΨTx + λx
(58)
CN β = Φx + λSx − (νTx )x
(59)
CN γ = Ψx + λTx + (νSx )x
(60)
Cω ω = νx − ΦT + ΨS
(61)
Under this flow the strains change according to Eqs. (47)-(48). The Lagrange multipliers
can now be determined from the constraints Eqs. (49)-(50). Substituting the solutions in
Eqs. (58)-(61) gives the equations for λ and ν,
CT 2
CT
CT 2
κ λ + (ΦSx + ΨTx )x +
(Sx Φx + Tx Ψx ) +
τκ ν
CN
CN
CN
i
Cω h
Sx Ψxx − Tx Φxx + λτ κ2 + Tx (νTx )xx + Sx (νSx )xx ,
= (ΦT − ΨS)x −
CN
λxx =
(62)
νxx
(63)
where I have used the expressions for curvature κ and torsion τ first derived in terms of S and
T in Eqs. (12)-(13). In the limit as Cω /CN → 0, the equation for ν simplifies considerably,
15
(May 20, 2009)
with solution
ν=
x
Z
0
(ΦT − ΨS) dx + c1 x + c2
(64)
The solution for ω is already explicit in terms of β and γ from the constraint Eq. (50),
ω=
Z
x
0
(Tx βx − Sx γx ) dx + const.
(65)
With one important exception, the above dynamical laws for ciliary matter are precisely
the usual phenomenological laws for three-dimensional motions of cilia, as derived by Gueron
and Liron [5]. It is remarkable that they emerge here without any appeal or reference to
the shape of the cilium (apart from the hydrodynamic interaction that one could introduce
into a more accurate dissipation function, Eq. (56)), but purely as a consequence of the
dynamics of S and T . The occurrence of κ and τ above is just an abbreviation for certain
combinations of derivatives of S and T that appeared. Of course the large scale shape of
the cilium could be reconstructed at any time from Eqs. (18)-(25).
The dynamical laws have been expressed here in terms of the force vector
F~ = λe1 + Φe2 + Ψe3
(66)
but the vector quantities in [5] were expressed in terms of the Frenet basis T̂ , N̂, B̂, where
F~ took the form
F~ = −FT T̂ − FN N̂ − FB B̂ .
(67)
From Eqs. (18)-(22) one has the transformation that connects these two descriptions,
T̂ = e1
(68)
N̂ = cos φ e2 + sin φ e3
(69)
B̂ = − sin φ e2 + cos φ e3
(70)
with φx = τ , the torsion of the curve. Using also Eqs. (12)-(13), it is straightforward
to verify that the dynamical laws of ciliary matter, as derived here, agree with standard
phenomenology in every respect but one: in ciliary matter there is an additional constraint,
and a corresponding additional term in the laws.
The additional constraint arises because the sliding filament model is constrained by more
than just the inextensibility of the filaments. Its constraints are expressed in the form of
the strain tensor, or equivalently in the form of the metric tensor in Lagrangian coordinates,
16
(May 20, 2009)
Eq. (8). It is to be expected that these additional constraints would have consequences for
the dynamics, just as inextensibility does, through the Lagrange multiplier λ. The dynamical
form that this constraint takes is a certain definite coupling of shear and twist, the last terms
in Eqs. (59)-(60), and more generally every reference to the Lagrange multiplier ν and the
twist velocity ω. Models of ciliary dynamics should either include these terms or justify the
violation of ciliary constraints.
V.
HELICES AND PROKARYOTIC FLAGELLAE
The flagellae of bacteria are unrelated to the cilia of eukaryotic cells. They are typically
helices of well defined curvature and torsion, although within each bacterial species they
exhibit polymorphism: more than one helix may exist as a metastable form. The helical
flagellum is turned by a rotary motor in the bacterial cell membrane like a screw propeller.
This mechanism of cell motility is clearly very different from the sliding filament mechanism
that causes cilia to beat.
The structure of the flagellum is nonetheless somewhat reminiscent of that of the cilium.
The protein flagellin polymerizes to form protofilaments, and typically 11 protofilaments
self-assemble side by side to form a hollow tube that is the flagellum. The protofilaments
constituting the flagellum may wrap around its helical form in either the left handed or
right handed sense. In a polymorphic transition, everything changes together, and may
even change from right handed to left handed, or the reverse. Attempts to understand
the existence of the helical form in the first place, and its polymorphism, have continued
since pioneering work of S. Asakura[18]. Recent work including a good introduction to the
problem is the article of S. Srigiriraju and T. Powers[19].
The behavior described above is not unlike that of ciliary matter. The necessarily cooperative motion of all filaments together, exhibited in the collective behavior of the protofilaments in flagellar polymorphic transitions, is the most characteristic feature of what I have
called ciliary flow. And as I show below, if one filament is a helix, the nearby ones wind
around it in the manner of the flagellin protofilaments around the centerline of the helical
flagellum. These similarities in the properties of abstract ciliary matter and real flagellae
motivate the following observations.
Ciliary matter is almost rigid, and there is almost nothing about it that is adjustable.
17
(May 20, 2009)
The only freedom is to choose the shape of one filament. Let that one filament be a helix
with curvature κ and torsion τ ,
√
κ
κ2 + τ 2 x
cos
κ2 + τ 2
√
κ
κ2 + τ 2 x
sin
Y = 2
κ + τ2
τx
Z = √ 2
,
κ + τ2
X =
(71)
(72)
(73)
where (X, Y, Z) are fixed Cartesian coordinates, and x is arclength along the helix. Then,
integrating Eqs. (18)-(20), the local shear strains between filaments are
κ
sin τ x
τ
κ
T = − cos τ x .
τ
S =
(74)
(75)
Inverting Eqs. (69)-(70), we see that along this helix the vectors e2 and e3 rotate with respect
to the Frenet frame with (spatial) period 2π/τ ,
e2 = cos(τ x)N̂ − sin(τ x)B̂
(76)
e3 = sin(τ x)N̂ + cos(τ x)B̂
(77)
Since e2 and e3 are defined by Lagrangian coordinates, and point to definite filaments, this
is not just the behavior of the coordinates, it means that the filaments of the ciliary matter
spiral about the central helix in this way, reminiscent of the spiraling of protofilaments in
the flagellum, as I noted above.
We are thus motivated to ask whether the protofilaments in the flagellum can be modeled
by the filaments of ciliary matter. It is immediate that sometimes the answer is ‘no.’ There
is a form of flagellin that forms a straight flagellar filament, i.e., a straight hollow cylinder, in
which the protofilaments that make up the cylinder spiral around the cylinder, like the second
cylinder in Fig. 3. By the rigidity of ciliary matter, if the protofilaments are helices, then
the flagellum itself should be a helix, and if the flagellum is straight, then the protofilaments
must be straight, like the first cylinder in Fig 3. Yet it is obvious that a straight cylinder
can be made of helical protofilaments, as happens in this case, just as it can presumably
happen in the 9+0 cilium [15].
The form of flagellin described above lacks a piece that extends inside the cylinder and
forms connections there among the protofilaments into what could be functionally a twelfth,
18
(May 20, 2009)
central protofilament. In Ref. [19] this central component was modeled as a central elastic
rod. I consider the possibility that it completes the three-dimensional structure, in the same
way that I have conjectured that the central microtubules complete it in the 9+2 cilium and
impose the ciliary constraints. With this (normal) flagellin the flagellum does behave more
like ciliary matter, as I have already noted. In Ref. [19] elastic stress imposed by the central
rod on the outer filaments was necessary to obtain the observed behavior: this may be an
equivalent way of saying that the central rod imposes the ciliary constrains.
Within the model of ciliary material it is easy to find the strain tensor in the cylindrical
surface of the flagellum, relative to the reference configuration of the straight flagellum, as the
flagellum varies through all helical shapes. It is this surface strain, and the associated elastic
energy, that is thought to control and select the observed helical shapes of flagella as local
elastic energy minima. I have already found in Eqs. (74)-(75) the two independent functions
S and T in the metric tensor g along a general helix, which I take to be the centerline of
the flagellum. The cylindrical surface surrounding this helix at a distance ρ ≈ 10 nm can
be coordinatized by x and θ, where (ρ, θ) are polar coordinates in normal planes, and the
direction of e2 is θ = 0. Note that θ is then a Lagrangian coordinate, labelling protofilaments
on the surface of the flagellum. In particular, θ = 0 labels the protofilament pointed to by
e2 . I restrict g to the surface by evaluating it on ∂/∂x and ∂/∂θ, where
∂
∂
∂
= −ρ sin θ
+ ρ cos θ .
∂θ
∂y
∂z
< 0.01, I find
Ignoring higher order corrections in κρ ∼

g=

1
2
−(ρκ/τ ) cos(τ x − θ)
(78)

2
−(ρκ/τ ) cos(τ x − θ) ρ [1 + ((ρκ/τ ) cos(τ x − θ)) ]


(79)
The two-dimensional strain tensor U is therefore, up to a factor of 2, the difference of this
tensor from its value at the reference configuration κ = 0, namely
U=


0
−(ρκ/τ ) cos(τ x − θ) 
1


2 −(ρκ/τ ) cos(τ x − θ) ((ρ2 κ/τ ) cos(τ x − θ))2
(80)
There is essentially just one independent component of shear strain,
Uxθ = −(ρκ/τ ) cos(τ x − θ) .
(81)
The challenge in modelling the flagellum is to understand what is the underlying elastic
energy that is minimized, at least locally, by this strain at the observed helical values of κ
19
(May 20, 2009)
2π
τ
x
0
θ
2π
FIG. 6: The coordinate plane for the surface of the flagellum has lines of constant θ that map onto
spiraling protofilaments. The coordinate x is arclength along protofilaments. The dotted line is a
line of constant shear strain, determining that the flagellum is a helix of torsion τ .
and τ . Any explanation for why the surface of the flagellum is strained in this way would
explain the shape of the prokaryotic flagellum, assuming ciliary constraints.
It is worth noting a point of common sense about this result, illustrated in Fig. 6, which
almost could be guessed without any computation. Since any fixed protofilament (fixed θ)
spirals around the helical flagellum, the shear strain in its neighborhood must be periodic
in x with period 2π/τ , as it visits the inside of the helix, then the outside, etc. Helical
symmetry then requires the strain to have the form of Eq. (81), at least the lowest order
Fourier component. Since protofilaments really do spiral around the flagellum, the actual
strain in the surface must look like this even if ciliary matter turns out not to be a good
model for it.
Detailed models of how strain is introduced into the surface, as the flagellum seeks its equilibrium configuration, suggest that there could be local conformational changes, rotations,
or displacements of the flagellin units that make up the protofilaments. In particular, these
changes, with their attendant shear strain, could be cooperative within a protofilament[19],
propagating shear strain along it. The picture that emerges from the ciliary model is similar,
20
(May 20, 2009)
but not identical to this. If the shear strain were propagated along a protofilament, it would
depend on the protofilament (i.e., on θ), but not on x, which is not the form of Eq. (81).
Of course the ultimate shear strain in Eq. (81) could be driven by an imposed strain of a
different form, with the flagellum eventually relaxing into Eq. (81) as a compromise involving other elastic energies, but the most straightforward idea is that the above shear strain is
simply imposed by whatever the mechanism is. In the language of cooperative interaction
between flagellin units, it says that the shear strain propagates along a protofilament, but
is gradually also communicated to the neighboring protofilament on one side. The distance
required for this strain to visit all protofilaments in turn and return to the original protofilament determines τ , the torsion of the eventual flagellum helix, and the maximum amplitude
of the shear strain determines κ, the curvature of the eventual helix. In this way nanoscopic
details of interaction would determine the mesoscopic form of the flagellum
Pursuing such details here would take me away from the original purpose of this paper,
which was to point out useful and previously unrecognized properties of a model that is
already widely used in biophysics, a continuum approximation to the sliding filament model.
Appendix A: Ciliary filaments as space curves
The rates of change of the vector fields e2 , e3 , along the integral curves of e1 , (the filaments
of the ciliary space) are given by the Lie derivatives (Lie brackets)
∂x e2 = [e1 , e2 ] = −Sx e1
(A1)
∂x e3 = [e1 , e3 ] = −Tx e1
(A2)
where the subscript x indicates ∂x , the derivative along a filament. Thus the curvature
vector, in the sense of the Frenet equations, is
κN̂ = Sx e2 + Tx e3
(A3)
where the curvature is the norm of this vector
κ=
q
Sx2 + Tx2 .
(A4)
The rate of change of the principal normal N̂ along the filament, dotted with the binormal
B̂ =
21
Sx e3 − Tx e2
κ
(May 20, 2009)
(A5)
is the torsion
τ = g([e1 , N̂], B̂) =
Txx Sx − Sxx Tx
κ2
(A6)
Appendix B: Consequences of flat geometry
Differential forms dual to the orthonormal vector fields e1 , e2 , e3 of Eqs. (9)-(11) are
σ 1 = dx + Sdy + T dz
(B1)
σ 2 = dy
(B2)
σ 3 = dz .
(B3)
Define
1
A = (T Sx − STx + Ty − Sz ) .
2
(B4)
It is straightforward to verify that the connection forms
ω 11 = ω 22 = ω 33 = 0
(B5)
ω 12 = −ω 21 = −Sx σ 1 + Aσ 3
(B6)
ω 13 = −ω 31 = −Tx σ 1 − Aσ 2
(B7)
ω 23 = −ω 32 = −Aσ 1
(B8)
satisfy the Cartan structure equations
dσ i + ω i j ∧ σ j = 0 .
(B9)
Now because the metric is flat, the curvature forms
θi j = dω i j + ω i k ∧ ω ki
(B10)
must vanish identically. In particular,
θ23 = −2A σ 2 ∧ σ 3 + other components
(B11)
so A must vanish, and this is Eq. (27). The remaining identities reduce to dω 12 = dω 13 = 0,
and this is Eq (26).
22
(May 20, 2009)
Appendix C: Geometrical calculations for ciliary flows
Under a ciliary flow V it is necessary that g11 keep the constant value 1, so that
0 = ∂t g11 = 2g([∂x , V ], ∂x ) = 2(αx + Sβx + T γx ) ,
(C1)
and this is Eq. (31). To preserve the form of g it is necessary that
∂t g22 = 2SSt = 2S∂t g12
(C2)
∂t g33 = 2T Tt = 2T ∂t g13
(C3)
∂t g23 = STt + T St = S∂t g13 + T ∂t g12
(C4)
and these conditions, using Eq. (4), are Eqs. (32)-(34). Finally, it is not enough that g keep
the form of Eq. (8), it must also continue to have zero Riemannian curvature. The most
efficient way to handle this computation is to use the moving orthonormal frame associated
with the changing g. The change in these vector fields is computed by the Lie derivative
(the Lie bracket for vector fields). Thus under the ciliary flow V of Eq. (30)
∂t e1 = [e1 , V ] = αx e1 + βx ∂y + γx ∂z = βx e2 + γx e3
∂t e2 = [e2 , V ] − (∂t S)e1 = σ 1 ([e2 , V ]) − ∂t S e1 + (e2 γ)e3
∂t e3 = [e3 , V ] − (∂t T )e1 = σ 1 ([e3 , V ]) − ∂t T e1 + (e3 β)e2
(C5)
(C6)
(C7)
As a check, one verifies what is required by orthonormality,
∂t S = σ 1 ([e2 , V ]) + βx
(C8)
∂t T = σ 1 ([e3 , V ]) + γx ,
(C9)
using Eq. (4), and also e3 β = −e2 γ by Eq. (34). To summarize,
∂t e1 = βx e2 + γx e3
(C10)
∂t e2 = −βx e1 − (e3 β)e3
(C11)
∂t e3 = −γx e1 − (e2 γ)e2
(C12)
For the Riemannian curvature to vanish it is necessary that
0 = ∂t [e2 , e3 ] = [∂t e2 , e3 ] + [e2 , ∂t e3 ]
= [(e3 β)x − (e2 γ)x + 2Tx βx − 2Sx γx ]e1 + (e2 e2 γ)e2 + (e3 e3 β)e3
23
(May 20, 2009)
(C13)
(C14)
The vanishing of this vector field is conditions Eqs. (35)-(37). It remains to show that
with these conditions the other components of the Riemannian curvature continue to vanish
identically. Dual to Eqs. (C10)-(C12) one has
∂t σ 1 = βx σ 2 + γx σ 3
(C15)
∂t σ 2 = −βx σ 1 − (e3 β)σ 3
(C16)
∂t σ 3 = −γx σ 1 − (e2 γ)σ 2
(C17)
The Cartan structure equations, which are identities, require
∂t ω 12 = (−βxx + Tx (e3 β))σ 1 − Sx βx σ 2 − Sx γx σ 3
(C18)
∂t ω 13 = (−γxx + Sx (e2 γ))σ 1 − Tx βx σ 2 − Tx γx σ 3
(C19)
∂t ω 23 = 0
(C20)
and a long but straightforward computation then shows that
∂t (dω i j + ω i k ∧ ω kj ) = 0
(C21)
for all components.
Appendix D: Making S and T constant in one normal plane by reptation
Let α be a reptation, i.e., V = α∂x and αx = 0. Then, rearranging Eqs. (38)-(39), the
convective derivative
St − αSx = αy
(D1)
Tt − αTx = αz
(D2)
is the rate of change of strain in a fixed normal plane. (The normal planes are not convected
with the material but are associated with the stationary pattern of the filaments, so to stay
in a normal plane, subtract the convective term). Thus in a fixed normal plane, since αx = 0,
(e2 S)t′ = e2 (St − αSx ) = e2 αy = e2 e2 α
(D3)
(e3 T )t′ = e3 (Tt − αTx ) = e3 αz = e3 e3 α
(D4)
(e3 S)t′ = (e2 T )t′ = e2 e3 α
(D5)
24
(May 20, 2009)
Here the subscript t′ means the time derivative at a fixed normal plane, not at fixed x. In
this normal plane, we can introduce coordinates (y ′ , z ′ ) such that e2 = ∂/∂y ′ , e3 = ∂/∂z ′ .
Then by the Poincaré lemma, Eq. (27) says that there exists locally a function F (y ′, z ′ , t)
such that in that plane S = e2 F , T = e3 F . Extend F to be constant along filaments, and
take α = −F . Then under the reptation V = α∂x ,
(e2 S)t′ = −e2 S
(D6)
(e3 T )t′ = −e3 T
(D7)
(e3 S)t′ = (e2 T )t′ = −e3 S = −e2 T .
(D8)
Thus these quantities decay to zero exponentially with time. In the limit as t → ∞, Eq. (29)
holds in one normal plane (and then by Eq. (26) it holds in all normal planes).
[1] J. Gray and G. Hancock, J. Exp. Biol. 32, 802-814 (1955).
[2] J. Lighthill, Soc. Ind. Appl. Math. Rev. 18, 161 (1976).
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[4] S. Gueron and N. Liron, Biophys. J. 63 1045-1058 (1992).
[5] S. Gueron and N. Liron, Biophys. J. 65 499-507 (1993).
[6] S. Gueron and K. Levit-Gurevich, Mathematical Methods in the Applied Sciences 24 15771603 (2001).
[7] A. Vilfan and F. Jülicher, Phys. Rev. Lett. 96 058102 (2006).
[8] C.J. Brokaw, Biophys. J. 12 564-586 (1972).
[9] J. Lubliner, J. Theor. Biol. 41 119-125 (1973).
[10] M. Hines and J.J. Blum, Biophys. J. 23 41-57 (1978).
[11] M. Hines and J.J. Blum, Biophys. J. 41 67-79 (1983).
[12] M. Hines and J.J. Blum, Biophys. J. 46 559-565 (1984).
[13] M. Hines and J.J. Blum, Biophys. J. 47 705-708 (1985).
[14] S. Camalet and F. Jülicher, New. J. Phys. 2 24 (2000).
[15] A. Hilfinger and F. Julicher, Phys. Biol 5 016003 (12pp) (2008).
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25
(May 20, 2009)
[17] T. Frankel, The Geometry of Physics, Cambridge Univ. Press, 1997.
[18] S. Asakura, Advances in Biophysics 1 99-155 (1970).
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26
(May 20, 2009)
Figure Captions
Fig. 1: Schematic cross-section of a cilium showing the 9+2 arrangement of microtubules
(dots) and microtubule doublets (double dots), the radial spokes (solid) and the dynein arms
(dotted).
Fig. 2: Under the ciliary constraints, the filaments in the neighborhood of a given filament
twist about it in a way that is determined by the geometry of the given filament as a
space curve (the displacement vector between the filaments is parallel transported along the
given filament). The above 9+2 cilium, which is assumed to have taken a helical shape
with curvature κ = 1 and torsion τ = 2, is shown in cross-section as it intersects a plane
containing the helix axis. It twists by about 38◦ on each turn.
Fig. 3: In a twist transition the microtubules shear along each other, but do not generate
bend, i.e., the cylinder remains straight. Such a motion violates the ciliary constraints. It
may be possible in a 9+0 axoneme, lacking the central 2 microtubules, but not in a 9+2
axoneme, in which the central microtubules enforce the ciliary constraints. The center of
the cylinder is deliberately shown empty here.
Fig. 4: In the skew coordinates (x,y) the distance between (0,0) and (0,1) is correctly given
√
as 1 + S 2 by the metric tensor of Eq. (7) .
Fig. 5: If filaments are bent into arcs of circles, with arclength x along the filaments and
plane polar coordinates (r, θ), then x = rθ, and hence shear strain S = −∂x/∂r = −θ =
−x/r is constant on lines perpendicular to the filaments, a special case of Eq. (29). Also
note Sx = −1/r is the curvature, a special case of Eq. (18). The minus sign is because the
curvature vector is opposite to e2 = ∂/∂r.
Fig. 6: The coordinate plane for the surface of the flagellum has lines of constant θ that
map onto spiraling protofilaments. The coordinate x is arclength along protofilaments. The
dotted line is a line of constant shear strain, determining that the flagellum is a helix of
torsion τ .
27
(May 20, 2009)
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